3.116 \(\int \frac {(c i+d i x) (A+B \log (e (\frac {a+b x}{c+d x})^n))}{(a g+b g x)^5} \, dx\)

Optimal. Leaf size=281 \[ -\frac {b^2 i (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 g^5 (a+b x)^4 (b c-a d)^3}-\frac {d^2 i (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^5 (a+b x)^2 (b c-a d)^3}+\frac {2 b d i (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 g^5 (a+b x)^3 (b c-a d)^3}-\frac {b^2 B i n (c+d x)^4}{16 g^5 (a+b x)^4 (b c-a d)^3}-\frac {B d^2 i n (c+d x)^2}{4 g^5 (a+b x)^2 (b c-a d)^3}+\frac {2 b B d i n (c+d x)^3}{9 g^5 (a+b x)^3 (b c-a d)^3} \]

[Out]

-1/4*B*d^2*i*n*(d*x+c)^2/(-a*d+b*c)^3/g^5/(b*x+a)^2+2/9*b*B*d*i*n*(d*x+c)^3/(-a*d+b*c)^3/g^5/(b*x+a)^3-1/16*b^
2*B*i*n*(d*x+c)^4/(-a*d+b*c)^3/g^5/(b*x+a)^4-1/2*d^2*i*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^3/
g^5/(b*x+a)^2+2/3*b*d*i*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^3/g^5/(b*x+a)^3-1/4*b^2*i*(d*x+c)
^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^3/g^5/(b*x+a)^4

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Rubi [A]  time = 0.41, antiderivative size = 269, normalized size of antiderivative = 0.96, 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} \[ -\frac {d i \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 b^2 g^5 (a+b x)^3}-\frac {i (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 b^2 g^5 (a+b x)^4}-\frac {B d^3 i n}{12 b^2 g^5 (a+b x) (b c-a d)^2}+\frac {B d^2 i n}{24 b^2 g^5 (a+b x)^2 (b c-a d)}-\frac {B d^4 i n \log (a+b x)}{12 b^2 g^5 (b c-a d)^3}+\frac {B d^4 i n \log (c+d x)}{12 b^2 g^5 (b c-a d)^3}-\frac {B i n (b c-a d)}{16 b^2 g^5 (a+b x)^4}-\frac {B d i n}{36 b^2 g^5 (a+b x)^3} \]

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)^5,x]

[Out]

-(B*(b*c - a*d)*i*n)/(16*b^2*g^5*(a + b*x)^4) - (B*d*i*n)/(36*b^2*g^5*(a + b*x)^3) + (B*d^2*i*n)/(24*b^2*(b*c
- a*d)*g^5*(a + b*x)^2) - (B*d^3*i*n)/(12*b^2*(b*c - a*d)^2*g^5*(a + b*x)) - (B*d^4*i*n*Log[a + b*x])/(12*b^2*
(b*c - a*d)^3*g^5) - ((b*c - a*d)*i*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*b^2*g^5*(a + b*x)^4) - (d*i*(A
+ B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b^2*g^5*(a + b*x)^3) + (B*d^4*i*n*Log[c + d*x])/(12*b^2*(b*c - a*d)^3*
g^5)

