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]

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

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Rubi [A]  time = 0.340598, antiderivative size = 236, normalized size of antiderivative = 1.3, 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 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{(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*}

Mathematica [A]  time = 0.460613, size = 196, normalized size = 1.08 \[ -\frac{i \left (\frac{12 A b c}{(a+b x)^3}+\frac{18 A d}{(a+b x)^2}-\frac{12 a A d}{(a+b x)^3}-\frac{6 B d^2 n}{(a+b x) (b c-a d)}-\frac{6 B d^3 n \log (a+b x)}{(b c-a d)^2}+\frac{6 B d^3 n \log (c+d x)}{(b c-a d)^2}+\frac{6 B (a d+2 b c+3 b d x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^3}+\frac{4 b B c n}{(a+b x)^3}+\frac{3 B d n}{(a+b x)^2}-\frac{4 a B d n}{(a+b x)^3}\right )}{36 b^2 g^4} \]

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]

-(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))/(36*b^2*g^4)

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Maple [F]  time = 0.531, size = 0, normalized size = 0. \begin{align*} \int{\frac{dix+ci}{ \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)*(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)*(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 = 1.49898, size = 1276, normalized size = 7.05 \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)^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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Fricas [B]  time = 0.559639, size = 988, normalized size = 5.46 \begin{align*} \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 \left (e\right ) + 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 )}} \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)^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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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)**4,x)

[Out]

Timed out

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Giac [B]  time = 1.27089, size = 624, normalized size = 3.45 \begin{align*} -\frac{B d^{3} n \log \left (b x + a\right )}{6 \,{\left (b^{4} c^{2} g^{4} i - 2 \, a b^{3} c d g^{4} i + a^{2} b^{2} d^{2} g^{4} i\right )}} + \frac{B d^{3} n \log \left (d x + c\right )}{6 \,{\left (b^{4} c^{2} g^{4} i - 2 \, a b^{3} c d g^{4} i + a^{2} b^{2} d^{2} g^{4} i\right )}} - \frac{{\left (3 \, B b d i n x + 2 \, B b c i n + B a d i n\right )} \log \left (\frac{b x + a}{d x + c}\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{6 \, B b^{2} d^{2} i n x^{2} - 3 \, B b^{2} c d i n x + 15 \, B a b d^{2} i n x - 4 \, B b^{2} c^{2} i n + 5 \, B a b c d i n + 5 \, B a^{2} d^{2} i n - 18 \, A b^{2} c d i x - 18 \, B b^{2} c d i x + 18 \, A a b d^{2} i x + 18 \, B a b d^{2} i x - 12 \, A b^{2} c^{2} i - 12 \, B b^{2} c^{2} i + 6 \, A a b c d i + 6 \, B a b c d i + 6 \, A a^{2} d^{2} i + 6 \, B a^{2} d^{2} i}{36 \,{\left (b^{6} c g^{4} x^{3} - a b^{5} d g^{4} x^{3} + 3 \, a b^{5} c g^{4} x^{2} - 3 \, a^{2} b^{4} d g^{4} x^{2} + 3 \, a^{2} b^{4} c g^{4} x - 3 \, a^{3} b^{3} d g^{4} x + a^{3} b^{3} c g^{4} - a^{4} b^{2} d g^{4}\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)^4,x, algorithm="giac")

[Out]

-1/6*B*d^3*n*log(b*x + a)/(b^4*c^2*g^4*i - 2*a*b^3*c*d*g^4*i + a^2*b^2*d^2*g^4*i) + 1/6*B*d^3*n*log(d*x + c)/(
b^4*c^2*g^4*i - 2*a*b^3*c*d*g^4*i + a^2*b^2*d^2*g^4*i) - 1/6*(3*B*b*d*i*n*x + 2*B*b*c*i*n + B*a*d*i*n)*log((b*
x + a)/(d*x + c))/(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) + 1/36*(6*B*b^2*d^2*i*n*x^2
- 3*B*b^2*c*d*i*n*x + 15*B*a*b*d^2*i*n*x - 4*B*b^2*c^2*i*n + 5*B*a*b*c*d*i*n + 5*B*a^2*d^2*i*n - 18*A*b^2*c*d*
i*x - 18*B*b^2*c*d*i*x + 18*A*a*b*d^2*i*x + 18*B*a*b*d^2*i*x - 12*A*b^2*c^2*i - 12*B*b^2*c^2*i + 6*A*a*b*c*d*i
+ 6*B*a*b*c*d*i + 6*A*a^2*d^2*i + 6*B*a^2*d^2*i)/(b^6*c*g^4*x^3 - a*b^5*d*g^4*x^3 + 3*a*b^5*c*g^4*x^2 - 3*a^2
*b^4*d*g^4*x^2 + 3*a^2*b^4*c*g^4*x - 3*a^3*b^3*d*g^4*x + a^3*b^3*c*g^4 - a^4*b^2*d*g^4)