3.294 \(\int (g+h x)^3 (A+B \log (e (a+b x)^n (c+d x)^{-n})) \, dx\)

Optimal. Leaf size=236 \[ -\frac{B h n x (b c-a d) \left (a^2 d^2 h^2-a b d h (4 d g-c h)+b^2 \left (c^2 h^2-4 c d g h+6 d^2 g^2\right )\right )}{4 b^3 d^3}+\frac{(g+h x)^4 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{4 h}-\frac{B h^2 n x^2 (b c-a d) (-a d h-b c h+4 b d g)}{8 b^2 d^2}-\frac{B n (b g-a h)^4 \log (a+b x)}{4 b^4 h}-\frac{B h^3 n x^3 (b c-a d)}{12 b d}+\frac{B n (d g-c h)^4 \log (c+d x)}{4 d^4 h} \]

[Out]

-(B*(b*c - a*d)*h*(a^2*d^2*h^2 - a*b*d*h*(4*d*g - c*h) + b^2*(6*d^2*g^2 - 4*c*d*g*h + c^2*h^2))*n*x)/(4*b^3*d^
3) - (B*(b*c - a*d)*h^2*(4*b*d*g - b*c*h - a*d*h)*n*x^2)/(8*b^2*d^2) - (B*(b*c - a*d)*h^3*n*x^3)/(12*b*d) - (B
*(b*g - a*h)^4*n*Log[a + b*x])/(4*b^4*h) + (B*(d*g - c*h)^4*n*Log[c + d*x])/(4*d^4*h) + ((g + h*x)^4*(A + B*Lo
g[(e*(a + b*x)^n)/(c + d*x)^n]))/(4*h)

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Rubi [A]  time = 0.455644, antiderivative size = 248, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {6742, 2492, 72} \[ -\frac{B h n x (b c-a d) \left (a^2 d^2 h^2-a b d h (4 d g-c h)+b^2 \left (c^2 h^2-4 c d g h+6 d^2 g^2\right )\right )}{4 b^3 d^3}-\frac{B h^2 n x^2 (b c-a d) (-a d h-b c h+4 b d g)}{8 b^2 d^2}-\frac{B n (b g-a h)^4 \log (a+b x)}{4 b^4 h}+\frac{B (g+h x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 h}-\frac{B h^3 n x^3 (b c-a d)}{12 b d}+\frac{A (g+h x)^4}{4 h}+\frac{B n (d g-c h)^4 \log (c+d x)}{4 d^4 h} \]

Antiderivative was successfully verified.

[In]

Int[(g + h*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]),x]

[Out]

-(B*(b*c - a*d)*h*(a^2*d^2*h^2 - a*b*d*h*(4*d*g - c*h) + b^2*(6*d^2*g^2 - 4*c*d*g*h + c^2*h^2))*n*x)/(4*b^3*d^
3) - (B*(b*c - a*d)*h^2*(4*b*d*g - b*c*h - a*d*h)*n*x^2)/(8*b^2*d^2) - (B*(b*c - a*d)*h^3*n*x^3)/(12*b*d) + (A
*(g + h*x)^4)/(4*h) - (B*(b*g - a*h)^4*n*Log[a + b*x])/(4*b^4*h) + (B*(d*g - c*h)^4*n*Log[c + d*x])/(4*d^4*h)
+ (B*(g + h*x)^4*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(4*h)

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 2492

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*((g_.) + (h_.)*(x_))^
(m_.), x_Symbol] :> Simp[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/(h*(m + 1)), x] - Dist[(p*
r*s*(b*c - a*d))/(h*(m + 1)), Int[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a + b*x)*
(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0]
&& IGtQ[s, 0] && NeQ[m, -1]

