3.118 \(\int \frac {1}{(a g+b g x)^3 (A+B \log (\frac {e (a+b x)}{c+d x}))^2} \, dx\)

Optimal. Leaf size=212 \[ -\frac {2 b e^2 e^{\frac {2 A}{B}} \text {Ei}\left (-\frac {2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{B}\right )}{B^2 g^3 (b c-a d)^2}+\frac {d e e^{A/B} \text {Ei}\left (-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{B}\right )}{B^2 g^3 (b c-a d)^2}-\frac {b (c+d x)^2}{B g^3 (a+b x)^2 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}+\frac {d (c+d x)}{B g^3 (a+b x) (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )} \]

[Out]

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

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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 = {} \[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((a*g + b*g*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2),x]

[Out]

Defer[Int][1/((a*g + b*g*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2), x]

Rubi steps

\begin {align*} \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx &=\int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.65, size = 136, normalized size = 0.64 \[ \frac {-2 b e^2 e^{\frac {2 A}{B}} \text {Ei}\left (-\frac {2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{B}\right )+d e e^{A/B} \text {Ei}\left (-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{B}\right )-\frac {B (c+d x) (b c-a d)}{(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}}{B^2 g^3 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a*g + b*g*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2),x]

[Out]

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

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fricas [B]  time = 0.62, size = 570, normalized size = 2.69 \[ -\frac {B b c^{2} - B a c d + {\left (B b c d - B a d^{2}\right )} x - {\left ({\left (B b^{2} d e x^{2} + 2 \, B a b d e x + B a^{2} d e\right )} e^{\frac {A}{B}} \log \left (\frac {b e x + a e}{d x + c}\right ) + {\left (A b^{2} d e x^{2} + 2 \, A a b d e x + A a^{2} d e\right )} e^{\frac {A}{B}}\right )} \operatorname {log\_integral}\left (\frac {{\left (d x + c\right )} e^{\left (-\frac {A}{B}\right )}}{b e x + a e}\right ) + 2 \, {\left ({\left (B b^{3} e^{2} x^{2} + 2 \, B a b^{2} e^{2} x + B a^{2} b e^{2}\right )} e^{\left (\frac {2 \, A}{B}\right )} \log \left (\frac {b e x + a e}{d x + c}\right ) + {\left (A b^{3} e^{2} x^{2} + 2 \, A a b^{2} e^{2} x + A a^{2} b e^{2}\right )} e^{\left (\frac {2 \, A}{B}\right )}\right )} \operatorname {log\_integral}\left (\frac {{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} e^{\left (-\frac {2 \, A}{B}\right )}}{b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}}\right )}{{\left (A B^{2} b^{4} c^{2} - 2 \, A B^{2} a b^{3} c d + A B^{2} a^{2} b^{2} d^{2}\right )} g^{3} x^{2} + 2 \, {\left (A B^{2} a b^{3} c^{2} - 2 \, A B^{2} a^{2} b^{2} c d + A B^{2} a^{3} b d^{2}\right )} g^{3} x + {\left (A B^{2} a^{2} b^{2} c^{2} - 2 \, A B^{2} a^{3} b c d + A B^{2} a^{4} d^{2}\right )} g^{3} + {\left ({\left (B^{3} b^{4} c^{2} - 2 \, B^{3} a b^{3} c d + B^{3} a^{2} b^{2} d^{2}\right )} g^{3} x^{2} + 2 \, {\left (B^{3} a b^{3} c^{2} - 2 \, B^{3} a^{2} b^{2} c d + B^{3} a^{3} b d^{2}\right )} g^{3} x + {\left (B^{3} a^{2} b^{2} c^{2} - 2 \, B^{3} a^{3} b c d + B^{3} a^{4} d^{2}\right )} g^{3}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*g*x+a*g)^3/(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algorithm="fricas")

[Out]

