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

Optimal. Leaf size=33 \[ \text{Unintegrable}\left (\frac{(f+g x)^2}{B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.174391, 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 = {} \[ \int \frac{(f+g x)^2}{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.176417, size = 0, normalized size = 0. \[ \int \frac{(f+g x)^2}{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.148, size = 0, normalized size = 0. \begin{align*} \int{ \left ( gx+f \right ) ^{2} \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) ^{2}}{ \left ( dx+c \right ) ^{2}}} \right ) \right ) ^{-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{g^{2} x^{2} + 2 \, f g x + f^{2}}{B \log \left (\frac{b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + A}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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