3.306 \(\int \frac {(f+g x^3)^2}{\log ^2(c (d+e x^2)^p)} \, dx\)

Optimal. Leaf size=27 \[ \text {Int}\left (\frac {\left (f+g x^3\right )^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )},x\right ) \]

[Out]

Unintegrable((g*x^3+f)^2/ln(c*(e*x^2+d)^p)^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 = {} \[ \int \frac {\left (f+g x^3\right )^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(f + g*x^3)^2/Log[c*(d + e*x^2)^p]^2,x]

[Out]

Defer[Int][(f + g*x^3)^2/Log[c*(d + e*x^2)^p]^2, x]

Rubi steps

\begin {align*} \int \frac {\left (f+g x^3\right )^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx &=\int \frac {\left (f+g x^3\right )^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.72, size = 0, normalized size = 0.00 \[ \int \frac {\left (f+g x^3\right )^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(f + g*x^3)^2/Log[c*(d + e*x^2)^p]^2,x]

[Out]

Integrate[(f + g*x^3)^2/Log[c*(d + e*x^2)^p]^2, x]

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fricas [A]  time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {g^{2} x^{6} + 2 \, f g x^{3} + f^{2}}{\log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^3+f)^2/log(c*(e*x^2+d)^p)^2,x, algorithm="fricas")

[Out]

integral((g^2*x^6 + 2*f*g*x^3 + f^2)/log((e*x^2 + d)^p*c)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^3+f)^2/log(c*(e*x^2+d)^p)^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 4.56, size = 0, normalized size = 0.00 \[ \int \frac {\left (g \,x^{3}+f \right )^{2}}{\ln \left (c \left (e \,x^{2}+d \right )^{p}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x^3+f)^2/ln(c*(e*x^2+d)^p)^2,x)

[Out]

int((g*x^3+f)^2/ln(c*(e*x^2+d)^p)^2,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {e g^{2} x^{8} + d g^{2} x^{6} + 2 \, e f g x^{5} + 2 \, d f g x^{3} + e f^{2} x^{2} + d f^{2}}{2 \, {\left (e p^{2} x \log \left (e x^{2} + d\right ) + e p x \log \relax (c)\right )}} + \int \frac {7 \, e g^{2} x^{8} + 5 \, d g^{2} x^{6} + 8 \, e f g x^{5} + 4 \, d f g x^{3} + e f^{2} x^{2} - d f^{2}}{2 \, {\left (e p^{2} x^{2} \log \left (e x^{2} + d\right ) + e p x^{2} \log \relax (c)\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^3+f)^2/log(c*(e*x^2+d)^p)^2,x, algorithm="maxima")

[Out]

-1/2*(e*g^2*x^8 + d*g^2*x^6 + 2*e*f*g*x^5 + 2*d*f*g*x^3 + e*f^2*x^2 + d*f^2)/(e*p^2*x*log(e*x^2 + d) + e*p*x*l
og(c)) + integrate(1/2*(7*e*g^2*x^8 + 5*d*g^2*x^6 + 8*e*f*g*x^5 + 4*d*f*g*x^3 + e*f^2*x^2 - d*f^2)/(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 \[ \int \frac {{\left (g\,x^3+f\right )}^2}{{\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f + g*x^3)^2/log(c*(d + e*x^2)^p)^2,x)

[Out]

int((f + g*x^3)^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 \[ \int \frac {\left (f + g x^{3}\right )^{2}}{\log {\left (c \left (d + e x^{2}\right )^{p} \right )}^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x**3+f)**2/ln(c*(e*x**2+d)**p)**2,x)

[Out]

Integral((f + g*x**3)**2/log(c*(d + e*x**2)**p)**2, x)

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