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

Optimal. Leaf size=181 \[ -\frac {b i (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 g^4 (a+b x)^3 (b c-a d)^2}+\frac {d i (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^4 (a+b x)^2 (b c-a d)^2}-\frac {b B i n (c+d x)^3}{9 g^4 (a+b x)^3 (b c-a d)^2}+\frac {B d i n (c+d x)^2}{4 g^4 (a+b x)^2 (b c-a d)^2} \]

[Out]

1/4*B*d*i*n*(d*x+c)^2/(-a*d+b*c)^2/g^4/(b*x+a)^2-1/9*b*B*i*n*(d*x+c)^3/(-a*d+b*c)^2/g^4/(b*x+a)^3+1/2*d*i*(d*x
+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^2/g^4/(b*x+a)^2-1/3*b*i*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))
^n))/(-a*d+b*c)^2/g^4/(b*x+a)^3

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Rubi [A]  time = 0.34, antiderivative size = 236, normalized size of antiderivative = 1.30, number of steps used = 10, number of rules used = 4, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {2528, 2525, 12, 44} \[ -\frac {d i \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^2 g^4 (a+b x)^2}-\frac {i (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 b^2 g^4 (a+b x)^3}+\frac {B d^2 i n}{6 b^2 g^4 (a+b x) (b c-a d)}+\frac {B d^3 i n \log (a+b x)}{6 b^2 g^4 (b c-a d)^2}-\frac {B d^3 i n \log (c+d x)}{6 b^2 g^4 (b c-a d)^2}-\frac {B i n (b c-a d)}{9 b^2 g^4 (a+b x)^3}-\frac {B d i n}{12 b^2 g^4 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(B*(b*c - a*d)*i*n)/(9*b^2*g^4*(a + b*x)^3) - (B*d*i*n)/(12*b^2*g^4*(a + b*x)^2) + (B*d^2*i*n)/(6*b^2*(b*c -
a*d)*g^4*(a + b*x)) + (B*d^3*i*n*Log[a + b*x])/(6*b^2*(b*c - a*d)^2*g^4) - ((b*c - a*d)*i*(A + B*Log[e*((a + b
*x)/(c + d*x))^n]))/(3*b^2*g^4*(a + b*x)^3) - (d*i*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*b^2*g^4*(a + b*x
)^2) - (B*d^3*i*n*Log[c + d*x])/(6*b^2*(b*c - a*d)^2*g^4)

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])

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 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]

Rubi steps

\begin {align*} \int \frac {(115 c+115 d x) \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 {115 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b g^4 (a+b x)^4}+\frac {115 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b g^4 (a+b x)^3}\right ) \, dx\\ &=\frac {(115 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 g^4}+\frac {(115 (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)^4} \, dx}{b g^4}\\ &=-\frac {115 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 g^4 (a+b x)^3}-\frac {115 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^2 g^4 (a+b x)^2}+\frac {(115 B d n) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2 b^2 g^4}+\frac {(115 B (b c-a d) n) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{3 b^2 g^4}\\ &=-\frac {115 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 g^4 (a+b x)^3}-\frac {115 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^2 g^4 (a+b x)^2}+\frac {(115 B d (b c-a d) n) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{2 b^2 g^4}+\frac {\left (115 B (b c-a d)^2 n\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{3 b^2 g^4}\\ &=-\frac {115 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 g^4 (a+b x)^3}-\frac {115 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^2 g^4 (a+b x)^2}+\frac {(115 B d (b c-a d) n) \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}{2 b^2 g^4}+\frac {\left (115 B (b c-a d)^2 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^2 g^4}\\ &=-\frac {115 B (b c-a d) n}{9 b^2 g^4 (a+b x)^3}-\frac {115 B d n}{12 b^2 g^4 (a+b x)^2}+\frac {115 B d^2 n}{6 b^2 (b c-a d) g^4 (a+b x)}+\frac {115 B d^3 n \log (a+b x)}{6 b^2 (b c-a d)^2 g^4}-\frac {115 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 g^4 (a+b x)^3}-\frac {115 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^2 g^4 (a+b x)^2}-\frac {115 B d^3 n \log (c+d x)}{6 b^2 (b c-a d)^2 g^4}\\ \end {align*}

