3.1.53 \(\int (f+g x)^2 (a+b \log (c (d+e x)^n))^3 \, dx\) [53]

Optimal. Leaf size=432 \[ \frac {6 a b^2 (e f-d g)^2 n^2 x}{e^2}-\frac {6 b^3 (e f-d g)^2 n^3 x}{e^2}-\frac {3 b^3 g (e f-d g) n^3 (d+e x)^2}{4 e^3}-\frac {2 b^3 g^2 n^3 (d+e x)^3}{27 e^3}+\frac {6 b^3 (e f-d g)^2 n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e^3}+\frac {3 b^2 g (e f-d g) n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^3}+\frac {2 b^2 g^2 n^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 e^3}-\frac {3 b (e f-d g)^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^3}-\frac {3 b g (e f-d g) n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^3}-\frac {b g^2 n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 e^3}+\frac {(e f-d g)^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac {g (e f-d g) (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac {g^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 e^3} \]

[Out]

6*a*b^2*(-d*g+e*f)^2*n^2*x/e^2-6*b^3*(-d*g+e*f)^2*n^3*x/e^2-3/4*b^3*g*(-d*g+e*f)*n^3*(e*x+d)^2/e^3-2/27*b^3*g^
2*n^3*(e*x+d)^3/e^3+6*b^3*(-d*g+e*f)^2*n^2*(e*x+d)*ln(c*(e*x+d)^n)/e^3+3/2*b^2*g*(-d*g+e*f)*n^2*(e*x+d)^2*(a+b
*ln(c*(e*x+d)^n))/e^3+2/9*b^2*g^2*n^2*(e*x+d)^3*(a+b*ln(c*(e*x+d)^n))/e^3-3*b*(-d*g+e*f)^2*n*(e*x+d)*(a+b*ln(c
*(e*x+d)^n))^2/e^3-3/2*b*g*(-d*g+e*f)*n*(e*x+d)^2*(a+b*ln(c*(e*x+d)^n))^2/e^3-1/3*b*g^2*n*(e*x+d)^3*(a+b*ln(c*
(e*x+d)^n))^2/e^3+(-d*g+e*f)^2*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^3/e^3+g*(-d*g+e*f)*(e*x+d)^2*(a+b*ln(c*(e*x+d)^n)
)^3/e^3+1/3*g^2*(e*x+d)^3*(a+b*ln(c*(e*x+d)^n))^3/e^3

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Rubi [A]
time = 0.27, antiderivative size = 432, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2448, 2436, 2333, 2332, 2437, 2342, 2341} \begin {gather*} \frac {3 b^2 g n^2 (d+e x)^2 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^3}+\frac {2 b^2 g^2 n^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 e^3}+\frac {6 a b^2 n^2 x (e f-d g)^2}{e^2}-\frac {3 b g n (d+e x)^2 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^3}-\frac {3 b n (d+e x) (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^3}+\frac {g (d+e x)^2 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac {(d+e x) (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}-\frac {b g^2 n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 e^3}+\frac {g^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 e^3}+\frac {6 b^3 n^2 (d+e x) (e f-d g)^2 \log \left (c (d+e x)^n\right )}{e^3}-\frac {3 b^3 g n^3 (d+e x)^2 (e f-d g)}{4 e^3}-\frac {2 b^3 g^2 n^3 (d+e x)^3}{27 e^3}-\frac {6 b^3 n^3 x (e f-d g)^2}{e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(f + g*x)^2*(a + b*Log[c*(d + e*x)^n])^3,x]

[Out]

(6*a*b^2*(e*f - d*g)^2*n^2*x)/e^2 - (6*b^3*(e*f - d*g)^2*n^3*x)/e^2 - (3*b^3*g*(e*f - d*g)*n^3*(d + e*x)^2)/(4
*e^3) - (2*b^3*g^2*n^3*(d + e*x)^3)/(27*e^3) + (6*b^3*(e*f - d*g)^2*n^2*(d + e*x)*Log[c*(d + e*x)^n])/e^3 + (3
*b^2*g*(e*f - d*g)*n^2*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^3) + (2*b^2*g^2*n^2*(d + e*x)^3*(a + b*Log
[c*(d + e*x)^n]))/(9*e^3) - (3*b*(e*f - d*g)^2*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e^3 - (3*b*g*(e*f - d
*g)*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(2*e^3) - (b*g^2*n*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])^2)/(
3*e^3) + ((e*f - d*g)^2*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/e^3 + (g*(e*f - d*g)*(d + e*x)^2*(a + b*Log[c*
(d + e*x)^n])^3)/e^3 + (g^2*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])^3)/(3*e^3)

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2437

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2448

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Int[Exp
andIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[
e*f - d*g, 0] && IGtQ[q, 0]

Rubi steps

\begin {align*} \int (f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx &=\int \left (\frac {(e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2}+\frac {2 g (e f-d g) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2}+\frac {g^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2}\right ) \, dx\\ &=\frac {g^2 \int (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx}{e^2}+\frac {(2 g (e f-d g)) \int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx}{e^2}+\frac {(e f-d g)^2 \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx}{e^2}\\ &=\frac {g^2 \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e^3}+\frac {(2 g (e f-d g)) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e^3}+\frac {(e f-d g)^2 \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e^3}\\ &=\frac {(e f-d g)^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac {g (e f-d g) (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac {g^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 e^3}-\frac {\left (b g^2 n\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e^3}-\frac {(3 b g (e f-d g) n) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e^3}-\frac {\left (3 b (e f-d g)^2 n\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e^3}\\ &=-\frac {3 b (e f-d g)^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^3}-\frac {3 b g (e f-d g) n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^3}-\frac {b g^2 n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 e^3}+\frac {(e f-d g)^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac {g (e f-d g) (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac {g^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 e^3}+\frac {\left (2 b^2 g^2 n^2\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{3 e^3}+\frac {\left (3 b^2 g (e f-d g) n^2\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^3}+\frac {\left (6 b^2 (e f-d g)^2 n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^3}\\ &=\frac {6 a b^2 (e f-d g)^2 n^2 x}{e^2}-\frac {3 b^3 g (e f-d g) n^3 (d+e x)^2}{4 e^3}-\frac {2 b^3 g^2 n^3 (d+e x)^3}{27 e^3}+\frac {3 b^2 g (e f-d g) n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^3}+\frac {2 b^2 g^2 n^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 e^3}-\frac {3 b (e f-d g)^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^3}-\frac {3 b g (e f-d g) n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^3}-\frac {b g^2 n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 e^3}+\frac {(e f-d g)^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac {g (e f-d g) (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac {g^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 e^3}+\frac {\left (6 b^3 (e f-d g)^2 n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e^3}\\ &=\frac {6 a b^2 (e f-d g)^2 n^2 x}{e^2}-\frac {6 b^3 (e f-d g)^2 n^3 x}{e^2}-\frac {3 b^3 g (e f-d g) n^3 (d+e x)^2}{4 e^3}-\frac {2 b^3 g^2 n^3 (d+e x)^3}{27 e^3}+\frac {6 b^3 (e f-d g)^2 n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e^3}+\frac {3 b^2 g (e f-d g) n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^3}+\frac {2 b^2 g^2 n^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 e^3}-\frac {3 b (e f-d g)^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^3}-\frac {3 b g (e f-d g) n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^3}-\frac {b g^2 n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 e^3}+\frac {(e f-d g)^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac {g (e f-d g) (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac {g^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 e^3}\\ \end {align*}

