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

Optimal. Leaf size=38 \[ \text {Int}\left (\frac {(a g+b g x)^2}{B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.23, 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 = {} \[ \int \frac {(a g+b g x)^2}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.77, size = 0, normalized size = 0.00 \[ \int \frac {(a g+b g x)^2}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} g^{2} x^{2} + 2 \, a b g^{2} x + a^{2} g^{2}}{B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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maple [A]  time = 0.26, size = 0, normalized size = 0.00 \[ \int \frac {\left (b g x +a g \right )^{2}}{B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (a\,g+b\,g\,x\right )}^2}{A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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