Optimal. Leaf size=64 \[ \frac{b \left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^p \, _2F_1\left (2,2 p+1;2 (p+1);\frac{b x^3}{a}+1\right )}{3 a^2 (2 p+1)} \]
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Rubi [A] time = 0.0411508, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1356, 266, 65} \[ \frac{b \left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^p \, _2F_1\left (2,2 p+1;2 (p+1);\frac{b x^3}{a}+1\right )}{3 a^2 (2 p+1)} \]
Antiderivative was successfully verified.
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Rule 1356
Rule 266
Rule 65
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^p}{x^4} \, 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^4} \, dx\\ &=\frac{1}{3} \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 ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{b x}{a}\right )^{2 p}}{x^2} \, dx,x,x^3\right )\\ &=\frac{b \left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^p \, _2F_1\left (2,1+2 p;2 (1+p);1+\frac{b x^3}{a}\right )}{3 a^2 (1+2 p)}\\ \end{align*}
Mathematica [A] time = 0.0114877, size = 55, normalized size = 0.86 \[ \frac{b \left (a+b x^3\right ) \left (\left (a+b x^3\right )^2\right )^p \, _2F_1\left (2,2 p+1;2 p+2;\frac{b x^3}{a}+1\right )}{3 a^2 (2 p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.039, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({b}^{2}{x}^{6}+2\,ab{x}^{3}+{a}^{2} \right ) ^{p}}{{x}^{4}}}\, 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^{4}}\,{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^{4}}, 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^{4}}\, 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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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