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Mathematica [A]
time = 0.53, size = 809, normalized size = 1.87 \begin {gather*} \frac {36 b^3 d \left (3 e^2 f^2-3 d e f g+d^2 g^2\right ) n^3 \log ^3(d+e x)-18 b^2 d n^2 \log ^2(d+e x) \left (6 a \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )+b \left (-18 e^2 f^2+27 d e f g-11 d^2 g^2\right ) n+6 b \left (3 e^2 f^2-3 d e f g+d^2 g^2\right ) \log \left (c (d+e x)^n\right )\right )+6 b d n \log (d+e x) \left (18 a^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )-6 a b \left (18 e^2 f^2-27 d e f g+11 d^2 g^2\right ) n+b^2 \left (108 e^2 f^2-189 d e f g+85 d^2 g^2\right ) n^2+6 b \left (6 a \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )+b \left (-18 e^2 f^2+27 d e f g-11 d^2 g^2\right ) n\right ) \log \left (c (d+e x)^n\right )+18 b^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right ) \log ^2\left (c (d+e x)^n\right )\right )+e x \left (36 a^3 e^2 \left (3 f^2+3 f g x+g^2 x^2\right )-18 a^2 b n \left (6 d^2 g^2-3 d e g (6 f+g x)+e^2 \left (18 f^2+9 f g x+2 g^2 x^2\right )\right )+6 a b^2 n^2 \left (66 d^2 g^2-3 d e g (54 f+5 g x)+e^2 \left (108 f^2+27 f g x+4 g^2 x^2\right )\right )-b^3 n^3 \left (510 d^2 g^2-3 d e g (378 f+19 g x)+e^2 \left (648 f^2+81 f g x+8 g^2 x^2\right )\right )+6 b \left (18 a^2 e^2 \left (3 f^2+3 f g x+g^2 x^2\right )-6 a b n \left (6 d^2 g^2-3 d e g (6 f+g x)+e^2 \left (18 f^2+9 f g x+2 g^2 x^2\right )\right )+b^2 n^2 \left (66 d^2 g^2-3 d e g (54 f+5 g x)+e^2 \left (108 f^2+27 f g x+4 g^2 x^2\right )\right )\right ) \log \left (c (d+e x)^n\right )+18 b^2 \left (6 a e^2 \left (3 f^2+3 f g x+g^2 x^2\right )-b n \left (6 d^2 g^2-3 d e g (6 f+g x)+e^2 \left (18 f^2+9 f g x+2 g^2 x^2\right )\right )\right ) \log ^2\left (c (d+e x)^n\right )+36 b^3 e^2 \left (3 f^2+3 f g x+g^2 x^2\right ) \log ^3\left (c (d+e x)^n\right )\right )}{108 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)^2*(a + b*Log[c*(d + e*x)^n])^3,x]

[Out]

(36*b^3*d*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2)*n^3*Log[d + e*x]^3 - 18*b^2*d*n^2*Log[d + e*x]^2*(6*a*(3*e^2*f^2 -
3*d*e*f*g + d^2*g^2) + b*(-18*e^2*f^2 + 27*d*e*f*g - 11*d^2*g^2)*n + 6*b*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2)*Lo
g[c*(d + e*x)^n]) + 6*b*d*n*Log[d + e*x]*(18*a^2*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2) - 6*a*b*(18*e^2*f^2 - 27*d*
e*f*g + 11*d^2*g^2)*n + b^2*(108*e^2*f^2 - 189*d*e*f*g + 85*d^2*g^2)*n^2 + 6*b*(6*a*(3*e^2*f^2 - 3*d*e*f*g + d
^2*g^2) + b*(-18*e^2*f^2 + 27*d*e*f*g - 11*d^2*g^2)*n)*Log[c*(d + e*x)^n] + 18*b^2*(3*e^2*f^2 - 3*d*e*f*g + d^
2*g^2)*Log[c*(d + e*x)^n]^2) + e*x*(36*a^3*e^2*(3*f^2 + 3*f*g*x + g^2*x^2) - 18*a^2*b*n*(6*d^2*g^2 - 3*d*e*g*(
6*f + g*x) + e^2*(18*f^2 + 9*f*g*x + 2*g^2*x^2)) + 6*a*b^2*n^2*(66*d^2*g^2 - 3*d*e*g*(54*f + 5*g*x) + e^2*(108
*f^2 + 27*f*g*x + 4*g^2*x^2)) - b^3*n^3*(510*d^2*g^2 - 3*d*e*g*(378*f + 19*g*x) + e^2*(648*f^2 + 81*f*g*x + 8*
g^2*x^2)) + 6*b*(18*a^2*e^2*(3*f^2 + 3*f*g*x + g^2*x^2) - 6*a*b*n*(6*d^2*g^2 - 3*d*e*g*(6*f + g*x) + e^2*(18*f
^2 + 9*f*g*x + 2*g^2*x^2)) + b^2*n^2*(66*d^2*g^2 - 3*d*e*g*(54*f + 5*g*x) + e^2*(108*f^2 + 27*f*g*x + 4*g^2*x^
2)))*Log[c*(d + e*x)^n] + 18*b^2*(6*a*e^2*(3*f^2 + 3*f*g*x + g^2*x^2) - b*n*(6*d^2*g^2 - 3*d*e*g*(6*f + g*x) +
e^2*(18*f^2 + 9*f*g*x + 2*g^2*x^2)))*Log[c*(d + e*x)^n]^2 + 36*b^3*e^2*(3*f^2 + 3*f*g*x + g^2*x^2)*Log[c*(d +
e*x)^n]^3))/(108*e^3)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.62, size = 20417, normalized size = 47.26

method	result	size
risch	\(\text {Expression too large to display}\)	\(20417\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^2*(a+b*ln(c*(e*x+d)^n))^3,x,method=_RETURNVERBOSE)

[Out]

result too large to display

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1172 vs. \(2 (435) = 870\).
time = 0.32, size = 1172, normalized size = 2.71 \begin {gather*} \frac {1}{3} \, b^{3} g^{2} x^{3} \log \left ({\left (x e + d\right )}^{n} c\right )^{3} + a b^{2} g^{2} x^{3} \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + b^{3} f g x^{2} \log \left ({\left (x e + d\right )}^{n} c\right )^{3} + a^{2} b g^{2} x^{3} \log \left ({\left (x e + d\right )}^{n} c\right ) + 3 \, a b^{2} f g x^{2} \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + b^{3} f^{2} x \log \left ({\left (x e + d\right )}^{n} c\right )^{3} + \frac {1}{3} \, a^{3} g^{2} x^{3} + 3 \, {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} a^{2} b f^{2} n e - \frac {3}{2} \, {\left (2 \, d^{2} e^{\left (-3\right )} \log \left (x e + d\right ) + {\left (x^{2} e - 2 \, d x\right )} e^{\left (-2\right )}\right )} a^{2} b f g n e + \frac {1}{6} \, {\left (6 \, d^{3} e^{\left (-4\right )} \log \left (x e + d\right ) - {\left (2 \, x^{3} e^{2} - 3 \, d x^{2} e + 6 \, d^{2} x\right )} e^{\left (-3\right )}\right )} a^{2} b g^{2} n e + 3 \, a^{2} b f g x^{2} \log \left ({\left (x e + d\right )}^{n} c\right ) + 3 \, a b^{2} f^{2} x \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + a^{3} f g x^{2} + 3 \, a^{2} b f^{2} x \log \left ({\left (x e + d\right )}^{n} c\right ) - 3 \, {\left ({\left (d \log \left (x e + d\right )^{2} - 2 \, x e + 2 \, d \log \left (x e + d\right )\right )} n^{2} e^{\left (-1\right )} - 2 \, {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} n e \log \left ({\left (x e + d\right )}^{n} c\right )\right )} a b^{2} f^{2} + {\left (3 \, {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} n e \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + {\left ({\left (d \log \left (x e + d\right )^{3} + 3 \, d \log \left (x e + d\right )^{2} - 6 \, x e + 6 \, d \log \left (x e + d\right )\right )} n^{2} e^{\left (-2\right )} - 3 \, {\left (d \log \left (x e + d\right )^{2} - 2 \, x e + 2 \, d \log \left (x e + d\right )\right )} n e^{\left (-2\right )} \log \left ({\left (x e + d\right )}^{n} c\right )\right )} n e\right )} b^{3} f^{2} + \frac {3}{2} \, {\left ({\left (2 \, d^{2} \log \left (x e + d\right )^{2} + x^{2} e^{2} - 6 \, d x e + 6 \, d^{2} \log \left (x e + d\right )\right )} n^{2} e^{\left (-2\right )} - 2 \, {\left (2 \, d^{2} e^{\left (-3\right )} \log \left (x e + d\right ) + {\left (x^{2} e - 2 \, d x\right )} e^{\left (-2\right )}\right )} n e \log \left ({\left (x e + d\right )}^{n} c\right )\right )} a b^{2} f g - \frac {1}{4} \, {\left (6 \, {\left (2 \, d^{2} e^{\left (-3\right )} \log \left (x e + d\right ) + {\left (x^{2} e - 2 \, d x\right )} e^{\left (-2\right )}\right )} n e \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + {\left ({\left (4 \, d^{2} \log \left (x e + d\right )^{3} + 18 \, d^{2} \log \left (x e + d\right )^{2} + 3 \, x^{2} e^{2} - 42 \, d x e + 42 \, d^{2} \log \left (x e + d\right )\right )} n^{2} e^{\left (-3\right )} - 6 \, {\left (2 \, d^{2} \log \left (x e + d\right )^{2} + x^{2} e^{2} - 6 \, d x e + 6 \, d^{2} \log \left (x e + d\right )\right )} n e^{\left (-3\right )} \log \left ({\left (x e + d\right )}^{n} c\right )\right )} n e\right )} b^{3} f g - \frac {1}{18} \, {\left ({\left (18 \, d^{3} \log \left (x e + d\right )^{2} - 4 \, x^{3} e^{3} + 15 \, d x^{2} e^{2} - 66 \, d^{2} x e + 66 \, d^{3} \log \left (x e + d\right )\right )} n^{2} e^{\left (-3\right )} - 6 \, {\left (6 \, d^{3} e^{\left (-4\right )} \log \left (x e + d\right ) - {\left (2 \, x^{3} e^{2} - 3 \, d x^{2} e + 6 \, d^{2} x\right )} e^{\left (-3\right )}\right )} n e \log \left ({\left (x e + d\right )}^{n} c\right )\right )} a b^{2} g^{2} + \frac {1}{108} \, {\left (18 \, {\left (6 \, d^{3} e^{\left (-4\right )} \log \left (x e + d\right ) - {\left (2 \, x^{3} e^{2} - 3 \, d x^{2} e + 6 \, d^{2} x\right )} e^{\left (-3\right )}\right )} n e \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + {\left ({\left (36 \, d^{3} \log \left (x e + d\right )^{3} + 198 \, d^{3} \log \left (x e + d\right )^{2} - 8 \, x^{3} e^{3} + 57 \, d x^{2} e^{2} - 510 \, d^{2} x e + 510 \, d^{3} \log \left (x e + d\right )\right )} n^{2} e^{\left (-4\right )} - 6 \, {\left (18 \, d^{3} \log \left (x e + d\right )^{2} - 4 \, x^{3} e^{3} + 15 \, d x^{2} e^{2} - 66 \, d^{2} x e + 66 \, d^{3} \log \left (x e + d\right )\right )} n e^{\left (-4\right )} \log \left ({\left (x e + d\right )}^{n} c\right )\right )} n e\right )} b^{3} g^{2} + a^{3} f^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*(a+b*log(c*(e*x+d)^n))^3,x, algorithm="maxima")

