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

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Rubi [F]  time = 0.0837941, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}

Mathematica [A]  time = 0.757335, 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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Maple [F]  time = 1.43, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bgx+ag \right ) ^{3}} \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) }{dx+c}} \right ) \right ) ^{-2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

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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Fricas [B]  time = 1.1091, size = 1187, normalized size = 5.6 \begin{align*} -\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 )} \logintegral \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 )} \logintegral \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 )} \end{align*}

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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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(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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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)