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Mathematica [A]  time = 0.45, size = 196, normalized size = 1.08 \[ -\frac {i \left (\frac {12 A b c}{(a+b x)^3}+\frac {18 A d}{(a+b x)^2}-\frac {12 a A d}{(a+b x)^3}-\frac {6 B d^3 n \log (a+b x)}{(b c-a d)^2}+\frac {6 B d^3 n \log (c+d x)}{(b c-a d)^2}-\frac {6 B d^2 n}{(a+b x) (b c-a d)}+\frac {6 B (a d+2 b c+3 b d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3}+\frac {4 b B c n}{(a+b x)^3}+\frac {3 B d n}{(a+b x)^2}-\frac {4 a B d n}{(a+b x)^3}\right )}{36 b^2 g^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/36*(i*((12*A*b*c)/(a + b*x)^3 - (12*a*A*d)/(a + b*x)^3 + (4*b*B*c*n)/(a + b*x)^3 - (4*a*B*d*n)/(a + b*x)^3
+ (18*A*d)/(a + b*x)^2 + (3*B*d*n)/(a + b*x)^2 - (6*B*d^2*n)/((b*c - a*d)*(a + b*x)) - (6*B*d^3*n*Log[a + b*x]
)/(b*c - a*d)^2 + (6*B*(2*b*c + a*d + 3*b*d*x)*Log[e*((a + b*x)/(c + d*x))^n])/(a + b*x)^3 + (6*B*d^3*n*Log[c
+ d*x])/(b*c - a*d)^2))/(b^2*g^4)

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fricas [B]  time = 0.63, size = 478, normalized size = 2.64 \[ \frac {6 \, {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} i n x^{2} - {\left (4 \, B b^{3} c^{3} - 9 \, B a b^{2} c^{2} d + 5 \, B a^{3} d^{3}\right )} i n - 6 \, {\left (2 \, A b^{3} c^{3} - 3 \, A a b^{2} c^{2} d + A a^{3} d^{3}\right )} i - 3 \, {\left ({\left (B b^{3} c^{2} d - 6 \, B a b^{2} c d^{2} + 5 \, B a^{2} b d^{3}\right )} i n + 6 \, {\left (A b^{3} c^{2} d - 2 \, A a b^{2} c d^{2} + A a^{2} b d^{3}\right )} i\right )} x - 6 \, {\left (3 \, {\left (B b^{3} c^{2} d - 2 \, B a b^{2} c d^{2} + B a^{2} b d^{3}\right )} i x + {\left (2 \, B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + B a^{3} d^{3}\right )} i\right )} \log \relax (e) + 6 \, {\left (B b^{3} d^{3} i n x^{3} + 3 \, B a b^{2} d^{3} i n x^{2} - 3 \, {\left (B b^{3} c^{2} d - 2 \, B a b^{2} c d^{2}\right )} i n x - {\left (2 \, B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d\right )} i n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{36 \, {\left ({\left (b^{7} c^{2} - 2 \, a b^{6} c d + a^{2} b^{5} d^{2}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{2} - 2 \, a^{2} b^{5} c d + a^{3} b^{4} d^{2}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{2} - 2 \, a^{3} b^{4} c d + a^{4} b^{3} d^{2}\right )} g^{4} x + {\left (a^{3} b^{4} c^{2} - 2 \, a^{4} b^{3} c d + a^{5} b^{2} d^{2}\right )} g^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*(6*(B*b^3*c*d^2 - B*a*b^2*d^3)*i*n*x^2 - (4*B*b^3*c^3 - 9*B*a*b^2*c^2*d + 5*B*a^3*d^3)*i*n - 6*(2*A*b^3*c
^3 - 3*A*a*b^2*c^2*d + A*a^3*d^3)*i - 3*((B*b^3*c^2*d - 6*B*a*b^2*c*d^2 + 5*B*a^2*b*d^3)*i*n + 6*(A*b^3*c^2*d
- 2*A*a*b^2*c*d^2 + A*a^2*b*d^3)*i)*x - 6*(3*(B*b^3*c^2*d - 2*B*a*b^2*c*d^2 + B*a^2*b*d^3)*i*x + (2*B*b^3*c^3
- 3*B*a*b^2*c^2*d + B*a^3*d^3)*i)*log(e) + 6*(B*b^3*d^3*i*n*x^3 + 3*B*a*b^2*d^3*i*n*x^2 - 3*(B*b^3*c^2*d - 2*B
*a*b^2*c*d^2)*i*n*x - (2*B*b^3*c^3 - 3*B*a*b^2*c^2*d)*i*n)*log((b*x + a)/(d*x + c)))/((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)