[Out]

1/3*b^3*g^2*x^3*log((x*e + d)^n*c)^3 + a*b^2*g^2*x^3*log((x*e + d)^n*c)^2 + b^3*f*g*x^2*log((x*e + d)^n*c)^3 +
a^2*b*g^2*x^3*log((x*e + d)^n*c) + 3*a*b^2*f*g*x^2*log((x*e + d)^n*c)^2 + b^3*f^2*x*log((x*e + d)^n*c)^3 + 1/
3*a^3*g^2*x^3 + 3*(d*e^(-2)*log(x*e + d) - x*e^(-1))*a^2*b*f^2*n*e - 3/2*(2*d^2*e^(-3)*log(x*e + d) + (x^2*e -
2*d*x)*e^(-2))*a^2*b*f*g*n*e + 1/6*(6*d^3*e^(-4)*log(x*e + d) - (2*x^3*e^2 - 3*d*x^2*e + 6*d^2*x)*e^(-3))*a^2
*b*g^2*n*e + 3*a^2*b*f*g*x^2*log((x*e + d)^n*c) + 3*a*b^2*f^2*x*log((x*e + d)^n*c)^2 + a^3*f*g*x^2 + 3*a^2*b*f
^2*x*log((x*e + d)^n*c) - 3*((d*log(x*e + d)^2 - 2*x*e + 2*d*log(x*e + d))*n^2*e^(-1) - 2*(d*e^(-2)*log(x*e +
d) - x*e^(-1))*n*e*log((x*e + d)^n*c))*a*b^2*f^2 + (3*(d*e^(-2)*log(x*e + d) - x*e^(-1))*n*e*log((x*e + d)^n*c
)^2 + ((d*log(x*e + d)^3 + 3*d*log(x*e + d)^2 - 6*x*e + 6*d*log(x*e + d))*n^2*e^(-2) - 3*(d*log(x*e + d)^2 - 2
*x*e + 2*d*log(x*e + d))*n*e^(-2)*log((x*e + d)^n*c))*n*e)*b^3*f^2 + 3/2*((2*d^2*log(x*e + d)^2 + x^2*e^2 - 6*
d*x*e + 6*d^2*log(x*e + d))*n^2*e^(-2) - 2*(2*d^2*e^(-3)*log(x*e + d) + (x^2*e - 2*d*x)*e^(-2))*n*e*log((x*e +
d)^n*c))*a*b^2*f*g - 1/4*(6*(2*d^2*e^(-3)*log(x*e + d) + (x^2*e - 2*d*x)*e^(-2))*n*e*log((x*e + d)^n*c)^2 + (
(4*d^2*log(x*e + d)^3 + 18*d^2*log(x*e + d)^2 + 3*x^2*e^2 - 42*d*x*e + 42*d^2*log(x*e + d))*n^2*e^(-3) - 6*(2*
d^2*log(x*e + d)^2 + x^2*e^2 - 6*d*x*e + 6*d^2*log(x*e + d))*n*e^(-3)*log((x*e + d)^n*c))*n*e)*b^3*f*g - 1/18*
((18*d^3*log(x*e + d)^2 - 4*x^3*e^3 + 15*d*x^2*e^2 - 66*d^2*x*e + 66*d^3*log(x*e + d))*n^2*e^(-3) - 6*(6*d^3*e
^(-4)*log(x*e + d) - (2*x^3*e^2 - 3*d*x^2*e + 6*d^2*x)*e^(-3))*n*e*log((x*e + d)^n*c))*a*b^2*g^2 + 1/108*(18*(
6*d^3*e^(-4)*log(x*e + d) - (2*x^3*e^2 - 3*d*x^2*e + 6*d^2*x)*e^(-3))*n*e*log((x*e + d)^n*c)^2 + ((36*d^3*log(
x*e + d)^3 + 198*d^3*log(x*e + d)^2 - 8*x^3*e^3 + 57*d*x^2*e^2 - 510*d^2*x*e + 510*d^3*log(x*e + d))*n^2*e^(-4
) - 6*(18*d^3*log(x*e + d)^2 - 4*x^3*e^3 + 15*d*x^2*e^2 - 66*d^2*x*e + 66*d^3*log(x*e + d))*n*e^(-4)*log((x*e
+ d)^n*c))*n*e)*b^3*g^2 + a^3*f^2*x

