3.2.9 \(\int \frac {(a g+b g x)^2}{A+B \log (\frac {e (a+b x)}{c+d x})} \, dx\) [109]

Optimal. Leaf size=35 \[ \text {Int}\left (\frac {(a g+b g x)^2}{A+B \log \left (\frac {e (a+b x)}{c+d x}\right )},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.02, 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 = {} \begin {gather*} \int \frac {(a g+b g x)^2}{A+B \log \left (\frac {e (a+b x)}{c+d x}\right )} \, dx \end {gather*}

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)]),x]

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.57, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a g+b g x)^2}{A+B \log \left (\frac {e (a+b x)}{c+d x}\right )} \, dx \end {gather*}

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)]),x]

[Out]

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

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Maple [A]
time = 0.76, size = 0, normalized size = 0.00 \[\int \frac {\left (b g x +a g \right )^{2}}{A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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))),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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))),x, algorithm="fricas")

[Out]

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

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

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))),x)

[Out]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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))),x, algorithm="giac")

[Out]

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

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

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))),x)

[Out]

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

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