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giac [A]  time = 11.76, size = 230, normalized size = 1.27 \[ -\frac {1}{36} \, {\left (\frac {6 \, {\left (2 \, B b i n - \frac {3 \, {\left (b x + a\right )} B d i n}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b x + a\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}} + \frac {4 \, B b i n - \frac {9 \, {\left (b x + a\right )} B d i n}{d x + c} + 12 \, A b i + 12 \, B b i - \frac {18 \, {\left (b x + a\right )} A d i}{d x + c} - \frac {18 \, {\left (b x + a\right )} B d i}{d x + c}}{\frac {{\left (b x + a\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b x + a\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/36*(6*(2*B*b*i*n - 3*(b*x + a)*B*d*i*n/(d*x + c))*log((b*x + a)/(d*x + c))/((b*x + a)^3*b*c*g^4/(d*x + c)^3
- (b*x + a)^3*a*d*g^4/(d*x + c)^3) + (4*B*b*i*n - 9*(b*x + a)*B*d*i*n/(d*x + c) + 12*A*b*i + 12*B*b*i - 18*(b
*x + a)*A*d*i/(d*x + c) - 18*(b*x + a)*B*d*i/(d*x + c))/((b*x + a)^3*b*c*g^4/(d*x + c)^3 - (b*x + a)^3*a*d*g^4
/(d*x + c)^3))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)

