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

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{1}{\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )^2},x\right ) \]

[Out]

Unintegrable[1/((d + e*x^2)^2*(a + b*Log[c*x^n])^2), x]

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Rubi [A]  time = 0.0324521, 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 = {} \[ \int \frac{1}{\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 12.0129, size = 0, normalized size = 0. \[ \int \frac{1}{\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 2.377, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( e{x}^{2}+d \right ) ^{2} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x^2+d)^2/(a+b*ln(c*x^n))^2,x)

[Out]

int(1/(e*x^2+d)^2/(a+b*ln(c*x^n))^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{x}{b^{2} d^{2} n \log \left (c\right ) + a b d^{2} n +{\left (b^{2} e^{2} n \log \left (c\right ) + a b e^{2} n\right )} x^{4} + 2 \,{\left (b^{2} d e n \log \left (c\right ) + a b d e n\right )} x^{2} +{\left (b^{2} e^{2} n x^{4} + 2 \, b^{2} d e n x^{2} + b^{2} d^{2} n\right )} \log \left (x^{n}\right )} - \int \frac{3 \, e x^{2} - d}{{\left (b^{2} e^{3} n \log \left (c\right ) + a b e^{3} n\right )} x^{6} + b^{2} d^{3} n \log \left (c\right ) + a b d^{3} n + 3 \,{\left (b^{2} d e^{2} n \log \left (c\right ) + a b d e^{2} n\right )} x^{4} + 3 \,{\left (b^{2} d^{2} e n \log \left (c\right ) + a b d^{2} e n\right )} x^{2} +{\left (b^{2} e^{3} n x^{6} + 3 \, b^{2} d e^{2} n x^{4} + 3 \, b^{2} d^{2} e n x^{2} + b^{2} d^{3} n\right )} \log \left (x^{n}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(b^2*d^2*n*log(c) + a*b*d^2*n + (b^2*e^2*n*log(c) + a*b*e^2*n)*x^4 + 2*(b^2*d*e*n*log(c) + a*b*d*e*n)*x^2 +
(b^2*e^2*n*x^4 + 2*b^2*d*e*n*x^2 + b^2*d^2*n)*log(x^n)) - integrate((3*e*x^2 - d)/((b^2*e^3*n*log(c) + a*b*e^
3*n)*x^6 + b^2*d^3*n*log(c) + a*b*d^3*n + 3*(b^2*d*e^2*n*log(c) + a*b*d*e^2*n)*x^4 + 3*(b^2*d^2*e*n*log(c) + a
*b*d^2*e*n)*x^2 + (b^2*e^3*n*x^6 + 3*b^2*d*e^2*n*x^4 + 3*b^2*d^2*e*n*x^2 + b^2*d^3*n)*log(x^n)), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{a^{2} e^{2} x^{4} + 2 \, a^{2} d e x^{2} + a^{2} d^{2} +{\left (b^{2} e^{2} x^{4} + 2 \, b^{2} d e x^{2} + b^{2} d^{2}\right )} \log \left (c x^{n}\right )^{2} + 2 \,{\left (a b e^{2} x^{4} + 2 \, a b d e x^{2} + a b d^{2}\right )} \log \left (c x^{n}\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/(a^2*e^2*x^4 + 2*a^2*d*e*x^2 + a^2*d^2 + (b^2*e^2*x^4 + 2*b^2*d*e*x^2 + b^2*d^2)*log(c*x^n)^2 + 2*(
a*b*e^2*x^4 + 2*a*b*d*e*x^2 + a*b*d^2)*log(c*x^n)), 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(1/(e*x**2+d)**2/(a+b*ln(c*x**n))**2,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (e x^{2} + d\right )}^{2}{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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