∫ (f + gx)3(a + b log(c(d + ex)n))3 dx [52]

3.1.52 \(\int (f+g x)^3 (a+b \log (c (d+e x)^n))^3 \, dx\) [52]

Optimal. Leaf size=598 \[ \frac {6 a b^2 (e f-d g)^3 n^2 x}{e^3}-\frac {6 b^3 (e f-d g)^3 n^3 x}{e^3}-\frac {9 b^3 g (e f-d g)^2 n^3 (d+e x)^2}{8 e^4}-\frac {2 b^3 g^2 (e f-d g) n^3 (d+e x)^3}{9 e^4}-\frac {3 b^3 g^3 n^3 (d+e x)^4}{128 e^4}+\frac {6 b^3 (e f-d g)^3 n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e^4}+\frac {9 b^2 g (e f-d g)^2 n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^4}+\frac {2 b^2 g^2 (e f-d g) n^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^4}+\frac {3 b^2 g^3 n^2 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{32 e^4}-\frac {3 b (e f-d g)^3 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^4}-\frac {9 b g (e f-d g)^2 n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^4}-\frac {b g^2 (e f-d g) n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^4}-\frac {3 b g^3 n (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{16 e^4}+\frac {(e f-d g)^3 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^4}+\frac {3 g (e f-d g)^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^4}+\frac {g^2 (e f-d g) (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^4}+\frac {g^3 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{4 e^4} \]

[Out]

6*a*b^2*(-d*g+e*f)^3*n^2*x/e^3-6*b^3*(-d*g+e*f)^3*n^3*x/e^3-9/8*b^3*g*(-d*g+e*f)^2*n^3*(e*x+d)^2/e^4-2/9*b^3*g

^2*(-d*g+e*f)*n^3*(e*x+d)^3/e^4-3/128*b^3*g^3*n^3*(e*x+d)^4/e^4+6*b^3*(-d*g+e*f)^3*n^2*(e*x+d)*ln(c*(e*x+d)^n)

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

ln(c*(e*x+d)^n))/e^4+3/32*b^2*g^3*n^2*(e*x+d)^4*(a+b*ln(c*(e*x+d)^n))/e^4-3*b*(-d*g+e*f)^3*n*(e*x+d)*(a+b*ln(c

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

a+b*ln(c*(e*x+d)^n))^2/e^4-3/16*b*g^3*n*(e*x+d)^4*(a+b*ln(c*(e*x+d)^n))^2/e^4+(-d*g+e*f)^3*(e*x+d)*(a+b*ln(c*(

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

e*x+d)^n))^3/e^4+1/4*g^3*(e*x+d)^4*(a+b*ln(c*(e*x+d)^n))^3/e^4

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Rubi [A]
time = 0.39, antiderivative size = 598, normalized size of antiderivative = 1.00, number of steps used = 19, 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 {2 b^2 g^2 n^2 (d+e x)^3 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^4}+\frac {9 b^2 g n^2 (d+e x)^2 (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^4}+\frac {3 b^2 g^3 n^2 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{32 e^4}+\frac {6 a b^2 n^2 x (e f-d g)^3}{e^3}+\frac {g^2 (d+e x)^3 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^4}-\frac {b g^2 n (d+e x)^3 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^4}+\frac {3 g (d+e x)^2 (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^4}-\frac {9 b g n (d+e x)^2 (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^4}+\frac {(d+e x) (e f-d g)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^4}-\frac {3 b n (d+e x) (e f-d g)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^4}+\frac {g^3 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{4 e^4}-\frac {3 b g^3 n (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{16 e^4}+\frac {6 b^3 n^2 (d+e x) (e f-d g)^3 \log \left (c (d+e x)^n\right )}{e^4}-\frac {2 b^3 g^2 n^3 (d+e x)^3 (e f-d g)}{9 e^4}-\frac {9 b^3 g n^3 (d+e x)^2 (e f-d g)^2}{8 e^4}-\frac {3 b^3 g^3 n^3 (d+e x)^4}{128 e^4}-\frac {6 b^3 n^3 x (e f-d g)^3}{e^3} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(6*a*b^2*(e*f - d*g)^3*n^2*x)/e^3 - (6*b^3*(e*f - d*g)^3*n^3*x)/e^3 - (9*b^3*g*(e*f - d*g)^2*n^3*(d + e*x)^2)/

(8*e^4) - (2*b^3*g^2*(e*f - d*g)*n^3*(d + e*x)^3)/(9*e^4) - (3*b^3*g^3*n^3*(d + e*x)^4)/(128*e^4) + (6*b^3*(e*

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

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

(d + e*x)^4*(a + b*Log[c*(d + e*x)^n]))/(32*e^4) - (3*b*(e*f - d*g)^3*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2

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

*x)^3*(a + b*Log[c*(d + e*x)^n])^2)/e^4 - (3*b*g^3*n*(d + e*x)^4*(a + b*Log[c*(d + e*x)^n])^2)/(16*e^4) + ((e*

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

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

[c*(d + e*x)^n])^3)/(4*e^4)

