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

Optimal. Leaf size=206 \[ \frac{d e^{-\frac{A}{2 B}} (c+d x) \text{Ei}\left (\frac{A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{2 B}\right )}{4 B^2 g^3 (a+b x) (b c-a d)^2 \sqrt{\frac{e (c+d x)^2}{(a+b x)^2}}}-\frac{b e^{-\frac{A}{B}} \text{Ei}\left (\frac{A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{B}\right )}{2 B^2 e g^3 (b c-a d)^2}+\frac{c+d x}{2 B g^3 (a+b x)^2 (b c-a d) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )} \]

[Out]

(d*(c + d*x)*ExpIntegralEi[(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])/(2*B)])/(4*B^2*(b*c - a*d)^2*E^(A/(2*B))*g
^3*(a + b*x)*Sqrt[(e*(c + d*x)^2)/(a + b*x)^2]) - (b*ExpIntegralEi[(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])/B]
)/(2*B^2*(b*c - a*d)^2*e*E^(A/B)*g^3) + (c + d*x)/(2*B*(b*c - a*d)*g^3*(a + b*x)^2*(A + B*Log[(e*(c + d*x)^2)/
(a + b*x)^2]))

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Rubi [F]  time = 0.0850021, 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 = {} \[ \int \frac{1}{(a g+b g x)^3 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [F]  time = 0.354644, size = 0, normalized size = 0. \[ \int \frac{1}{(a g+b g x)^3 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 1.622, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bgx+ag \right ) ^{3}} \left ( A+B\ln \left ({\frac{e \left ( dx+c \right ) ^{2}}{ \left ( bx+a \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)^3/(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))^2,x)

[Out]

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

[Out]

1/2*(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 - 2*((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) + 2*((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(-1/2*(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 - 2*((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) + 2*((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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{A^{2} b^{3} g^{3} x^{3} + 3 \, A^{2} a b^{2} g^{3} x^{2} + 3 \, A^{2} a^{2} b g^{3} x + A^{2} a^{3} g^{3} +{\left (B^{2} b^{3} g^{3} x^{3} + 3 \, B^{2} a b^{2} g^{3} x^{2} + 3 \, B^{2} a^{2} b g^{3} x + B^{2} a^{3} g^{3}\right )} \log \left (\frac{d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )^{2} + 2 \,{\left (A B b^{3} g^{3} x^{3} + 3 \, A B a b^{2} g^{3} x^{2} + 3 \, A B a^{2} b g^{3} x + A B a^{3} g^{3}\right )} \log \left (\frac{d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}, x\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*(d*x+c)^2/(b*x+a)^2))^2,x, algorithm="fricas")

[Out]

integral(1/(A^2*b^3*g^3*x^3 + 3*A^2*a*b^2*g^3*x^2 + 3*A^2*a^2*b*g^3*x + A^2*a^3*g^3 + (B^2*b^3*g^3*x^3 + 3*B^2
*a*b^2*g^3*x^2 + 3*B^2*a^2*b*g^3*x + B^2*a^3*g^3)*log((d^2*e*x^2 + 2*c*d*e*x + c^2*e)/(b^2*x^2 + 2*a*b*x + a^2
))^2 + 2*(A*B*b^3*g^3*x^3 + 3*A*B*a*b^2*g^3*x^2 + 3*A*B*a^2*b*g^3*x + A*B*a^3*g^3)*log((d^2*e*x^2 + 2*c*d*e*x
+ c^2*e)/(b^2*x^2 + 2*a*b*x + a^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)**3/(A+B*ln(e*(d*x+c)**2/(b*x+a)**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 )}^{3}{\left (B \log \left (\frac{{\left (d x + c\right )}^{2} e}{{\left (b x + a\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)^3/(A+B*log(e*(d*x+c)^2/(b*x+a)^2))^2,x, algorithm="giac")

[Out]

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