Optimal. Leaf size=137 \[ \frac {\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^5}{2 b^3}-\frac {6 a \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^4}{5 b^3}+\frac {3 a^2 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^3}{4 b^3} \]
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Rubi [A] time = 0.06, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1341, 645} \[ \frac {\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^5}{2 b^3}-\frac {6 a \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^4}{5 b^3}+\frac {3 a^2 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^3}{4 b^3} \]
Antiderivative was successfully verified.
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Rule 645
Rule 1341
Rubi steps
\begin {align*} \int \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{3/2} \, dx &=3 \operatorname {Subst}\left (\int x^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {\left (3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}\right ) \operatorname {Subst}\left (\int \left (\frac {a^2 \left (a b+b^2 x\right )^3}{b^2}-\frac {2 a \left (a b+b^2 x\right )^4}{b^3}+\frac {\left (a b+b^2 x\right )^5}{b^4}\right ) \, dx,x,\sqrt [3]{x}\right )}{b^3 \left (a+b \sqrt [3]{x}\right )}\\ &=\frac {3 a^2 \left (a+b \sqrt [3]{x}\right )^3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}{4 b^3}-\frac {6 a \left (a+b \sqrt [3]{x}\right )^4 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}{5 b^3}+\frac {\left (a+b \sqrt [3]{x}\right )^5 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}{2 b^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 65, normalized size = 0.47 \[ \frac {x \sqrt {\left (a+b \sqrt [3]{x}\right )^2} \left (20 a^3+45 a^2 b \sqrt [3]{x}+36 a b^2 x^{2/3}+10 b^3 x\right )}{20 \left (a+b \sqrt [3]{x}\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.39, size = 32, normalized size = 0.23 \[ \frac {1}{2} \, b^{3} x^{2} + \frac {9}{5} \, a b^{2} x^{\frac {5}{3}} + \frac {9}{4} \, a^{2} b x^{\frac {4}{3}} + a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 64, normalized size = 0.47 \[ \frac {1}{2} \, b^{3} x^{2} \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right ) + \frac {9}{5} \, a b^{2} x^{\frac {5}{3}} \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right ) + \frac {9}{4} \, a^{2} b x^{\frac {4}{3}} \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right ) + a^{3} x \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 65, normalized size = 0.47 \[ \frac {\sqrt {b^{2} x^{\frac {2}{3}}+2 a b \,x^{\frac {1}{3}}+a^{2}}\, \left (10 b^{3} x^{2}+36 a \,b^{2} x^{\frac {5}{3}}+45 a^{2} b \,x^{\frac {4}{3}}+20 a^{3} x \right )}{20 b \,x^{\frac {1}{3}}+20 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 114, normalized size = 0.83 \[ \frac {3 \, {\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{\frac {3}{2}} a^{2} x^{\frac {1}{3}}}{4 \, b^{2}} + \frac {3 \, {\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{\frac {3}{2}} a^{3}}{4 \, b^{3}} + \frac {{\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{\frac {5}{2}} x^{\frac {1}{3}}}{2 \, b^{2}} - \frac {7 \, {\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{\frac {5}{2}} a}{10 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a^2+b^2\,x^{2/3}+2\,a\,b\,x^{1/3}\right )}^{3/2} \,d x \]
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 \[ \int \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac {2}{3}}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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