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1676 vs. \(2 (435) = 870\).
time = 0.39, size = 1676, normalized size = 3.88 \begin {gather*} \frac {1}{108} \, {\left (36 \, {\left (b^{3} g^{2} x^{3} + 3 \, b^{3} f g x^{2} + 3 \, b^{3} f^{2} x\right )} e^{3} \log \left (c\right )^{3} + 36 \, {\left (b^{3} d^{3} g^{2} n^{3} - 3 \, b^{3} d^{2} f g n^{3} e + 3 \, b^{3} d f^{2} n^{3} e^{2} + {\left (b^{3} g^{2} n^{3} x^{3} + 3 \, b^{3} f g n^{3} x^{2} + 3 \, b^{3} f^{2} n^{3} x\right )} e^{3}\right )} \log \left (x e + d\right )^{3} - 6 \, {\left (85 \, b^{3} d^{2} g^{2} n^{3} - 66 \, a b^{2} d^{2} g^{2} n^{2} + 18 \, a^{2} b d^{2} g^{2} n\right )} x e - 18 \, {\left (11 \, b^{3} d^{3} g^{2} n^{3} - 6 \, a b^{2} d^{3} g^{2} n^{2} + {\left (2 \, {\left (b^{3} g^{2} n^{3} - 3 \, a b^{2} g^{2} n^{2}\right )} x^{3} + 9 \, {\left (b^{3} f g n^{3} - 2 \, a b^{2} f g n^{2}\right )} x^{2} + 18 \, {\left (b^{3} f^{2} n^{3} - a b^{2} f^{2} n^{2}\right )} x\right )} e^{3} - 3 \, {\left (b^{3} d g^{2} n^{3} x^{2} + 6 \, b^{3} d f g n^{3} x - 6 \, b^{3} d f^{2} n^{3} + 6 \, a b^{2} d f^{2} n^{2}\right )} e^{2} + 3 \, {\left (2 \, b^{3} d^{2} g^{2} n^{3} x - 9 \, b^{3} d^{2} f g n^{3} + 6 \, a b^{2} d^{2} f g n^{2}\right )} e - 6 \, {\left (b^{3} d^{3} g^{2} n^{2} - 3 \, b^{3} d^{2} f g n^{2} e + 3 \, b^{3} d f^{2} n^{2} e^{2} + {\left (b^{3} g^{2} n^{2} x^{3} + 3 \, b^{3} f g n^{2} x^{2} + 3 \, b^{3} f^{2} n^{2} x\right )} e^{3}\right )} \log \left (c\right )\right )} \log \left (x e + d\right )^{2} - 18 \, {\left (6 \, b^{3} d^{2} g^{2} n x e + {\left (2 \, {\left (b^{3} g^{2} n - 3 \, a b^{2} g^{2}\right )} x^{3} + 9 \, {\left (b^{3} f g n - 2 \, a b^{2} f g\right )} x^{2} + 18 \, {\left (b^{3} f^{2} n - a b^{2} f^{2}\right )} x\right )} e^{3} - 3 \, {\left (b^{3} d g^{2} n x^{2} + 6 \, b^{3} d f g n x\right )} e^{2}\right )} \log \left (c\right )^{2} - {\left (4 \, {\left (2 \, b^{3} g^{2} n^{3} - 6 \, a b^{2} g^{2} n^{2} + 9 \, a^{2} b g^{2} n - 9 \, a^{3} g^{2}\right )} x^{3} + 27 \, {\left (3 \, b^{3} f g n^{3} - 6 \, a b^{2} f g n^{2} + 6 \, a^{2} b f g n - 4 \, a^{3} f g\right )} x^{2} + 108 \, {\left (6 \, b^{3} f^{2} n^{3} - 6 \, a b^{2} f^{2} n^{2} + 3 \, a^{2} b f^{2} n - a^{3} f^{2}\right )} x\right )} e^{3} + 3 \, {\left ({\left (19 \, b^{3} d g^{2} n^{3} - 30 \, a b^{2} d g^{2} n^{2} + 18 \, a^{2} b d g^{2} n\right )} x^{2} + 54 \, {\left (7 \, b^{3} d f g n^{3} - 6 \, a b^{2} d f g n^{2} + 2 \, a^{2} b d f g n\right )} x\right )} e^{2} + 6 \, {\left (85 \, b^{3} d^{3} g^{2} n^{3} - 66 \, a b^{2} d^{3} g^{2} n^{2} + 18 \, a^{2} b d^{3} g^{2} n + 18 \, {\left (b^{3} d^{3} g^{2} n - 3 \, b^{3} d^{2} f g n e + 3 \, b^{3} d f^{2} n e^{2} + {\left (b^{3} g^{2} n x^{3} + 3 \, b^{3} f g n x^{2} + 3 \, b^{3} f^{2} n x\right )} e^{3}\right )} \log \left (c\right )^{2} + {\left (2 \, {\left (2 \, b^{3} g^{2} n^{3} - 6 \, a b^{2} g^{2} n^{2} + 9 \, a^{2} b g^{2} n\right )} x^{3} + 27 \, {\left (b^{3} f g n^{3} - 2 \, a b^{2} f g n^{2} + 2 \, a^{2} b f g n\right )} x^{2} + 54 \, {\left (2 \, b^{3} f^{2} n^{3} - 2 \, a b^{2} f^{2} n^{2} + a^{2} b f^{2} n\right )} x\right )} e^{3} + 3 \, {\left (36 \, b^{3} d f^{2} n^{3} - 36 \, a b^{2} d f^{2} n^{2} + 18 \, a^{2} b d f^{2} n - {\left (5 \, b^{3} d g^{2} n^{3} - 6 \, a b^{2} d g^{2} n^{2}\right )} x^{2} - 18 \, {\left (3 \, b^{3} d f g n^{3} - 2 \, a b^{2} d f g n^{2}\right )} x\right )} e^{2} - 3 \, {\left (63 \, b^{3} d^{2} f g n^{3} - 54 \, a b^{2} d^{2} f g n^{2} + 18 \, a^{2} b d^{2} f g n - 2 \, {\left (11 \, b^{3} d^{2} g^{2} n^{3} - 6 \, a b^{2} d^{2} g^{2} n^{2}\right )} x\right )} e - 6 \, {\left (11 \, b^{3} d^{3} g^{2} n^{2} - 6 \, a b^{2} d^{3} g^{2} n + {\left (2 \, {\left (b^{3} g^{2} n^{2} - 3 \, a b^{2} g^{2} n\right )} x^{3} + 9 \, {\left (b^{3} f g n^{2} - 2 \, a b^{2} f g n\right )} x^{2} + 18 \, {\left (b^{3} f^{2} n^{2} - a b^{2} f^{2} n\right )} x\right )} e^{3} - 3 \, {\left (b^{3} d g^{2} n^{2} x^{2} + 6 \, b^{3} d f g n^{2} x - 6 \, b^{3} d f^{2} n^{2} + 6 \, a b^{2} d f^{2} n\right )} e^{2} + 3 \, {\left (2 \, b^{3} d^{2} g^{2} n^{2} x - 9 \, b^{3} d^{2} f g n^{2} + 6 \, a b^{2} d^{2} f g n\right )} e\right )} \log \left (c\right )\right )} \log \left (x e + d\right ) + 6 \, {\left (6 \, {\left (11 \, b^{3} d^{2} g^{2} n^{2} - 6 \, a b^{2} d^{2} g^{2} n\right )} x e + {\left (2 \, {\left (2 \, b^{3} g^{2} n^{2} - 6 \, a b^{2} g^{2} n + 9 \, a^{2} b g^{2}\right )} x^{3} + 27 \, {\left (b^{3} f g n^{2} - 2 \, a b^{2} f g n + 2 \, a^{2} b f g\right )} x^{2} + 54 \, {\left (2 \, b^{3} f^{2} n^{2} - 2 \, a b^{2} f^{2} n + a^{2} b f^{2}\right )} x\right )} e^{3} - 3 \, {\left ({\left (5 \, b^{3} d g^{2} n^{2} - 6 \, a b^{2} d g^{2} n\right )} x^{2} + 18 \, {\left (3 \, b^{3} d f g n^{2} - 2 \, a b^{2} d f g n\right )} x\right )} e^{2}\right )} \log \left (c\right )\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*(a+b*log(c*(e*x+d)^n))^3,x, algorithm="fricas")

[Out]

