3.223 \(\int \frac {(a g+b g x)^{-2-m} (c i+d i x)^m}{(A+B \log (e (\frac {a+b x}{c+d x})^n))^3} \, dx\)

Optimal. Leaf size=306 \[ \frac {(m+1)^2 (a+b x) e^{\frac {A (m+1)}{B n}} (g (a+b x))^{-m-2} (i (c+d x))^{m+2} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{\frac {m+1}{n}} \text {Ei}\left (-\frac {(m+1) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{2 B^3 i^2 n^3 (c+d x) (b c-a d)}+\frac {(m+1) (a+b x) (g (a+b x))^{-m-2} (i (c+d x))^{m+2}}{2 B^2 i^2 n^2 (c+d x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}-\frac {(a+b x) (g (a+b x))^{-m-2} (i (c+d x))^{m+2}}{2 B i^2 n (c+d x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2} \]

[Out]

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

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Rubi [F]  time = 0.76, 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 = {} \[ \int \frac {(a g+b g x)^{-2-m} (c i+d i x)^m}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3} \, dx \]

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])^3,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])^3, x]

Rubi steps

\begin {align*} \int \frac {(223 c+223 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 )^3} \, dx &=\int \frac {(223 c+223 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 )^3} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {(a g+b g x)^{-2-m} (c i+d i x)^m}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3} \, dx \]

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])^3,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])^3, x]

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fricas [B]  time = 0.93, size = 815, normalized size = 2.66 \[ -\frac {{\left (B^{2} a c g^{2} n^{2} + {\left (B^{2} b d g^{2} n^{2} - {\left (A B b d g^{2} m + A B b d g^{2}\right )} n\right )} x^{2} - {\left (A B a c g^{2} m + A B a c g^{2}\right )} n + {\left ({\left (B^{2} b c + B^{2} a d\right )} g^{2} n^{2} - {\left ({\left (A B b c + A B a d\right )} g^{2} m + {\left (A B b c + A B a d\right )} g^{2}\right )} n\right )} x - {\left ({\left (B^{2} b d g^{2} m + B^{2} b d g^{2}\right )} n x^{2} + {\left ({\left (B^{2} b c + B^{2} a d\right )} g^{2} m + {\left (B^{2} b c + B^{2} a d\right )} g^{2}\right )} n x + {\left (B^{2} a c g^{2} m + B^{2} a c g^{2}\right )} n\right )} \log \relax (e) - {\left ({\left (B^{2} b d g^{2} m + B^{2} b d g^{2}\right )} n^{2} x^{2} + {\left ({\left (B^{2} b c + B^{2} a d\right )} g^{2} m + {\left (B^{2} b c + B^{2} a d\right )} g^{2}\right )} n^{2} x + {\left (B^{2} a c g^{2} m + B^{2} a c g^{2}\right )} n^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )\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^{2} m^{2} + 2 \, B^{2} m + B^{2}\right )} n^{2} \log \left (\frac {b x + a}{d x + c}\right )^{2} + A^{2} m^{2} + 2 \, A^{2} m + {\left (B^{2} m^{2} + 2 \, B^{2} m + B^{2}\right )} \log \relax (e)^{2} + 2 \, {\left (A B m^{2} + 2 \, A B m + A B\right )} n \log \left (\frac {b x + a}{d x + c}\right ) + A^{2} + 2 \, {\left (A B m^{2} + 2 \, A B m + {\left (B^{2} m^{2} + 2 \, B^{2} m + B^{2}\right )} n \log \left (\frac {b x + a}{d x + c}\right ) + A B\right )} \log \relax (e)\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 )}}{2 \, {\left ({\left (B^{5} b c - B^{5} a d\right )} g^{2} n^{5} \log \left (\frac {b x + a}{d x + c}\right )^{2} + {\left (B^{5} b c - B^{5} a d\right )} g^{2} n^{3} \log \relax (e)^{2} + 2 \, {\left (A B^{4} b c - A B^{4} a d\right )} g^{2} n^{4} \log \left (\frac {b x + a}{d x + c}\right ) + {\left (A^{2} B^{3} b c - A^{2} B^{3} a d\right )} g^{2} n^{3} + 2 \, {\left ({\left (B^{5} b c - B^{5} a d\right )} g^{2} n^{4} \log \left (\frac {b x + a}{d x + c}\right ) + {\left (A B^{4} b c - A B^{4} a d\right )} g^{2} n^{3}\right )} \log \relax (e)\right )}} \]

