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

Optimal. Leaf size=147 \[ -\frac{e^{\frac{A}{2 B}} (c+d x) \sqrt{\frac{e (a+b x)^2}{(c+d x)^2}} \text{Ei}\left (-\frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{2 B}\right )}{4 B^2 g^2 (a+b x) (b c-a d)}-\frac{c+d x}{2 B g^2 (a+b x) (b c-a d) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )} \]

[Out]

-(E^(A/(2*B))*Sqrt[(e*(a + b*x)^2)/(c + d*x)^2]*(c + d*x)*ExpIntegralEi[-(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^
2])/(2*B)])/(4*B^2*(b*c - a*d)*g^2*(a + b*x)) - (c + d*x)/(2*B*(b*c - a*d)*g^2*(a + b*x)*(A + B*Log[(e*(a + b*
x)^2)/(c + d*x)^2]))

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Rubi [F]  time = 0.0935896, 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 = {} \[ \int \frac{1}{(a g+b g x)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [F]  time = 0.178894, size = 0, normalized size = 0. \[ \int \frac{1}{(a g+b g x)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 1.224, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bgx+ag \right ) ^{2}} \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) ^{2}}{ \left ( dx+c \right ) ^{2}}} \right ) \right ) ^{-2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{d x + c}{2 \,{\left ({\left (a b c g^{2} - a^{2} d g^{2}\right )} A B +{\left (a b c g^{2} \log \left (e\right ) - a^{2} d g^{2} \log \left (e\right )\right )} B^{2} +{\left ({\left (b^{2} c g^{2} - a b d g^{2}\right )} A B +{\left (b^{2} c g^{2} \log \left (e\right ) - a b d g^{2} \log \left (e\right )\right )} B^{2}\right )} x + 2 \,{\left ({\left (b^{2} c g^{2} - a b d g^{2}\right )} B^{2} x +{\left (a b c g^{2} - a^{2} d g^{2}\right )} B^{2}\right )} \log \left (b x + a\right ) - 2 \,{\left ({\left (b^{2} c g^{2} - a b d g^{2}\right )} B^{2} x +{\left (a b c g^{2} - a^{2} d g^{2}\right )} B^{2}\right )} \log \left (d x + c\right )\right )}} + \int -\frac{1}{2 \,{\left (B^{2} a^{2} g^{2} \log \left (e\right ) + A B a^{2} g^{2} +{\left (B^{2} b^{2} g^{2} \log \left (e\right ) + A B b^{2} g^{2}\right )} x^{2} + 2 \,{\left (B^{2} a b g^{2} \log \left (e\right ) + A B a b g^{2}\right )} x + 2 \,{\left (B^{2} b^{2} g^{2} x^{2} + 2 \, B^{2} a b g^{2} x + B^{2} a^{2} g^{2}\right )} \log \left (b x + a\right ) - 2 \,{\left (B^{2} b^{2} g^{2} x^{2} + 2 \, B^{2} a b g^{2} x + B^{2} a^{2} g^{2}\right )} \log \left (d x + c\right )\right )}}\,{d x} \end{align*}

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)^2/(d*x+c)^2))^2,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{A^{2} b^{2} g^{2} x^{2} + 2 \, A^{2} a b g^{2} x + A^{2} a^{2} g^{2} +{\left (B^{2} b^{2} g^{2} x^{2} + 2 \, B^{2} a b g^{2} x + B^{2} a^{2} g^{2}\right )} \log \left (\frac{b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2} + 2 \,{\left (A B b^{2} g^{2} x^{2} + 2 \, A B a b g^{2} x + A B a^{2} g^{2}\right )} \log \left (\frac{b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}, x\right ) \end{align*}

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)^2/(d*x+c)^2))^2,x, algorithm="fricas")

[Out]

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

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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)**2/(A+B*ln(e*(b*x+a)**2/(d*x+c)**2))**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 )}^{2}{\left (B \log \left (\frac{{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\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)^2/(A+B*log(e*(b*x+a)^2/(d*x+c)^2))^2,x, algorithm="giac")

[Out]

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