-(B*b*c^2 - B*a*c*d + (B*b*c*d - B*a*d^2)*x - ((B*b^2*d*e*x^2 + 2*B*a*b*d*e*x + B*a^2*d*e)*e^(A/B)*log((b*e*x
+ a*e)/(d*x + c)) + (A*b^2*d*e*x^2 + 2*A*a*b*d*e*x + A*a^2*d*e)*e^(A/B))*log_integral((d*x + c)*e^(-A/B)/(b*e*
x + a*e)) + 2*((B*b^3*e^2*x^2 + 2*B*a*b^2*e^2*x + B*a^2*b*e^2)*e^(2*A/B)*log((b*e*x + a*e)/(d*x + c)) + (A*b^3
*e^2*x^2 + 2*A*a*b^2*e^2*x + A*a^2*b*e^2)*e^(2*A/B))*log_integral((d^2*x^2 + 2*c*d*x + c^2)*e^(-2*A/B)/(b^2*e^
2*x^2 + 2*a*b*e^2*x + a^2*e^2)))/((A*B^2*b^4*c^2 - 2*A*B^2*a*b^3*c*d + A*B^2*a^2*b^2*d^2)*g^3*x^2 + 2*(A*B^2*a
*b^3*c^2 - 2*A*B^2*a^2*b^2*c*d + A*B^2*a^3*b*d^2)*g^3*x + (A*B^2*a^2*b^2*c^2 - 2*A*B^2*a^3*b*c*d + A*B^2*a^4*d
^2)*g^3 + ((B^3*b^4*c^2 - 2*B^3*a*b^3*c*d + B^3*a^2*b^2*d^2)*g^3*x^2 + 2*(B^3*a*b^3*c^2 - 2*B^3*a^2*b^2*c*d +
B^3*a^3*b*d^2)*g^3*x + (B^3*a^2*b^2*c^2 - 2*B^3*a^3*b*c*d + B^3*a^4*d^2)*g^3)*log((b*e*x + a*e)/(d*x + c)))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b g x + a g\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*g*x+a*g)^3/(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algorithm="giac")

[Out]

integrate(1/((b*g*x + a*g)^3*(B*log((b*x + a)*e/(d*x + c)) + A)^2), x)

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maple [F]  time = 1.51, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b g x +a g \right )^{3} \left (B \ln \left (\frac {\left (b x +a \right ) e}{d x +c}\right )+A \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*g*x+a*g)^3/(B*ln((b*x+a)/(d*x+c)*e)+A)^2,x)

[Out]

int(1/(b*g*x+a*g)^3/(B*ln((b*x+a)/(d*x+c)*e)+A)^2,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {d x + c}{{\left (a^{2} b c g^{3} - a^{3} d g^{3}\right )} A B + {\left (a^{2} b c g^{3} \log \relax (e) - a^{3} d g^{3} \log \relax (e)\right )} B^{2} + {\left ({\left (b^{3} c g^{3} - a b^{2} d g^{3}\right )} A B + {\left (b^{3} c g^{3} \log \relax (e) - a b^{2} d g^{3} \log \relax (e)\right )} B^{2}\right )} x^{2} + 2 \, {\left ({\left (a b^{2} c g^{3} - a^{2} b d g^{3}\right )} A B + {\left (a b^{2} c g^{3} \log \relax (e) - a^{2} b d g^{3} \log \relax (e)\right )} B^{2}\right )} x + {\left ({\left (b^{3} c g^{3} - a b^{2} d g^{3}\right )} B^{2} x^{2} + 2 \, {\left (a b^{2} c g^{3} - a^{2} b d g^{3}\right )} B^{2} x + {\left (a^{2} b c g^{3} - a^{3} d g^{3}\right )} B^{2}\right )} \log \left (b x + a\right ) - {\left ({\left (b^{3} c g^{3} - a b^{2} d g^{3}\right )} B^{2} x^{2} + 2 \, {\left (a b^{2} c g^{3} - a^{2} b d g^{3}\right )} B^{2} x + {\left (a^{2} b c g^{3} - a^{3} d g^{3}\right )} B^{2}\right )} \log \left (d x + c\right )} - \int \frac {b d x + 2 \, b c - a d}{{\left ({\left (b^{4} c g^{3} - a b^{3} d g^{3}\right )} A B + {\left (b^{4} c g^{3} \log \relax (e) - a b^{3} d g^{3} \log \relax (e)\right )} B^{2}\right )} x^{3} + {\left (a^{3} b c g^{3} - a^{4} d g^{3}\right )} A B + {\left (a^{3} b c g^{3} \log \relax (e) - a^{4} d g^{3} \log \relax (e)\right )} B^{2} + 3 \, {\left ({\left (a b^{3} c g^{3} - a^{2} b^{2} d g^{3}\right )} A B + {\left (a b^{3} c g^{3} \log \relax (e) - a^{2} b^{2} d g^{3} \log \relax (e)\right )} B^{2}\right )} x^{2} + 3 \, {\left ({\left (a^{2} b^{2} c g^{3} - a^{3} b d g^{3}\right )} A B + {\left (a^{2} b^{2} c g^{3} \log \relax (e) - a^{3} b d g^{3} \log \relax (e)\right )} B^{2}\right )} x + {\left ({\left (b^{4} c g^{3} - a b^{3} d g^{3}\right )} B^{2} x^{3} + 3 \, {\left (a b^{3} c g^{3} - a^{2} b^{2} d g^{3}\right )} B^{2} x^{2} + 3 \, {\left (a^{2} b^{2} c g^{3} - a^{3} b d g^{3}\right )} B^{2} x + {\left (a^{3} b c g^{3} - a^{4} d g^{3}\right )} B^{2}\right )} \log \left (b x + a\right ) - {\left ({\left (b^{4} c g^{3} - a b^{3} d g^{3}\right )} B^{2} x^{3} + 3 \, {\left (a b^{3} c g^{3} - a^{2} b^{2} d g^{3}\right )} B^{2} x^{2} + 3 \, {\left (a^{2} b^{2} c g^{3} - a^{3} b d g^{3}\right )} B^{2} x + {\left (a^{3} b c g^{3} - a^{4} d g^{3}\right )} B^{2}\right )} \log \left (d x + c\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*g*x+a*g)^3/(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algorithm="maxima")

