3.124 \(\int \frac{(c i+d i x)^2 (A+B \log (e (\frac{a+b x}{c+d x})^n))}{(a g+b g x)^4} \, dx\)

Optimal. Leaf size=93 \[ -\frac{i^2 (c+d x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 g^4 (a+b x)^3 (b c-a d)}-\frac{B i^2 n (c+d x)^3}{9 g^4 (a+b x)^3 (b c-a d)} \]

[Out]

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

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Rubi [B]  time = 0.515931, antiderivative size = 301, normalized size of antiderivative = 3.24, number of steps used = 14, number of rules used = 4, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.093, Rules used = {2528, 2525, 12, 44} \[ -\frac{d^2 i^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g^4 (a+b x)}-\frac{d i^2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g^4 (a+b x)^2}-\frac{i^2 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 b^3 g^4 (a+b x)^3}-\frac{B d^3 i^2 n \log (a+b x)}{3 b^3 g^4 (b c-a d)}+\frac{B d^3 i^2 n \log (c+d x)}{3 b^3 g^4 (b c-a d)}-\frac{B d i^2 n (b c-a d)}{3 b^3 g^4 (a+b x)^2}-\frac{B i^2 n (b c-a d)^2}{9 b^3 g^4 (a+b x)^3}-\frac{B d^2 i^2 n}{3 b^3 g^4 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[((c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a*g + b*g*x)^4,x]

[Out]

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

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(124 c+124 d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^4} \, dx &=\int \left (\frac{15376 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^4 (a+b x)^4}+\frac{30752 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^4 (a+b x)^3}+\frac{15376 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^4 (a+b x)^2}\right ) \, dx\\ &=\frac{\left (15376 d^2\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{b^2 g^4}+\frac{(30752 d (b c-a d)) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{b^2 g^4}+\frac{\left (15376 (b c-a d)^2\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^4} \, dx}{b^2 g^4}\\ &=-\frac{15376 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^4 (a+b x)^3}-\frac{15376 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^4 (a+b x)^2}-\frac{15376 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^4 (a+b x)}+\frac{\left (15376 B d^2 n\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^4}+\frac{(15376 B d (b c-a d) n) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^4}+\frac{\left (15376 B (b c-a d)^2 n\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^4}\\ &=-\frac{15376 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^4 (a+b x)^3}-\frac{15376 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^4 (a+b x)^2}-\frac{15376 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^4 (a+b x)}+\frac{\left (15376 B d^2 (b c-a d) n\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^4}+\frac{\left (15376 B d (b c-a d)^2 n\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^4}+\frac{\left (15376 B (b c-a d)^3 n\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^4}\\ &=-\frac{15376 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^4 (a+b x)^3}-\frac{15376 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^4 (a+b x)^2}-\frac{15376 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^4 (a+b x)}+\frac{\left (15376 B d^2 (b c-a d) n\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^2}-\frac{b d}{(b c-a d)^2 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^3 g^4}+\frac{\left (15376 B d (b c-a d)^2 n\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^3 g^4}+\frac{\left (15376 B (b c-a d)^3 n\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^4}-\frac{b d}{(b c-a d)^2 (a+b x)^3}+\frac{b d^2}{(b c-a d)^3 (a+b x)^2}-\frac{b d^3}{(b c-a d)^4 (a+b x)}+\frac{d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b^3 g^4}\\ &=-\frac{15376 B (b c-a d)^2 n}{9 b^3 g^4 (a+b x)^3}-\frac{15376 B d (b c-a d) n}{3 b^3 g^4 (a+b x)^2}-\frac{15376 B d^2 n}{3 b^3 g^4 (a+b x)}-\frac{15376 B d^3 n \log (a+b x)}{3 b^3 (b c-a d) g^4}-\frac{15376 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^4 (a+b x)^3}-\frac{15376 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^4 (a+b x)^2}-\frac{15376 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^4 (a+b x)}+\frac{15376 B d^3 n \log (c+d x)}{3 b^3 (b c-a d) g^4}\\ \end{align*}