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

\begin{align*} \int (g+h x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx &=\int \left (A (g+h x)^3+B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx\\ &=\frac{A (g+h x)^4}{4 h}+B \int (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx\\ &=\frac{A (g+h x)^4}{4 h}+\frac{B (g+h x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 h}-\frac{(B (b c-a d) n) \int \frac{(g+h x)^4}{(a+b x) (c+d x)} \, dx}{4 h}\\ &=\frac{A (g+h x)^4}{4 h}+\frac{B (g+h x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 h}-\frac{(B (b c-a d) n) \int \left (\frac{h^2 \left (a^2 d^2 h^2-a b d h (4 d g-c h)+b^2 \left (6 d^2 g^2-4 c d g h+c^2 h^2\right )\right )}{b^3 d^3}+\frac{h^3 (4 b d g-b c h-a d h) x}{b^2 d^2}+\frac{h^4 x^2}{b d}+\frac{(b g-a h)^4}{b^3 (b c-a d) (a+b x)}+\frac{(d g-c h)^4}{d^3 (-b c+a d) (c+d x)}\right ) \, dx}{4 h}\\ &=-\frac{B (b c-a d) h \left (a^2 d^2 h^2-a b d h (4 d g-c h)+b^2 \left (6 d^2 g^2-4 c d g h+c^2 h^2\right )\right ) n x}{4 b^3 d^3}-\frac{B (b c-a d) h^2 (4 b d g-b c h-a d h) n x^2}{8 b^2 d^2}-\frac{B (b c-a d) h^3 n x^3}{12 b d}+\frac{A (g+h x)^4}{4 h}-\frac{B (b g-a h)^4 n \log (a+b x)}{4 b^4 h}+\frac{B (d g-c h)^4 n \log (c+d x)}{4 d^4 h}+\frac{B (g+h x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 h}\\ \end{align*}

Mathematica [A]  time = 0.566985, size = 314, normalized size = 1.33 \[ \frac{b d x \left (6 A b^3 d^3 \left (6 g^2 h x+4 g^3+4 g h^2 x^2+h^3 x^3\right )-B h n (b c-a d) \left (6 a^2 d^2 h^2-3 a b d h (-2 c h+8 d g+d h x)+b^2 \left (6 c^2 h^2-3 c d h (8 g+h x)+2 d^2 \left (18 g^2+6 g h x+h^2 x^2\right )\right )\right )\right )-6 a^2 B d^4 h n \left (a^2 h^2-4 a b g h+6 b^2 g^2\right ) \log (a+b x)+6 b^3 B n \log (c+d x) \left (4 a d^4 g^3+b c \left (-4 c^2 d g h^2+c^3 h^3+6 c d^2 g^2 h-4 d^3 g^3\right )\right )+6 b^3 B d^4 \left (4 a g^3+b x \left (6 g^2 h x+4 g^3+4 g h^2 x^2+h^3 x^3\right )\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{24 b^4 d^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(g + h*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]),x]

[Out]

(b*d*x*(6*A*b^3*d^3*(4*g^3 + 6*g^2*h*x + 4*g*h^2*x^2 + h^3*x^3) - B*(b*c - a*d)*h*n*(6*a^2*d^2*h^2 - 3*a*b*d*h
*(8*d*g - 2*c*h + d*h*x) + b^2*(6*c^2*h^2 - 3*c*d*h*(8*g + h*x) + 2*d^2*(18*g^2 + 6*g*h*x + h^2*x^2)))) - 6*a^
2*B*d^4*h*(6*b^2*g^2 - 4*a*b*g*h + a^2*h^2)*n*Log[a + b*x] + 6*b^3*B*(4*a*d^4*g^3 + b*c*(-4*d^3*g^3 + 6*c*d^2*
g^2*h - 4*c^2*d*g*h^2 + c^3*h^3))*n*Log[c + d*x] + 6*b^3*B*d^4*(4*a*g^3 + b*x*(4*g^3 + 6*g^2*h*x + 4*g*h^2*x^2
+ h^3*x^3))*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(24*b^4*d^4)

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Maple [C]  time = 0.613, size = 1967, normalized size = 8.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((h*x+g)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n))),x)

[Out]