[Out]

-(d*x + c)/((a^2*b*c*g^3 - a^3*d*g^3)*A*B + (a^2*b*c*g^3*log(e) - a^3*d*g^3*log(e))*B^2 + ((b^3*c*g^3 - a*b^2*
d*g^3)*A*B + (b^3*c*g^3*log(e) - a*b^2*d*g^3*log(e))*B^2)*x^2 + 2*((a*b^2*c*g^3 - a^2*b*d*g^3)*A*B + (a*b^2*c*
g^3*log(e) - a^2*b*d*g^3*log(e))*B^2)*x + ((b^3*c*g^3 - a*b^2*d*g^3)*B^2*x^2 + 2*(a*b^2*c*g^3 - a^2*b*d*g^3)*B
^2*x + (a^2*b*c*g^3 - a^3*d*g^3)*B^2)*log(b*x + a) - ((b^3*c*g^3 - a*b^2*d*g^3)*B^2*x^2 + 2*(a*b^2*c*g^3 - a^2
*b*d*g^3)*B^2*x + (a^2*b*c*g^3 - a^3*d*g^3)*B^2)*log(d*x + c)) - integrate((b*d*x + 2*b*c - a*d)/(((b^4*c*g^3
- a*b^3*d*g^3)*A*B + (b^4*c*g^3*log(e) - a*b^3*d*g^3*log(e))*B^2)*x^3 + (a^3*b*c*g^3 - a^4*d*g^3)*A*B + (a^3*b
*c*g^3*log(e) - a^4*d*g^3*log(e))*B^2 + 3*((a*b^3*c*g^3 - a^2*b^2*d*g^3)*A*B + (a*b^3*c*g^3*log(e) - a^2*b^2*d
*g^3*log(e))*B^2)*x^2 + 3*((a^2*b^2*c*g^3 - a^3*b*d*g^3)*A*B + (a^2*b^2*c*g^3*log(e) - a^3*b*d*g^3*log(e))*B^2
)*x + ((b^4*c*g^3 - a*b^3*d*g^3)*B^2*x^3 + 3*(a*b^3*c*g^3 - a^2*b^2*d*g^3)*B^2*x^2 + 3*(a^2*b^2*c*g^3 - a^3*b*
d*g^3)*B^2*x + (a^3*b*c*g^3 - a^4*d*g^3)*B^2)*log(b*x + a) - ((b^4*c*g^3 - a*b^3*d*g^3)*B^2*x^3 + 3*(a*b^3*c*g
^3 - a^2*b^2*d*g^3)*B^2*x^2 + 3*(a^2*b^2*c*g^3 - a^3*b*d*g^3)*B^2*x + (a^3*b*c*g^3 - a^4*d*g^3)*B^2)*log(d*x +
c)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a\,g+b\,g\,x\right )}^3\,{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*g + b*g*x)^3*(A + B*log((e*(a + b*x))/(c + d*x)))^2),x)

[Out]

int(1/((a*g + b*g*x)^3*(A + B*log((e*(a + b*x))/(c + d*x)))^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)**3/(A+B*ln(e*(b*x+a)/(d*x+c)))**2,x)

[Out]

Timed out

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