1/108*(36*(b^3*g^2*x^3 + 3*b^3*f*g*x^2 + 3*b^3*f^2*x)*e^3*log(c)^3 + 36*(b^3*d^3*g^2*n^3 - 3*b^3*d^2*f*g*n^3*e
+ 3*b^3*d*f^2*n^3*e^2 + (b^3*g^2*n^3*x^3 + 3*b^3*f*g*n^3*x^2 + 3*b^3*f^2*n^3*x)*e^3)*log(x*e + d)^3 - 6*(85*b
^3*d^2*g^2*n^3 - 66*a*b^2*d^2*g^2*n^2 + 18*a^2*b*d^2*g^2*n)*x*e - 18*(11*b^3*d^3*g^2*n^3 - 6*a*b^2*d^3*g^2*n^2
+ (2*(b^3*g^2*n^3 - 3*a*b^2*g^2*n^2)*x^3 + 9*(b^3*f*g*n^3 - 2*a*b^2*f*g*n^2)*x^2 + 18*(b^3*f^2*n^3 - a*b^2*f^
2*n^2)*x)*e^3 - 3*(b^3*d*g^2*n^3*x^2 + 6*b^3*d*f*g*n^3*x - 6*b^3*d*f^2*n^3 + 6*a*b^2*d*f^2*n^2)*e^2 + 3*(2*b^3
*d^2*g^2*n^3*x - 9*b^3*d^2*f*g*n^3 + 6*a*b^2*d^2*f*g*n^2)*e - 6*(b^3*d^3*g^2*n^2 - 3*b^3*d^2*f*g*n^2*e + 3*b^3
*d*f^2*n^2*e^2 + (b^3*g^2*n^2*x^3 + 3*b^3*f*g*n^2*x^2 + 3*b^3*f^2*n^2*x)*e^3)*log(c))*log(x*e + d)^2 - 18*(6*b
^3*d^2*g^2*n*x*e + (2*(b^3*g^2*n - 3*a*b^2*g^2)*x^3 + 9*(b^3*f*g*n - 2*a*b^2*f*g)*x^2 + 18*(b^3*f^2*n - a*b^2*
f^2)*x)*e^3 - 3*(b^3*d*g^2*n*x^2 + 6*b^3*d*f*g*n*x)*e^2)*log(c)^2 - (4*(2*b^3*g^2*n^3 - 6*a*b^2*g^2*n^2 + 9*a^
2*b*g^2*n - 9*a^3*g^2)*x^3 + 27*(3*b^3*f*g*n^3 - 6*a*b^2*f*g*n^2 + 6*a^2*b*f*g*n - 4*a^3*f*g)*x^2 + 108*(6*b^3
*f^2*n^3 - 6*a*b^2*f^2*n^2 + 3*a^2*b*f^2*n - a^3*f^2)*x)*e^3 + 3*((19*b^3*d*g^2*n^3 - 30*a*b^2*d*g^2*n^2 + 18*
a^2*b*d*g^2*n)*x^2 + 54*(7*b^3*d*f*g*n^3 - 6*a*b^2*d*f*g*n^2 + 2*a^2*b*d*f*g*n)*x)*e^2 + 6*(85*b^3*d^3*g^2*n^3
- 66*a*b^2*d^3*g^2*n^2 + 18*a^2*b*d^3*g^2*n + 18*(b^3*d^3*g^2*n - 3*b^3*d^2*f*g*n*e + 3*b^3*d*f^2*n*e^2 + (b^
3*g^2*n*x^3 + 3*b^3*f*g*n*x^2 + 3*b^3*f^2*n*x)*e^3)*log(c)^2 + (2*(2*b^3*g^2*n^3 - 6*a*b^2*g^2*n^2 + 9*a^2*b*g
^2*n)*x^3 + 27*(b^3*f*g*n^3 - 2*a*b^2*f*g*n^2 + 2*a^2*b*f*g*n)*x^2 + 54*(2*b^3*f^2*n^3 - 2*a*b^2*f^2*n^2 + a^2
*b*f^2*n)*x)*e^3 + 3*(36*b^3*d*f^2*n^3 - 36*a*b^2*d*f^2*n^2 + 18*a^2*b*d*f^2*n - (5*b^3*d*g^2*n^3 - 6*a*b^2*d*
g^2*n^2)*x^2 - 18*(3*b^3*d*f*g*n^3 - 2*a*b^2*d*f*g*n^2)*x)*e^2 - 3*(63*b^3*d^2*f*g*n^3 - 54*a*b^2*d^2*f*g*n^2
+ 18*a^2*b*d^2*f*g*n - 2*(11*b^3*d^2*g^2*n^3 - 6*a*b^2*d^2*g^2*n^2)*x)*e - 6*(11*b^3*d^3*g^2*n^2 - 6*a*b^2*d^3
*g^2*n + (2*(b^3*g^2*n^2 - 3*a*b^2*g^2*n)*x^3 + 9*(b^3*f*g*n^2 - 2*a*b^2*f*g*n)*x^2 + 18*(b^3*f^2*n^2 - a*b^2*
f^2*n)*x)*e^3 - 3*(b^3*d*g^2*n^2*x^2 + 6*b^3*d*f*g*n^2*x - 6*b^3*d*f^2*n^2 + 6*a*b^2*d*f^2*n)*e^2 + 3*(2*b^3*d
^2*g^2*n^2*x - 9*b^3*d^2*f*g*n^2 + 6*a*b^2*d^2*f*g*n)*e)*log(c))*log(x*e + d) + 6*(6*(11*b^3*d^2*g^2*n^2 - 6*a
*b^2*d^2*g^2*n)*x*e + (2*(2*b^3*g^2*n^2 - 6*a*b^2*g^2*n + 9*a^2*b*g^2)*x^3 + 27*(b^3*f*g*n^2 - 2*a*b^2*f*g*n +
2*a^2*b*f*g)*x^2 + 54*(2*b^3*f^2*n^2 - 2*a*b^2*f^2*n + a^2*b*f^2)*x)*e^3 - 3*((5*b^3*d*g^2*n^2 - 6*a*b^2*d*g^
2*n)*x^2 + 18*(3*b^3*d*f*g*n^2 - 2*a*b^2*d*f*g*n)*x)*e^2)*log(c))*e^(-3)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1578 vs. \(2 (422) = 844\).
time = 2.65, size = 1578, normalized size = 3.65 \begin {gather*} \begin {cases} a^{3} f^{2} x + a^{3} f g x^{2} + \frac {a^{3} g^{2} x^{3}}{3} + \frac {a^{2} b d^{3} g^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{e^{3}} - \frac {3 a^{2} b d^{2} f g \log {\left (c \left (d + e x\right )^{n} \right )}}{e^{2}} - \frac {a^{2} b d^{2} g^{2} n x}{e^{2}} + \frac {3 a^{2} b d f^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + \frac {3 a^{2} b d f g n x}{e} + \frac {a^{2} b d g^{2} n x^{2}}{2 e} - 3 a^{2} b f^{2} n x + 3 a^{2} b f^{2} x \log {\left (c \left (d + e x\right )^{n} \right )} - \frac {3 a^{2} b f g n x^{2}}{2} + 3 a^{2} b f g x^{2} \log {\left (c \left (d + e x\right )^{n} \right )} - \frac {a^{2} b g^{2} n x^{3}}{3} + a^{2} b g^{2} x^{3} \log {\left (c \left (d + e x\right )^{n} \right )} - \frac {11 a b^{2} d^{3} g^{2} n \log {\left (c \left (d + e x\right )^{n} \right )}}{3 e^{3}} + \frac {a b^{2} d^{3} g^{2} \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e^{3}} + \frac {9 a b^{2} d^{2} f g n \log {\left (c \left (d + e x\right )^{n} \right )}}{e^{2}} - \frac {3 a b^{2} d^{2} f g \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e^{2}} + \frac {11 a b^{2} d^{2} g^{2} n^{2} x}{3 e^{2}} - \frac {2 a b^{2} d^{2} g^{2} n x \log {\left (c \left (d + e x\right )^{n} \right )}}{e^{2}} - \frac {6 a b^{2} d f^{2} n \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + \frac {3 a b^{2} d f^{2} \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} - \frac {9 a b^{2} d f g n^{2} x}{e} + \frac {6 a b^{2} d f g n x \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - \frac {5 a b^{2} d g^{2} n^{2} x^{2}}{6 e} + \frac {a b^{2} d g^{2} n x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + 6 a b^{2} f^{2} n^{2} x - 6 a b^{2} f^{2} n x \log {\left (c \left (d + e x\right )^{n} \right )} + 3 a b^{2} f^{2} x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} + \frac {3 a b^{2} f g n^{2} x^{2}}{2} - 3 a b^{2} f g n x^{2} \log {\left (c \left (d + e x\right )^{n} \right )} + 3 a b^{2} f g x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}^{2} + \frac {2 a b^{2} g^{2} n^{2} x^{3}}{9} - \frac {2 a b^{2} g^{2} n x^{3} \log {\left (c \left (d + e x\right )^{n} \right )}}{3} + a b^{2} g^{2} x^{3} \log {\left (c \left (d + e x\right )^{n} \right )}^{2} + \frac {85 b^{3} d^{3} g^{2} n^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{18 e^{3}} - \frac {11 b^{3} d^{3} g^{2} n \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{6 e^{3}} + \frac {b^{3} d^{3} g^{2} \log {\left (c \left (d + e x\right )^{n} \right )}^{3}}{3 e^{3}} - \frac {21 b^{3} d^{2} f g n^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{2 e^{2}} + \frac {9 b^{3} d^{2} f g n \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{2 e^{2}} - \frac {b^{3} d^{2} f g \log {\left (c \left (d + e x\right )^{n} \right )}^{3}}{e^{2}} - \frac {85 b^{3} d^{2} g^{2} n^{3} x}{18 e^{2}} + \frac {11 b^{3} d^{2} g^{2} n^{2} x \log {\left (c \left (d + e x\right )^{n} \right )}}{3 e^{2}} - \frac {b^{3} d^{2} g^{2} n x \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e^{2}} + \frac {6 b^{3} d f^{2} n^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - \frac {3 b^{3} d f^{2} n \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} + \frac {b^{3} d f^{2} \log {\left (c \left (d + e x\right )^{n} \right )}^{3}}{e} + \frac {21 b^{3} d f g n^{3} x}{2 e} - \frac {9 b^{3} d f g n^{2} x \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + \frac {3 b^{3} d f g n x \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} + \frac {19 b^{3} d g^{2} n^{3} x^{2}}{36 e} - \frac {5 b^{3} d g^{2} n^{2} x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{6 e} + \frac {b^{3} d g^{2} n x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{2 e} - 6 b^{3} f^{2} n^{3} x + 6 b^{3} f^{2} n^{2} x \log {\left (c \left (d + e x\right )^{n} \right )} - 3 b^{3} f^{2} n x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} + b^{3} f^{2} x \log {\left (c \left (d + e x\right )^{n} \right )}^{3} - \frac {3 b^{3} f g n^{3} x^{2}}{4} + \frac {3 b^{3} f g n^{2} x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{2} - \frac {3 b^{3} f g n x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{2} + b^{3} f g x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}^{3} - \frac {2 b^{3} g^{2} n^{3} x^{3}}{27} + \frac {2 b^{3} g^{2} n^{2} x^{3} \log {\left (c \left (d + e x\right )^{n} \right )}}{9} - \frac {b^{3} g^{2} n x^{3} \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{3} + \frac {b^{3} g^{2} x^{3} \log {\left (c \left (d + e x\right )^{n} \right )}^{3}}{3} & \text {for}\: e \neq 0 \\\left (a + b \log {\left (c d^{n} \right )}\right )^{3} \left (f^{2} x + f g x^{2} + \frac {g^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**2*(a+b*ln(c*(e*x+d)**n))**3,x)

