3.55 \(\int \frac{1}{(c g+d g x)^2 (A+B \log (e (\frac{a+b x}{c+d x})^n))^2} \, dx\)

Optimal. Leaf size=154 \[ \frac{(a+b x) e^{-\frac{A}{B n}} \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{-1/n} \text{Ei}\left (\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{B n}\right )}{B^2 g^2 n^2 (c+d x) (b c-a d)}-\frac{a+b x}{B g^2 n (c+d x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )} \]

[Out]

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

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Rubi [F]  time = 0.103582, 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 = {} \[ \int \frac{1}{(c g+d g x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((c*g + d*g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2),x]

[Out]

Defer[Int][1/((c*g + d*g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2), x]

Rubi steps

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

Mathematica [A]  time = 0.171179, size = 180, normalized size = 1.17 \[ -\frac{(a+b x) e^{-\frac{A}{B n}} \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{-1/n} \left (B n e^{\frac{A}{B n}} \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{\frac{1}{n}}-\left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right ) \text{Ei}\left (\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{B n}\right )\right )}{B^2 g^2 n^2 (c+d x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((c*g + d*g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2),x]

[Out]

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

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Maple [F]  time = 0.438, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dgx+cg \right ) ^{2}} \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(1/(d*g*x+c*g)^2/(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2,x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*g*x+c*g)^2/(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.756198, size = 635, normalized size = 4.12 \begin{align*} -\frac{{\left ({\left (B b n x + B a n\right )} e^{\left (\frac{B \log \left (e\right ) + A}{B n}\right )} -{\left (A d x + A c +{\left (B d x + B c\right )} \log \left (e\right ) +{\left (B d n x + B c n\right )} \log \left (\frac{b x + a}{d x + c}\right )\right )} \logintegral \left (\frac{{\left (b x + a\right )} e^{\left (\frac{B \log \left (e\right ) + A}{B n}\right )}}{d x + c}\right )\right )} e^{\left (-\frac{B \log \left (e\right ) + A}{B n}\right )}}{{\left (A B^{2} b c d - A B^{2} a d^{2}\right )} g^{2} n^{2} x +{\left (A B^{2} b c^{2} - A B^{2} a c d\right )} g^{2} n^{2} +{\left ({\left (B^{3} b c d - B^{3} a d^{2}\right )} g^{2} n^{2} x +{\left (B^{3} b c^{2} - B^{3} a c d\right )} g^{2} n^{2}\right )} \log \left (e\right ) +{\left ({\left (B^{3} b c d - B^{3} a d^{2}\right )} g^{2} n^{3} x +{\left (B^{3} b c^{2} - B^{3} a c d\right )} g^{2} n^{3}\right )} \log \left (\frac{b x + a}{d x + c}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*g*x+c*g)^2/(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="fricas")

[Out]

-((B*b*n*x + B*a*n)*e^((B*log(e) + A)/(B*n)) - (A*d*x + A*c + (B*d*x + B*c)*log(e) + (B*d*n*x + B*c*n)*log((b*
x + a)/(d*x + c)))*log_integral((b*x + a)*e^((B*log(e) + A)/(B*n))/(d*x + c)))*e^(-(B*log(e) + A)/(B*n))/((A*B
^2*b*c*d - A*B^2*a*d^2)*g^2*n^2*x + (A*B^2*b*c^2 - A*B^2*a*c*d)*g^2*n^2 + ((B^3*b*c*d - B^3*a*d^2)*g^2*n^2*x +
(B^3*b*c^2 - B^3*a*c*d)*g^2*n^2)*log(e) + ((B^3*b*c*d - B^3*a*d^2)*g^2*n^3*x + (B^3*b*c^2 - B^3*a*c*d)*g^2*n^
3)*log((b*x + a)/(d*x + c)))

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

[Out]

Timed out

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Giac [A]  time = 1.45868, size = 396, normalized size = 2.57 \begin{align*} \frac{{\rm Ei}\left (\frac{A}{B n} + \frac{1}{n} + \log \left (\frac{b x + a}{d x + c}\right )\right ) e^{\left (-\frac{A}{B n} - \frac{1}{n}\right )}}{B^{2} b c g^{2} n^{2} - B^{2} a d g^{2} n^{2}} - \frac{b x + a}{B^{2} b c d g^{2} n^{2} x \log \left (\frac{b x + a}{d x + c}\right ) - B^{2} a d^{2} g^{2} n^{2} x \log \left (\frac{b x + a}{d x + c}\right ) + B^{2} b c^{2} g^{2} n^{2} \log \left (\frac{b x + a}{d x + c}\right ) - B^{2} a c d g^{2} n^{2} \log \left (\frac{b x + a}{d x + c}\right ) + A B b c d g^{2} n x + B^{2} b c d g^{2} n x - A B a d^{2} g^{2} n x - B^{2} a d^{2} g^{2} n x + A B b c^{2} g^{2} n + B^{2} b c^{2} g^{2} n - A B a c d g^{2} n - B^{2} a c d g^{2} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*g*x+c*g)^2/(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="giac")

[Out]

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