Optimal. Leaf size=142 \[ \frac {3 \left (a+b \sqrt [3]{x}\right )^3 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^3 (2 p+3)}-\frac {3 a \left (a+b \sqrt [3]{x}\right )^2 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^3 (p+1)}+\frac {3 a^2 \left (a+b \sqrt [3]{x}\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^3 (2 p+1)} \]
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Rubi [A] time = 0.07, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1341, 646, 43} \[ \frac {3 \left (a+b \sqrt [3]{x}\right )^3 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^3 (2 p+3)}-\frac {3 a \left (a+b \sqrt [3]{x}\right )^2 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^3 (p+1)}+\frac {3 a^2 \left (a+b \sqrt [3]{x}\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^3 (2 p+1)} \]
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rule 1341
Rubi steps
\begin {align*} \int \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p \, dx &=3 \operatorname {Subst}\left (\int x^2 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx,x,\sqrt [3]{x}\right )\\ &=\left (3 \left (b \left (a+b \sqrt [3]{x}\right )\right )^{-2 p} \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p\right ) \operatorname {Subst}\left (\int x^2 \left (a b+b^2 x\right )^{2 p} \, dx,x,\sqrt [3]{x}\right )\\ &=\left (3 \left (b \left (a+b \sqrt [3]{x}\right )\right )^{-2 p} \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p\right ) \operatorname {Subst}\left (\int \left (\frac {a^2 \left (a b+b^2 x\right )^{2 p}}{b^2}-\frac {2 a \left (a b+b^2 x\right )^{1+2 p}}{b^3}+\frac {\left (a b+b^2 x\right )^{2+2 p}}{b^4}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3 a^2 \left (a+b \sqrt [3]{x}\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^3 (1+2 p)}-\frac {3 a \left (a+b \sqrt [3]{x}\right )^2 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^3 (1+p)}+\frac {3 \left (a+b \sqrt [3]{x}\right )^3 \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{b^3 (3+2 p)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 83, normalized size = 0.58 \[ \frac {3 \left (a+b \sqrt [3]{x}\right ) \left (\left (a+b \sqrt [3]{x}\right )^2\right )^p \left (a^2-a b (2 p+1) \sqrt [3]{x}+b^2 \left (2 p^2+3 p+1\right ) x^{2/3}\right )}{b^3 (p+1) (2 p+1) (2 p+3)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.11, size = 110, normalized size = 0.77 \[ -\frac {3 \, {\left (2 \, a^{2} b p x^{\frac {1}{3}} - a^{3} - {\left (2 \, b^{3} p^{2} + 3 \, b^{3} p + b^{3}\right )} x - {\left (2 \, a b^{2} p^{2} + a b^{2} p\right )} x^{\frac {2}{3}}\right )} {\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{p}}{4 \, b^{3} p^{3} + 12 \, b^{3} p^{2} + 11 \, b^{3} p + 3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.48, size = 229, normalized size = 1.61 \[ \frac {3 \, {\left (2 \, {\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{p} b^{3} p^{2} x + 2 \, {\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{p} a b^{2} p^{2} x^{\frac {2}{3}} + 3 \, {\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{p} b^{3} p x + {\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{p} a b^{2} p x^{\frac {2}{3}} - 2 \, {\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{p} a^{2} b p x^{\frac {1}{3}} + {\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{p} b^{3} x + {\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{p} a^{3}\right )}}{4 \, b^{3} p^{3} + 12 \, b^{3} p^{2} + 11 \, b^{3} p + 3 \, b^{3}} \]
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 \[ \int \left (b^{2} x^{\frac {2}{3}}+2 a b \,x^{\frac {1}{3}}+a^{2}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.80, size = 77, normalized size = 0.54 \[ \frac {3 \, {\left ({\left (2 \, p^{2} + 3 \, p + 1\right )} b^{3} x + {\left (2 \, p^{2} + p\right )} a b^{2} x^{\frac {2}{3}} - 2 \, a^{2} b p x^{\frac {1}{3}} + a^{3}\right )} {\left (b x^{\frac {1}{3}} + a\right )}^{2 \, p}}{{\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.54, size = 138, normalized size = 0.97 \[ {\left (a^2+b^2\,x^{2/3}+2\,a\,b\,x^{1/3}\right )}^p\,\left (\frac {3\,x\,\left (2\,p^2+3\,p+1\right )}{4\,p^3+12\,p^2+11\,p+3}+\frac {3\,a^3}{b^3\,\left (4\,p^3+12\,p^2+11\,p+3\right )}-\frac {6\,a^2\,p\,x^{1/3}}{b^2\,\left (4\,p^3+12\,p^2+11\,p+3\right )}+\frac {3\,a\,p\,x^{2/3}\,\left (2\,p+1\right )}{b\,\left (4\,p^3+12\,p^2+11\,p+3\right )}\right ) \]
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 \[ \int \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac {2}{3}}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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