[Out]

Piecewise((a**3*f**2*x + a**3*f*g*x**2 + a**3*g**2*x**3/3 + a**2*b*d**3*g**2*log(c*(d + e*x)**n)/e**3 - 3*a**2
*b*d**2*f*g*log(c*(d + e*x)**n)/e**2 - a**2*b*d**2*g**2*n*x/e**2 + 3*a**2*b*d*f**2*log(c*(d + e*x)**n)/e + 3*a
**2*b*d*f*g*n*x/e + a**2*b*d*g**2*n*x**2/(2*e) - 3*a**2*b*f**2*n*x + 3*a**2*b*f**2*x*log(c*(d + e*x)**n) - 3*a
**2*b*f*g*n*x**2/2 + 3*a**2*b*f*g*x**2*log(c*(d + e*x)**n) - a**2*b*g**2*n*x**3/3 + a**2*b*g**2*x**3*log(c*(d
+ e*x)**n) - 11*a*b**2*d**3*g**2*n*log(c*(d + e*x)**n)/(3*e**3) + a*b**2*d**3*g**2*log(c*(d + e*x)**n)**2/e**3
+ 9*a*b**2*d**2*f*g*n*log(c*(d + e*x)**n)/e**2 - 3*a*b**2*d**2*f*g*log(c*(d + e*x)**n)**2/e**2 + 11*a*b**2*d*
*2*g**2*n**2*x/(3*e**2) - 2*a*b**2*d**2*g**2*n*x*log(c*(d + e*x)**n)/e**2 - 6*a*b**2*d*f**2*n*log(c*(d + e*x)*
*n)/e + 3*a*b**2*d*f**2*log(c*(d + e*x)**n)**2/e - 9*a*b**2*d*f*g*n**2*x/e + 6*a*b**2*d*f*g*n*x*log(c*(d + e*x
)**n)/e - 5*a*b**2*d*g**2*n**2*x**2/(6*e) + a*b**2*d*g**2*n*x**2*log(c*(d + e*x)**n)/e + 6*a*b**2*f**2*n**2*x
- 6*a*b**2*f**2*n*x*log(c*(d + e*x)**n) + 3*a*b**2*f**2*x*log(c*(d + e*x)**n)**2 + 3*a*b**2*f*g*n**2*x**2/2 -
3*a*b**2*f*g*n*x**2*log(c*(d + e*x)**n) + 3*a*b**2*f*g*x**2*log(c*(d + e*x)**n)**2 + 2*a*b**2*g**2*n**2*x**3/9
- 2*a*b**2*g**2*n*x**3*log(c*(d + e*x)**n)/3 + a*b**2*g**2*x**3*log(c*(d + e*x)**n)**2 + 85*b**3*d**3*g**2*n*
*2*log(c*(d + e*x)**n)/(18*e**3) - 11*b**3*d**3*g**2*n*log(c*(d + e*x)**n)**2/(6*e**3) + b**3*d**3*g**2*log(c*
(d + e*x)**n)**3/(3*e**3) - 21*b**3*d**2*f*g*n**2*log(c*(d + e*x)**n)/(2*e**2) + 9*b**3*d**2*f*g*n*log(c*(d +
e*x)**n)**2/(2*e**2) - b**3*d**2*f*g*log(c*(d + e*x)**n)**3/e**2 - 85*b**3*d**2*g**2*n**3*x/(18*e**2) + 11*b**
3*d**2*g**2*n**2*x*log(c*(d + e*x)**n)/(3*e**2) - b**3*d**2*g**2*n*x*log(c*(d + e*x)**n)**2/e**2 + 6*b**3*d*f*
*2*n**2*log(c*(d + e*x)**n)/e - 3*b**3*d*f**2*n*log(c*(d + e*x)**n)**2/e + b**3*d*f**2*log(c*(d + e*x)**n)**3/
e + 21*b**3*d*f*g*n**3*x/(2*e) - 9*b**3*d*f*g*n**2*x*log(c*(d + e*x)**n)/e + 3*b**3*d*f*g*n*x*log(c*(d + e*x)*
*n)**2/e + 19*b**3*d*g**2*n**3*x**2/(36*e) - 5*b**3*d*g**2*n**2*x**2*log(c*(d + e*x)**n)/(6*e) + b**3*d*g**2*n
*x**2*log(c*(d + e*x)**n)**2/(2*e) - 6*b**3*f**2*n**3*x + 6*b**3*f**2*n**2*x*log(c*(d + e*x)**n) - 3*b**3*f**2
*n*x*log(c*(d + e*x)**n)**2 + b**3*f**2*x*log(c*(d + e*x)**n)**3 - 3*b**3*f*g*n**3*x**2/4 + 3*b**3*f*g*n**2*x*
*2*log(c*(d + e*x)**n)/2 - 3*b**3*f*g*n*x**2*log(c*(d + e*x)**n)**2/2 + b**3*f*g*x**2*log(c*(d + e*x)**n)**3 -
2*b**3*g**2*n**3*x**3/27 + 2*b**3*g**2*n**2*x**3*log(c*(d + e*x)**n)/9 - b**3*g**2*n*x**3*log(c*(d + e*x)**n)
**2/3 + b**3*g**2*x**3*log(c*(d + e*x)**n)**3/3, Ne(e, 0)), ((a + b*log(c*d**n))**3*(f**2*x + f*g*x**2 + g**2*
x**3/3), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2992 vs. \(2 (435) = 870\).
time = 4.13, size = 2992, normalized size = 6.93 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*(a+b*log(c*(e*x+d)^n))^3,x, algorithm="giac")

[Out]