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)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx &=\int \left (\frac {(e f-d g)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac {3 g (e f-d g)^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac {3 g^2 (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^3 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}\right ) \, dx\\ &=\frac {g^3 \int (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx}{e^3}+\frac {\left (3 g^2 (e f-d g)\right ) \int (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx}{e^3}+\frac {\left (3 g (e f-d g)^2\right ) \int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx}{e^3}+\frac {(e f-d g)^3 \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx}{e^3}\\ &=\frac {g^3 \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e^4}+\frac {\left (3 g^2 (e f-d g)\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e^4}+\frac {\left (3 g (e f-d g)^2\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e^4}+\frac {(e f-d g)^3 \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e^4}\\ &=\frac {(e f-d g)^3 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^4}+\frac {3 g (e f-d g)^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^4}+\frac {g^2 (e f-d g) (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^4}+\frac {g^3 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{4 e^4}-\frac {\left (3 b g^3 n\right ) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{4 e^4}-\frac {\left (3 b g^2 (e f-d g) 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^4}-\frac {\left (9 b g (e f-d g)^2 n\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{2 e^4}-\frac {\left (3 b (e f-d g)^3 n\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e^4}\\ &=-\frac {3 b (e f-d g)^3 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^4}-\frac {9 b g (e f-d g)^2 n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^4}-\frac {b g^2 (e f-d g) n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^4}-\frac {3 b g^3 n (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{16 e^4}+\frac {(e f-d g)^3 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^4}+\frac {3 g (e f-d g)^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^4}+\frac {g^2 (e f-d g) (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^4}+\frac {g^3 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{4 e^4}+\frac {\left (3 b^2 g^3 n^2\right ) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{8 e^4}+\frac {\left (2 b^2 g^2 (e f-d g) 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 )}{e^4}+\frac {\left (9 b^2 g (e f-d g)^2 n^2\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{2 e^4}+\frac {\left (6 b^2 (e f-d g)^3 n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^4}\\ &=\frac {6 a b^2 (e f-d g)^3 n^2 x}{e^3}-\frac {9 b^3 g (e f-d g)^2 n^3 (d+e x)^2}{8 e^4}-\frac {2 b^3 g^2 (e f-d g) n^3 (d+e x)^3}{9 e^4}-\frac {3 b^3 g^3 n^3 (d+e x)^4}{128 e^4}+\frac {9 b^2 g (e f-d g)^2 n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^4}+\frac {2 b^2 g^2 (e f-d g) n^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^4}+\frac {3 b^2 g^3 n^2 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{32 e^4}-\frac {3 b (e f-d g)^3 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^4}-\frac {9 b g (e f-d g)^2 n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^4}-\frac {b g^2 (e f-d g) n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^4}-\frac {3 b g^3 n (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{16 e^4}+\frac {(e f-d g)^3 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^4}+\frac {3 g (e f-d g)^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^4}+\frac {g^2 (e f-d g) (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^4}+\frac {g^3 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{4 e^4}+\frac {\left (6 b^3 (e f-d g)^3 n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e^4}\\ &=\frac {6 a b^2 (e f-d g)^3 n^2 x}{e^3}-\frac {6 b^3 (e f-d g)^3 n^3 x}{e^3}-\frac {9 b^3 g (e f-d g)^2 n^3 (d+e x)^2}{8 e^4}-\frac {2 b^3 g^2 (e f-d g) n^3 (d+e x)^3}{9 e^4}-\frac {3 b^3 g^3 n^3 (d+e x)^4}{128 e^4}+\frac {6 b^3 (e f-d g)^3 n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e^4}+\frac {9 b^2 g (e f-d g)^2 n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^4}+\frac {2 b^2 g^2 (e f-d g) n^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^4}+\frac {3 b^2 g^3 n^2 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{32 e^4}-\frac {3 b (e f-d g)^3 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^4}-\frac {9 b g (e f-d g)^2 n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^4}-\frac {b g^2 (e f-d g) n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^4}-\frac {3 b g^3 n (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{16 e^4}+\frac {(e f-d g)^3 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^4}+\frac {3 g (e f-d g)^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^4}+\frac {g^2 (e f-d g) (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^4}+\frac {g^3 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{4 e^4}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1241\) vs. \(2(598)=1196\).
time = 0.83, size = 1241, normalized size = 2.08 \begin {gather*} \frac {-288 b^3 d \left (-4 e^3 f^3+6 d e^2 f^2 g-4 d^2 e f g^2+d^3 g^3\right ) n^3 \log ^3(d+e x)+72 b^2 d n^2 \log ^2(d+e x) \left (-12 a \left (4 e^3 f^3-6 d e^2 f^2 g+4 d^2 e f g^2-d^3 g^3\right )+b \left (48 e^3 f^3-108 d e^2 f^2 g+88 d^2 e f g^2-25 d^3 g^3\right ) n-12 b \left (4 e^3 f^3-6 d e^2 f^2 g+4 d^2 e f g^2-d^3 g^3\right ) \log \left (c (d+e x)^n\right )\right )-12 b d n \log (d+e x) \left (-72 a^2 \left (4 e^3 f^3-6 d e^2 f^2 g+4 d^2 e f g^2-d^3 g^3\right )+12 a b \left (48 e^3 f^3-108 d e^2 f^2 g+88 d^2 e f g^2-25 d^3 g^3\right ) n+b^2 \left (-576 e^3 f^3+1512 d e^2 f^2 g-1360 d^2 e f g^2+415 d^3 g^3\right ) n^2-12 b \left (12 a \left (4 e^3 f^3-6 d e^2 f^2 g+4 d^2 e f g^2-d^3 g^3\right )+b \left (-48 e^3 f^3+108 d e^2 f^2 g-88 d^2 e f g^2+25 d^3 g^3\right ) n\right ) \log \left (c (d+e x)^n\right )-72 b^2 \left (4 e^3 f^3-6 d e^2 f^2 g+4 d^2 e f g^2-d^3 g^3\right ) \log ^2\left (c (d+e x)^n\right )\right )+e x \left (288 a^3 e^3 \left (4 f^3+6 f^2 g x+4 f g^2 x^2+g^3 x^3\right )-72 a^2 b n \left (-12 d^3 g^3+6 d^2 e g^2 (8 f+g x)-4 d e^2 g \left (18 f^2+6 f g x+g^2 x^2\right )+e^3 \left (48 f^3+36 f^2 g x+16 f g^2 x^2+3 g^3 x^3\right )\right )+12 a b^2 n^2 \left (-300 d^3 g^3+6 d^2 e g^2 (176 f+13 g x)-4 d e^2 g \left (324 f^2+60 f g x+7 g^2 x^2\right )+e^3 \left (576 f^3+216 f^2 g x+64 f g^2 x^2+9 g^3 x^3\right )\right )-b^3 n^3 \left (-4980 d^3 g^3+30 d^2 e g^2 (544 f+23 g x)-4 d e^2 g \left (4536 f^2+456 f g x+37 g^2 x^2\right )+e^3 \left (6912 f^3+1296 f^2 g x+256 f g^2 x^2+27 g^3 x^3\right )\right )+12 b \left (72 a^2 e^3 \left (4 f^3+6 f^2 g x+4 f g^2 x^2+g^3 x^3\right )-12 a b n \left (-12 d^3 g^3+6 d^2 e g^2 (8 f+g x)-4 d e^2 g \left (18 f^2+6 f g x+g^2 x^2\right )+e^3 \left (48 f^3+36 f^2 g x+16 f g^2 x^2+3 g^3 x^3\right )\right )+b^2 n^2 \left (-300 d^3 g^3+6 d^2 e g^2 (176 f+13 g x)-4 d e^2 g \left (324 f^2+60 f g x+7 g^2 x^2\right )+e^3 \left (576 f^3+216 f^2 g x+64 f g^2 x^2+9 g^3 x^3\right )\right )\right ) \log \left (c (d+e x)^n\right )+72 b^2 \left (12 a e^3 \left (4 f^3+6 f^2 g x+4 f g^2 x^2+g^3 x^3\right )-b n \left (-12 d^3 g^3+6 d^2 e g^2 (8 f+g x)-4 d e^2 g \left (18 f^2+6 f g x+g^2 x^2\right )+e^3 \left (48 f^3+36 f^2 g x+16 f g^2 x^2+3 g^3 x^3\right )\right )\right ) \log ^2\left (c (d+e x)^n\right )+288 b^3 e^3 \left (4 f^3+6 f^2 g x+4 f g^2 x^2+g^3 x^3\right ) \log ^3\left (c (d+e x)^n\right )\right )}{1152 e^4} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-288*b^3*d*(-4*e^3*f^3 + 6*d*e^2*f^2*g - 4*d^2*e*f*g^2 + d^3*g^3)*n^3*Log[d + e*x]^3 + 72*b^2*d*n^2*Log[d + e

*x]^2*(-12*a*(4*e^3*f^3 - 6*d*e^2*f^2*g + 4*d^2*e*f*g^2 - d^3*g^3) + b*(48*e^3*f^3 - 108*d*e^2*f^2*g + 88*d^2*

e*f*g^2 - 25*d^3*g^3)*n - 12*b*(4*e^3*f^3 - 6*d*e^2*f^2*g + 4*d^2*e*f*g^2 - d^3*g^3)*Log[c*(d + e*x)^n]) - 12*

b*d*n*Log[d + e*x]*(-72*a^2*(4*e^3*f^3 - 6*d*e^2*f^2*g + 4*d^2*e*f*g^2 - d^3*g^3) + 12*a*b*(48*e^3*f^3 - 108*d

*e^2*f^2*g + 88*d^2*e*f*g^2 - 25*d^3*g^3)*n + b^2*(-576*e^3*f^3 + 1512*d*e^2*f^2*g - 1360*d^2*e*f*g^2 + 415*d^

3*g^3)*n^2 - 12*b*(12*a*(4*e^3*f^3 - 6*d*e^2*f^2*g + 4*d^2*e*f*g^2 - d^3*g^3) + b*(-48*e^3*f^3 + 108*d*e^2*f^2

*g - 88*d^2*e*f*g^2 + 25*d^3*g^3)*n)*Log[c*(d + e*x)^n] - 72*b^2*(4*e^3*f^3 - 6*d*e^2*f^2*g + 4*d^2*e*f*g^2 -

d^3*g^3)*Log[c*(d + e*x)^n]^2) + e*x*(288*a^3*e^3*(4*f^3 + 6*f^2*g*x + 4*f*g^2*x^2 + g^3*x^3) - 72*a^2*b*n*(-1

2*d^3*g^3 + 6*d^2*e*g^2*(8*f + g*x) - 4*d*e^2*g*(18*f^2 + 6*f*g*x + g^2*x^2) + e^3*(48*f^3 + 36*f^2*g*x + 16*f

*g^2*x^2 + 3*g^3*x^3)) + 12*a*b^2*n^2*(-300*d^3*g^3 + 6*d^2*e*g^2*(176*f + 13*g*x) - 4*d*e^2*g*(324*f^2 + 60*f

*g*x + 7*g^2*x^2) + e^3*(576*f^3 + 216*f^2*g*x + 64*f*g^2*x^2 + 9*g^3*x^3)) - b^3*n^3*(-4980*d^3*g^3 + 30*d^2*

e*g^2*(544*f + 23*g*x) - 4*d*e^2*g*(4536*f^2 + 456*f*g*x + 37*g^2*x^2) + e^3*(6912*f^3 + 1296*f^2*g*x + 256*f*

g^2*x^2 + 27*g^3*x^3)) + 12*b*(72*a^2*e^3*(4*f^3 + 6*f^2*g*x + 4*f*g^2*x^2 + g^3*x^3) - 12*a*b*n*(-12*d^3*g^3

+ 6*d^2*e*g^2*(8*f + g*x) - 4*d*e^2*g*(18*f^2 + 6*f*g*x + g^2*x^2) + e^3*(48*f^3 + 36*f^2*g*x + 16*f*g^2*x^2 +

