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 = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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 245
Rule 246
Rule 1343
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*}
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Mathematica [C] time = 0.17, 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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fricas [F] time = 1.10, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int \left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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