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maple [F]  time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {\left (d i x +c i \right ) \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )}{\left (b g x +a g \right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 2.12, size = 945, normalized size = 5.22 \[ -\frac {1}{18} \, B c i n {\left (\frac {6 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{2} - 7 \, a b c d + 11 \, a^{2} d^{2} - 3 \, {\left (b^{2} c d - 5 \, a b d^{2}\right )} x}{{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} g^{4} x^{3} + 3 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} g^{4} x + {\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} g^{4}} + \frac {6 \, d^{3} \log \left (b x + a\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}} - \frac {6 \, d^{3} \log \left (d x + c\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}}\right )} - \frac {1}{36} \, B d i n {\left (\frac {5 \, a b^{2} c^{2} - 22 \, a^{2} b c d + 5 \, a^{3} d^{2} - 6 \, {\left (3 \, b^{3} c d - a b^{2} d^{2}\right )} x^{2} + 3 \, {\left (3 \, b^{3} c^{2} - 16 \, a b^{2} c d + 5 \, a^{2} b d^{2}\right )} x}{{\left (b^{7} c^{2} - 2 \, a b^{6} c d + a^{2} b^{5} d^{2}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{2} - 2 \, a^{2} b^{5} c d + a^{3} b^{4} d^{2}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{2} - 2 \, a^{3} b^{4} c d + a^{4} b^{3} d^{2}\right )} g^{4} x + {\left (a^{3} b^{4} c^{2} - 2 \, a^{4} b^{3} c d + a^{5} b^{2} d^{2}\right )} g^{4}} - \frac {6 \, {\left (3 \, b c d^{2} - a d^{3}\right )} \log \left (b x + a\right )}{{\left (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}\right )} g^{4}} + \frac {6 \, {\left (3 \, b c d^{2} - a d^{3}\right )} \log \left (d x + c\right )}{{\left (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}\right )} g^{4}}\right )} - \frac {{\left (3 \, b x + a\right )} B d i \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{6 \, {\left (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}\right )}} - \frac {{\left (3 \, b x + a\right )} A d i}{6 \, {\left (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}\right )}} - \frac {B c i \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{3 \, {\left (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}\right )}} - \frac {A c i}{3 \, {\left (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}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*B*c*i*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)) - 1/36*B*d*i*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)) - 1/6*(3*b*x + a)*B*d*i*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) - 1/6*(3*b*x + a)*A*d*i/(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*B*c*i*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*i/(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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mupad [B]  time = 5.28, size = 374, normalized size = 2.07 \[ -\frac {\frac {6\,A\,a^2\,d^2\,i-12\,A\,b^2\,c^2\,i+5\,B\,a^2\,d^2\,i\,n-4\,B\,b^2\,c^2\,i\,n+6\,A\,a\,b\,c\,d\,i+5\,B\,a\,b\,c\,d\,i\,n}{6\,\left (a\,d-b\,c\right )}+\frac {x\,\left (6\,A\,a\,b\,d^2\,i-6\,A\,b^2\,c\,d\,i-B\,b^2\,c\,d\,i\,n+5\,B\,a\,b\,d^2\,i\,n\right )}{2\,\left (a\,d-b\,c\right )}+\frac {B\,b^2\,d^2\,i\,n\,x^2}{a\,d-b\,c}}{6\,a^3\,b^2\,g^4+18\,a^2\,b^3\,g^4\,x+18\,a\,b^4\,g^4\,x^2+6\,b^5\,g^4\,x^3}-\frac {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {B\,c\,i}{3\,b}+\frac {B\,a\,d\,i}{6\,b^2}+\frac {B\,d\,i\,x}{2\,b}\right )}{a^3\,g^4+3\,a^2\,b\,g^4\,x+3\,a\,b^2\,g^4\,x^2+b^3\,g^4\,x^3}-\frac {B\,d^3\,i\,n\,\mathrm {atanh}\left (\frac {6\,b^4\,c^2\,g^4-6\,a^2\,b^2\,d^2\,g^4}{6\,b^2\,g^4\,{\left (a\,d-b\,c\right )}^2}-\frac {2\,b\,d\,x}{a\,d-b\,c}\right )}{3\,b^2\,g^4\,{\left (a\,d-b\,c\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((6*A*a^2*d^2*i - 12*A*b^2*c^2*i + 5*B*a^2*d^2*i*n - 4*B*b^2*c^2*i*n + 6*A*a*b*c*d*i + 5*B*a*b*c*d*i*n)/(6*(
a*d - b*c)) + (x*(6*A*a*b*d^2*i - 6*A*b^2*c*d*i - B*b^2*c*d*i*n + 5*B*a*b*d^2*i*n))/(2*(a*d - b*c)) + (B*b^2*d
^2*i*n*x^2)/(a*d - b*c))/(6*a^3*b^2*g^4 + 6*b^5*g^4*x^3 + 18*a^2*b^3*g^4*x + 18*a*b^4*g^4*x^2) - (log(e*((a +
b*x)/(c + d*x))^n)*((B*c*i)/(3*b) + (B*a*d*i)/(6*b^2) + (B*d*i*x)/(2*b)))/(a^3*g^4 + b^3*g^4*x^3 + 3*a*b^2*g^4
*x^2 + 3*a^2*b*g^4*x) - (B*d^3*i*n*atanh((6*b^4*c^2*g^4 - 6*a^2*b^2*d^2*g^4)/(6*b^2*g^4*(a*d - b*c)^2) - (2*b*
d*x)/(a*d - b*c)))/(3*b^2*g^4*(a*d - b*c)^2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)*(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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