3.122 \(\int \frac{1}{x^2 \log ^3(c (a+b x^2)^p)} \, dx\)

Optimal. Leaf size=20 \[ \text{Unintegrable}\left (\frac{1}{x^2 \log ^3\left (c \left (a+b x^2\right )^p\right )},x\right ) \]

[Out]

Unintegrable[1/(x^2*Log[c*(a + b*x^2)^p]^3), x]

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Rubi [A]  time = 0.0167578, 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 = {} \[ \int \frac{1}{x^2 \log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x^2*Log[c*(a + b*x^2)^p]^3),x]

[Out]

Defer[Int][1/(x^2*Log[c*(a + b*x^2)^p]^3), x]

Rubi steps

\begin{align*} \int \frac{1}{x^2 \log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx &=\int \frac{1}{x^2 \log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx\\ \end{align*}

Mathematica [A]  time = 2.15763, size = 0, normalized size = 0. \[ \int \frac{1}{x^2 \log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x^2*Log[c*(a + b*x^2)^p]^3),x]

[Out]

Integrate[1/(x^2*Log[c*(a + b*x^2)^p]^3), x]

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Maple [A]  time = 3.457, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ( \ln \left ( c \left ( b{x}^{2}+a \right ) ^{p} \right ) \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/ln(c*(b*x^2+a)^p)^3,x)

[Out]

int(1/x^2/ln(c*(b*x^2+a)^p)^3,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{b^{2}{\left (2 \, p - \log \left (c\right )\right )} x^{4} + 2 \, a b{\left (p - 2 \, \log \left (c\right )\right )} x^{2} - 3 \, a^{2} \log \left (c\right ) -{\left (b^{2} x^{4} + 4 \, a b x^{2} + 3 \, a^{2}\right )} \log \left ({\left (b x^{2} + a\right )}^{p}\right )}{8 \,{\left (b^{2} p^{2} x^{5} \log \left ({\left (b x^{2} + a\right )}^{p}\right )^{2} + 2 \, b^{2} p^{2} x^{5} \log \left ({\left (b x^{2} + a\right )}^{p}\right ) \log \left (c\right ) + b^{2} p^{2} x^{5} \log \left (c\right )^{2}\right )}} + \int \frac{b^{2} x^{4} + 12 \, a b x^{2} + 15 \, a^{2}}{8 \,{\left (b^{2} p^{2} x^{6} \log \left ({\left (b x^{2} + a\right )}^{p}\right ) + b^{2} p^{2} x^{6} \log \left (c\right )\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/log(c*(b*x^2+a)^p)^3,x, algorithm="maxima")

[Out]

-1/8*(b^2*(2*p - log(c))*x^4 + 2*a*b*(p - 2*log(c))*x^2 - 3*a^2*log(c) - (b^2*x^4 + 4*a*b*x^2 + 3*a^2)*log((b*
x^2 + a)^p))/(b^2*p^2*x^5*log((b*x^2 + a)^p)^2 + 2*b^2*p^2*x^5*log((b*x^2 + a)^p)*log(c) + b^2*p^2*x^5*log(c)^
2) + integrate(1/8*(b^2*x^4 + 12*a*b*x^2 + 15*a^2)/(b^2*p^2*x^6*log((b*x^2 + a)^p) + b^2*p^2*x^6*log(c)), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{x^{2} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/log(c*(b*x^2+a)^p)^3,x, algorithm="fricas")

[Out]

integral(1/(x^2*log((b*x^2 + a)^p*c)^3), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \log{\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/ln(c*(b*x**2+a)**p)**3,x)

[Out]

Integral(1/(x**2*log(c*(a + b*x**2)**p)**3), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/log(c*(b*x^2+a)^p)^3,x, algorithm="giac")

[Out]

integrate(1/(x^2*log((b*x^2 + a)^p*c)^3), x)