Mathematica [B]  time = 0.337913, size = 329, normalized size = 3.54 \[ -\frac{i^2 \left (-9 a^2 A b d^3 x-3 a^3 A d^3+3 B (b c-a d) \left (a^2 d^2+a b d (c+3 d x)+b^2 \left (c^2+3 c d x+3 d^2 x^2\right )\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-9 a^2 b B d^3 n x \log (c+d x)-3 a^2 b B d^3 n x-3 a^3 B d^3 n \log (c+d x)-a^3 B d^3 n-9 a A b^2 d^3 x^2-9 a b^2 B d^3 n x^2 \log (c+d x)-3 a b^2 B d^3 n x^2+3 B d^3 n (a+b x)^3 \log (a+b x)+9 A b^3 c^2 d x+3 A b^3 c^3+9 A b^3 c d^2 x^2+3 b^3 B c^2 d n x+b^3 B c^3 n+3 b^3 B c d^2 n x^2-3 b^3 B d^3 n x^3 \log (c+d x)\right )}{9 b^3 g^4 (a+b x)^3 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Integrate[((c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a*g + b*g*x)^4,x]

[Out]

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

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Maple [F]  time = 0.673, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dix+ci \right ) ^{2}}{ \left ( bgx+ag \right ) ^{4}} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*i*x+c*i)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4,x)

[Out]

int((d*i*x+c*i)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4,x)

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Maxima [B]  time = 1.61, size = 2084, normalized size = 22.41 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4,x, algorithm="maxima")

[Out]

-1/18*B*d^2*i^2*n*((11*a^2*b^2*c^2 - 7*a^3*b*c*d + 2*a^4*d^2 + 6*(3*b^4*c^2 - 3*a*b^3*c*d + a^2*b^2*d^2)*x^2 +
3*(9*a*b^3*c^2 - 7*a^2*b^2*c*d + 2*a^3*b*d^2)*x)/((b^8*c^2 - 2*a*b^7*c*d + a^2*b^6*d^2)*g^4*x^3 + 3*(a*b^7*c^
2 - 2*a^2*b^6*c*d + a^3*b^5*d^2)*g^4*x^2 + 3*(a^2*b^6*c^2 - 2*a^3*b^5*c*d + a^4*b^4*d^2)*g^4*x + (a^3*b^5*c^2
- 2*a^4*b^4*c*d + a^5*b^3*d^2)*g^4) + 6*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*log(b*x + a)/((b^6*c^3 - 3*a*b^5
*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*g^4) - 6*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*log(d*x + c)/((b^6*c^3
- 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*g^4)) - 1/18*B*c^2*i^2*n*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*
c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*g^4*x^3 + 3*(a*b^5*c^2 -
2*a^2*b^4*c*d + a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3*c^2 - 2*
a^4*b^2*c*d + a^5*b*d^2)*g^4) + 6*d^3*log(b*x + a)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^
4) - 6*d^3*log(d*x + c)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4)) - 1/18*B*c*d*i^2*n*((5*
a*b^2*c^2 - 22*a^2*b*c*d + 5*a^3*d^2 - 6*(3*b^3*c*d - a*b^2*d^2)*x^2 + 3*(3*b^3*c^2 - 16*a*b^2*c*d + 5*a^2*b*d
^2)*x)/((b^7*c^2 - 2*a*b^6*c*d + a^2*b^5*d^2)*g^4*x^3 + 3*(a*b^6*c^2 - 2*a^2*b^5*c*d + a^3*b^4*d^2)*g^4*x^2 +
3*(a^2*b^5*c^2 - 2*a^3*b^4*c*d + a^4*b^3*d^2)*g^4*x + (a^3*b^4*c^2 - 2*a^4*b^3*c*d + a^5*b^2*d^2)*g^4) - 6*(3*
b*c*d^2 - a*d^3)*log(b*x + a)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4) + 6*(3*b*c*d^2 -
a*d^3)*log(d*x + c)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4)) - 1/3*(3*b*x + a)*B*c*d*
i^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(b^5*g^4*x^3 + 3*a*b^4*g^4*x^2 + 3*a^2*b^3*g^4*x + a^3*b^2*g^4) - 1
/3*(3*b^2*x^2 + 3*a*b*x + a^2)*B*d^2*i^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(b^6*g^4*x^3 + 3*a*b^5*g^4*x^2
+ 3*a^2*b^4*g^4*x + a^3*b^3*g^4) - 1/3*(3*b*x + a)*A*c*d*i^2/(b^5*g^4*x^3 + 3*a*b^4*g^4*x^2 + 3*a^2*b^3*g^4*x
+ a^3*b^2*g^4) - 1/3*(3*b^2*x^2 + 3*a*b*x + a^2)*A*d^2*i^2/(b^6*g^4*x^3 + 3*a*b^5*g^4*x^2 + 3*a^2*b^4*g^4*x +
a^3*b^3*g^4) - 1/3*B*c^2*i^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^
2*g^4*x + a^3*b*g^4) - 1/3*A*c^2*i^2/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4)