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 {(116 c+116 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)^5} \, dx &=\int \left (\frac {116 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b g^5 (a+b x)^5}+\frac {116 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b g^5 (a+b x)^4}\right ) \, dx\\ &=\frac {(116 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^5}+\frac {(116 (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)^5} \, dx}{b g^5}\\ &=-\frac {29 (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^5 (a+b x)^4}-\frac {116 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 g^5 (a+b x)^3}+\frac {(116 B d n) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{3 b^2 g^5}+\frac {(29 B (b c-a d) n) \int \frac {b c-a d}{(a+b x)^5 (c+d x)} \, dx}{b^2 g^5}\\ &=-\frac {29 (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^5 (a+b x)^4}-\frac {116 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 g^5 (a+b x)^3}+\frac {(116 B d (b c-a d) n) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{3 b^2 g^5}+\frac {\left (29 B (b c-a d)^2 n\right ) \int \frac {1}{(a+b x)^5 (c+d x)} \, dx}{b^2 g^5}\\ &=-\frac {29 (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^5 (a+b x)^4}-\frac {116 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 g^5 (a+b x)^3}+\frac {(116 B d (b c-a d) n) \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^5}+\frac {\left (29 B (b c-a d)^2 n\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^5}-\frac {b d}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2}{(b c-a d)^3 (a+b x)^3}-\frac {b d^3}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4}{(b c-a d)^5 (a+b x)}-\frac {d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{b^2 g^5}\\ &=-\frac {29 B (b c-a d) n}{4 b^2 g^5 (a+b x)^4}-\frac {29 B d n}{9 b^2 g^5 (a+b x)^3}+\frac {29 B d^2 n}{6 b^2 (b c-a d) g^5 (a+b x)^2}-\frac {29 B d^3 n}{3 b^2 (b c-a d)^2 g^5 (a+b x)}-\frac {29 B d^4 n \log (a+b x)}{3 b^2 (b c-a d)^3 g^5}-\frac {29 (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^5 (a+b x)^4}-\frac {116 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 g^5 (a+b x)^3}+\frac {29 B d^4 n \log (c+d x)}{3 b^2 (b c-a d)^3 g^5}\\ \end {align*}

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Mathematica [A]  time = 0.50, size = 220, normalized size = 0.78 \[ -\frac {i \left (\frac {36 A b c}{(a+b x)^4}+\frac {48 A d}{(a+b x)^3}-\frac {36 a A d}{(a+b x)^4}+\frac {12 B d^4 n \log (a+b x)}{(b c-a d)^3}-\frac {12 B d^4 n \log (c+d x)}{(b c-a d)^3}+\frac {12 B d^3 n}{(a+b x) (b c-a d)^2}-\frac {6 B d^2 n}{(a+b x)^2 (b c-a d)}+\frac {12 B (a d+3 b c+4 b d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^4}+\frac {9 b B c n}{(a+b x)^4}+\frac {4 B d n}{(a+b x)^3}-\frac {9 a B d n}{(a+b x)^4}\right )}{144 b^2 g^5} \]

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)^5,x]

[Out]

-1/144*(i*((36*A*b*c)/(a + b*x)^4 - (36*a*A*d)/(a + b*x)^4 + (9*b*B*c*n)/(a + b*x)^4 - (9*a*B*d*n)/(a + b*x)^4
+ (48*A*d)/(a + b*x)^3 + (4*B*d*n)/(a + b*x)^3 - (6*B*d^2*n)/((b*c - a*d)*(a + b*x)^2) + (12*B*d^3*n)/((b*c -
a*d)^2*(a + b*x)) + (12*B*d^4*n*Log[a + b*x])/(b*c - a*d)^3 + (12*B*(3*b*c + a*d + 4*b*d*x)*Log[e*((a + b*x)/
(c + d*x))^n])/(a + b*x)^4 - (12*B*d^4*n*Log[c + d*x])/(b*c - a*d)^3))/(b^2*g^5)

