3.27 \(\int \frac {1}{(a g+b g x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2} \, dx\)

Optimal. Leaf size=153 \[ -\frac {e^{\frac {A}{B n}} (c+d x) \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{\frac {1}{n}} \text {Ei}\left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{B n}\right )}{B^2 g^2 n^2 (a+b x) (b c-a d)}-\frac {c+d x}{B g^2 n (a+b x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )} \]

[Out]

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

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Rubi [F]  time = 0.10, 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 = {} \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((a*g + b*g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2),x]

[Out]

Defer[Int][1/((a*g + b*g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2), x]

Rubi steps

\begin {align*} \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx &=\int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.19, size = 146, normalized size = 0.95 \[ -\frac {(c+d x) \left (e^{\frac {A}{B n}} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{\frac {1}{n}} \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right ) \text {Ei}\left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{B n}\right )+B n\right )}{B^2 g^2 n^2 (a+b x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a*g + b*g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2),x]

[Out]

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

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fricas [A]  time = 0.88, size = 274, normalized size = 1.79 \[ -\frac {B d n x + B c n + {\left (A b x + A a + {\left (B b x + B a\right )} \log \relax (e) + {\left (B b n x + B a n\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} e^{\left (\frac {B \log \relax (e) + A}{B n}\right )} \operatorname {log\_integral}\left (\frac {{\left (d x + c\right )} e^{\left (-\frac {B \log \relax (e) + A}{B n}\right )}}{b x + a}\right )}{{\left (A B^{2} b^{2} c - A B^{2} a b d\right )} g^{2} n^{2} x + {\left (A B^{2} a b c - A B^{2} a^{2} d\right )} g^{2} n^{2} + {\left ({\left (B^{3} b^{2} c - B^{3} a b d\right )} g^{2} n^{2} x + {\left (B^{3} a b c - B^{3} a^{2} d\right )} g^{2} n^{2}\right )} \log \relax (e) + {\left ({\left (B^{3} b^{2} c - B^{3} a b d\right )} g^{2} n^{3} x + {\left (B^{3} a b c - B^{3} a^{2} d\right )} g^{2} n^{3}\right )} \log \left (\frac {b x + a}{d x + c}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*g*x+a*g)^2/(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b g x + a g\right )}^{2} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*g*x+a*g)^2/(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="giac")

[Out]

integrate(1/((b*g*x + a*g)^2*(B*log(e*((b*x + a)/(d*x + c))^n) + A)^2), x)

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maple [F]  time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b g x +a g \right )^{2} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*g*x+a*g)^2/(B*ln(e*((b*x+a)/(d*x+c))^n)+A)^2,x)

[Out]

int(1/(b*g*x+a*g)^2/(B*ln(e*((b*x+a)/(d*x+c))^n)+A)^2,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {d x + c}{{\left (a b c g^{2} n - a^{2} d g^{2} n\right )} A B + {\left (a b c g^{2} n \log \relax (e) - a^{2} d g^{2} n \log \relax (e)\right )} B^{2} + {\left ({\left (b^{2} c g^{2} n - a b d g^{2} n\right )} A B + {\left (b^{2} c g^{2} n \log \relax (e) - a b d g^{2} n \log \relax (e)\right )} B^{2}\right )} x + {\left ({\left (b^{2} c g^{2} n - a b d g^{2} n\right )} B^{2} x + {\left (a b c g^{2} n - a^{2} d g^{2} n\right )} B^{2}\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left ({\left (b^{2} c g^{2} n - a b d g^{2} n\right )} B^{2} x + {\left (a b c g^{2} n - a^{2} d g^{2} n\right )} B^{2}\right )} \log \left ({\left (d x + c\right )}^{n}\right )} + \int -\frac {1}{B^{2} a^{2} g^{2} n \log \relax (e) + A B a^{2} g^{2} n + {\left (B^{2} b^{2} g^{2} n \log \relax (e) + A B b^{2} g^{2} n\right )} x^{2} + 2 \, {\left (B^{2} a b g^{2} n \log \relax (e) + A B a b g^{2} n\right )} x + {\left (B^{2} b^{2} g^{2} n x^{2} + 2 \, B^{2} a b g^{2} n x + B^{2} a^{2} g^{2} n\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left (B^{2} b^{2} g^{2} n x^{2} + 2 \, B^{2} a b g^{2} n x + B^{2} a^{2} g^{2} n\right )} \log \left ({\left (d x + c\right )}^{n}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*g*x+a*g)^2/(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="maxima")

[Out]

-(d*x + c)/((a*b*c*g^2*n - a^2*d*g^2*n)*A*B + (a*b*c*g^2*n*log(e) - a^2*d*g^2*n*log(e))*B^2 + ((b^2*c*g^2*n -
a*b*d*g^2*n)*A*B + (b^2*c*g^2*n*log(e) - a*b*d*g^2*n*log(e))*B^2)*x + ((b^2*c*g^2*n - a*b*d*g^2*n)*B^2*x + (a*
b*c*g^2*n - a^2*d*g^2*n)*B^2)*log((b*x + a)^n) - ((b^2*c*g^2*n - a*b*d*g^2*n)*B^2*x + (a*b*c*g^2*n - a^2*d*g^2
*n)*B^2)*log((d*x + c)^n)) + integrate(-1/(B^2*a^2*g^2*n*log(e) + A*B*a^2*g^2*n + (B^2*b^2*g^2*n*log(e) + A*B*
b^2*g^2*n)*x^2 + 2*(B^2*a*b*g^2*n*log(e) + A*B*a*b*g^2*n)*x + (B^2*b^2*g^2*n*x^2 + 2*B^2*a*b*g^2*n*x + B^2*a^2
*g^2*n)*log((b*x + a)^n) - (B^2*b^2*g^2*n*x^2 + 2*B^2*a*b*g^2*n*x + B^2*a^2*g^2*n)*log((d*x + c)^n)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a\,g+b\,g\,x\right )}^2\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*g + b*g*x)^2*(A + B*log(e*((a + b*x)/(c + d*x))^n))^2),x)

[Out]

int(1/((a*g + b*g*x)^2*(A + B*log(e*((a + b*x)/(c + d*x))^n))^2), 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/(b*g*x+a*g)**2/(A+B*ln(e*((b*x+a)/(d*x+c))**n))**2,x)

[Out]

Timed out

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