B*g^3*x*ln((b*x+a)^n)+B*ln(e)*g^3*x+1/4*h^3*B*x^4*ln((b*x+a)^n)+1/4*h^3*B*ln(e)*x^4-1/4*(h*x+g)^4*B/h*ln((d*x+
c)^n)+1/4*h^3*A*x^4+3/2*h*B*ln(e)*g^2*x^2+1/4/h*B*ln(-d*x-c)*g^4*n+h^2*B*g*x^3*ln((b*x+a)^n)+3/2*h*B*g^2*x^2*l
n((b*x+a)^n)+h^2*B*ln(e)*g*x^3+1/12*h^3/b*B*a*n*x^3-1/12/d*h^3*B*c*n*x^3-1/8*h^3/b^2*B*a^2*n*x^2+1/8/d^2*h^3*B
*c^2*n*x^2+1/4*h^3/b^3*B*a^3*n*x-1/4/d^3*h^3*B*c^3*n*x-1/d^3*h^2*B*ln(-d*x-c)*c^3*g*n+3/2/d^2*h*B*ln(-d*x-c)*c
^2*g^2*n+h^2/b^3*B*ln(b*x+a)*a^3*g*n-3/2*h/b^2*B*ln(b*x+a)*a^2*g^2*n+1/2*I*B*Pi*g^3*x*csgn(I*(b*x+a)^n/((d*x+c
)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+1/2*I*B*Pi*g^3*x*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+1/8*I*h^3
*B*Pi*x^4*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2-1/2*I*h^2*B*Pi*g*x^3*csgn(I*(b*x+a)^n/((d*x+c)^n))^3-1/2
*I*h^2*B*Pi*g*x^3*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3-3/4*I*h*B*Pi*g^2*x^2*csgn(I*(b*x+a)^n/((d*x+c)^n))^3-3/4*I
*h*B*Pi*g^2*x^2*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3+1/8*I*h^3*B*Pi*x^4*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*
x+c)^n))^2+1/8*I*h^3*B*Pi*x^4*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+1/2*I*B*Pi*g^3*x
*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+1/2*I*B*Pi*g^3*x*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x
+c)^n))^2+1/8*I*h^3*B*Pi*x^4*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2-1/2*I*B*Pi*g^3*x*csgn(I*e/((d*x
+c)^n)*(b*x+a)^n)^3-1/8*I*h^3*B*Pi*x^4*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3-1/8*I*h^3*B*Pi*x^4*csgn(I*(b*x+a)^n/(
(d*x+c)^n))^3-1/2*I*B*Pi*g^3*x*csgn(I*(b*x+a)^n/((d*x+c)^n))^3+1/2*h^2/b*B*a*g*n*x^2-1/2/d*h^2*B*c*g*n*x^2-h^2
/b^2*B*a^2*g*n*x+3/2*h/b*B*a*g^2*n*x+1/d^2*h^2*B*c^2*g*n*x-3/2/d*h*B*c*g^2*n*x+1/2*I*h^2*B*Pi*g*x^3*csgn(I*(b*
x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+3/4*I*h*B*Pi*g^2*x^2*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/
((d*x+c)^n))^2-1/8*I*h^3*B*Pi*x^4*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))+3/4*I*h*
B*Pi*g^2*x^2*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2-1/2*I*B*Pi*g^3*x*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^
n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)-1/2*I*B*Pi*g^3*x*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((
d*x+c)^n))+1/2*I*h^2*B*Pi*g*x^3*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+1/2*I*h^2*B*Pi*g*x^3*csgn(I/
((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+3/4*I*h*B*Pi*g^2*x^2*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x
+c)^n)*(b*x+a)^n)^2+1/2*I*h^2*B*Pi*g*x^3*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+3/4*I*h*B*Pi*g^2*x^2*csgn
(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2-1/8*I*h^3*B*Pi*x^4*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^n))*csg
n(I*e/((d*x+c)^n)*(b*x+a)^n)+h^2*A*g*x^3+3/2*h*A*g^2*x^2+A*g^3*x+1/4/d^4*h^3*B*ln(-d*x-c)*c^4*n-1/4*h^3/b^4*B*
ln(b*x+a)*a^4*n-1/d*B*ln(-d*x-c)*c*g^3*n+1/b*B*ln(b*x+a)*a*g^3*n-3/4*I*h*B*Pi*g^2*x^2*csgn(I*e)*csgn(I*(b*x+a)
^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)-1/2*I*h^2*B*Pi*g*x^3*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csg
n(I*(b*x+a)^n/((d*x+c)^n))-3/4*I*h*B*Pi*g^2*x^2*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c
)^n))-1/2*I*h^2*B*Pi*g*x^3*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)

