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

Optimal result

Integrand size = 24, antiderivative size = 432 \[ \int (f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\frac {6 a b^2 (e f-d g)^2 n^2 x}{e^2}-\frac {6 b^3 (e f-d g)^2 n^3 x}{e^2}-\frac {3 b^3 g (e f-d g) n^3 (d+e x)^2}{4 e^3}-\frac {2 b^3 g^2 n^3 (d+e x)^3}{27 e^3}+\frac {6 b^3 (e f-d g)^2 n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e^3}+\frac {3 b^2 g (e f-d g) n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^3}+\frac {2 b^2 g^2 n^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 e^3}-\frac {3 b (e f-d g)^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^3}-\frac {3 b g (e f-d g) n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^3}-\frac {b g^2 n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 e^3}+\frac {(e f-d g)^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac {g (e f-d g) (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac {g^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 e^3} \]

[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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2448, 2436, 2333, 2332, 2437, 2342, 2341} \[ \int (f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\frac {3 b^2 g n^2 (d+e x)^2 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^3}+\frac {2 b^2 g^2 n^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 e^3}+\frac {6 a b^2 n^2 x (e f-d g)^2}{e^2}-\frac {3 b g n (d+e x)^2 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^3}-\frac {3 b n (d+e x) (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^3}+\frac {g (d+e x)^2 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac {(d+e x) (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}-\frac {b g^2 n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 e^3}+\frac {g^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 e^3}+\frac {6 b^3 n^2 (d+e x) (e f-d g)^2 \log \left (c (d+e x)^n\right )}{e^3}-\frac {3 b^3 g n^3 (d+e x)^2 (e f-d g)}{4 e^3}-\frac {2 b^3 g^2 n^3 (d+e x)^3}{27 e^3}-\frac {6 b^3 n^3 x (e f-d g)^2}{e^2} \]

[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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 333, normalized size of antiderivative = 0.77 \[ \int (f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\frac {108 (e f-d g)^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3+108 g (e f-d g) (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3+36 g^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3-324 b (e f-d g)^2 n \left ((d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-2 b n \left (e (a-b n) x+b (d+e x) \log \left (c (d+e x)^n\right )\right )\right )-81 b g (e f-d g) n \left (2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2+b n \left (b e n x (2 d+e x)-2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )\right )\right )-4 b g^2 n \left (9 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2+2 b n \left (b e n x \left (3 d^2+3 d e x+e^2 x^2\right )-3 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )\right )\right )}{108 e^3} \]

[In]

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

[Out]

(108*(e*f - d*g)^2*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3 + 108*g*(e*f - d*g)*(d + e*x)^2*(a + b*Log[c*(d + e*
x)^n])^3 + 36*g^2*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])^3 - 324*b*(e*f - d*g)^2*n*((d + e*x)*(a + b*Log[c*(d
+ e*x)^n])^2 - 2*b*n*(e*(a - b*n)*x + b*(d + e*x)*Log[c*(d + e*x)^n])) - 81*b*g*(e*f - d*g)*n*(2*(d + e*x)^2*(
a + b*Log[c*(d + e*x)^n])^2 + b*n*(b*e*n*x*(2*d + e*x) - 2*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))) - 4*b*g^2*
n*(9*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])^2 + 2*b*n*(b*e*n*x*(3*d^2 + 3*d*e*x + e^2*x^2) - 3*(d + e*x)^3*(a
+ b*Log[c*(d + e*x)^n]))))/(108*e^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1732\) vs. \(2(418)=836\).

Time = 3.55 (sec) , antiderivative size = 1733, normalized size of antiderivative = 4.01

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1733\)
risch	\(\text {Expression too large to display}\)	\(20417\)

[In]

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

[Out]

