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]

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

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Rubi [A]  time = 0.409461, 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 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{(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*}

Mathematica [A]  time = 0.530503, size = 220, normalized size = 0.78 \[ -\frac{i \left (\frac{36 A b c}{(a+b x)^4}+\frac{48 A d}{(a+b x)^3}-\frac{36 a A d}{(a+b x)^4}+\frac{12 B d^3 n}{(a+b x) (b c-a d)^2}-\frac{6 B d^2 n}{(a+b x)^2 (b c-a d)}+\frac{12 B d^4 n \log (a+b x)}{(b c-a d)^3}-\frac{12 B d^4 n \log (c+d x)}{(b c-a d)^3}+\frac{12 B (a d+3 b c+4 b d x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^4}+\frac{9 b B c n}{(a+b x)^4}+\frac{4 B d n}{(a+b x)^3}-\frac{9 a B d n}{(a+b x)^4}\right )}{144 b^2 g^5} \]

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]

-(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))/(144*b^2*g^5)

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Maple [F]  time = 0.535, size = 0, normalized size = 0. \begin{align*} \int{\frac{dix+ci}{ \left ( bgx+ag \right ) ^{5}} \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)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^5,x)

[Out]

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

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Maxima [B]  time = 1.62016, size = 1887, normalized size = 6.72 \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)*(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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Fricas [B]  time = 0.547895, size = 1574, normalized size = 5.6 \begin{align*} -\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 \left (e\right ) + 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 )}} \end{align*}

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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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)*(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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Giac [B]  time = 1.39094, size = 1025, normalized size = 3.65 \begin{align*} \frac{B d^{4} n \log \left (b x + a\right )}{12 \,{\left (b^{5} c^{3} g^{5} i - 3 \, a b^{4} c^{2} d g^{5} i + 3 \, a^{2} b^{3} c d^{2} g^{5} i - a^{3} b^{2} d^{3} g^{5} i\right )}} - \frac{B d^{4} n \log \left (d x + c\right )}{12 \,{\left (b^{5} c^{3} g^{5} i - 3 \, a b^{4} c^{2} d g^{5} i + 3 \, a^{2} b^{3} c d^{2} g^{5} i - a^{3} b^{2} d^{3} g^{5} i\right )}} - \frac{{\left (4 \, B b d i n x + 3 \, B b c i n + B a d i n\right )} \log \left (\frac{b x + a}{d x + c}\right )}{12 \,{\left (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}\right )}} + \frac{12 \, B b^{3} d^{3} n x^{3} - 6 \, B b^{3} c d^{2} n x^{2} + 42 \, B a b^{2} d^{3} n x^{2} + 4 \, B b^{3} c^{2} d n x - 20 \, B a b^{2} c d^{2} n x + 52 \, B a^{2} b d^{3} n x + 9 \, B b^{3} c^{3} n - 23 \, B a b^{2} c^{2} d n + 13 \, B a^{2} b c d^{2} n + 13 \, B a^{3} d^{3} n + 48 \, A b^{3} c^{2} d x + 48 \, B b^{3} c^{2} d x - 96 \, A a b^{2} c d^{2} x - 96 \, B a b^{2} c d^{2} x + 48 \, A a^{2} b d^{3} x + 48 \, B a^{2} b d^{3} x + 36 \, A b^{3} c^{3} + 36 \, B b^{3} c^{3} - 60 \, A a b^{2} c^{2} d - 60 \, B a b^{2} c^{2} d + 12 \, A a^{2} b c d^{2} + 12 \, B a^{2} b c d^{2} + 12 \, A a^{3} d^{3} + 12 \, B a^{3} d^{3}}{144 \,{\left (b^{8} c^{2} g^{5} i x^{4} - 2 \, a b^{7} c d g^{5} i x^{4} + a^{2} b^{6} d^{2} g^{5} i x^{4} + 4 \, a b^{7} c^{2} g^{5} i x^{3} - 8 \, a^{2} b^{6} c d g^{5} i x^{3} + 4 \, a^{3} b^{5} d^{2} g^{5} i x^{3} + 6 \, a^{2} b^{6} c^{2} g^{5} i x^{2} - 12 \, a^{3} b^{5} c d g^{5} i x^{2} + 6 \, a^{4} b^{4} d^{2} g^{5} i x^{2} + 4 \, a^{3} b^{5} c^{2} g^{5} i x - 8 \, a^{4} b^{4} c d g^{5} i x + 4 \, a^{5} b^{3} d^{2} g^{5} i x + a^{4} b^{4} c^{2} g^{5} i - 2 \, a^{5} b^{3} c d g^{5} i + a^{6} b^{2} d^{2} g^{5} i\right )}} \end{align*}

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]

1/12*B*d^4*n*log(b*x + a)/(b^5*c^3*g^5*i - 3*a*b^4*c^2*d*g^5*i + 3*a^2*b^3*c*d^2*g^5*i - a^3*b^2*d^3*g^5*i) -
1/12*B*d^4*n*log(d*x + c)/(b^5*c^3*g^5*i - 3*a*b^4*c^2*d*g^5*i + 3*a^2*b^3*c*d^2*g^5*i - a^3*b^2*d^3*g^5*i) -
1/12*(4*B*b*d*i*n*x + 3*B*b*c*i*n + B*a*d*i*n)*log((b*x + a)/(d*x + c))/(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) + 1/144*(12*B*b^3*d^3*n*x^3 - 6*B*b^3*c*d^2*n*x^2 + 42*B*a*b^2*d
^3*n*x^2 + 4*B*b^3*c^2*d*n*x - 20*B*a*b^2*c*d^2*n*x + 52*B*a^2*b*d^3*n*x + 9*B*b^3*c^3*n - 23*B*a*b^2*c^2*d*n
+ 13*B*a^2*b*c*d^2*n + 13*B*a^3*d^3*n + 48*A*b^3*c^2*d*x + 48*B*b^3*c^2*d*x - 96*A*a*b^2*c*d^2*x - 96*B*a*b^2*
c*d^2*x + 48*A*a^2*b*d^3*x + 48*B*a^2*b*d^3*x + 36*A*b^3*c^3 + 36*B*b^3*c^3 - 60*A*a*b^2*c^2*d - 60*B*a*b^2*c^
2*d + 12*A*a^2*b*c*d^2 + 12*B*a^2*b*c*d^2 + 12*A*a^3*d^3 + 12*B*a^3*d^3)/(b^8*c^2*g^5*i*x^4 - 2*a*b^7*c*d*g^5*
i*x^4 + a^2*b^6*d^2*g^5*i*x^4 + 4*a*b^7*c^2*g^5*i*x^3 - 8*a^2*b^6*c*d*g^5*i*x^3 + 4*a^3*b^5*d^2*g^5*i*x^3 + 6*
a^2*b^6*c^2*g^5*i*x^2 - 12*a^3*b^5*c*d*g^5*i*x^2 + 6*a^4*b^4*d^2*g^5*i*x^2 + 4*a^3*b^5*c^2*g^5*i*x - 8*a^4*b^4
*c*d*g^5*i*x + 4*a^5*b^3*d^2*g^5*i*x + a^4*b^4*c^2*g^5*i - 2*a^5*b^3*c*d*g^5*i + a^6*b^2*d^2*g^5*i)