3*g^3*x^3)) + b^2*n^2*(-300*d^3*g^3 + 6*d^2*e*g^2*(176*f + 13*g*x) - 4*d*e^2*g*(324*f^2 + 60*f*g*x + 7*g^2*x^

2) + e^3*(576*f^3 + 216*f^2*g*x + 64*f*g^2*x^2 + 9*g^3*x^3)))*Log[c*(d + e*x)^n] + 72*b^2*(12*a*e^3*(4*f^3 + 6

*f^2*g*x + 4*f*g^2*x^2 + g^3*x^3) - b*n*(-12*d^3*g^3 + 6*d^2*e*g^2*(8*f + g*x) - 4*d*e^2*g*(18*f^2 + 6*f*g*x +

g^2*x^2) + e^3*(48*f^3 + 36*f^2*g*x + 16*f*g^2*x^2 + 3*g^3*x^3)))*Log[c*(d + e*x)^n]^2 + 288*b^3*e^3*(4*f^3 +

6*f^2*g*x + 4*f*g^2*x^2 + g^3*x^3)*Log[c*(d + e*x)^n]^3))/(1152*e^4)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 3.49, size = 30495, normalized size = 50.99


method	result	size

risch	\(\text {Expression too large to display}\)	\(30495\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1720 vs. \(2 (602) = 1204\).
time = 0.35, size = 1720, normalized size = 2.88 \begin {gather*} \text {Too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b^3*g^3*x^4*log((x*e + d)^n*c)^3 + 3/4*a*b^2*g^3*x^4*log((x*e + d)^n*c)^2 + b^3*f*g^2*x^3*log((x*e + d)^n*

c)^3 + 3/4*a^2*b*g^3*x^4*log((x*e + d)^n*c) + 3*a*b^2*f*g^2*x^3*log((x*e + d)^n*c)^2 + 3/2*b^3*f^2*g*x^2*log((

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

*c)^2 + b^3*f^3*x*log((x*e + d)^n*c)^3 + a^3*f*g^2*x^3 + 3*(d*e^(-2)*log(x*e + d) - x*e^(-1))*a^2*b*f^3*n*e -

9/4*(2*d^2*e^(-3)*log(x*e + d) + (x^2*e - 2*d*x)*e^(-2))*a^2*b*f^2*g*n*e + 1/2*(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*f*g^2*n*e - 1/16*(12*d^4*e^(-5)*log(x*e + d) + (3*x^4*e^3 - 4*d*

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

*x*log((x*e + d)^n*c)^2 + 3/2*a^3*f^2*g*x^2 + 3*a^2*b*f^3*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^3 + (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^3 + 9/4*((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^2*g - 3/8*(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^2*g - 1/6*((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^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*f*g^2 + 1/36*(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 + 5

7*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*f*g^2 + 1/96*((72*d^4*log(x*e +

d)^2 + 9*x^4*e^4 - 28*d*x^3*e^3 + 78*d^2*x^2*e^2 - 300*d^3*x*e + 300*d^4*log(x*e + d))*n^2*e^(-4) - 12*(12*d^

4*e^(-5)*log(x*e + d) + (3*x^4*e^3 - 4*d*x^3*e^2 + 6*d^2*x^2*e - 12*d^3*x)*e^(-4))*n*e*log((x*e + d)^n*c))*a*b

^2*g^3 - 1/1152*(72*(12*d^4*e^(-5)*log(x*e + d) + (3*x^4*e^3 - 4*d*x^3*e^2 + 6*d^2*x^2*e - 12*d^3*x)*e^(-4))*n

*e*log((x*e + d)^n*c)^2 + ((288*d^4*log(x*e + d)^3 + 1800*d^4*log(x*e + d)^2 + 27*x^4*e^4 - 148*d*x^3*e^3 + 69

0*d^2*x^2*e^2 - 4980*d^3*x*e + 4980*d^4*log(x*e + d))*n^2*e^(-5) - 12*(72*d^4*log(x*e + d)^2 + 9*x^4*e^4 - 28*

d*x^3*e^3 + 78*d^2*x^2*e^2 - 300*d^3*x*e + 300*d^4*log(x*e + d))*n*e^(-5)*log((x*e + d)^n*c))*n*e)*b^3*g^3 + a

^3*f^3*x

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2694 vs. \(2 (602) = 1204\).
time = 0.41, size = 2694, normalized size = 4.51 \begin {gather*} \text {Too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1152*(288*(b^3*g^3*x^4 + 4*b^3*f*g^2*x^3 + 6*b^3*f^2*g*x^2 + 4*b^3*f^3*x)*e^4*log(c)^3 - 288*(b^3*d^4*g^3*n^

3 - 4*b^3*d^3*f*g^2*n^3*e + 6*b^3*d^2*f^2*g*n^3*e^2 - 4*b^3*d*f^3*n^3*e^3 - (b^3*g^3*n^3*x^4 + 4*b^3*f*g^2*n^3

*x^3 + 6*b^3*f^2*g*n^3*x^2 + 4*b^3*f^3*n^3*x)*e^4)*log(x*e + d)^3 + 12*(415*b^3*d^3*g^3*n^3 - 300*a*b^2*d^3*g^

3*n^2 + 72*a^2*b*d^3*g^3*n)*x*e + 72*(25*b^3*d^4*g^3*n^3 - 12*a*b^2*d^4*g^3*n^2 - (3*(b^3*g^3*n^3 - 4*a*b^2*g^

3*n^2)*x^4 + 16*(b^3*f*g^2*n^3 - 3*a*b^2*f*g^2*n^2)*x^3 + 36*(b^3*f^2*g*n^3 - 2*a*b^2*f^2*g*n^2)*x^2 + 48*(b^3

*f^3*n^3 - a*b^2*f^3*n^2)*x)*e^4 + 4*(b^3*d*g^3*n^3*x^3 + 6*b^3*d*f*g^2*n^3*x^2 + 18*b^3*d*f^2*g*n^3*x - 12*b^

3*d*f^3*n^3 + 12*a*b^2*d*f^3*n^2)*e^3 - 6*(b^3*d^2*g^3*n^3*x^2 + 8*b^3*d^2*f*g^2*n^3*x - 18*b^3*d^2*f^2*g*n^3

+ 12*a*b^2*d^2*f^2*g*n^2)*e^2 + 4*(3*b^3*d^3*g^3*n^3*x - 22*b^3*d^3*f*g^2*n^3 + 12*a*b^2*d^3*f*g^2*n^2)*e - 12

*(b^3*d^4*g^3*n^2 - 4*b^3*d^3*f*g^2*n^2*e + 6*b^3*d^2*f^2*g*n^2*e^2 - 4*b^3*d*f^3*n^2*e^3 - (b^3*g^3*n^2*x^4 +

4*b^3*f*g^2*n^2*x^3 + 6*b^3*f^2*g*n^2*x^2 + 4*b^3*f^3*n^2*x)*e^4)*log(c))*log(x*e + d)^2 + 72*(12*b^3*d^3*g^3

*n*x*e - (3*(b^3*g^3*n - 4*a*b^2*g^3)*x^4 + 16*(b^3*f*g^2*n - 3*a*b^2*f*g^2)*x^3 + 36*(b^3*f^2*g*n - 2*a*b^2*f

^2*g)*x^2 + 48*(b^3*f^3*n - a*b^2*f^3)*x)*e^4 + 4*(b^3*d*g^3*n*x^3 + 6*b^3*d*f*g^2*n*x^2 + 18*b^3*d*f^2*g*n*x)

*e^3 - 6*(b^3*d^2*g^3*n*x^2 + 8*b^3*d^2*f*g^2*n*x)*e^2)*log(c)^2 - (9*(3*b^3*g^3*n^3 - 12*a*b^2*g^3*n^2 + 24*a

^2*b*g^3*n - 32*a^3*g^3)*x^4 + 128*(2*b^3*f*g^2*n^3 - 6*a*b^2*f*g^2*n^2 + 9*a^2*b*f*g^2*n - 9*a^3*f*g^2)*x^3 +

432*(3*b^3*f^2*g*n^3 - 6*a*b^2*f^2*g*n^2 + 6*a^2*b*f^2*g*n - 4*a^3*f^2*g)*x^2 + 1152*(6*b^3*f^3*n^3 - 6*a*b^2

*f^3*n^2 + 3*a^2*b*f^3*n - a^3*f^3)*x)*e^4 + 4*((37*b^3*d*g^3*n^3 - 84*a*b^2*d*g^3*n^2 + 72*a^2*b*d*g^3*n)*x^3