1/3*(x*e + d)^3*b^3*g^2*n^3*e^(-3)*log(x*e + d)^3 - (x*e + d)^2*b^3*d*g^2*n^3*e^(-3)*log(x*e + d)^3 + (x*e + d
)*b^3*d^2*g^2*n^3*e^(-3)*log(x*e + d)^3 - 1/3*(x*e + d)^3*b^3*g^2*n^3*e^(-3)*log(x*e + d)^2 + 3/2*(x*e + d)^2*
b^3*d*g^2*n^3*e^(-3)*log(x*e + d)^2 - 3*(x*e + d)*b^3*d^2*g^2*n^3*e^(-3)*log(x*e + d)^2 + (x*e + d)^2*b^3*f*g*
n^3*e^(-2)*log(x*e + d)^3 - 2*(x*e + d)*b^3*d*f*g*n^3*e^(-2)*log(x*e + d)^3 + (x*e + d)^3*b^3*g^2*n^2*e^(-3)*l
og(x*e + d)^2*log(c) - 3*(x*e + d)^2*b^3*d*g^2*n^2*e^(-3)*log(x*e + d)^2*log(c) + 3*(x*e + d)*b^3*d^2*g^2*n^2*
e^(-3)*log(x*e + d)^2*log(c) + 2/9*(x*e + d)^3*b^3*g^2*n^3*e^(-3)*log(x*e + d) - 3/2*(x*e + d)^2*b^3*d*g^2*n^3
*e^(-3)*log(x*e + d) + 6*(x*e + d)*b^3*d^2*g^2*n^3*e^(-3)*log(x*e + d) - 3/2*(x*e + d)^2*b^3*f*g*n^3*e^(-2)*lo
g(x*e + d)^2 + 6*(x*e + d)*b^3*d*f*g*n^3*e^(-2)*log(x*e + d)^2 + (x*e + d)^3*a*b^2*g^2*n^2*e^(-3)*log(x*e + d)
^2 - 3*(x*e + d)^2*a*b^2*d*g^2*n^2*e^(-3)*log(x*e + d)^2 + 3*(x*e + d)*a*b^2*d^2*g^2*n^2*e^(-3)*log(x*e + d)^2
+ (x*e + d)*b^3*f^2*n^3*e^(-1)*log(x*e + d)^3 - 2/3*(x*e + d)^3*b^3*g^2*n^2*e^(-3)*log(x*e + d)*log(c) + 3*(x
*e + d)^2*b^3*d*g^2*n^2*e^(-3)*log(x*e + d)*log(c) - 6*(x*e + d)*b^3*d^2*g^2*n^2*e^(-3)*log(x*e + d)*log(c) +
3*(x*e + d)^2*b^3*f*g*n^2*e^(-2)*log(x*e + d)^2*log(c) - 6*(x*e + d)*b^3*d*f*g*n^2*e^(-2)*log(x*e + d)^2*log(c
) + (x*e + d)^3*b^3*g^2*n*e^(-3)*log(x*e + d)*log(c)^2 - 3*(x*e + d)^2*b^3*d*g^2*n*e^(-3)*log(x*e + d)*log(c)^
2 + 3*(x*e + d)*b^3*d^2*g^2*n*e^(-3)*log(x*e + d)*log(c)^2 - 2/27*(x*e + d)^3*b^3*g^2*n^3*e^(-3) + 3/4*(x*e +
d)^2*b^3*d*g^2*n^3*e^(-3) - 6*(x*e + d)*b^3*d^2*g^2*n^3*e^(-3) + 3/2*(x*e + d)^2*b^3*f*g*n^3*e^(-2)*log(x*e +
d) - 12*(x*e + d)*b^3*d*f*g*n^3*e^(-2)*log(x*e + d) - 2/3*(x*e + d)^3*a*b^2*g^2*n^2*e^(-3)*log(x*e + d) + 3*(x
*e + d)^2*a*b^2*d*g^2*n^2*e^(-3)*log(x*e + d) - 6*(x*e + d)*a*b^2*d^2*g^2*n^2*e^(-3)*log(x*e + d) - 3*(x*e + d
)*b^3*f^2*n^3*e^(-1)*log(x*e + d)^2 + 3*(x*e + d)^2*a*b^2*f*g*n^2*e^(-2)*log(x*e + d)^2 - 6*(x*e + d)*a*b^2*d*
f*g*n^2*e^(-2)*log(x*e + d)^2 + 2/9*(x*e + d)^3*b^3*g^2*n^2*e^(-3)*log(c) - 3/2*(x*e + d)^2*b^3*d*g^2*n^2*e^(-
3)*log(c) + 6*(x*e + d)*b^3*d^2*g^2*n^2*e^(-3)*log(c) - 3*(x*e + d)^2*b^3*f*g*n^2*e^(-2)*log(x*e + d)*log(c) +
12*(x*e + d)*b^3*d*f*g*n^2*e^(-2)*log(x*e + d)*log(c) + 2*(x*e + d)^3*a*b^2*g^2*n*e^(-3)*log(x*e + d)*log(c)
- 6*(x*e + d)^2*a*b^2*d*g^2*n*e^(-3)*log(x*e + d)*log(c) + 6*(x*e + d)*a*b^2*d^2*g^2*n*e^(-3)*log(x*e + d)*log
(c) + 3*(x*e + d)*b^3*f^2*n^2*e^(-1)*log(x*e + d)^2*log(c) - 1/3*(x*e + d)^3*b^3*g^2*n*e^(-3)*log(c)^2 + 3/2*(
x*e + d)^2*b^3*d*g^2*n*e^(-3)*log(c)^2 - 3*(x*e + d)*b^3*d^2*g^2*n*e^(-3)*log(c)^2 + 3*(x*e + d)^2*b^3*f*g*n*e
^(-2)*log(x*e + d)*log(c)^2 - 6*(x*e + d)*b^3*d*f*g*n*e^(-2)*log(x*e + d)*log(c)^2 + 1/3*(x*e + d)^3*b^3*g^2*e
^(-3)*log(c)^3 - (x*e + d)^2*b^3*d*g^2*e^(-3)*log(c)^3 + (x*e + d)*b^3*d^2*g^2*e^(-3)*log(c)^3 - 3/4*(x*e + d)
^2*b^3*f*g*n^3*e^(-2) + 12*(x*e + d)*b^3*d*f*g*n^3*e^(-2) + 2/9*(x*e + d)^3*a*b^2*g^2*n^2*e^(-3) - 3/2*(x*e +
d)^2*a*b^2*d*g^2*n^2*e^(-3) + 6*(x*e + d)*a*b^2*d^2*g^2*n^2*e^(-3) + 6*(x*e + d)*b^3*f^2*n^3*e^(-1)*log(x*e +
d) - 3*(x*e + d)^2*a*b^2*f*g*n^2*e^(-2)*log(x*e + d) + 12*(x*e + d)*a*b^2*d*f*g*n^2*e^(-2)*log(x*e + d) + (x*e
+ d)^3*a^2*b*g^2*n*e^(-3)*log(x*e + d) - 3*(x*e + d)^2*a^2*b*d*g^2*n*e^(-3)*log(x*e + d) + 3*(x*e + d)*a^2*b*
d^2*g^2*n*e^(-3)*log(x*e + d) + 3*(x*e + d)*a*b^2*f^2*n^2*e^(-1)*log(x*e + d)^2 + 3/2*(x*e + d)^2*b^3*f*g*n^2*
e^(-2)*log(c) - 12*(x*e + d)*b^3*d*f*g*n^2*e^(-2)*log(c) - 2/3*(x*e + d)^3*a*b^2*g^2*n*e^(-3)*log(c) + 3*(x*e
+ d)^2*a*b^2*d*g^2*n*e^(-3)*log(c) - 6*(x*e + d)*a*b^2*d^2*g^2*n*e^(-3)*log(c) - 6*(x*e + d)*b^3*f^2*n^2*e^(-1
)*log(x*e + d)*log(c) + 6*(x*e + d)^2*a*b^2*f*g*n*e^(-2)*log(x*e + d)*log(c) - 12*(x*e + d)*a*b^2*d*f*g*n*e^(-
2)*log(x*e + d)*log(c) - 3/2*(x*e + d)^2*b^3*f*g*n*e^(-2)*log(c)^2 + 6*(x*e + d)*b^3*d*f*g*n*e^(-2)*log(c)^2 +
(x*e + d)^3*a*b^2*g^2*e^(-3)*log(c)^2 - 3*(x*e + d)^2*a*b^2*d*g^2*e^(-3)*log(c)^2 + 3*(x*e + d)*a*b^2*d^2*g^2
*e^(-3)*log(c)^2 + 3*(x*e + d)*b^3*f^2*n*e^(-1)*log(x*e + d)*log(c)^2 + (x*e + d)^2*b^3*f*g*e^(-2)*log(c)^3 -
2*(x*e + d)*b^3*d*f*g*e^(-2)*log(c)^3 - 6*(x*e + d)*b^3*f^2*n^3*e^(-1) + 3/2*(x*e + d)^2*a*b^2*f*g*n^2*e^(-2)
- 12*(x*e + d)*a*b^2*d*f*g*n^2*e^(-2) - 1/3*(x*e + d)^3*a^2*b*g^2*n*e^(-3) + 3/2*(x*e + d)^2*a^2*b*d*g^2*n*e^(
-3) - 3*(x*e + d)*a^2*b*d^2*g^2*n*e^(-3) - 6*(x*e + d)*a*b^2*f^2*n^2*e^(-1)*log(x*e + d) + 3*(x*e + d)^2*a^2*b
*f*g*n*e^(-2)*log(x*e + d) - 6*(x*e + d)*a^2*b*d*f*g*n*e^(-2)*log(x*e + d) + 6*(x*e + d)*b^3*f^2*n^2*e^(-1)*lo
g(c) - 3*(x*e + d)^2*a*b^2*f*g*n*e^(-2)*log(c) + 12*(x*e + d)*a*b^2*d*f*g*n*e^(-2)*log(c) + (x*e + d)^3*a^2*b*
g^2*e^(-3)*log(c) - 3*(x*e + d)^2*a^2*b*d*g^2*e^(-3)*log(c) + 3*(x*e + d)*a^2*b*d^2*g^2*e^(-3)*log(c) + 6*(x*e
+ d)*a*b^2*f^2*n*e^(-1)*log(x*e + d)*log(c) - 3*(x*e + d)*b^3*f^2*n*e^(-1)*log(c)^2 + 3*(x*e + d)^2*a*b^2*f*g
*e^(-2)*log(c)^2 - 6*(x*e + d)*a*b^2*d*f*g*e^(-2)*log(c)^2 + (x*e + d)*b^3*f^2*e^(-1)*log(c)^3 + 6*(x*e + d)*a
*b^2*f^2*n^2*e^(-1) - 3/2*(x*e + d)^2*a^2*b*f*g...

