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]

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

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Rubi [F]  time = 0.78, 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 = {} \[ \int \frac {(a g+b g x)^{-2-m} (c i+d i x)^m}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]

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*}

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Mathematica [F]  time = 0.27, size = 0, normalized size = 0.00 \[ \int \frac {(a g+b g x)^{-2-m} (c i+d i x)^m}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]

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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fricas [A]  time = 0.62, size = 284, normalized size = 1.33 \[ -\frac {{\left (B b d g^{2} n x^{2} + B a c g^{2} n + {\left (B b c + B a d\right )} g^{2} n x\right )} {\left (b g x + a g\right )}^{-m - 2} e^{\left (m \log \left (b g x + a g\right ) - m \log \left (\frac {b x + a}{d x + c}\right ) + m \log \left (\frac {i}{g}\right )\right )} + {\left ({\left (B m + B\right )} n \log \left (\frac {b x + a}{d x + c}\right ) + A m + {\left (B m + B\right )} \log \relax (e) + A\right )} {\rm Ei}\left (-\frac {{\left (B m + B\right )} n \log \left (\frac {b x + a}{d x + c}\right ) + A m + {\left (B m + B\right )} \log \relax (e) + A}{B n}\right ) e^{\left (\frac {B m n \log \left (\frac {i}{g}\right ) + A m + {\left (B m + B\right )} \log \relax (e) + A}{B n}\right )}}{{\left (B^{3} b c - B^{3} a d\right )} g^{2} n^{3} \log \left (\frac {b x + a}{d x + c}\right ) + {\left (B^{3} b c - B^{3} a d\right )} g^{2} n^{2} \log \relax (e) + {\left (A B^{2} b c - A B^{2} a d\right )} g^{2} n^{2}} \]

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

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)

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maple [F(-1)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (b g x +a g \right )^{-m -2} \left (d i x +c i \right )^{m}}{\left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ i^{m} {\left (m + 1\right )} \int -\frac {{\left (d x + c\right )}^{m}}{{\left (B^{2} b^{2} g^{m + 2} n x^{2} + 2 \, B^{2} a b g^{m + 2} n x + B^{2} a^{2} g^{m + 2} n\right )} {\left (b x + a\right )}^{m} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left (B^{2} b^{2} g^{m + 2} n x^{2} + 2 \, B^{2} a b g^{m + 2} n x + B^{2} a^{2} g^{m + 2} n\right )} {\left (b x + a\right )}^{m} \log \left ({\left (d x + c\right )}^{n}\right ) + {\left (B^{2} a^{2} g^{m + 2} n \log \relax (e) + A B a^{2} g^{m + 2} n + {\left (B^{2} b^{2} g^{m + 2} n \log \relax (e) + A B b^{2} g^{m + 2} n\right )} x^{2} + 2 \, {\left (B^{2} a b g^{m + 2} n \log \relax (e) + A B a b g^{m + 2} n\right )} x\right )} {\left (b x + a\right )}^{m}}\,{d x} - \frac {{\left (d i^{m} x + c i^{m}\right )} {\left (d x + c\right )}^{m}}{{\left ({\left (b^{2} c g^{m + 2} n - a b d g^{m + 2} n\right )} B^{2} x + {\left (a b c g^{m + 2} n - a^{2} d g^{m + 2} n\right )} B^{2}\right )} {\left (b x + a\right )}^{m} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left ({\left (b^{2} c g^{m + 2} n - a b d g^{m + 2} n\right )} B^{2} x + {\left (a b c g^{m + 2} n - a^{2} d g^{m + 2} n\right )} B^{2}\right )} {\left (b x + a\right )}^{m} \log \left ({\left (d x + c\right )}^{n}\right ) + {\left ({\left (a b c g^{m + 2} n - a^{2} d g^{m + 2} n\right )} A B + {\left (a b c g^{m + 2} n \log \relax (e) - a^{2} d g^{m + 2} n \log \relax (e)\right )} B^{2} + {\left ({\left (b^{2} c g^{m + 2} n - a b d g^{m + 2} n\right )} A B + {\left (b^{2} c g^{m + 2} n \log \relax (e) - a b d g^{m + 2} n \log \relax (e)\right )} B^{2}\right )} x\right )} {\left (b x + a\right )}^{m}} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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