+ 24*(19*b^3*d*f*g^2*n^3 - 30*a*b^2*d*f*g^2*n^2 + 18*a^2*b*d*f*g^2*n)*x^2 + 648*(7*b^3*d*f^2*g*n^3 - 6*a*b^2*

d*f^2*g*n^2 + 2*a^2*b*d*f^2*g*n)*x)*e^3 - 6*((115*b^3*d^2*g^3*n^3 - 156*a*b^2*d^2*g^3*n^2 + 72*a^2*b*d^2*g^3*n

)*x^2 + 32*(85*b^3*d^2*f*g^2*n^3 - 66*a*b^2*d^2*f*g^2*n^2 + 18*a^2*b*d^2*f*g^2*n)*x)*e^2 - 12*(415*b^3*d^4*g^3

*n^3 - 300*a*b^2*d^4*g^3*n^2 + 72*a^2*b*d^4*g^3*n + 72*(b^3*d^4*g^3*n - 4*b^3*d^3*f*g^2*n*e + 6*b^3*d^2*f^2*g*

n*e^2 - 4*b^3*d*f^3*n*e^3 - (b^3*g^3*n*x^4 + 4*b^3*f*g^2*n*x^3 + 6*b^3*f^2*g*n*x^2 + 4*b^3*f^3*n*x)*e^4)*log(c

)^2 - (9*(b^3*g^3*n^3 - 4*a*b^2*g^3*n^2 + 8*a^2*b*g^3*n)*x^4 + 32*(2*b^3*f*g^2*n^3 - 6*a*b^2*f*g^2*n^2 + 9*a^2

*b*f*g^2*n)*x^3 + 216*(b^3*f^2*g*n^3 - 2*a*b^2*f^2*g*n^2 + 2*a^2*b*f^2*g*n)*x^2 + 288*(2*b^3*f^3*n^3 - 2*a*b^2

*f^3*n^2 + a^2*b*f^3*n)*x)*e^4 - 4*(144*b^3*d*f^3*n^3 - 144*a*b^2*d*f^3*n^2 + 72*a^2*b*d*f^3*n - (7*b^3*d*g^3*

n^3 - 12*a*b^2*d*g^3*n^2)*x^3 - 12*(5*b^3*d*f*g^2*n^3 - 6*a*b^2*d*f*g^2*n^2)*x^2 - 108*(3*b^3*d*f^2*g*n^3 - 2*

a*b^2*d*f^2*g*n^2)*x)*e^3 + 6*(252*b^3*d^2*f^2*g*n^3 - 216*a*b^2*d^2*f^2*g*n^2 + 72*a^2*b*d^2*f^2*g*n - (13*b^

3*d^2*g^3*n^3 - 12*a*b^2*d^2*g^3*n^2)*x^2 - 16*(11*b^3*d^2*f*g^2*n^3 - 6*a*b^2*d^2*f*g^2*n^2)*x)*e^2 - 4*(340*

b^3*d^3*f*g^2*n^3 - 264*a*b^2*d^3*f*g^2*n^2 + 72*a^2*b*d^3*f*g^2*n - 3*(25*b^3*d^3*g^3*n^3 - 12*a*b^2*d^3*g^3*

n^2)*x)*e - 12*(25*b^3*d^4*g^3*n^2 - 12*a*b^2*d^4*g^3*n - (3*(b^3*g^3*n^2 - 4*a*b^2*g^3*n)*x^4 + 16*(b^3*f*g^2

*n^2 - 3*a*b^2*f*g^2*n)*x^3 + 36*(b^3*f^2*g*n^2 - 2*a*b^2*f^2*g*n)*x^2 + 48*(b^3*f^3*n^2 - a*b^2*f^3*n)*x)*e^4

+ 4*(b^3*d*g^3*n^2*x^3 + 6*b^3*d*f*g^2*n^2*x^2 + 18*b^3*d*f^2*g*n^2*x - 12*b^3*d*f^3*n^2 + 12*a*b^2*d*f^3*n)*

e^3 - 6*(b^3*d^2*g^3*n^2*x^2 + 8*b^3*d^2*f*g^2*n^2*x - 18*b^3*d^2*f^2*g*n^2 + 12*a*b^2*d^2*f^2*g*n)*e^2 + 4*(3

*b^3*d^3*g^3*n^2*x - 22*b^3*d^3*f*g^2*n^2 + 12*a*b^2*d^3*f*g^2*n)*e)*log(c))*log(x*e + d) - 12*(12*(25*b^3*d^3

*g^3*n^2 - 12*a*b^2*d^3*g^3*n)*x*e - (9*(b^3*g^3*n^2 - 4*a*b^2*g^3*n + 8*a^2*b*g^3)*x^4 + 32*(2*b^3*f*g^2*n^2

- 6*a*b^2*f*g^2*n + 9*a^2*b*f*g^2)*x^3 + 216*(b^3*f^2*g*n^2 - 2*a*b^2*f^2*g*n + 2*a^2*b*f^2*g)*x^2 + 288*(2*b^

3*f^3*n^2 - 2*a*b^2*f^3*n + a^2*b*f^3)*x)*e^4 + 4*((7*b^3*d*g^3*n^2 - 12*a*b^2*d*g^3*n)*x^3 + 12*(5*b^3*d*f*g^

2*n^2 - 6*a*b^2*d*f*g^2*n)*x^2 + 108*(3*b^3*d*f^2*g*n^2 - 2*a*b^2*d*f^2*g*n)*x)*e^3 - 6*((13*b^3*d^2*g^3*n^2 -

12*a*b^2*d^2*g^3*n)*x^2 + 16*(11*b^3*d^2*f*g^2*n^2 - 6*a*b^2*d^2*f*g^2*n)*x)*e^2)*log(c))*e^(-4)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2594 vs. \(2 (583) = 1166\).
time = 5.21, size = 2594, normalized size = 4.34 \begin {gather*} \text {Too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**3*f**3*x + 3*a**3*f**2*g*x**2/2 + a**3*f*g**2*x**3 + a**3*g**3*x**4/4 - 3*a**2*b*d**4*g**3*log(c

*(d + e*x)**n)/(4*e**4) + 3*a**2*b*d**3*f*g**2*log(c*(d + e*x)**n)/e**3 + 3*a**2*b*d**3*g**3*n*x/(4*e**3) - 9*

a**2*b*d**2*f**2*g*log(c*(d + e*x)**n)/(2*e**2) - 3*a**2*b*d**2*f*g**2*n*x/e**2 - 3*a**2*b*d**2*g**3*n*x**2/(8

*e**2) + 3*a**2*b*d*f**3*log(c*(d + e*x)**n)/e + 9*a**2*b*d*f**2*g*n*x/(2*e) + 3*a**2*b*d*f*g**2*n*x**2/(2*e)

+ a**2*b*d*g**3*n*x**3/(4*e) - 3*a**2*b*f**3*n*x + 3*a**2*b*f**3*x*log(c*(d + e*x)**n) - 9*a**2*b*f**2*g*n*x**

2/4 + 9*a**2*b*f**2*g*x**2*log(c*(d + e*x)**n)/2 - a**2*b*f*g**2*n*x**3 + 3*a**2*b*f*g**2*x**3*log(c*(d + e*x)

**n) - 3*a**2*b*g**3*n*x**4/16 + 3*a**2*b*g**3*x**4*log(c*(d + e*x)**n)/4 + 25*a*b**2*d**4*g**3*n*log(c*(d + e

*x)**n)/(8*e**4) - 3*a*b**2*d**4*g**3*log(c*(d + e*x)**n)**2/(4*e**4) - 11*a*b**2*d**3*f*g**2*n*log(c*(d + e*x

)**n)/e**3 + 3*a*b**2*d**3*f*g**2*log(c*(d + e*x)**n)**2/e**3 - 25*a*b**2*d**3*g**3*n**2*x/(8*e**3) + 3*a*b**2

*d**3*g**3*n*x*log(c*(d + e*x)**n)/(2*e**3) + 27*a*b**2*d**2*f**2*g*n*log(c*(d + e*x)**n)/(2*e**2) - 9*a*b**2*

d**2*f**2*g*log(c*(d + e*x)**n)**2/(2*e**2) + 11*a*b**2*d**2*f*g**2*n**2*x/e**2 - 6*a*b**2*d**2*f*g**2*n*x*log

(c*(d + e*x)**n)/e**2 + 13*a*b**2*d**2*g**3*n**2*x**2/(16*e**2) - 3*a*b**2*d**2*g**3*n*x**2*log(c*(d + e*x)**n