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Mupad [B]
time = 0.86, size = 1157, normalized size = 2.68 \begin {gather*} {\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2\,\left (x^2\,\left (\frac {3\,b^2\,g\,\left (a\,d\,g+2\,a\,e\,f-b\,e\,f\,n\right )}{2\,e}-\frac {b^2\,d\,g^2\,\left (3\,a-b\,n\right )}{2\,e}\right )-x\,\left (\frac {d\,\left (\frac {3\,b^2\,g\,\left (a\,d\,g+2\,a\,e\,f-b\,e\,f\,n\right )}{e}-\frac {b^2\,d\,g^2\,\left (3\,a-b\,n\right )}{e}\right )}{e}-\frac {3\,b^2\,f\,\left (2\,a\,d\,g+a\,e\,f-b\,e\,f\,n\right )}{e}\right )+\frac {d\,\left (-11\,n\,b^3\,d^2\,g^2+27\,n\,b^3\,d\,e\,f\,g-18\,n\,b^3\,e^2\,f^2+6\,a\,b^2\,d^2\,g^2-18\,a\,b^2\,d\,e\,f\,g+18\,a\,b^2\,e^2\,f^2\right )}{6\,e^3}+\frac {b^2\,g^2\,x^3\,\left (3\,a-b\,n\right )}{3}\right )+x\,\left (\frac {36\,a^3\,d\,e\,f\,g+18\,a^3\,e^2\,f^2-54\,a^2\,b\,e^2\,f^2\,n+36\,a\,b^2\,d^2\,g^2\,n^2-108\,a\,b^2\,d\,e\,f\,g\,n^2+108\,a\,b^2\,e^2\,f^2\,n^2-66\,b^3\,d^2\,g^2\,n^3+162\,b^3\,d\,e\,f\,g\,n^3-108\,b^3\,e^2\,f^2\,n^3}{18\,e^2}-\frac {d\,\left (\frac {g\,\left (6\,a^3\,d\,g+12\,a^3\,e\,f+5\,b^3\,d\,g\,n^3-9\,b^3\,e\,f\,n^3-6\,a\,b^2\,d\,g\,n^2+18\,a\,b^2\,e\,f\,n^2-18\,a^2\,b\,e\,f\,n\right )}{6\,e}-\frac {d\,g^2\,\left (9\,a^3-9\,a^2\,b\,n+6\,a\,b^2\,n^2-2\,b^3\,n^3\right )}{9\,e}\right )}{e}\right )+x^2\,\left (\frac {g\,\left (6\,a^3\,d\,g+12\,a^3\,e\,f+5\,b^3\,d\,g\,n^3-9\,b^3\,e\,f\,n^3-6\,a\,b^2\,d\,g\,n^2+18\,a\,b^2\,e\,f\,n^2-18\,a^2\,b\,e\,f\,n\right )}{12\,e}-\frac {d\,g^2\,\left (9\,a^3-9\,a^2\,b\,n+6\,a\,b^2\,n^2-2\,b^3\,n^3\right )}{18\,e}\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^3\,\left (b^3\,f^2\,x+\frac {b^3\,g^2\,x^3}{3}+\frac {d\,\left (b^3\,d^2\,g^2-3\,b^3\,d\,e\,f\,g+3\,b^3\,e^2\,f^2\right )}{3\,e^3}+b^3\,f\,g\,x^2\right )+\frac {g^2\,x^3\,\left (9\,a^3-9\,a^2\,b\,n+6\,a\,b^2\,n^2-2\,b^3\,n^3\right )}{27}+\frac {\ln \left (d+e\,x\right )\,\left (18\,a^2\,b\,d^3\,g^2\,n-54\,a^2\,b\,d^2\,e\,f\,g\,n+54\,a^2\,b\,d\,e^2\,f^2\,n-66\,a\,b^2\,d^3\,g^2\,n^2+162\,a\,b^2\,d^2\,e\,f\,g\,n^2-108\,a\,b^2\,d\,e^2\,f^2\,n^2+85\,b^3\,d^3\,g^2\,n^3-189\,b^3\,d^2\,e\,f\,g\,n^3+108\,b^3\,d\,e^2\,f^2\,n^3\right )}{18\,e^3}+\frac {\ln \left (c\,{\left (d+e\,x\right )}^n\right )\,\left (\frac {x^2\,\left (9\,b\,e\,g\,\left (3\,a^2\,d\,g+6\,a^2\,e\,f-b^2\,d\,g\,n^2+3\,b^2\,e\,f\,n^2-6\,a\,b\,e\,f\,n\right )-3\,b\,d\,e\,g^2\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )\right )}{6\,e}+\frac {x\,\left (\frac {54\,a^2\,b\,d\,e^2\,f\,g+27\,a^2\,b\,e^3\,f^2-54\,a\,b^2\,e^3\,f^2\,n+18\,b^3\,d^2\,e\,g^2\,n^2-54\,b^3\,d\,e^2\,f\,g\,n^2+54\,b^3\,e^3\,f^2\,n^2}{e}-\frac {d\,\left (9\,b\,e\,g\,\left (3\,a^2\,d\,g+6\,a^2\,e\,f-b^2\,d\,g\,n^2+3\,b^2\,e\,f\,n^2-6\,a\,b\,e\,f\,n\right )-3\,b\,d\,e\,g^2\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )\right )}{e}\right )}{3\,e}+\frac {b\,e\,g^2\,x^3\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )}{3}\right )}{3\,e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f + g*x)^2*(a + b*log(c*(d + e*x)^n))^3,x)

[Out]

log(c*(d + e*x)^n)^2*(x^2*((3*b^2*g*(a*d*g + 2*a*e*f - b*e*f*n))/(2*e) - (b^2*d*g^2*(3*a - b*n))/(2*e)) - x*((
d*((3*b^2*g*(a*d*g + 2*a*e*f - b*e*f*n))/e - (b^2*d*g^2*(3*a - b*n))/e))/e - (3*b^2*f*(2*a*d*g + a*e*f - b*e*f
*n))/e) + (d*(6*a*b^2*d^2*g^2 + 18*a*b^2*e^2*f^2 - 11*b^3*d^2*g^2*n - 18*b^3*e^2*f^2*n - 18*a*b^2*d*e*f*g + 27
*b^3*d*e*f*g*n))/(6*e^3) + (b^2*g^2*x^3*(3*a - b*n))/3) + x*((18*a^3*e^2*f^2 - 66*b^3*d^2*g^2*n^3 - 108*b^3*e^
2*f^2*n^3 - 54*a^2*b*e^2*f^2*n + 36*a^3*d*e*f*g + 36*a*b^2*d^2*g^2*n^2 + 108*a*b^2*e^2*f^2*n^2 + 162*b^3*d*e*f
*g*n^3 - 108*a*b^2*d*e*f*g*n^2)/(18*e^2) - (d*((g*(6*a^3*d*g + 12*a^3*e*f + 5*b^3*d*g*n^3 - 9*b^3*e*f*n^3 - 6*
a*b^2*d*g*n^2 + 18*a*b^2*e*f*n^2 - 18*a^2*b*e*f*n))/(6*e) - (d*g^2*(9*a^3 - 2*b^3*n^3 + 6*a*b^2*n^2 - 9*a^2*b*
n))/(9*e)))/e) + x^2*((g*(6*a^3*d*g + 12*a^3*e*f + 5*b^3*d*g*n^3 - 9*b^3*e*f*n^3 - 6*a*b^2*d*g*n^2 + 18*a*b^2*
e*f*n^2 - 18*a^2*b*e*f*n))/(12*e) - (d*g^2*(9*a^3 - 2*b^3*n^3 + 6*a*b^2*n^2 - 9*a^2*b*n))/(18*e)) + log(c*(d +
e*x)^n)^3*(b^3*f^2*x + (b^3*g^2*x^3)/3 + (d*(b^3*d^2*g^2 + 3*b^3*e^2*f^2 - 3*b^3*d*e*f*g))/(3*e^3) + b^3*f*g*
x^2) + (g^2*x^3*(9*a^3 - 2*b^3*n^3 + 6*a*b^2*n^2 - 9*a^2*b*n))/27 + (log(d + e*x)*(85*b^3*d^3*g^2*n^3 + 18*a^2
*b*d^3*g^2*n - 66*a*b^2*d^3*g^2*n^2 + 108*b^3*d*e^2*f^2*n^3 - 108*a*b^2*d*e^2*f^2*n^2 + 54*a^2*b*d*e^2*f^2*n -
189*b^3*d^2*e*f*g*n^3 + 162*a*b^2*d^2*e*f*g*n^2 - 54*a^2*b*d^2*e*f*g*n))/(18*e^3) + (log(c*(d + e*x)^n)*((x^2
*(9*b*e*g*(3*a^2*d*g + 6*a^2*e*f - b^2*d*g*n^2 + 3*b^2*e*f*n^2 - 6*a*b*e*f*n) - 3*b*d*e*g^2*(9*a^2 + 2*b^2*n^2
- 6*a*b*n)))/(6*e) + (x*((27*a^2*b*e^3*f^2 + 54*b^3*e^3*f^2*n^2 - 54*a*b^2*e^3*f^2*n + 18*b^3*d^2*e*g^2*n^2 +
54*a^2*b*d*e^2*f*g - 54*b^3*d*e^2*f*g*n^2)/e - (d*(9*b*e*g*(3*a^2*d*g + 6*a^2*e*f - b^2*d*g*n^2 + 3*b^2*e*f*n
^2 - 6*a*b*e*f*n) - 3*b*d*e*g^2*(9*a^2 + 2*b^2*n^2 - 6*a*b*n)))/e))/(3*e) + (b*e*g^2*x^3*(9*a^2 + 2*b^2*n^2 -
6*a*b*n))/3))/(3*e)

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