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fricas [B]  time = 0.65, size = 773, normalized size = 2.75 \[ -\frac {12 \, {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} i n x^{3} - 6 \, {\left (B b^{4} c^{2} d^{2} - 8 \, B a b^{3} c d^{3} + 7 \, B a^{2} b^{2} d^{4}\right )} i n x^{2} + {\left (9 \, B b^{4} c^{4} - 32 \, B a b^{3} c^{3} d + 36 \, B a^{2} b^{2} c^{2} d^{2} - 13 \, B a^{4} d^{4}\right )} i n + 12 \, {\left (3 \, A b^{4} c^{4} - 8 \, A a b^{3} c^{3} d + 6 \, A a^{2} b^{2} c^{2} d^{2} - A a^{4} d^{4}\right )} i + 4 \, {\left ({\left (B b^{4} c^{3} d - 6 \, B a b^{3} c^{2} d^{2} + 18 \, B a^{2} b^{2} c d^{3} - 13 \, B a^{3} b d^{4}\right )} i n + 12 \, {\left (A b^{4} c^{3} d - 3 \, A a b^{3} c^{2} d^{2} + 3 \, A a^{2} b^{2} c d^{3} - A a^{3} b d^{4}\right )} i\right )} x + 12 \, {\left (4 \, {\left (B b^{4} c^{3} d - 3 \, B a b^{3} c^{2} d^{2} + 3 \, B a^{2} b^{2} c d^{3} - B a^{3} b d^{4}\right )} i x + {\left (3 \, B b^{4} c^{4} - 8 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2} - B a^{4} d^{4}\right )} i\right )} \log \relax (e) + 12 \, {\left (B b^{4} d^{4} i n x^{4} + 4 \, B a b^{3} d^{4} i n x^{3} + 6 \, B a^{2} b^{2} d^{4} i n x^{2} + 4 \, {\left (B b^{4} c^{3} d - 3 \, B a b^{3} c^{2} d^{2} + 3 \, B a^{2} b^{2} c d^{3}\right )} i n x + {\left (3 \, B b^{4} c^{4} - 8 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2}\right )} i n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{144 \, {\left ({\left (b^{9} c^{3} - 3 \, a b^{8} c^{2} d + 3 \, a^{2} b^{7} c d^{2} - a^{3} b^{6} d^{3}\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c^{3} - 3 \, a^{2} b^{7} c^{2} d + 3 \, a^{3} b^{6} c d^{2} - a^{4} b^{5} d^{3}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c^{3} - 3 \, a^{3} b^{6} c^{2} d + 3 \, a^{4} b^{5} c d^{2} - a^{5} b^{4} d^{3}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c^{3} - 3 \, a^{4} b^{5} c^{2} d + 3 \, a^{5} b^{4} c d^{2} - a^{6} b^{3} d^{3}\right )} g^{5} x + {\left (a^{4} b^{5} c^{3} - 3 \, a^{5} b^{4} c^{2} d + 3 \, a^{6} b^{3} c d^{2} - a^{7} b^{2} d^{3}\right )} g^{5}\right )}} \]

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)^5,x, algorithm="fricas")

[Out]

-1/144*(12*(B*b^4*c*d^3 - B*a*b^3*d^4)*i*n*x^3 - 6*(B*b^4*c^2*d^2 - 8*B*a*b^3*c*d^3 + 7*B*a^2*b^2*d^4)*i*n*x^2
+ (9*B*b^4*c^4 - 32*B*a*b^3*c^3*d + 36*B*a^2*b^2*c^2*d^2 - 13*B*a^4*d^4)*i*n + 12*(3*A*b^4*c^4 - 8*A*a*b^3*c^
3*d + 6*A*a^2*b^2*c^2*d^2 - A*a^4*d^4)*i + 4*((B*b^4*c^3*d - 6*B*a*b^3*c^2*d^2 + 18*B*a^2*b^2*c*d^3 - 13*B*a^3
*b*d^4)*i*n + 12*(A*b^4*c^3*d - 3*A*a*b^3*c^2*d^2 + 3*A*a^2*b^2*c*d^3 - A*a^3*b*d^4)*i)*x + 12*(4*(B*b^4*c^3*d
- 3*B*a*b^3*c^2*d^2 + 3*B*a^2*b^2*c*d^3 - B*a^3*b*d^4)*i*x + (3*B*b^4*c^4 - 8*B*a*b^3*c^3*d + 6*B*a^2*b^2*c^2
*d^2 - B*a^4*d^4)*i)*log(e) + 12*(B*b^4*d^4*i*n*x^4 + 4*B*a*b^3*d^4*i*n*x^3 + 6*B*a^2*b^2*d^4*i*n*x^2 + 4*(B*b
^4*c^3*d - 3*B*a*b^3*c^2*d^2 + 3*B*a^2*b^2*c*d^3)*i*n*x + (3*B*b^4*c^4 - 8*B*a*b^3*c^3*d + 6*B*a^2*b^2*c^2*d^2
)*i*n)*log((b*x + a)/(d*x + c)))/((b^9*c^3 - 3*a*b^8*c^2*d + 3*a^2*b^7*c*d^2 - a^3*b^6*d^3)*g^5*x^4 + 4*(a*b^8
*c^3 - 3*a^2*b^7*c^2*d + 3*a^3*b^6*c*d^2 - a^4*b^5*d^3)*g^5*x^3 + 6*(a^2*b^7*c^3 - 3*a^3*b^6*c^2*d + 3*a^4*b^5
*c*d^2 - a^5*b^4*d^3)*g^5*x^2 + 4*(a^3*b^6*c^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2 - a^6*b^3*d^3)*g^5*x + (a^4
*b^5*c^3 - 3*a^5*b^4*c^2*d + 3*a^6*b^3*c*d^2 - a^7*b^2*d^3)*g^5)