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Fricas [B]  time = 0.514387, size = 799, normalized size = 8.59 \begin{align*} -\frac{{\left (B b^{3} c^{3} - B a^{3} d^{3}\right )} i^{2} n + 3 \,{\left (A b^{3} c^{3} - A a^{3} d^{3}\right )} i^{2} + 3 \,{\left ({\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} i^{2} n + 3 \,{\left (A b^{3} c d^{2} - A a b^{2} d^{3}\right )} i^{2}\right )} x^{2} + 3 \,{\left ({\left (B b^{3} c^{2} d - B a^{2} b d^{3}\right )} i^{2} n + 3 \,{\left (A b^{3} c^{2} d - A a^{2} b d^{3}\right )} i^{2}\right )} x + 3 \,{\left (3 \,{\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} i^{2} x^{2} + 3 \,{\left (B b^{3} c^{2} d - B a^{2} b d^{3}\right )} i^{2} x +{\left (B b^{3} c^{3} - B a^{3} d^{3}\right )} i^{2}\right )} \log \left (e\right ) + 3 \,{\left (B b^{3} d^{3} i^{2} n x^{3} + 3 \, B b^{3} c d^{2} i^{2} n x^{2} + 3 \, B b^{3} c^{2} d i^{2} n x + B b^{3} c^{3} i^{2} n\right )} \log \left (\frac{b x + a}{d x + c}\right )}{9 \,{\left ({\left (b^{7} c - a b^{6} d\right )} g^{4} x^{3} + 3 \,{\left (a b^{6} c - a^{2} b^{5} d\right )} g^{4} x^{2} + 3 \,{\left (a^{2} b^{5} c - a^{3} b^{4} d\right )} g^{4} x +{\left (a^{3} b^{4} c - a^{4} b^{3} d\right )} g^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4,x, algorithm="fricas")

[Out]

-1/9*((B*b^3*c^3 - B*a^3*d^3)*i^2*n + 3*(A*b^3*c^3 - A*a^3*d^3)*i^2 + 3*((B*b^3*c*d^2 - B*a*b^2*d^3)*i^2*n + 3
*(A*b^3*c*d^2 - A*a*b^2*d^3)*i^2)*x^2 + 3*((B*b^3*c^2*d - B*a^2*b*d^3)*i^2*n + 3*(A*b^3*c^2*d - A*a^2*b*d^3)*i
^2)*x + 3*(3*(B*b^3*c*d^2 - B*a*b^2*d^3)*i^2*x^2 + 3*(B*b^3*c^2*d - B*a^2*b*d^3)*i^2*x + (B*b^3*c^3 - B*a^3*d^
3)*i^2)*log(e) + 3*(B*b^3*d^3*i^2*n*x^3 + 3*B*b^3*c*d^2*i^2*n*x^2 + 3*B*b^3*c^2*d*i^2*n*x + B*b^3*c^3*i^2*n)*l
og((b*x + a)/(d*x + c)))/((b^7*c - a*b^6*d)*g^4*x^3 + 3*(a*b^6*c - a^2*b^5*d)*g^4*x^2 + 3*(a^2*b^5*c - a^3*b^4
*d)*g^4*x + (a^3*b^4*c - a^4*b^3*d)*g^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((d*i*x+c*i)**2*(A+B*ln(e*((b*x+a)/(d*x+c))**n))/(b*g*x+a*g)**4,x)

