3.51 \(\int \frac {1}{(c g+d g x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n))} \, dx\)

Optimal. Leaf size=199 \[ \frac {b (a+b x) e^{-\frac {A}{B n}} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-1/n} \text {Ei}\left (\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{B n}\right )}{B g^3 n (c+d x) (b c-a d)^2}-\frac {d (a+b x)^2 e^{-\frac {2 A}{B n}} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-2/n} \text {Ei}\left (\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{B g^3 n (c+d x)^2 (b c-a d)^2} \]

[Out]

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

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Rubi [F]  time = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{(c g+d g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((c*g + d*g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])),x]

[Out]

Defer[Int][1/((c*g + d*g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])), x]

Rubi steps

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

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Mathematica [A]  time = 0.29, size = 174, normalized size = 0.87 \[ \frac {(a+b x) e^{-\frac {2 A}{B n}} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-2/n} \left (b e^{\frac {A}{B n}} (c+d x) \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{\frac {1}{n}} \text {Ei}\left (\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{B n}\right )-d (a+b x) \text {Ei}\left (\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )\right )}{B g^3 n (c+d x)^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((c*g + d*g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])),x]

[Out]

((a + b*x)*(b*E^(A/(B*n))*(e*((a + b*x)/(c + d*x))^n)^n^(-1)*(c + d*x)*ExpIntegralEi[(A + B*Log[e*((a + b*x)/(
c + d*x))^n])/(B*n)] - d*(a + b*x)*ExpIntegralEi[(2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(B*n)]))/(B*(b*c -
a*d)^2*E^((2*A)/(B*n))*g^3*n*(e*((a + b*x)/(c + d*x))^n)^(2/n)*(c + d*x)^2)

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fricas [A]  time = 0.90, size = 147, normalized size = 0.74 \[ \frac {{\left (b e^{\left (\frac {B \log \relax (e) + A}{B n}\right )} \operatorname {log\_integral}\left (\frac {{\left (b x + a\right )} e^{\left (\frac {B \log \relax (e) + A}{B n}\right )}}{d x + c}\right ) - d \operatorname {log\_integral}\left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} e^{\left (\frac {2 \, {\left (B \log \relax (e) + A\right )}}{B n}\right )}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )\right )} e^{\left (-\frac {2 \, {\left (B \log \relax (e) + A\right )}}{B n}\right )}}{{\left (B b^{2} c^{2} - 2 \, B a b c d + B a^{2} d^{2}\right )} g^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*g*x+c*g)^3/(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="fricas")

[Out]

(b*e^((B*log(e) + A)/(B*n))*log_integral((b*x + a)*e^((B*log(e) + A)/(B*n))/(d*x + c)) - d*log_integral((b^2*x
^2 + 2*a*b*x + a^2)*e^(2*(B*log(e) + A)/(B*n))/(d^2*x^2 + 2*c*d*x + c^2)))*e^(-2*(B*log(e) + A)/(B*n))/((B*b^2
*c^2 - 2*B*a*b*c*d + B*a^2*d^2)*g^3*n)

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giac [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(1/(d*g*x+c*g)^3/(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d g x +c g \right )^{3} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*g*x+c*g)^3/(B*ln(e*((b*x+a)/(d*x+c))^n)+A),x)

[Out]

int(1/(d*g*x+c*g)^3/(B*ln(e*((b*x+a)/(d*x+c))^n)+A),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (d g x + c g\right )}^{3} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*g*x+c*g)^3/(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="maxima")

[Out]

integrate(1/((d*g*x + c*g)^3*(B*log(e*((b*x + a)/(d*x + c))^n) + A)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (c\,g+d\,g\,x\right )}^3\,\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((c*g + d*g*x)^3*(A + B*log(e*((a + b*x)/(c + d*x))^n))),x)

[Out]

int(1/((c*g + d*g*x)^3*(A + B*log(e*((a + b*x)/(c + d*x))^n))), x)

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

[Out]

Timed out

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