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))^3,x, algorithm="fricas")

[Out]

-1/2*((B^2*a*c*g^2*n^2 + (B^2*b*d*g^2*n^2 - (A*B*b*d*g^2*m + A*B*b*d*g^2)*n)*x^2 - (A*B*a*c*g^2*m + A*B*a*c*g^
2)*n + ((B^2*b*c + B^2*a*d)*g^2*n^2 - ((A*B*b*c + A*B*a*d)*g^2*m + (A*B*b*c + A*B*a*d)*g^2)*n)*x - ((B^2*b*d*g
^2*m + B^2*b*d*g^2)*n*x^2 + ((B^2*b*c + B^2*a*d)*g^2*m + (B^2*b*c + B^2*a*d)*g^2)*n*x + (B^2*a*c*g^2*m + B^2*a
*c*g^2)*n)*log(e) - ((B^2*b*d*g^2*m + B^2*b*d*g^2)*n^2*x^2 + ((B^2*b*c + B^2*a*d)*g^2*m + (B^2*b*c + B^2*a*d)*
g^2)*n^2*x + (B^2*a*c*g^2*m + B^2*a*c*g^2)*n^2)*log((b*x + a)/(d*x + c)))*(b*g*x + a*g)^(-m - 2)*e^(m*log(b*g*
x + a*g) - m*log((b*x + a)/(d*x + c)) + m*log(i/g)) - ((B^2*m^2 + 2*B^2*m + B^2)*n^2*log((b*x + a)/(d*x + c))^
2 + A^2*m^2 + 2*A^2*m + (B^2*m^2 + 2*B^2*m + B^2)*log(e)^2 + 2*(A*B*m^2 + 2*A*B*m + A*B)*n*log((b*x + a)/(d*x
+ c)) + A^2 + 2*(A*B*m^2 + 2*A*B*m + (B^2*m^2 + 2*B^2*m + B^2)*n*log((b*x + a)/(d*x + c)) + A*B)*log(e))*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^5*b*c - B^5*a*d)*g^2*n^5*log((b*x + a)/(d*x + c))^2 + (B^5*b*c - B^5*a*d)*g^2*n^3*log
(e)^2 + 2*(A*B^4*b*c - A*B^4*a*d)*g^2*n^4*log((b*x + a)/(d*x + c)) + (A^2*B^3*b*c - A^2*B^3*a*d)*g^2*n^3 + 2*(
(B^5*b*c - B^5*a*d)*g^2*n^4*log((b*x + a)/(d*x + c)) + (A*B^4*b*c - A*B^4*a*d)*g^2*n^3)*log(e))

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

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))^3,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)^3, x)

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maple [F(-1)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (b g x +a g \right )^{-m -2} \left (d i x +c i \right )^{m}}{\left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{3}}\, dx \]

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)^3,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)^3,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

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))^3,x, algorithm="maxima")

[Out]

