3.285 \(\int \frac {f+g x^2}{\log ^2(c (d+e x^2)^p)} \, dx\)

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {f+g x^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )},x\right ) \]

[Out]

Unintegrable((g*x^2+f)/ln(c*(e*x^2+d)^p)^2,x)

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Rubi [A]  time = 0.01, 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 = {} \[ \int \frac {f+g x^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(f + g*x^2)/Log[c*(d + e*x^2)^p]^2,x]

[Out]

Defer[Int][(f + g*x^2)/Log[c*(d + e*x^2)^p]^2, x]

Rubi steps

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

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Mathematica [A]  time = 0.65, size = 0, normalized size = 0.00 \[ \int \frac {f+g x^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(f + g*x^2)/Log[c*(d + e*x^2)^p]^2,x]

[Out]

Integrate[(f + g*x^2)/Log[c*(d + e*x^2)^p]^2, x]

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fricas [A]  time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {g x^{2} + f}{\log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((g*x^2 + f)/log((e*x^2 + d)^p*c)^2, x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {g x^{2} + f}{\log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((g*x^2 + f)/log((e*x^2 + d)^p*c)^2, x)

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maple [A]  time = 4.25, size = 0, normalized size = 0.00 \[ \int \frac {g \,x^{2}+f}{\ln \left (c \left (e \,x^{2}+d \right )^{p}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x^2+f)/ln(c*(e*x^2+d)^p)^2,x)

[Out]

int((g*x^2+f)/ln(c*(e*x^2+d)^p)^2,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {e g x^{4} + {\left (e f + d g\right )} x^{2} + d f}{2 \, {\left (e p^{2} x \log \left (e x^{2} + d\right ) + e p x \log \relax (c)\right )}} + \int \frac {3 \, e g x^{4} + {\left (e f + d g\right )} x^{2} - d f}{2 \, {\left (e p^{2} x^{2} \log \left (e x^{2} + d\right ) + e p x^{2} \log \relax (c)\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(e*g*x^4 + (e*f + d*g)*x^2 + d*f)/(e*p^2*x*log(e*x^2 + d) + e*p*x*log(c)) + integrate(1/2*(3*e*g*x^4 + (e
*f + d*g)*x^2 - d*f)/(e*p^2*x^2*log(e*x^2 + d) + e*p*x^2*log(c)), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {g\,x^2+f}{{\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f + g*x^2)/log(c*(d + e*x^2)^p)^2,x)

[Out]

int((f + g*x^2)/log(c*(d + e*x^2)^p)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x**2+f)/ln(c*(e*x**2+d)**p)**2,x)

[Out]

Integral((f + g*x**2)/log(c*(d + e*x**2)**p)**2, x)

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