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Maxima [B]  time = 1.20409, size = 630, normalized size = 2.67 \begin{align*} \frac{1}{4} \, B h^{3} x^{4} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + \frac{1}{4} \, A h^{3} x^{4} + B g h^{2} x^{3} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A g h^{2} x^{3} + \frac{3}{2} \, B g^{2} h x^{2} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + \frac{3}{2} \, A g^{2} h x^{2} + B g^{3} x \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A g^{3} x + \frac{{\left (\frac{a e n \log \left (b x + a\right )}{b} - \frac{c e n \log \left (d x + c\right )}{d}\right )} B g^{3}}{e} - \frac{3 \,{\left (\frac{a^{2} e n \log \left (b x + a\right )}{b^{2}} - \frac{c^{2} e n \log \left (d x + c\right )}{d^{2}} + \frac{{\left (b c e n - a d e n\right )} x}{b d}\right )} B g^{2} h}{2 \, e} + \frac{{\left (\frac{2 \, a^{3} e n \log \left (b x + a\right )}{b^{3}} - \frac{2 \, c^{3} e n \log \left (d x + c\right )}{d^{3}} - \frac{{\left (b^{2} c d e n - a b d^{2} e n\right )} x^{2} - 2 \,{\left (b^{2} c^{2} e n - a^{2} d^{2} e n\right )} x}{b^{2} d^{2}}\right )} B g h^{2}}{2 \, e} - \frac{{\left (\frac{6 \, a^{4} e n \log \left (b x + a\right )}{b^{4}} - \frac{6 \, c^{4} e n \log \left (d x + c\right )}{d^{4}} + \frac{2 \,{\left (b^{3} c d^{2} e n - a b^{2} d^{3} e n\right )} x^{3} - 3 \,{\left (b^{3} c^{2} d e n - a^{2} b d^{3} e n\right )} x^{2} + 6 \,{\left (b^{3} c^{3} e n - a^{3} d^{3} e n\right )} x}{b^{3} d^{3}}\right )} B h^{3}}{24 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*B*h^3*x^4*log((b*x + a)^n*e/(d*x + c)^n) + 1/4*A*h^3*x^4 + B*g*h^2*x^3*log((b*x + a)^n*e/(d*x + c)^n) + A*
g*h^2*x^3 + 3/2*B*g^2*h*x^2*log((b*x + a)^n*e/(d*x + c)^n) + 3/2*A*g^2*h*x^2 + B*g^3*x*log((b*x + a)^n*e/(d*x
+ c)^n) + A*g^3*x + (a*e*n*log(b*x + a)/b - c*e*n*log(d*x + c)/d)*B*g^3/e - 3/2*(a^2*e*n*log(b*x + a)/b^2 - c^
2*e*n*log(d*x + c)/d^2 + (b*c*e*n - a*d*e*n)*x/(b*d))*B*g^2*h/e + 1/2*(2*a^3*e*n*log(b*x + a)/b^3 - 2*c^3*e*n*
log(d*x + c)/d^3 - ((b^2*c*d*e*n - a*b*d^2*e*n)*x^2 - 2*(b^2*c^2*e*n - a^2*d^2*e*n)*x)/(b^2*d^2))*B*g*h^2/e -
1/24*(6*a^4*e*n*log(b*x + a)/b^4 - 6*c^4*e*n*log(d*x + c)/d^4 + (2*(b^3*c*d^2*e*n - a*b^2*d^3*e*n)*x^3 - 3*(b^
3*c^2*d*e*n - a^2*b*d^3*e*n)*x^2 + 6*(b^3*c^3*e*n - a^3*d^3*e*n)*x)/(b^3*d^3))*B*h^3/e

