3.3.87 \(\int \frac {(f+g x)^2}{(A+B \log (\frac {e (a+b x)^2}{(c+d x)^2}))^2} \, dx\) [287]

Optimal. Leaf size=34 \[ \text {Int}\left (\frac {(f+g x)^2}{\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2},x\right ) \]

[Out]

Unintegrable((g*x+f)^2/(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2,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 {(f+g x)^2}{\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2} \, dx \end {gather*}

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])^2,x]

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.35, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(f+g x)^2}{\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2} \, dx \end {gather*}

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])^2,x]

[Out]

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

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Maple [A]
time = 0.88, size = 0, normalized size = 0.00 \[\int \frac {\left (g x +f \right )^{2}}{\left (A +B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )\right )^{2}}\, dx\]

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))^2,x)

[Out]

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

[Out]

-1/2*(b*d*g^2*x^4 + a*c*f^2 + (a*d*g^2 + (2*d*f*g + c*g^2)*b)*x^3 + ((2*d*f*g + c*g^2)*a + (d*f^2 + 2*c*f*g)*b
)*x^2 + (b*c*f^2 + (d*f^2 + 2*c*f*g)*a)*x)/(2*(b*c - a*d)*B^2*log(b*x + a) - 2*(b*c - a*d)*B^2*log(d*x + c) +
(b*c - a*d)*A*B + (b*c - a*d)*B^2) + integrate(1/2*(4*b*d*g^2*x^3 + b*c*f^2 + 3*(a*d*g^2 + (2*d*f*g + c*g^2)*b
)*x^2 + (d*f^2 + 2*c*f*g)*a + 2*((2*d*f*g + c*g^2)*a + (d*f^2 + 2*c*f*g)*b)*x)/(2*(b*c - a*d)*B^2*log(b*x + a)
- 2*(b*c - a*d)*B^2*log(d*x + c) + (b*c - a*d)*A*B + (b*c - a*d)*B^2), 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((g*x+f)^2/(A+B*log(e*(b*x+a)^2/(d*x+c)^2))^2,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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))**2,x)

[Out]

Timed out

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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