)/(4*e**2) - 6*a*b**2*d*f**3*n*log(c*(d + e*x)**n)/e + 3*a*b**2*d*f**3*log(c*(d + e*x)**n)**2/e - 27*a*b**2*d*

f**2*g*n**2*x/(2*e) + 9*a*b**2*d*f**2*g*n*x*log(c*(d + e*x)**n)/e - 5*a*b**2*d*f*g**2*n**2*x**2/(2*e) + 3*a*b*

*2*d*f*g**2*n*x**2*log(c*(d + e*x)**n)/e - 7*a*b**2*d*g**3*n**2*x**3/(24*e) + a*b**2*d*g**3*n*x**3*log(c*(d +

e*x)**n)/(2*e) + 6*a*b**2*f**3*n**2*x - 6*a*b**2*f**3*n*x*log(c*(d + e*x)**n) + 3*a*b**2*f**3*x*log(c*(d + e*x

)**n)**2 + 9*a*b**2*f**2*g*n**2*x**2/4 - 9*a*b**2*f**2*g*n*x**2*log(c*(d + e*x)**n)/2 + 9*a*b**2*f**2*g*x**2*l

og(c*(d + e*x)**n)**2/2 + 2*a*b**2*f*g**2*n**2*x**3/3 - 2*a*b**2*f*g**2*n*x**3*log(c*(d + e*x)**n) + 3*a*b**2*

f*g**2*x**3*log(c*(d + e*x)**n)**2 + 3*a*b**2*g**3*n**2*x**4/32 - 3*a*b**2*g**3*n*x**4*log(c*(d + e*x)**n)/8 +

3*a*b**2*g**3*x**4*log(c*(d + e*x)**n)**2/4 - 415*b**3*d**4*g**3*n**2*log(c*(d + e*x)**n)/(96*e**4) + 25*b**3

*d**4*g**3*n*log(c*(d + e*x)**n)**2/(16*e**4) - b**3*d**4*g**3*log(c*(d + e*x)**n)**3/(4*e**4) + 85*b**3*d**3*

f*g**2*n**2*log(c*(d + e*x)**n)/(6*e**3) - 11*b**3*d**3*f*g**2*n*log(c*(d + e*x)**n)**2/(2*e**3) + b**3*d**3*f

*g**2*log(c*(d + e*x)**n)**3/e**3 + 415*b**3*d**3*g**3*n**3*x/(96*e**3) - 25*b**3*d**3*g**3*n**2*x*log(c*(d +

e*x)**n)/(8*e**3) + 3*b**3*d**3*g**3*n*x*log(c*(d + e*x)**n)**2/(4*e**3) - 63*b**3*d**2*f**2*g*n**2*log(c*(d +

e*x)**n)/(4*e**2) + 27*b**3*d**2*f**2*g*n*log(c*(d + e*x)**n)**2/(4*e**2) - 3*b**3*d**2*f**2*g*log(c*(d + e*x

)**n)**3/(2*e**2) - 85*b**3*d**2*f*g**2*n**3*x/(6*e**2) + 11*b**3*d**2*f*g**2*n**2*x*log(c*(d + e*x)**n)/e**2

- 3*b**3*d**2*f*g**2*n*x*log(c*(d + e*x)**n)**2/e**2 - 115*b**3*d**2*g**3*n**3*x**2/(192*e**2) + 13*b**3*d**2*

g**3*n**2*x**2*log(c*(d + e*x)**n)/(16*e**2) - 3*b**3*d**2*g**3*n*x**2*log(c*(d + e*x)**n)**2/(8*e**2) + 6*b**

3*d*f**3*n**2*log(c*(d + e*x)**n)/e - 3*b**3*d*f**3*n*log(c*(d + e*x)**n)**2/e + b**3*d*f**3*log(c*(d + e*x)**

n)**3/e + 63*b**3*d*f**2*g*n**3*x/(4*e) - 27*b**3*d*f**2*g*n**2*x*log(c*(d + e*x)**n)/(2*e) + 9*b**3*d*f**2*g*

n*x*log(c*(d + e*x)**n)**2/(2*e) + 19*b**3*d*f*g**2*n**3*x**2/(12*e) - 5*b**3*d*f*g**2*n**2*x**2*log(c*(d + e*

x)**n)/(2*e) + 3*b**3*d*f*g**2*n*x**2*log(c*(d + e*x)**n)**2/(2*e) + 37*b**3*d*g**3*n**3*x**3/(288*e) - 7*b**3

*d*g**3*n**2*x**3*log(c*(d + e*x)**n)/(24*e) + b**3*d*g**3*n*x**3*log(c*(d + e*x)**n)**2/(4*e) - 6*b**3*f**3*n

**3*x + 6*b**3*f**3*n**2*x*log(c*(d + e*x)**n) - 3*b**3*f**3*n*x*log(c*(d + e*x)**n)**2 + b**3*f**3*x*log(c*(d

+ e*x)**n)**3 - 9*b**3*f**2*g*n**3*x**2/8 + 9*b**3*f**2*g*n**2*x**2*log(c*(d + e*x)**n)/4 - 9*b**3*f**2*g*n*x

**2*log(c*(d + e*x)**n)**2/4 + 3*b**3*f**2*g*x**2*log(c*(d + e*x)**n)**3/2 - 2*b**3*f*g**2*n**3*x**3/9 + 2*b**

3*f*g**2*n**2*x**3*log(c*(d + e*x)**n)/3 - b**3*f*g**2*n*x**3*log(c*(d + e*x)**n)**2 + b**3*f*g**2*x**3*log(c*

(d + e*x)**n)**3 - 3*b**3*g**3*n**3*x**4/128 + 3*b**3*g**3*n**2*x**4*log(c*(d + e*x)**n)/32 - 3*b**3*g**3*n*x*

*4*log(c*(d + e*x)**n)**2/16 + b**3*g**3*x**4*log(c*(d + e*x)**n)**3/4, Ne(e, 0)), ((a + b*log(c*d**n))**3*(f*

*3*x + 3*f**2*g*x**2/2 + f*g**2*x**3 + g**3*x**4/4), True))

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 5282 vs. \(2 (602) = 1204\).
time = 3.98, size = 5282, normalized size = 8.83 \begin {gather*} \text {Too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(x*e + d)^4*b^3*g^3*n^3*e^(-4)*log(x*e + d)^3 - (x*e + d)^3*b^3*d*g^3*n^3*e^(-4)*log(x*e + d)^3 + 3/2*(x*e

+ d)^2*b^3*d^2*g^3*n^3*e^(-4)*log(x*e + d)^3 - (x*e + d)*b^3*d^3*g^3*n^3*e^(-4)*log(x*e + d)^3 - 3/16*(x*e +

d)^4*b^3*g^3*n^3*e^(-4)*log(x*e + d)^2 + (x*e + d)^3*b^3*d*g^3*n^3*e^(-4)*log(x*e + d)^2 - 9/4*(x*e + d)^2*b^3

*d^2*g^3*n^3*e^(-4)*log(x*e + d)^2 + 3*(x*e + d)*b^3*d^3*g^3*n^3*e^(-4)*log(x*e + d)^2 + (x*e + d)^3*b^3*f*g^2

*n^3*e^(-3)*log(x*e + d)^3 - 3*(x*e + d)^2*b^3*d*f*g^2*n^3*e^(-3)*log(x*e + d)^3 + 3*(x*e + d)*b^3*d^2*f*g^2*n

^3*e^(-3)*log(x*e + d)^3 + 3/4*(x*e + d)^4*b^3*g^3*n^2*e^(-4)*log(x*e + d)^2*log(c) - 3*(x*e + d)^3*b^3*d*g^3*

n^2*e^(-4)*log(x*e + d)^2*log(c) + 9/2*(x*e + d)^2*b^3*d^2*g^3*n^2*e^(-4)*log(x*e + d)^2*log(c) - 3*(x*e + d)*

b^3*d^3*g^3*n^2*e^(-4)*log(x*e + d)^2*log(c) + 3/32*(x*e + d)^4*b^3*g^3*n^3*e^(-4)*log(x*e + d) - 2/3*(x*e + d

)^3*b^3*d*g^3*n^3*e^(-4)*log(x*e + d) + 9/4*(x*e + d)^2*b^3*d^2*g^3*n^3*e^(-4)*log(x*e + d) - 6*(x*e + d)*b^3*

d^3*g^3*n^3*e^(-4)*log(x*e + d) - (x*e + d)^3*b^3*f*g^2*n^3*e^(-3)*log(x*e + d)^2 + 9/2*(x*e + d)^2*b^3*d*f*g^