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giac [A]  time = 17.57, size = 388, normalized size = 1.38 \[ -\frac {1}{144} \, {\left (\frac {12 \, {\left (3 \, B b^{2} i n - \frac {8 \, {\left (b x + a\right )} B b d i n}{d x + c} + \frac {6 \, {\left (b x + a\right )}^{2} B d^{2} i n}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{4} b^{2} c^{2} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {2 \, {\left (b x + a\right )}^{4} a b c d g^{5}}{{\left (d x + c\right )}^{4}} + \frac {{\left (b x + a\right )}^{4} a^{2} d^{2} g^{5}}{{\left (d x + c\right )}^{4}}} + \frac {9 \, B b^{2} i n - \frac {32 \, {\left (b x + a\right )} B b d i n}{d x + c} + \frac {36 \, {\left (b x + a\right )}^{2} B d^{2} i n}{{\left (d x + c\right )}^{2}} + 36 \, A b^{2} i + 36 \, B b^{2} i - \frac {96 \, {\left (b x + a\right )} A b d i}{d x + c} - \frac {96 \, {\left (b x + a\right )} B b d i}{d x + c} + \frac {72 \, {\left (b x + a\right )}^{2} A d^{2} i}{{\left (d x + c\right )}^{2}} + \frac {72 \, {\left (b x + a\right )}^{2} B d^{2} i}{{\left (d x + c\right )}^{2}}}{\frac {{\left (b x + a\right )}^{4} b^{2} c^{2} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {2 \, {\left (b x + a\right )}^{4} a b c d g^{5}}{{\left (d x + c\right )}^{4}} + \frac {{\left (b x + a\right )}^{4} a^{2} d^{2} g^{5}}{{\left (d x + c\right )}^{4}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]

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)^5,x, algorithm="giac")

[Out]

-1/144*(12*(3*B*b^2*i*n - 8*(b*x + a)*B*b*d*i*n/(d*x + c) + 6*(b*x + a)^2*B*d^2*i*n/(d*x + c)^2)*log((b*x + a)
/(d*x + c))/((b*x + a)^4*b^2*c^2*g^5/(d*x + c)^4 - 2*(b*x + a)^4*a*b*c*d*g^5/(d*x + c)^4 + (b*x + a)^4*a^2*d^2
*g^5/(d*x + c)^4) + (9*B*b^2*i*n - 32*(b*x + a)*B*b*d*i*n/(d*x + c) + 36*(b*x + a)^2*B*d^2*i*n/(d*x + c)^2 + 3
6*A*b^2*i + 36*B*b^2*i - 96*(b*x + a)*A*b*d*i/(d*x + c) - 96*(b*x + a)*B*b*d*i/(d*x + c) + 72*(b*x + a)^2*A*d^
2*i/(d*x + c)^2 + 72*(b*x + a)^2*B*d^2*i/(d*x + c)^2)/((b*x + a)^4*b^2*c^2*g^5/(d*x + c)^4 - 2*(b*x + a)^4*a*b
*c*d*g^5/(d*x + c)^4 + (b*x + a)^4*a^2*d^2*g^5/(d*x + c)^4))*(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 \[ \int \frac {\left (d i x +c i \right ) \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )}{\left (b g x +a g \right )^{5}}\, dx \]

