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

Optimal. Leaf size=206 \[ \frac{(m+1) (a+b x) e^{-\frac{A (m+1)}{B n}} (g (a+b x))^m (i (c+d x))^{-m} \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{-\frac{m+1}{n}} \text{Ei}\left (\frac{(m+1) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{B^2 i^2 n^2 (c+d x) (b c-a d)}-\frac{(a+b x) (g (a+b x))^m (i (c+d x))^{-m}}{B i^2 n (c+d x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )} \]

[Out]

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

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Rubi [F]  time = 0.826472, 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 = {} \[ \int \frac{(a g+b g x)^m (c i+d i x)^{-2-m}}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{(216 c+216 d x)^{-2-m} (a g+b g x)^m}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx &=\int \frac{(216 c+216 d x)^{-2-m} (a g+b g x)^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 = 0.257235, size = 0, normalized size = 0. \[ \int \frac{(a g+b g x)^m (c i+d i x)^{-2-m}}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 26.166, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bgx+ag \right ) ^{m} \left ( dix+ci \right ) ^{-2-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)^m*(d*i*x+c*i)^(-2-m)/(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2,x)

[Out]

int((b*g*x+a*g)^m*(d*i*x+c*i)^(-2-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*} -g^{m}{\left (m + 1\right )} \int -\frac{{\left (b x + a\right )}^{m}}{{\left (B^{2} d^{2} i^{m + 2} n x^{2} + 2 \, B^{2} c d i^{m + 2} n x + B^{2} c^{2} i^{m + 2} n\right )}{\left (d x + c\right )}^{m} \log \left ({\left (b x + a\right )}^{n}\right ) -{\left (B^{2} d^{2} i^{m + 2} n x^{2} + 2 \, B^{2} c d i^{m + 2} n x + B^{2} c^{2} i^{m + 2} n\right )}{\left (d x + c\right )}^{m} \log \left ({\left (d x + c\right )}^{n}\right ) +{\left (B^{2} c^{2} i^{m + 2} n \log \left (e\right ) + A B c^{2} i^{m + 2} n +{\left (B^{2} d^{2} i^{m + 2} n \log \left (e\right ) + A B d^{2} i^{m + 2} n\right )} x^{2} + 2 \,{\left (B^{2} c d i^{m + 2} n \log \left (e\right ) + A B c d i^{m + 2} n\right )} x\right )}{\left (d x + c\right )}^{m}}\,{d x} - \frac{{\left (b g^{m} x + a g^{m}\right )}{\left (b x + a\right )}^{m}}{{\left ({\left (b c d i^{m + 2} n - a d^{2} i^{m + 2} n\right )} B^{2} x +{\left (b c^{2} i^{m + 2} n - a c d i^{m + 2} n\right )} B^{2}\right )}{\left (d x + c\right )}^{m} \log \left ({\left (b x + a\right )}^{n}\right ) -{\left ({\left (b c d i^{m + 2} n - a d^{2} i^{m + 2} n\right )} B^{2} x +{\left (b c^{2} i^{m + 2} n - a c d i^{m + 2} n\right )} B^{2}\right )}{\left (d x + c\right )}^{m} \log \left ({\left (d x + c\right )}^{n}\right ) +{\left ({\left (b c^{2} i^{m + 2} n - a c d i^{m + 2} n\right )} A B +{\left (b c^{2} i^{m + 2} n \log \left (e\right ) - a c d i^{m + 2} n \log \left (e\right )\right )} B^{2} +{\left ({\left (b c d i^{m + 2} n - a d^{2} i^{m + 2} n\right )} A B +{\left (b c d i^{m + 2} n \log \left (e\right ) - a d^{2} i^{m + 2} n \log \left (e\right )\right )} B^{2}\right )} x\right )}{\left (d x + c\right )}^{m}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.544517, size = 676, normalized size = 3.28 \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} e^{\left (-{\left (m + 2\right )} \log \left (b g x + a g\right ) +{\left (m + 2\right )} \log \left (\frac{b x + a}{d x + c}\right ) -{\left (m + 2\right )} \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{{\left (B m + 2 \, B\right )} 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)^m*(d*i*x+c*i)^(-2-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*e^(-(m + 2)*log(b*g*x + a*g) + (m
+ 2)*log((b*x + a)/(d*x + c)) - (m + 2)*log(i/g)) - ((B*m + B)*n*log((b*x + a)/(d*x + c)) + A*m + (B*m + B)*lo
g(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 + 2*B)*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*log(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)**m*(d*i*x+c*i)**(-2-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}{\left (d i x + c i\right )}^{-m - 2}}{{\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)^m*(d*i*x+c*i)^(-2-m)/(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="giac")

[Out]

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