2*n^3*e^(-3)*log(x*e + d)^2 - 9*(x*e + d)*b^3*d^2*f*g^2*n^3*e^(-3)*log(x*e + d)^2 + 3/4*(x*e + d)^4*a*b^2*g^3*

n^2*e^(-4)*log(x*e + d)^2 - 3*(x*e + d)^3*a*b^2*d*g^3*n^2*e^(-4)*log(x*e + d)^2 + 9/2*(x*e + d)^2*a*b^2*d^2*g^

3*n^2*e^(-4)*log(x*e + d)^2 - 3*(x*e + d)*a*b^2*d^3*g^3*n^2*e^(-4)*log(x*e + d)^2 + 3/2*(x*e + d)^2*b^3*f^2*g*

n^3*e^(-2)*log(x*e + d)^3 - 3*(x*e + d)*b^3*d*f^2*g*n^3*e^(-2)*log(x*e + d)^3 - 3/8*(x*e + d)^4*b^3*g^3*n^2*e^

(-4)*log(x*e + d)*log(c) + 2*(x*e + d)^3*b^3*d*g^3*n^2*e^(-4)*log(x*e + d)*log(c) - 9/2*(x*e + d)^2*b^3*d^2*g^

3*n^2*e^(-4)*log(x*e + d)*log(c) + 6*(x*e + d)*b^3*d^3*g^3*n^2*e^(-4)*log(x*e + d)*log(c) + 3*(x*e + d)^3*b^3*

f*g^2*n^2*e^(-3)*log(x*e + d)^2*log(c) - 9*(x*e + d)^2*b^3*d*f*g^2*n^2*e^(-3)*log(x*e + d)^2*log(c) + 9*(x*e +

d)*b^3*d^2*f*g^2*n^2*e^(-3)*log(x*e + d)^2*log(c) + 3/4*(x*e + d)^4*b^3*g^3*n*e^(-4)*log(x*e + d)*log(c)^2 -

3*(x*e + d)^3*b^3*d*g^3*n*e^(-4)*log(x*e + d)*log(c)^2 + 9/2*(x*e + d)^2*b^3*d^2*g^3*n*e^(-4)*log(x*e + d)*log

(c)^2 - 3*(x*e + d)*b^3*d^3*g^3*n*e^(-4)*log(x*e + d)*log(c)^2 - 3/128*(x*e + d)^4*b^3*g^3*n^3*e^(-4) + 2/9*(x

*e + d)^3*b^3*d*g^3*n^3*e^(-4) - 9/8*(x*e + d)^2*b^3*d^2*g^3*n^3*e^(-4) + 6*(x*e + d)*b^3*d^3*g^3*n^3*e^(-4) +

2/3*(x*e + d)^3*b^3*f*g^2*n^3*e^(-3)*log(x*e + d) - 9/2*(x*e + d)^2*b^3*d*f*g^2*n^3*e^(-3)*log(x*e + d) + 18*

(x*e + d)*b^3*d^2*f*g^2*n^3*e^(-3)*log(x*e + d) - 3/8*(x*e + d)^4*a*b^2*g^3*n^2*e^(-4)*log(x*e + d) + 2*(x*e +

d)^3*a*b^2*d*g^3*n^2*e^(-4)*log(x*e + d) - 9/2*(x*e + d)^2*a*b^2*d^2*g^3*n^2*e^(-4)*log(x*e + d) + 6*(x*e + d

)*a*b^2*d^3*g^3*n^2*e^(-4)*log(x*e + d) - 9/4*(x*e + d)^2*b^3*f^2*g*n^3*e^(-2)*log(x*e + d)^2 + 9*(x*e + d)*b^

3*d*f^2*g*n^3*e^(-2)*log(x*e + d)^2 + 3*(x*e + d)^3*a*b^2*f*g^2*n^2*e^(-3)*log(x*e + d)^2 - 9*(x*e + d)^2*a*b^

2*d*f*g^2*n^2*e^(-3)*log(x*e + d)^2 + 9*(x*e + d)*a*b^2*d^2*f*g^2*n^2*e^(-3)*log(x*e + d)^2 + (x*e + d)*b^3*f^

3*n^3*e^(-1)*log(x*e + d)^3 + 3/32*(x*e + d)^4*b^3*g^3*n^2*e^(-4)*log(c) - 2/3*(x*e + d)^3*b^3*d*g^3*n^2*e^(-4

)*log(c) + 9/4*(x*e + d)^2*b^3*d^2*g^3*n^2*e^(-4)*log(c) - 6*(x*e + d)*b^3*d^3*g^3*n^2*e^(-4)*log(c) - 2*(x*e

+ d)^3*b^3*f*g^2*n^2*e^(-3)*log(x*e + d)*log(c) + 9*(x*e + d)^2*b^3*d*f*g^2*n^2*e^(-3)*log(x*e + d)*log(c) - 1

8*(x*e + d)*b^3*d^2*f*g^2*n^2*e^(-3)*log(x*e + d)*log(c) + 3/2*(x*e + d)^4*a*b^2*g^3*n*e^(-4)*log(x*e + d)*log

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

x*e + d)^2*log(c) - 9*(x*e + d)*b^3*d*f^2*g*n^2*e^(-2)*log(x*e + d)^2*log(c) - 3/16*(x*e + d)^4*b^3*g^3*n*e^(-

4)*log(c)^2 + (x*e + d)^3*b^3*d*g^3*n*e^(-4)*log(c)^2 - 9/4*(x*e + d)^2*b^3*d^2*g^3*n*e^(-4)*log(c)^2 + 3*(x*e

+ d)*b^3*d^3*g^3*n*e^(-4)*log(c)^2 + 3*(x*e + d)^3*b^3*f*g^2*n*e^(-3)*log(x*e + d)*log(c)^2 - 9*(x*e + d)^2*b

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

e + d)^4*b^3*g^3*e^(-4)*log(c)^3 - (x*e + d)^3*b^3*d*g^3*e^(-4)*log(c)^3 + 3/2*(x*e + d)^2*b^3*d^2*g^3*e^(-4)*

log(c)^3 - (x*e + d)*b^3*d^3*g^3*e^(-4)*log(c)^3 - 2/9*(x*e + d)^3*b^3*f*g^2*n^3*e^(-3) + 9/4*(x*e + d)^2*b^3*

d*f*g^2*n^3*e^(-3) - 18*(x*e + d)*b^3*d^2*f*g^2*n^3*e^(-3) + 3/32*(x*e + d)^4*a*b^2*g^3*n^2*e^(-4) - 2/3*(x*e

+ d)^3*a*b^2*d*g^3*n^2*e^(-4) + 9/4*(x*e + d)^2*a*b^2*d^2*g^3*n^2*e^(-4) - 6*(x*e + d)*a*b^2*d^3*g^3*n^2*e^(-4

) + 9/4*(x*e + d)^2*b^3*f^2*g*n^3*e^(-2)*log(x*e + d) - 18*(x*e + d)*b^3*d*f^2*g*n^3*e^(-2)*log(x*e + d) - 2*(

x*e + d)^3*a*b^2*f*g^2*n^2*e^(-3)*log(x*e + d) + 9*(x*e + d)^2*a*b^2*d*f*g^2*n^2*e^(-3)*log(x*e + d) - 18*(x*e

+ d)*a*b^2*d^2*f*g^2*n^2*e^(-3)*log(x*e + d) + 3/4*(x*e + d)^4*a^2*b*g^3*n*e^(-4)*log(x*e + d) - 3*(x*e + d)^

3*a^2*b*d*g^3*n*e^(-4)*log(x*e + d) + 9/2*(x*e + d)^2*a^2*b*d^2*g^3*n*e^(-4)*log(x*e + d) - 3*(x*e + d)*a^2*b*

d^3*g^3*n*e^(-4)*log(x*e + d) - 3*(x*e + d)*b^3...