1/108*(-108*a^2*b*d^2*e*g^2*n*x-81*b^3*e^3*f*g*n^3*x^2-36*a^2*b*e^3*g^2*n*x^3-510*b^3*d^2*e*g^2*n^3*x+36*x^3*l
n(c*(e*x+d)^n)^3*b^3*e^3*g^2+108*x*ln(c*(e*x+d)^n)^3*b^3*e^3*f^2+108*ln(c*(e*x+d)^n)^3*b^3*d*e^2*f^2-198*ln(c*
(e*x+d)^n)^2*b^3*d^3*g^2*n-396*ln(c*(e*x+d)^n)*b^3*d^3*g^2*n^2+108*ln(c*(e*x+d)^n)^2*a*b^2*d^3*g^2-90*a*b^2*d*
e^2*g^2*n^2*x^2+162*a*b^2*e^3*f*g*n^2*x^2+1134*b^3*d*e^2*f*g*n^3*x-162*a^2*b*e^3*f*g*n*x^2+396*a*b^2*d^2*e*g^2
*n^2*x-648*b^3*e^3*f^2*n^3*x+648*b^3*d*e^2*f^2*n^3-396*a*b^2*d^3*g^2*n^2+648*a*b^2*e^3*f^2*n^2*x-324*a^2*b*e^3
*f^2*n*x+108*a^2*b*d^3*g^2*n-1134*b^3*d^2*e*f*g*n^3-648*a*b^2*d*e^2*f^2*n^2+36*a^3*e^3*g^2*x^3+324*a^2*b*d*e^2
*f*g*n*x-8*b^3*e^3*g^2*n^3*x^3+108*a^3*e^3*f*g*x^2+24*a*b^2*e^3*g^2*n^2*x^3+57*b^3*d*e^2*g^2*n^3*x^2-108*a^3*d
*e^2*f^2+108*a^3*e^3*f^2*x+510*b^3*d^3*g^2*n^3+906*ln(e*x+d)*b^3*d^3*g^2*n^3-972*a*b^2*d*e^2*f*g*n^2*x+324*a^2
*b*d*e^2*f^2*n+36*ln(c*(e*x+d)^n)^3*b^3*d^3*g^2+972*a*b^2*d^2*e*f*g*n^2+108*x^2*ln(c*(e*x+d)^n)*a*b^2*d*e^2*g^
2*n+1296*ln(e*x+d)*b^3*d*e^2*f^2*n^3-612*ln(e*x+d)*a*b^2*d^3*g^2*n^2+108*ln(e*x+d)*a^2*b*d^3*g^2*n+324*x*ln(c*
(e*x+d)^n)^2*a*b^2*e^3*f^2-108*ln(c*(e*x+d)^n)^3*b^3*d^2*e*f*g-324*ln(c*(e*x+d)^n)^2*b^3*d*e^2*f^2*n-648*ln(c*
(e*x+d)^n)*b^3*d*e^2*f^2*n^2+324*x*ln(c*(e*x+d)^n)*a^2*b*e^3*f^2+324*ln(c*(e*x+d)^n)^2*a*b^2*d*e^2*f^2+216*ln(
c*(e*x+d)^n)*a*b^2*d^3*g^2*n-324*ln(c*(e*x+d)^n)*a^2*b*d*e^2*f^2-324*a^2*b*d^2*e*f*g*n-324*x^2*ln(c*(e*x+d)^n)
*a*b^2*e^3*f*g*n+324*x*ln(c*(e*x+d)^n)^2*b^3*d*e^2*f*g*n-972*x*ln(c*(e*x+d)^n)*b^3*d*e^2*f*g*n^2-216*x*ln(c*(e
*x+d)^n)*a*b^2*d^2*e*g^2*n-648*ln(c*(e*x+d)^n)*a*b^2*d^2*e*f*g*n-36*x^3*ln(c*(e*x+d)^n)^2*b^3*e^3*g^2*n+24*x^3
*ln(c*(e*x+d)^n)*b^3*e^3*g^2*n^2+108*x^3*ln(c*(e*x+d)^n)^2*a*b^2*e^3*g^2+108*x^2*ln(c*(e*x+d)^n)^3*b^3*e^3*f*g
+108*x^3*ln(c*(e*x+d)^n)*a^2*b*e^3*g^2-324*x*ln(c*(e*x+d)^n)^2*b^3*e^3*f^2*n+648*x*ln(c*(e*x+d)^n)*b^3*e^3*f^2
*n^2-72*x^3*ln(c*(e*x+d)^n)*a*b^2*e^3*g^2*n+54*x^2*ln(c*(e*x+d)^n)^2*b^3*d*e^2*g^2*n-162*x^2*ln(c*(e*x+d)^n)^2
*b^3*e^3*f*g*n-90*x^2*ln(c*(e*x+d)^n)*b^3*d*e^2*g^2*n^2+162*x^2*ln(c*(e*x+d)^n)*b^3*e^3*f*g*n^2+324*x^2*ln(c*(
e*x+d)^n)^2*a*b^2*e^3*f*g-108*x*ln(c*(e*x+d)^n)^2*b^3*d^2*e*g^2*n+396*x*ln(c*(e*x+d)^n)*b^3*d^2*e*g^2*n^2+324*
x^2*ln(c*(e*x+d)^n)*a^2*b*e^3*f*g-648*x*ln(c*(e*x+d)^n)*a*b^2*e^3*f^2*n+486*ln(c*(e*x+d)^n)^2*b^3*d^2*e*f*g*n+
972*ln(c*(e*x+d)^n)*b^3*d^2*e*f*g*n^2-324*ln(c*(e*x+d)^n)^2*a*b^2*d^2*e*f*g+648*ln(c*(e*x+d)^n)*a*b^2*d*e^2*f^
2*n+648*x*ln(c*(e*x+d)^n)*a*b^2*d*e^2*f*g*n+1620*ln(e*x+d)*a*b^2*d^2*e*f*g*n^2-324*ln(e*x+d)*a^2*b*d^2*e*f*g*n
+54*a^2*b*d*e^2*g^2*n*x^2-2106*ln(e*x+d)*b^3*d^2*e*f*g*n^3-1296*ln(e*x+d)*a*b^2*d*e^2*f^2*n^2+648*ln(e*x+d)*a^
2*b*d*e^2*f^2*n)/e^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1771 vs. \(2 (418) = 836\).

