3.154 \(\int \frac{A+B \log (e (\frac{a+b x}{c+d x})^n)}{(c i+d i x)^3} \, dx\)

Optimal. Leaf size=151 \[ -\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A}{2 d i^3 (c+d x)^2}+\frac{b^2 B n \log (a+b x)}{2 d i^3 (b c-a d)^2}-\frac{b^2 B n \log (c+d x)}{2 d i^3 (b c-a d)^2}+\frac{b B n}{2 d i^3 (c+d x) (b c-a d)}+\frac{B n}{4 d i^3 (c+d x)^2} \]

[Out]

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

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Rubi [A]  time = 0.104282, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2525, 12, 44} \[ -\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A}{2 d i^3 (c+d x)^2}+\frac{b^2 B n \log (a+b x)}{2 d i^3 (b c-a d)^2}-\frac{b^2 B n \log (c+d x)}{2 d i^3 (b c-a d)^2}+\frac{b B n}{2 d i^3 (c+d x) (b c-a d)}+\frac{B n}{4 d i^3 (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(c*i + d*i*x)^3,x]

[Out]

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

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{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(154 c+154 d x)^3} \, dx &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{7304528 d (c+d x)^2}+\frac{(B n) \int \frac{b c-a d}{23716 (a+b x) (c+d x)^3} \, dx}{308 d}\\ &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{7304528 d (c+d x)^2}+\frac{(B (b c-a d) n) \int \frac{1}{(a+b x) (c+d x)^3} \, dx}{7304528 d}\\ &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{7304528 d (c+d x)^2}+\frac{(B (b c-a d) n) \int \left (\frac{b^3}{(b c-a d)^3 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^3}-\frac{b d}{(b c-a d)^2 (c+d x)^2}-\frac{b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{7304528 d}\\ &=\frac{B n}{14609056 d (c+d x)^2}+\frac{b B n}{7304528 d (b c-a d) (c+d x)}+\frac{b^2 B n \log (a+b x)}{7304528 d (b c-a d)^2}-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{7304528 d (c+d x)^2}-\frac{b^2 B n \log (c+d x)}{7304528 d (b c-a d)^2}\\ \end{align*}

Mathematica [A]  time = 0.132717, size = 115, normalized size = 0.76 \[ \frac{\frac{B n \left (2 b^2 (c+d x)^2 \log (a+b x)+(b c-a d) (-a d+3 b c+2 b d x)-2 b^2 (c+d x)^2 \log (c+d x)\right )}{(b c-a d)^2}-2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{4 d i^3 (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(c*i + d*i*x)^3,x]

[Out]

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

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Maple [F]  time = 0.538, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dix+ci \right ) ^{3}} \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((A+B*ln(e*((b*x+a)/(d*x+c))^n))/(d*i*x+c*i)^3,x)

[Out]

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))/(d*i*x+c*i)^3,x)

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Maxima [A]  time = 1.20476, size = 350, normalized size = 2.32 \begin{align*} \frac{1}{4} \, B n{\left (\frac{2 \, b d x + 3 \, b c - a d}{{\left (b c d^{3} - a d^{4}\right )} i^{3} x^{2} + 2 \,{\left (b c^{2} d^{2} - a c d^{3}\right )} i^{3} x +{\left (b c^{3} d - a c^{2} d^{2}\right )} i^{3}} + \frac{2 \, b^{2} \log \left (b x + a\right )}{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} i^{3}} - \frac{2 \, b^{2} \log \left (d x + c\right )}{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} i^{3}}\right )} - \frac{B \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right )}{2 \,{\left (d^{3} i^{3} x^{2} + 2 \, c d^{2} i^{3} x + c^{2} d i^{3}\right )}} - \frac{A}{2 \,{\left (d^{3} i^{3} x^{2} + 2 \, c d^{2} i^{3} x + c^{2} d i^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(d*i*x+c*i)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.524769, size = 562, normalized size = 3.72 \begin{align*} -\frac{2 \, A b^{2} c^{2} - 4 \, A a b c d + 2 \, A a^{2} d^{2} - 2 \,{\left (B b^{2} c d - B a b d^{2}\right )} n x -{\left (3 \, B b^{2} c^{2} - 4 \, B a b c d + B a^{2} d^{2}\right )} n + 2 \,{\left (B b^{2} c^{2} - 2 \, B a b c d + B a^{2} d^{2}\right )} \log \left (e\right ) - 2 \,{\left (B b^{2} d^{2} n x^{2} + 2 \, B b^{2} c d n x +{\left (2 \, B a b c d - B a^{2} d^{2}\right )} n\right )} \log \left (\frac{b x + a}{d x + c}\right )}{4 \,{\left ({\left (b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}\right )} i^{3} x^{2} + 2 \,{\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} i^{3} x +{\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} i^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(d*i*x+c*i)^3,x, algorithm="fricas")

[Out]

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

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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((A+B*ln(e*((b*x+a)/(d*x+c))**n))/(d*i*x+c*i)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.30498, size = 309, normalized size = 2.05 \begin{align*} -\frac{B b^{2} n \log \left (b x + a\right )}{2 \,{\left (b^{2} c^{2} d i - 2 \, a b c d^{2} i + a^{2} d^{3} i\right )}} + \frac{B b^{2} n \log \left (d x + c\right )}{2 \,{\left (b^{2} c^{2} d i - 2 \, a b c d^{2} i + a^{2} d^{3} i\right )}} - \frac{B i n \log \left (\frac{b x + a}{d x + c}\right )}{2 \,{\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} + \frac{2 \, B b d i n x + 3 \, B b c i n - B a d i n - 2 \, A b c i - 2 \, B b c i + 2 \, A a d i + 2 \, B a d i}{4 \,{\left (b c d^{3} x^{2} - a d^{4} x^{2} + 2 \, b c^{2} d^{2} x - 2 \, a c d^{3} x + b c^{3} d - a c^{2} d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(d*i*x+c*i)^3,x, algorithm="giac")

[Out]

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