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Fricas [B]  time = 1.10268, size = 1149, normalized size = 4.87 \begin{align*} \frac{6 \, A b^{4} d^{4} h^{3} x^{4} + 2 \,{\left (12 \, A b^{4} d^{4} g h^{2} -{\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} h^{3} n\right )} x^{3} + 3 \,{\left (12 \, A b^{4} d^{4} g^{2} h -{\left (4 \,{\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} g h^{2} -{\left (B b^{4} c^{2} d^{2} - B a^{2} b^{2} d^{4}\right )} h^{3}\right )} n\right )} x^{2} + 6 \,{\left (4 \, A b^{4} d^{4} g^{3} -{\left (6 \,{\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} g^{2} h - 4 \,{\left (B b^{4} c^{2} d^{2} - B a^{2} b^{2} d^{4}\right )} g h^{2} +{\left (B b^{4} c^{3} d - B a^{3} b d^{4}\right )} h^{3}\right )} n\right )} x + 6 \,{\left (B b^{4} d^{4} h^{3} n x^{4} + 4 \, B b^{4} d^{4} g h^{2} n x^{3} + 6 \, B b^{4} d^{4} g^{2} h n x^{2} + 4 \, B b^{4} d^{4} g^{3} n x +{\left (4 \, B a b^{3} d^{4} g^{3} - 6 \, B a^{2} b^{2} d^{4} g^{2} h + 4 \, B a^{3} b d^{4} g h^{2} - B a^{4} d^{4} h^{3}\right )} n\right )} \log \left (b x + a\right ) - 6 \,{\left (B b^{4} d^{4} h^{3} n x^{4} + 4 \, B b^{4} d^{4} g h^{2} n x^{3} + 6 \, B b^{4} d^{4} g^{2} h n x^{2} + 4 \, B b^{4} d^{4} g^{3} n x +{\left (4 \, B b^{4} c d^{3} g^{3} - 6 \, B b^{4} c^{2} d^{2} g^{2} h + 4 \, B b^{4} c^{3} d g h^{2} - B b^{4} c^{4} h^{3}\right )} n\right )} \log \left (d x + c\right ) + 6 \,{\left (B b^{4} d^{4} h^{3} x^{4} + 4 \, B b^{4} d^{4} g h^{2} x^{3} + 6 \, B b^{4} d^{4} g^{2} h x^{2} + 4 \, B b^{4} d^{4} g^{3} x\right )} \log \left (e\right )}{24 \, b^{4} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(6*A*b^4*d^4*h^3*x^4 + 2*(12*A*b^4*d^4*g*h^2 - (B*b^4*c*d^3 - B*a*b^3*d^4)*h^3*n)*x^3 + 3*(12*A*b^4*d^4*g
^2*h - (4*(B*b^4*c*d^3 - B*a*b^3*d^4)*g*h^2 - (B*b^4*c^2*d^2 - B*a^2*b^2*d^4)*h^3)*n)*x^2 + 6*(4*A*b^4*d^4*g^3
- (6*(B*b^4*c*d^3 - B*a*b^3*d^4)*g^2*h - 4*(B*b^4*c^2*d^2 - B*a^2*b^2*d^4)*g*h^2 + (B*b^4*c^3*d - B*a^3*b*d^4
)*h^3)*n)*x + 6*(B*b^4*d^4*h^3*n*x^4 + 4*B*b^4*d^4*g*h^2*n*x^3 + 6*B*b^4*d^4*g^2*h*n*x^2 + 4*B*b^4*d^4*g^3*n*x
+ (4*B*a*b^3*d^4*g^3 - 6*B*a^2*b^2*d^4*g^2*h + 4*B*a^3*b*d^4*g*h^2 - B*a^4*d^4*h^3)*n)*log(b*x + a) - 6*(B*b^
4*d^4*h^3*n*x^4 + 4*B*b^4*d^4*g*h^2*n*x^3 + 6*B*b^4*d^4*g^2*h*n*x^2 + 4*B*b^4*d^4*g^3*n*x + (4*B*b^4*c*d^3*g^3
- 6*B*b^4*c^2*d^2*g^2*h + 4*B*b^4*c^3*d*g*h^2 - B*b^4*c^4*h^3)*n)*log(d*x + c) + 6*(B*b^4*d^4*h^3*x^4 + 4*B*b
^4*d^4*g*h^2*x^3 + 6*B*b^4*d^4*g^2*h*x^2 + 4*B*b^4*d^4*g^3*x)*log(e))/(b^4*d^4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)**3*(A+B*ln(e*(b*x+a)**n/((d*x+c)**n))),x)

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out