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)^5,x)

[Out]

int((d*i*x+c*i)*(B*ln(e*((b*x+a)/(d*x+c))^n)+A)/(b*g*x+a*g)^5,x)

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maxima [B]  time = 1.89, size = 1398, normalized size = 4.98 \[ \text {result too large to display} \]

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)^5,x, algorithm="maxima")

[Out]

1/48*B*c*i*n*((12*b^3*d^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*
b^2*d^3)*x^2 + 4*(b^3*c^2*d - 5*a*b^2*c*d^2 + 13*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a
^3*b^5*d^3)*g^5*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*g^5*x^3 + 6*(a^2*b^6*c^3
- 3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d
^2 - a^6*b^2*d^3)*g^5*x + (a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3)*g^5) + 12*d^4*log(b*x
+ a)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12*d^4*log(d*x + c)/(
(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5)) - 1/144*B*d*i*n*((7*a*b^3*c^
3 - 33*a^2*b^2*c^2*d + 75*a^3*b*c*d^2 - 13*a^4*d^3 + 12*(4*b^4*c*d^2 - a*b^3*d^3)*x^3 - 6*(4*b^4*c^2*d - 29*a*
b^3*c*d^2 + 7*a^2*b^2*d^3)*x^2 + 4*(4*b^4*c^3 - 21*a*b^3*c^2*d + 57*a^2*b^2*c*d^2 - 13*a^3*b*d^3)*x)/((b^9*c^3
- 3*a*b^8*c^2*d + 3*a^2*b^7*c*d^2 - a^3*b^6*d^3)*g^5*x^4 + 4*(a*b^8*c^3 - 3*a^2*b^7*c^2*d + 3*a^3*b^6*c*d^2 -
a^4*b^5*d^3)*g^5*x^3 + 6*(a^2*b^7*c^3 - 3*a^3*b^6*c^2*d + 3*a^4*b^5*c*d^2 - a^5*b^4*d^3)*g^5*x^2 + 4*(a^3*b^6
*c^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2 - a^6*b^3*d^3)*g^5*x + (a^4*b^5*c^3 - 3*a^5*b^4*c^2*d + 3*a^6*b^3*c*d
^2 - a^7*b^2*d^3)*g^5) + 12*(4*b*c*d^3 - a*d^4)*log(b*x + a)/((b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4
*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5) - 12*(4*b*c*d^3 - a*d^4)*log(d*x + c)/((b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4
*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5)) - 1/12*(4*b*x + a)*B*d*i*log(e*(b*x/(d*x + c) + a/(d*x + c))^n
)/(b^6*g^5*x^4 + 4*a*b^5*g^5*x^3 + 6*a^2*b^4*g^5*x^2 + 4*a^3*b^3*g^5*x + a^4*b^2*g^5) - 1/12*(4*b*x + a)*A*d*i
/(b^6*g^5*x^4 + 4*a*b^5*g^5*x^3 + 6*a^2*b^4*g^5*x^2 + 4*a^3*b^3*g^5*x + a^4*b^2*g^5) - 1/4*B*c*i*log(e*(b*x/(d
*x + c) + a/(d*x + c))^n)/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5) -
1/4*A*c*i/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5)

