Optimal. Leaf size=53 \[ \frac{x \left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^p \, _2F_1\left (1,2 p+\frac{4}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{a} \]
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Rubi [A] time = 0.019247, antiderivative size = 55, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {1343, 246, 245} \[ x \left (\frac{b x^3}{a}+1\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p \, _2F_1\left (\frac{1}{3},-2 p;\frac{4}{3};-\frac{b x^3}{a}\right ) \]
Antiderivative was successfully verified.
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Rule 1343
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx &=\left (\left (2 a b+2 b^2 x^3\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p\right ) \int \left (2 a b+2 b^2 x^3\right )^{2 p} \, 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 \left (1+\frac{b x^3}{a}\right )^{2 p} \, dx\\ &=x \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{1}{3},-2 p;\frac{4}{3};-\frac{b x^3}{a}\right )\\ \end{align*}
Mathematica [C] time = 0.166889, size = 204, normalized size = 3.85 \[ \frac{4^{-p} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}\right )^{-2 p} \left (\frac{i \left (\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )}{\sqrt{3}+3 i}\right )^{-2 p} \left (\left (a+b x^3\right )^2\right )^p F_1\left (2 p+1;-2 p,-2 p;2 (p+1);-\frac{i \left (\sqrt [3]{b} x+(-1)^{2/3} \sqrt [3]{a}\right )}{\sqrt{3} \sqrt [3]{a}},\frac{-\frac{2 i \sqrt [3]{b} x}{\sqrt [3]{a}}+\sqrt{3}+i}{3 i+\sqrt{3}}\right )}{\sqrt [3]{b} (2 p+1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.013, size = 0, normalized size = 0. \begin{align*} \int \left ({b}^{2}{x}^{6}+2\,ab{x}^{3}+{a}^{2} \right ) ^{p}\, 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{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p}\,{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 ({\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p}, 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 \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{p}\, 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{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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