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Mupad [B]
time = 1.21, size = 2133, normalized size = 3.57 \begin {gather*} x^3\,\left (\frac {g^2\,\left (24\,a^3\,d\,g+72\,a^3\,e\,f+7\,b^3\,d\,g\,n^3-16\,b^3\,e\,f\,n^3-12\,a\,b^2\,d\,g\,n^2+48\,a\,b^2\,e\,f\,n^2-72\,a^2\,b\,e\,f\,n\right )}{72\,e}-\frac {d\,g^3\,\left (32\,a^3-24\,a^2\,b\,n+12\,a\,b^2\,n^2-3\,b^3\,n^3\right )}{96\,e}\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^3\,\left (b^3\,f^3\,x-\frac {d\,\left (b^3\,d^3\,g^3-4\,b^3\,d^2\,e\,f\,g^2+6\,b^3\,d\,e^2\,f^2\,g-4\,b^3\,e^3\,f^3\right )}{4\,e^4}+\frac {b^3\,g^3\,x^4}{4}+\frac {3\,b^3\,f^2\,g\,x^2}{2}+b^3\,f\,g^2\,x^3\right )-x^2\,\left (\frac {d\,\left (\frac {g^2\,\left (24\,a^3\,d\,g+72\,a^3\,e\,f+7\,b^3\,d\,g\,n^3-16\,b^3\,e\,f\,n^3-12\,a\,b^2\,d\,g\,n^2+48\,a\,b^2\,e\,f\,n^2-72\,a^2\,b\,e\,f\,n\right )}{24\,e}-\frac {d\,g^3\,\left (32\,a^3-24\,a^2\,b\,n+12\,a\,b^2\,n^2-3\,b^3\,n^3\right )}{32\,e}\right )}{2\,e}-\frac {g\,\left (48\,a^3\,d\,e\,f\,g+48\,a^3\,e^2\,f^2-72\,a^2\,b\,e^2\,f^2\,n+12\,a\,b^2\,d^2\,g^2\,n^2-48\,a\,b^2\,d\,e\,f\,g\,n^2+72\,a\,b^2\,e^2\,f^2\,n^2-13\,b^3\,d^2\,g^2\,n^3+40\,b^3\,d\,e\,f\,g\,n^3-36\,b^3\,e^2\,f^2\,n^3\right )}{32\,e^2}\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2\,\left (\frac {x^3\,\left (\frac {4\,b^2\,g^2\,\left (a\,d\,g+3\,a\,e\,f-b\,e\,f\,n\right )}{e}-\frac {b^2\,d\,g^3\,\left (4\,a-b\,n\right )}{e}\right )}{4}-\frac {x^2\,\left (\frac {d\,\left (\frac {48\,b^2\,g^2\,\left (a\,d\,g+3\,a\,e\,f-b\,e\,f\,n\right )}{e}-\frac {12\,b^2\,d\,g^3\,\left (4\,a-b\,n\right )}{e}\right )}{8\,e}-\frac {9\,b^2\,f\,g\,\left (2\,a\,d\,g+2\,a\,e\,f-b\,e\,f\,n\right )}{e}\right )}{4}+\frac {x\,\left (\frac {d\,\left (\frac {d\,\left (\frac {48\,b^2\,g^2\,\left (a\,d\,g+3\,a\,e\,f-b\,e\,f\,n\right )}{e}-\frac {12\,b^2\,d\,g^3\,\left (4\,a-b\,n\right )}{e}\right )}{e}-\frac {72\,b^2\,f\,g\,\left (2\,a\,d\,g+2\,a\,e\,f-b\,e\,f\,n\right )}{e}\right )}{4\,e}+\frac {12\,b^2\,f^2\,\left (3\,a\,d\,g+a\,e\,f-b\,e\,f\,n\right )}{e}\right )}{4}-\frac {-25\,n\,b^3\,d^4\,g^3+88\,n\,b^3\,d^3\,e\,f\,g^2-108\,n\,b^3\,d^2\,e^2\,f^2\,g+48\,n\,b^3\,d\,e^3\,f^3+12\,a\,b^2\,d^4\,g^3-48\,a\,b^2\,d^3\,e\,f\,g^2+72\,a\,b^2\,d^2\,e^2\,f^2\,g-48\,a\,b^2\,d\,e^3\,f^3}{16\,e^4}+\frac {3\,b^2\,g^3\,x^4\,\left (4\,a-b\,n\right )}{16}\right )+x\,\left (\frac {288\,a^3\,d\,e^2\,f^2\,g+96\,a^3\,e^3\,f^3-288\,a^2\,b\,e^3\,f^3\,n-144\,a\,b^2\,d^3\,g^3\,n^2+576\,a\,b^2\,d^2\,e\,f\,g^2\,n^2-864\,a\,b^2\,d\,e^2\,f^2\,g\,n^2+576\,a\,b^2\,e^3\,f^3\,n^2+300\,b^3\,d^3\,g^3\,n^3-1056\,b^3\,d^2\,e\,f\,g^2\,n^3+1296\,b^3\,d\,e^2\,f^2\,g\,n^3-576\,b^3\,e^3\,f^3\,n^3}{96\,e^3}+\frac {d\,\left (\frac {d\,\left (\frac {g^2\,\left (24\,a^3\,d\,g+72\,a^3\,e\,f+7\,b^3\,d\,g\,n^3-16\,b^3\,e\,f\,n^3-12\,a\,b^2\,d\,g\,n^2+48\,a\,b^2\,e\,f\,n^2-72\,a^2\,b\,e\,f\,n\right )}{24\,e}-\frac {d\,g^3\,\left (32\,a^3-24\,a^2\,b\,n+12\,a\,b^2\,n^2-3\,b^3\,n^3\right )}{32\,e}\right )}{e}-\frac {g\,\left (48\,a^3\,d\,e\,f\,g+48\,a^3\,e^2\,f^2-72\,a^2\,b\,e^2\,f^2\,n+12\,a\,b^2\,d^2\,g^2\,n^2-48\,a\,b^2\,d\,e\,f\,g\,n^2+72\,a\,b^2\,e^2\,f^2\,n^2-13\,b^3\,d^2\,g^2\,n^3+40\,b^3\,d\,e\,f\,g\,n^3-36\,b^3\,e^2\,f^2\,n^3\right )}{16\,e^2}\right )}{e}\right )-\frac {\ln \left (d+e\,x\right )\,\left (72\,a^2\,b\,d^4\,g^3\,n-288\,a^2\,b\,d^3\,e\,f\,g^2\,n+432\,a^2\,b\,d^2\,e^2\,f^2\,g\,n-288\,a^2\,b\,d\,e^3\,f^3\,n-300\,a\,b^2\,d^4\,g^3\,n^2+1056\,a\,b^2\,d^3\,e\,f\,g^2\,n^2-1296\,a\,b^2\,d^2\,e^2\,f^2\,g\,n^2+576\,a\,b^2\,d\,e^3\,f^3\,n^2+415\,b^3\,d^4\,g^3\,n^3-1360\,b^3\,d^3\,e\,f\,g^2\,n^3+1512\,b^3\,d^2\,e^2\,f^2\,g\,n^3-576\,b^3\,d\,e^3\,f^3\,n^3\right )}{96\,e^4}+\frac {g^3\,x^4\,\left (32\,a^3-24\,a^2\,b\,n+12\,a\,b^2\,n^2-3\,b^3\,n^3\right )}{128}+\frac {\ln \left (c\,{\left (d+e\,x\right )}^n\right )\,\left (\frac {x^3\,\left (32\,b\,e^3\,g^2\,\left (6\,a^2\,d\,g+18\,a^2\,e\,f-b^2\,d\,g\,n^2+4\,b^2\,e\,f\,n^2-12\,a\,b\,e\,f\,n\right )-24\,b\,d\,e^3\,g^3\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )\right )}{24\,e^2}-\frac {x^2\,\left (\frac {d\,\left (32\,b\,e^3\,g^2\,\left (6\,a^2\,d\,g+18\,a^2\,e\,f-b^2\,d\,g\,n^2+4\,b^2\,e\,f\,n^2-12\,a\,b\,e\,f\,n\right )-24\,b\,d\,e^3\,g^3\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )\right )}{e}-48\,b\,e^2\,g\,\left (12\,a^2\,d\,e\,f\,g+12\,a^2\,e^2\,f^2-12\,a\,b\,e^2\,f^2\,n+b^2\,d^2\,g^2\,n^2-4\,b^2\,d\,e\,f\,g\,n^2+6\,b^2\,e^2\,f^2\,n^2\right )\right )}{16\,e^2}+\frac {x\,\left (\frac {576\,a^2\,b\,d\,e^4\,f^2\,g+192\,a^2\,b\,e^5\,f^3-384\,a\,b^2\,e^5\,f^3\,n-96\,b^3\,d^3\,e^2\,g^3\,n^2+384\,b^3\,d^2\,e^3\,f\,g^2\,n^2-576\,b^3\,d\,e^4\,f^2\,g\,n^2+384\,b^3\,e^5\,f^3\,n^2}{e}+\frac {d\,\left (\frac {d\,\left (32\,b\,e^3\,g^2\,\left (6\,a^2\,d\,g+18\,a^2\,e\,f-b^2\,d\,g\,n^2+4\,b^2\,e\,f\,n^2-12\,a\,b\,e\,f\,n\right )-24\,b\,d\,e^3\,g^3\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )\right )}{e}-48\,b\,e^2\,g\,\left (12\,a^2\,d\,e\,f\,g+12\,a^2\,e^2\,f^2-12\,a\,b\,e^2\,f^2\,n+b^2\,d^2\,g^2\,n^2-4\,b^2\,d\,e\,f\,g\,n^2+6\,b^2\,e^2\,f^2\,n^2\right )\right )}{e}\right )}{8\,e^2}+\frac {3\,b\,e^2\,g^3\,x^4\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )}{4}\right )}{8\,e^2} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^3*((g^2*(24*a^3*d*g + 72*a^3*e*f + 7*b^3*d*g*n^3 - 16*b^3*e*f*n^3 - 12*a*b^2*d*g*n^2 + 48*a*b^2*e*f*n^2 - 72