Time = 0.34 (sec) , antiderivative size = 1771, normalized size of antiderivative = 4.10 \[ \int (f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/108*(4*(2*b^3*e^3*g^2*n^3 - 6*a*b^2*e^3*g^2*n^2 + 9*a^2*b*e^3*g^2*n - 9*a^3*e^3*g^2)*x^3 - 36*(b^3*e^3*g^2*
n^3*x^3 + 3*b^3*e^3*f*g*n^3*x^2 + 3*b^3*e^3*f^2*n^3*x + (3*b^3*d*e^2*f^2 - 3*b^3*d^2*e*f*g + b^3*d^3*g^2)*n^3)
*log(e*x + d)^3 - 36*(b^3*e^3*g^2*x^3 + 3*b^3*e^3*f*g*x^2 + 3*b^3*e^3*f^2*x)*log(c)^3 - 3*(36*a^3*e^3*f*g - (2
7*b^3*e^3*f*g - 19*b^3*d*e^2*g^2)*n^3 + 6*(9*a*b^2*e^3*f*g - 5*a*b^2*d*e^2*g^2)*n^2 - 18*(3*a^2*b*e^3*f*g - a^
2*b*d*e^2*g^2)*n)*x^2 + 18*((18*b^3*d*e^2*f^2 - 27*b^3*d^2*e*f*g + 11*b^3*d^3*g^2)*n^3 + 2*(b^3*e^3*g^2*n^3 -
3*a*b^2*e^3*g^2*n^2)*x^3 - 6*(3*a*b^2*d*e^2*f^2 - 3*a*b^2*d^2*e*f*g + a*b^2*d^3*g^2)*n^2 - 3*(6*a*b^2*e^3*f*g*
n^2 - (3*b^3*e^3*f*g - b^3*d*e^2*g^2)*n^3)*x^2 - 6*(3*a*b^2*e^3*f^2*n^2 - (3*b^3*e^3*f^2 - 3*b^3*d*e^2*f*g + b
^3*d^2*e*g^2)*n^3)*x - 6*(b^3*e^3*g^2*n^2*x^3 + 3*b^3*e^3*f*g*n^2*x^2 + 3*b^3*e^3*f^2*n^2*x + (3*b^3*d*e^2*f^2
- 3*b^3*d^2*e*f*g + b^3*d^3*g^2)*n^2)*log(c))*log(e*x + d)^2 + 18*(2*(b^3*e^3*g^2*n - 3*a*b^2*e^3*g^2)*x^3 -
3*(6*a*b^2*e^3*f*g - (3*b^3*e^3*f*g - b^3*d*e^2*g^2)*n)*x^2 - 6*(3*a*b^2*e^3*f^2 - (3*b^3*e^3*f^2 - 3*b^3*d*e^
2*f*g + b^3*d^2*e*g^2)*n)*x)*log(c)^2 - 6*(18*a^3*e^3*f^2 - (108*b^3*e^3*f^2 - 189*b^3*d*e^2*f*g + 85*b^3*d^2*
e*g^2)*n^3 + 6*(18*a*b^2*e^3*f^2 - 27*a*b^2*d*e^2*f*g + 11*a*b^2*d^2*e*g^2)*n^2 - 18*(3*a^2*b*e^3*f^2 - 3*a^2*
b*d*e^2*f*g + a^2*b*d^2*e*g^2)*n)*x - 6*((108*b^3*d*e^2*f^2 - 189*b^3*d^2*e*f*g + 85*b^3*d^3*g^2)*n^3 + 2*(2*b
^3*e^3*g^2*n^3 - 6*a*b^2*e^3*g^2*n^2 + 9*a^2*b*e^3*g^2*n)*x^3 - 6*(18*a*b^2*d*e^2*f^2 - 27*a*b^2*d^2*e*f*g + 1
1*a*b^2*d^3*g^2)*n^2 + 3*(18*a^2*b*e^3*f*g*n + (9*b^3*e^3*f*g - 5*b^3*d*e^2*g^2)*n^3 - 6*(3*a*b^2*e^3*f*g - a*
b^2*d*e^2*g^2)*n^2)*x^2 + 18*(b^3*e^3*g^2*n*x^3 + 3*b^3*e^3*f*g*n*x^2 + 3*b^3*e^3*f^2*n*x + (3*b^3*d*e^2*f^2 -
