3.222 \(\int \frac{(a g+b g x)^{-2-m} (c i+d i x)^m}{(A+B \log (e (\frac{a+b x}{c+d x})^n))^2} \, dx\)

Optimal. Leaf size=214 \[ -\frac{(m+1) (a+b x) e^{\frac{A (m+1)}{B n}} (g (a+b x))^{-m-2} (i (c+d x))^{m+2} \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{\frac{m+1}{n}} \text{Ei}\left (-\frac{(m+1) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{B^2 i^2 n^2 (c+d x) (b c-a d)}-\frac{(a+b x) (g (a+b x))^{-m-2} (i (c+d x))^{m+2}}{B i^2 n (c+d x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )} \]

[Out]

-((E^((A*(1 + m))/(B*n))*(1 + m)*(a + b*x)*(g*(a + b*x))^(-2 - m)*(e*((a + b*x)/(c + d*x))^n)^((1 + m)/n)*(i*(
c + d*x))^(2 + m)*ExpIntegralEi[-(((1 + m)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(B*n))])/(B^2*(b*c - a*d)*i
^2*n^2*(c + d*x))) - ((a + b*x)*(g*(a + b*x))^(-2 - m)*(i*(c + d*x))^(2 + m))/(B*(b*c - a*d)*i^2*n*(c + d*x)*(
A + B*Log[e*((a + b*x)/(c + d*x))^n]))

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Rubi [F]  time = 0.775964, 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 = {} \[ \int \frac{(a g+b g x)^{-2-m} (c i+d i x)^m}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{(222 c+222 d x)^m (a g+b g x)^{-2-m}}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx &=\int \frac{(222 c+222 d x)^m (a g+b g x)^{-2-m}}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx\\ \end{align*}

Mathematica [F]  time = 0.260073, size = 0, normalized size = 0. \[ \int \frac{(a g+b g x)^{-2-m} (c i+d i x)^m}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 25.665, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bgx+ag \right ) ^{-2-m} \left ( dix+ci \right ) ^{m} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{-2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*g*x+a*g)^(-2-m)*(d*i*x+c*i)^m/(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2,x)

[Out]

int((b*g*x+a*g)^(-2-m)*(d*i*x+c*i)^m/(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^(-2-m)*(d*i*x+c*i)^m/(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.538907, size = 649, normalized size = 3.03 \begin{align*} -\frac{{\left (B b d g^{2} n x^{2} + B a c g^{2} n +{\left (B b c + B a d\right )} g^{2} n x\right )}{\left (b g x + a g\right )}^{-m - 2} e^{\left (m \log \left (b g x + a g\right ) - m \log \left (\frac{b x + a}{d x + c}\right ) + m \log \left (\frac{i}{g}\right )\right )} +{\left ({\left (B m + B\right )} n \log \left (\frac{b x + a}{d x + c}\right ) + A m +{\left (B m + B\right )} \log \left (e\right ) + A\right )}{\rm Ei}\left (-\frac{{\left (B m + B\right )} n \log \left (\frac{b x + a}{d x + c}\right ) + A m +{\left (B m + B\right )} \log \left (e\right ) + A}{B n}\right ) e^{\left (\frac{B m n \log \left (\frac{i}{g}\right ) + A m +{\left (B m + B\right )} \log \left (e\right ) + A}{B n}\right )}}{{\left (B^{3} b c - B^{3} a d\right )} g^{2} n^{3} \log \left (\frac{b x + a}{d x + c}\right ) +{\left (B^{3} b c - B^{3} a d\right )} g^{2} n^{2} \log \left (e\right ) +{\left (A B^{2} b c - A B^{2} a d\right )} g^{2} n^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^(-2-m)*(d*i*x+c*i)^m/(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="fricas")

[Out]

-((B*b*d*g^2*n*x^2 + B*a*c*g^2*n + (B*b*c + B*a*d)*g^2*n*x)*(b*g*x + a*g)^(-m - 2)*e^(m*log(b*g*x + a*g) - m*l
og((b*x + a)/(d*x + c)) + m*log(i/g)) + ((B*m + B)*n*log((b*x + a)/(d*x + c)) + A*m + (B*m + B)*log(e) + A)*Ei
(-((B*m + B)*n*log((b*x + a)/(d*x + c)) + A*m + (B*m + B)*log(e) + A)/(B*n))*e^((B*m*n*log(i/g) + A*m + (B*m +
B)*log(e) + A)/(B*n)))/((B^3*b*c - B^3*a*d)*g^2*n^3*log((b*x + a)/(d*x + c)) + (B^3*b*c - B^3*a*d)*g^2*n^2*lo
g(e) + (A*B^2*b*c - A*B^2*a*d)*g^2*n^2)

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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((b*g*x+a*g)**(-2-m)*(d*i*x+c*i)**m/(A+B*ln(e*((b*x+a)/(d*x+c))**n))**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^(-2-m)*(d*i*x+c*i)^m/(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="giac")

[Out]

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