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mupad [B]  time = 5.87, size = 610, normalized size = 2.17 \[ \frac {B\,d^4\,i\,n\,\mathrm {atanh}\left (\frac {12\,a^3\,b^2\,d^3\,g^5-12\,a^2\,b^3\,c\,d^2\,g^5-12\,a\,b^4\,c^2\,d\,g^5+12\,b^5\,c^3\,g^5}{12\,b^2\,g^5\,{\left (a\,d-b\,c\right )}^3}+\frac {2\,b\,d\,x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3}\right )}{6\,b^2\,g^5\,{\left (a\,d-b\,c\right )}^3}-\frac {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {B\,c\,i}{4\,b}+\frac {B\,a\,d\,i}{12\,b^2}+\frac {B\,d\,i\,x}{3\,b}\right )}{a^4\,g^5+4\,a^3\,b\,g^5\,x+6\,a^2\,b^2\,g^5\,x^2+4\,a\,b^3\,g^5\,x^3+b^4\,g^5\,x^4}-\frac {\frac {12\,A\,a^3\,d^3\,i+36\,A\,b^3\,c^3\,i+13\,B\,a^3\,d^3\,i\,n+9\,B\,b^3\,c^3\,i\,n-60\,A\,a\,b^2\,c^2\,d\,i+12\,A\,a^2\,b\,c\,d^2\,i-23\,B\,a\,b^2\,c^2\,d\,i\,n+13\,B\,a^2\,b\,c\,d^2\,i\,n}{12\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x\,\left (12\,A\,a^2\,b\,d^3\,i+12\,A\,b^3\,c^2\,d\,i-24\,A\,a\,b^2\,c\,d^2\,i+13\,B\,a^2\,b\,d^3\,i\,n+B\,b^3\,c^2\,d\,i\,n-5\,B\,a\,b^2\,c\,d^2\,i\,n\right )}{3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {d\,x^2\,\left (B\,b^3\,c\,d\,i\,n-7\,B\,a\,b^2\,d^2\,i\,n\right )}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {B\,b^3\,d^3\,i\,n\,x^3}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}}{12\,a^4\,b^2\,g^5+48\,a^3\,b^3\,g^5\,x+72\,a^2\,b^4\,g^5\,x^2+48\,a\,b^5\,g^5\,x^3+12\,b^6\,g^5\,x^4} \]

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)^5,x)

[Out]

(B*d^4*i*n*atanh((12*b^5*c^3*g^5 + 12*a^3*b^2*d^3*g^5 - 12*a*b^4*c^2*d*g^5 - 12*a^2*b^3*c*d^2*g^5)/(12*b^2*g^5
*(a*d - b*c)^3) + (2*b*d*x*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(a*d - b*c)^3))/(6*b^2*g^5*(a*d - b*c)^3) - (log(e
*((a + b*x)/(c + d*x))^n)*((B*c*i)/(4*b) + (B*a*d*i)/(12*b^2) + (B*d*i*x)/(3*b)))/(a^4*g^5 + b^4*g^5*x^4 + 4*a
*b^3*g^5*x^3 + 6*a^2*b^2*g^5*x^2 + 4*a^3*b*g^5*x) - ((12*A*a^3*d^3*i + 36*A*b^3*c^3*i + 13*B*a^3*d^3*i*n + 9*B
*b^3*c^3*i*n - 60*A*a*b^2*c^2*d*i + 12*A*a^2*b*c*d^2*i - 23*B*a*b^2*c^2*d*i*n + 13*B*a^2*b*c*d^2*i*n)/(12*(a^2
*d^2 + b^2*c^2 - 2*a*b*c*d)) + (x*(12*A*a^2*b*d^3*i + 12*A*b^3*c^2*d*i - 24*A*a*b^2*c*d^2*i + 13*B*a^2*b*d^3*i
*n + B*b^3*c^2*d*i*n - 5*B*a*b^2*c*d^2*i*n))/(3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - (d*x^2*(B*b^3*c*d*i*n - 7*B
*a*b^2*d^2*i*n))/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (B*b^3*d^3*i*n*x^3)/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(1
2*a^4*b^2*g^5 + 12*b^6*g^5*x^4 + 48*a^3*b^3*g^5*x + 48*a*b^5*g^5*x^3 + 72*a^2*b^4*g^5*x^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)**5,x)

[Out]

Timed out

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