3*b^3*d^2*e*f*g + b^3*d^3*g^2)*n)*log(c)^2 + 18*(3*a^2*b*d*e^2*f^2 - 3*a^2*b*d^2*e*f*g + a^2*b*d^3*g^2)*n + 6
*(9*a^2*b*e^3*f^2*n + (18*b^3*e^3*f^2 - 27*b^3*d*e^2*f*g + 11*b^3*d^2*e*g^2)*n^3 - 6*(3*a*b^2*e^3*f^2 - 3*a*b^
2*d*e^2*f*g + a*b^2*d^2*e*g^2)*n^2)*x - 6*(2*(b^3*e^3*g^2*n^2 - 3*a*b^2*e^3*g^2*n)*x^3 + (18*b^3*d*e^2*f^2 - 2
7*b^3*d^2*e*f*g + 11*b^3*d^3*g^2)*n^2 - 3*(6*a*b^2*e^3*f*g*n - (3*b^3*e^3*f*g - b^3*d*e^2*g^2)*n^2)*x^2 - 6*(3
*a*b^2*d*e^2*f^2 - 3*a*b^2*d^2*e*f*g + a*b^2*d^3*g^2)*n - 6*(3*a*b^2*e^3*f^2*n - (3*b^3*e^3*f^2 - 3*b^3*d*e^2*
f*g + b^3*d^2*e*g^2)*n^2)*x)*log(c))*log(e*x + d) - 6*(2*(2*b^3*e^3*g^2*n^2 - 6*a*b^2*e^3*g^2*n + 9*a^2*b*e^3*
g^2)*x^3 + 3*(18*a^2*b*e^3*f*g + (9*b^3*e^3*f*g - 5*b^3*d*e^2*g^2)*n^2 - 6*(3*a*b^2*e^3*f*g - a*b^2*d*e^2*g^2)
*n)*x^2 + 6*(9*a^2*b*e^3*f^2 + (18*b^3*e^3*f^2 - 27*b^3*d*e^2*f*g + 11*b^3*d^2*e*g^2)*n^2 - 6*(3*a*b^2*e^3*f^2
- 3*a*b^2*d*e^2*f*g + a*b^2*d^2*e*g^2)*n)*x)*log(c))/e^3

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1578 vs. \(2 (422) = 844\).

Time = 2.41 (sec) , antiderivative size = 1578, normalized size of antiderivative = 3.65 \[ \int (f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\text {Too large to display} \]

[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))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1140 vs. \(2 (418) = 836\).

Time = 0.24 (sec) , antiderivative size = 1140, normalized size of antiderivative = 2.64 \[ \int (f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\text {Too large to display} \]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2932 vs. \(2 (418) = 836\).

Time = 0.38 (sec) , antiderivative size = 2932, normalized size of antiderivative = 6.79 \[ \int (f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.97 (sec) , antiderivative size = 1157, normalized size of antiderivative = 2.68 \[ \int (f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx={\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} \]

[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)