Optimal. Leaf size=60 \[ -\frac{\left (\frac{b x^3}{a}+1\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p \, _2F_1\left (-\frac{2}{3},-2 p;\frac{1}{3};-\frac{b x^3}{a}\right )}{2 x^2} \]
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Rubi [A] time = 0.0188288, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {1356, 364} \[ -\frac{\left (\frac{b x^3}{a}+1\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p \, _2F_1\left (-\frac{2}{3},-2 p;\frac{1}{3};-\frac{b x^3}{a}\right )}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 1356
Rule 364
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^p}{x^3} \, dx &=\left (\left (1+\frac{b x^3}{a}\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p\right ) \int \frac{\left (1+\frac{b x^3}{a}\right )^{2 p}}{x^3} \, dx\\ &=-\frac{\left (1+\frac{b x^3}{a}\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p \, _2F_1\left (-\frac{2}{3},-2 p;\frac{1}{3};-\frac{b x^3}{a}\right )}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0060824, size = 51, normalized size = 0.85 \[ -\frac{\left (\left (a+b x^3\right )^2\right )^p \left (\frac{b x^3}{a}+1\right )^{-2 p} \, _2F_1\left (-\frac{2}{3},-2 p;\frac{1}{3};-\frac{b x^3}{a}\right )}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({b}^{2}{x}^{6}+2\,ab{x}^{3}+{a}^{2} \right ) ^{p}}{{x}^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p}}{x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{p}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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