-(m^2 + 2*m + 1)*i^m*integrate(-1/2*(d*x + c)^m/((B^3*b^2*g^(m + 2)*n^2*x^2 + 2*B^3*a*b*g^(m + 2)*n^2*x + B^3*
a^2*g^(m + 2)*n^2)*(b*x + a)^m*log((b*x + a)^n) - (B^3*b^2*g^(m + 2)*n^2*x^2 + 2*B^3*a*b*g^(m + 2)*n^2*x + B^3
*a^2*g^(m + 2)*n^2)*(b*x + a)^m*log((d*x + c)^n) + (B^3*a^2*g^(m + 2)*n^2*log(e) + A*B^2*a^2*g^(m + 2)*n^2 + (
B^3*b^2*g^(m + 2)*n^2*log(e) + A*B^2*b^2*g^(m + 2)*n^2)*x^2 + 2*(B^3*a*b*g^(m + 2)*n^2*log(e) + A*B^2*a*b*g^(m
+ 2)*n^2)*x)*(b*x + a)^m), x) + 1/2*((B*d*i^m*(m + 1)*x + B*c*i^m*(m + 1))*(d*x + c)^m*log((b*x + a)^n) - (B*
d*i^m*(m + 1)*x + B*c*i^m*(m + 1))*(d*x + c)^m*log((d*x + c)^n) + (A*c*i^m*(m + 1) + (i^m*(m + 1)*log(e) - i^m
*n)*B*c + (A*d*i^m*(m + 1) + (i^m*(m + 1)*log(e) - i^m*n)*B*d)*x)*(d*x + c)^m)/(((b^2*c*g^(m + 2)*n^2 - a*b*d*
g^(m + 2)*n^2)*B^4*x + (a*b*c*g^(m + 2)*n^2 - a^2*d*g^(m + 2)*n^2)*B^4)*(b*x + a)^m*log((b*x + a)^n)^2 + ((b^2
*c*g^(m + 2)*n^2 - a*b*d*g^(m + 2)*n^2)*B^4*x + (a*b*c*g^(m + 2)*n^2 - a^2*d*g^(m + 2)*n^2)*B^4)*(b*x + a)^m*l
og((d*x + c)^n)^2 + 2*((a*b*c*g^(m + 2)*n^2 - a^2*d*g^(m + 2)*n^2)*A*B^3 + (a*b*c*g^(m + 2)*n^2*log(e) - a^2*d
*g^(m + 2)*n^2*log(e))*B^4 + ((b^2*c*g^(m + 2)*n^2 - a*b*d*g^(m + 2)*n^2)*A*B^3 + (b^2*c*g^(m + 2)*n^2*log(e)
- a*b*d*g^(m + 2)*n^2*log(e))*B^4)*x)*(b*x + a)^m*log((b*x + a)^n) + ((a*b*c*g^(m + 2)*n^2 - a^2*d*g^(m + 2)*n
^2)*A^2*B^2 + 2*(a*b*c*g^(m + 2)*n^2*log(e) - a^2*d*g^(m + 2)*n^2*log(e))*A*B^3 + (a*b*c*g^(m + 2)*n^2*log(e)^
2 - a^2*d*g^(m + 2)*n^2*log(e)^2)*B^4 + ((b^2*c*g^(m + 2)*n^2 - a*b*d*g^(m + 2)*n^2)*A^2*B^2 + 2*(b^2*c*g^(m +
2)*n^2*log(e) - a*b*d*g^(m + 2)*n^2*log(e))*A*B^3 + (b^2*c*g^(m + 2)*n^2*log(e)^2 - a*b*d*g^(m + 2)*n^2*log(e
)^2)*B^4)*x)*(b*x + a)^m - 2*(((b^2*c*g^(m + 2)*n^2 - a*b*d*g^(m + 2)*n^2)*B^4*x + (a*b*c*g^(m + 2)*n^2 - a^2*
d*g^(m + 2)*n^2)*B^4)*(b*x + a)^m*log((b*x + a)^n) + ((a*b*c*g^(m + 2)*n^2 - a^2*d*g^(m + 2)*n^2)*A*B^3 + (a*b
*c*g^(m + 2)*n^2*log(e) - a^2*d*g^(m + 2)*n^2*log(e))*B^4 + ((b^2*c*g^(m + 2)*n^2 - a*b*d*g^(m + 2)*n^2)*A*B^3
+ (b^2*c*g^(m + 2)*n^2*log(e) - a*b*d*g^(m + 2)*n^2*log(e))*B^4)*x)*(b*x + a)^m)*log((d*x + c)^n))

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

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))^3),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))^3), 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))**3,x)

[Out]

Timed out

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