[Out]

Timed out

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Giac [B]  time = 1.22472, size = 552, normalized size = 5.94 \begin{align*} \frac{B d^{3} n \log \left (b x + a\right )}{3 \,{\left (b^{4} c g^{4} - a b^{3} d g^{4}\right )}} - \frac{B d^{3} n \log \left (d x + c\right )}{3 \,{\left (b^{4} c g^{4} - a b^{3} d g^{4}\right )}} + \frac{{\left (3 \, B b^{2} d^{2} n x^{2} + 3 \, B b^{2} c d n x + 3 \, B a b d^{2} n x + B b^{2} c^{2} n + B a b c d n + B a^{2} d^{2} n\right )} \log \left (\frac{b x + a}{d x + c}\right )}{3 \,{\left (b^{6} g^{4} x^{3} + 3 \, a b^{5} g^{4} x^{2} + 3 \, a^{2} b^{4} g^{4} x + a^{3} b^{3} g^{4}\right )}} + \frac{3 \, B b^{2} d^{2} n x^{2} + 3 \, B b^{2} c d n x + 3 \, B a b d^{2} n x + 9 \, A b^{2} d^{2} x^{2} + 9 \, B b^{2} d^{2} x^{2} + B b^{2} c^{2} n + B a b c d n + B a^{2} d^{2} n + 9 \, A b^{2} c d x + 9 \, B b^{2} c d x + 9 \, A a b d^{2} x + 9 \, B a b d^{2} x + 3 \, A b^{2} c^{2} + 3 \, B b^{2} c^{2} + 3 \, A a b c d + 3 \, B a b c d + 3 \, A a^{2} d^{2} + 3 \, B a^{2} d^{2}}{9 \,{\left (b^{6} g^{4} x^{3} + 3 \, a b^{5} g^{4} x^{2} + 3 \, a^{2} b^{4} g^{4} x + a^{3} b^{3} g^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4,x, algorithm="giac")

[Out]

1/3*B*d^3*n*log(b*x + a)/(b^4*c*g^4 - a*b^3*d*g^4) - 1/3*B*d^3*n*log(d*x + c)/(b^4*c*g^4 - a*b^3*d*g^4) + 1/3*
(3*B*b^2*d^2*n*x^2 + 3*B*b^2*c*d*n*x + 3*B*a*b*d^2*n*x + B*b^2*c^2*n + B*a*b*c*d*n + B*a^2*d^2*n)*log((b*x + a
)/(d*x + c))/(b^6*g^4*x^3 + 3*a*b^5*g^4*x^2 + 3*a^2*b^4*g^4*x + a^3*b^3*g^4) + 1/9*(3*B*b^2*d^2*n*x^2 + 3*B*b^
2*c*d*n*x + 3*B*a*b*d^2*n*x + 9*A*b^2*d^2*x^2 + 9*B*b^2*d^2*x^2 + B*b^2*c^2*n + B*a*b*c*d*n + B*a^2*d^2*n + 9*
A*b^2*c*d*x + 9*B*b^2*c*d*x + 9*A*a*b*d^2*x + 9*B*a*b*d^2*x + 3*A*b^2*c^2 + 3*B*b^2*c^2 + 3*A*a*b*c*d + 3*B*a*
b*c*d + 3*A*a^2*d^2 + 3*B*a^2*d^2)/(b^6*g^4*x^3 + 3*a*b^5*g^4*x^2 + 3*a^2*b^4*g^4*x + a^3*b^3*g^4)