*a^2*b*e*f*n))/(72*e) - (d*g^3*(32*a^3 - 3*b^3*n^3 + 12*a*b^2*n^2 - 24*a^2*b*n))/(96*e)) + log(c*(d + e*x)^n)^

3*(b^3*f^3*x - (d*(b^3*d^3*g^3 - 4*b^3*e^3*f^3 + 6*b^3*d*e^2*f^2*g - 4*b^3*d^2*e*f*g^2))/(4*e^4) + (b^3*g^3*x^

4)/4 + (3*b^3*f^2*g*x^2)/2 + b^3*f*g^2*x^3) - x^2*((d*((g^2*(24*a^3*d*g + 72*a^3*e*f + 7*b^3*d*g*n^3 - 16*b^3*

e*f*n^3 - 12*a*b^2*d*g*n^2 + 48*a*b^2*e*f*n^2 - 72*a^2*b*e*f*n))/(24*e) - (d*g^3*(32*a^3 - 3*b^3*n^3 + 12*a*b^

2*n^2 - 24*a^2*b*n))/(32*e)))/(2*e) - (g*(48*a^3*e^2*f^2 - 13*b^3*d^2*g^2*n^3 - 36*b^3*e^2*f^2*n^3 - 72*a^2*b*

e^2*f^2*n + 48*a^3*d*e*f*g + 12*a*b^2*d^2*g^2*n^2 + 72*a*b^2*e^2*f^2*n^2 + 40*b^3*d*e*f*g*n^3 - 48*a*b^2*d*e*f

*g*n^2))/(32*e^2)) + log(c*(d + e*x)^n)^2*((x^3*((4*b^2*g^2*(a*d*g + 3*a*e*f - b*e*f*n))/e - (b^2*d*g^3*(4*a -

b*n))/e))/4 - (x^2*((d*((48*b^2*g^2*(a*d*g + 3*a*e*f - b*e*f*n))/e - (12*b^2*d*g^3*(4*a - b*n))/e))/(8*e) - (

9*b^2*f*g*(2*a*d*g + 2*a*e*f - b*e*f*n))/e))/4 + (x*((d*((d*((48*b^2*g^2*(a*d*g + 3*a*e*f - b*e*f*n))/e - (12*

b^2*d*g^3*(4*a - b*n))/e))/e - (72*b^2*f*g*(2*a*d*g + 2*a*e*f - b*e*f*n))/e))/(4*e) + (12*b^2*f^2*(3*a*d*g + a

*e*f - b*e*f*n))/e))/4 - (12*a*b^2*d^4*g^3 - 25*b^3*d^4*g^3*n - 48*a*b^2*d*e^3*f^3 + 48*b^3*d*e^3*f^3*n + 72*a

*b^2*d^2*e^2*f^2*g - 108*b^3*d^2*e^2*f^2*g*n - 48*a*b^2*d^3*e*f*g^2 + 88*b^3*d^3*e*f*g^2*n)/(16*e^4) + (3*b^2*

g^3*x^4*(4*a - b*n))/16) + x*((96*a^3*e^3*f^3 + 300*b^3*d^3*g^3*n^3 - 576*b^3*e^3*f^3*n^3 + 288*a^3*d*e^2*f^2*

g - 288*a^2*b*e^3*f^3*n - 144*a*b^2*d^3*g^3*n^2 + 576*a*b^2*e^3*f^3*n^2 + 1296*b^3*d*e^2*f^2*g*n^3 - 1056*b^3*

d^2*e*f*g^2*n^3 - 864*a*b^2*d*e^2*f^2*g*n^2 + 576*a*b^2*d^2*e*f*g^2*n^2)/(96*e^3) + (d*((d*((g^2*(24*a^3*d*g +

72*a^3*e*f + 7*b^3*d*g*n^3 - 16*b^3*e*f*n^3 - 12*a*b^2*d*g*n^2 + 48*a*b^2*e*f*n^2 - 72*a^2*b*e*f*n))/(24*e) -

(d*g^3*(32*a^3 - 3*b^3*n^3 + 12*a*b^2*n^2 - 24*a^2*b*n))/(32*e)))/e - (g*(48*a^3*e^2*f^2 - 13*b^3*d^2*g^2*n^3

- 36*b^3*e^2*f^2*n^3 - 72*a^2*b*e^2*f^2*n + 48*a^3*d*e*f*g + 12*a*b^2*d^2*g^2*n^2 + 72*a*b^2*e^2*f^2*n^2 + 40

*b^3*d*e*f*g*n^3 - 48*a*b^2*d*e*f*g*n^2))/(16*e^2)))/e) - (log(d + e*x)*(415*b^3*d^4*g^3*n^3 + 72*a^2*b*d^4*g^

3*n - 300*a*b^2*d^4*g^3*n^2 - 576*b^3*d*e^3*f^3*n^3 + 576*a*b^2*d*e^3*f^3*n^2 - 1360*b^3*d^3*e*f*g^2*n^3 + 151

2*b^3*d^2*e^2*f^2*g*n^3 - 288*a^2*b*d*e^3*f^3*n - 1296*a*b^2*d^2*e^2*f^2*g*n^2 - 288*a^2*b*d^3*e*f*g^2*n + 432

*a^2*b*d^2*e^2*f^2*g*n + 1056*a*b^2*d^3*e*f*g^2*n^2))/(96*e^4) + (g^3*x^4*(32*a^3 - 3*b^3*n^3 + 12*a*b^2*n^2 -

24*a^2*b*n))/128 + (log(c*(d + e*x)^n)*((x^3*(32*b*e^3*g^2*(6*a^2*d*g + 18*a^2*e*f - b^2*d*g*n^2 + 4*b^2*e*f*

n^2 - 12*a*b*e*f*n) - 24*b*d*e^3*g^3*(8*a^2 + b^2*n^2 - 4*a*b*n)))/(24*e^2) - (x^2*((d*(32*b*e^3*g^2*(6*a^2*d*

g + 18*a^2*e*f - b^2*d*g*n^2 + 4*b^2*e*f*n^2 - 12*a*b*e*f*n) - 24*b*d*e^3*g^3*(8*a^2 + b^2*n^2 - 4*a*b*n)))/e

- 48*b*e^2*g*(12*a^2*e^2*f^2 + b^2*d^2*g^2*n^2 + 6*b^2*e^2*f^2*n^2 - 12*a*b*e^2*f^2*n + 12*a^2*d*e*f*g - 4*b^2

*d*e*f*g*n^2)))/(16*e^2) + (x*((192*a^2*b*e^5*f^3 + 384*b^3*e^5*f^3*n^2 - 96*b^3*d^3*e^2*g^3*n^2 - 384*a*b^2*e

^5*f^3*n - 576*b^3*d*e^4*f^2*g*n^2 + 384*b^3*d^2*e^3*f*g^2*n^2 + 576*a^2*b*d*e^4*f^2*g)/e + (d*((d*(32*b*e^3*g

^2*(6*a^2*d*g + 18*a^2*e*f - b^2*d*g*n^2 + 4*b^2*e*f*n^2 - 12*a*b*e*f*n) - 24*b*d*e^3*g^3*(8*a^2 + b^2*n^2 - 4

*a*b*n)))/e - 48*b*e^2*g*(12*a^2*e^2*f^2 + b^2*d^2*g^2*n^2 + 6*b^2*e^2*f^2*n^2 - 12*a*b*e^2*f^2*n + 12*a^2*d*e

*f*g - 4*b^2*d*e*f*g*n^2)))/e))/(8*e^2) + (3*b*e^2*g^3*x^4*(8*a^2 + b^2*n^2 - 4*a*b*n))/4))/(8*e^2)

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