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

Optimal. Leaf size=125 \[ \frac {(a+b x) e^{-\frac {A (m+1)}{B n}} (g (a+b x))^m (i (c+d x))^{-m} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-\frac {m+1}{n}} \text {Ei}\left (\frac {(m+1) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{B i^2 n (c+d x) (b c-a d)} \]

[Out]

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

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Rubi [F]  time = 0.74, 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 = {} \[ \int \frac {(a g+b g x)^m (c i+d i x)^{-2-m}}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx \]

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]),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]), x]

Rubi steps

\begin {align*} \int \frac {(215 c+215 d x)^{-2-m} (a g+b g x)^m}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx &=\int \frac {(215 c+215 d x)^{-2-m} (a g+b g x)^m}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.25, size = 0, normalized size = 0.00 \[ \int \frac {(a g+b g x)^m (c i+d i x)^{-2-m}}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx \]

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]),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]), x]

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fricas [A]  time = 0.83, size = 98, normalized size = 0.78 \[ \frac {{\rm Ei}\left (\frac {{\left (B m + B\right )} n \log \left (\frac {b x + a}{d x + c}\right ) + A m + {\left (B m + B\right )} \log \relax (e) + A}{B n}\right ) e^{\left (-\frac {{\left (B m + 2 \, B\right )} n \log \left (\frac {i}{g}\right ) + A m + {\left (B m + B\right )} \log \relax (e) + A}{B n}\right )}}{{\left (B b c - B a d\right )} g^{2} n} \]

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)),x, algorithm="fricas")

[Out]

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*b*c - B*a*d)*g^2*n)

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

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)),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), x)

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maple [F]  time = 4.21, size = 0, normalized size = 0.00 \[ \int \frac {\left (b g x +a g \right )^{m} \left (d i x +c i \right )^{-m -2}}{B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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)),x, algorithm="maxima")

[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), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a\,g+b\,g\,x\right )}^m}{{\left (c\,i+d\,i\,x\right )}^{m+2}\,\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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