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

Optimal. Leaf size=266 \[ \frac {d e^{\frac {A}{2 B}} (c+d x) \sqrt {\frac {e (a+b x)^2}{(c+d x)^2}} \text {Ei}\left (\frac {-A-B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{2 B}\right )}{4 B^2 g^3 (a+b x) (b c-a d)^2}-\frac {b e e^{A/B} \text {Ei}\left (-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{B}\right )}{2 B^2 g^3 (b c-a d)^2}-\frac {b (c+d x)^2}{2 B g^3 (a+b x)^2 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}+\frac {d (c+d x)}{2 B g^3 (a+b x) (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )} \]

[Out]

-1/2*b*e*exp(A/B)*Ei((-A-B*ln(e*(b*x+a)^2/(d*x+c)^2))/B)/B^2/(-a*d+b*c)^2/g^3+1/2*d*(d*x+c)/B/(-a*d+b*c)^2/g^3
/(b*x+a)/(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))-1/2*b*(d*x+c)^2/B/(-a*d+b*c)^2/g^3/(b*x+a)^2/(A+B*ln(e*(b*x+a)^2/(d*x
+c)^2))+1/4*d*exp(1/2*A/B)*(d*x+c)*Ei(1/2*(-A-B*ln(e*(b*x+a)^2/(d*x+c)^2))/B)*(e*(b*x+a)^2/(d*x+c)^2)^(1/2)/B^
2/(-a*d+b*c)^2/g^3/(b*x+a)

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

Verification is Not applicable to the result.

[In]

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

[Out]

Defer[Int][1/((a*g + b*g*x)^3*(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)^3 \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)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F]  time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{A^{2} b^{3} g^{3} x^{3} + 3 \, A^{2} a b^{2} g^{3} x^{2} + 3 \, A^{2} a^{2} b g^{3} x + A^{2} a^{3} g^{3} + {\left (B^{2} b^{3} g^{3} x^{3} + 3 \, B^{2} a b^{2} g^{3} x^{2} + 3 \, B^{2} a^{2} b g^{3} x + B^{2} a^{3} g^{3}\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2} + 2 \, {\left (A B b^{3} g^{3} x^{3} + 3 \, A B a b^{2} g^{3} x^{2} + 3 \, A B a^{2} b g^{3} x + A B a^{3} g^{3}\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}, x\right ) \]

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

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

[Out]

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

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maple [F]  time = 1.32, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b g x +a g \right )^{3} \left (B \ln \left (\frac {\left (b x +a \right )^{2} e}{\left (d x +c \right )^{2}}\right )+A \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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)^2)/(c + d*x)^2))^2),x)

[Out]

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

[Out]

Timed out

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