Optimal. Leaf size=137 \[ \frac {3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^7}{8 b^3}-\frac {6 a \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^6}{7 b^3}+\frac {a^2 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^5}{2 b^3} \]
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Rubi [A] time = 0.07, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1341, 646, 43} \[ \frac {3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^7}{8 b^3}-\frac {6 a \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^6}{7 b^3}+\frac {a^2 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^5}{2 b^3} \]
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 )^{5/2} \, dx &=3 \operatorname {Subst}\left (\int x^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, 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 x^2 \left (a b+b^2 x\right )^5 \, dx,x,\sqrt [3]{x}\right )}{b^5 \left (a+b \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 )^5}{b^2}-\frac {2 a \left (a b+b^2 x\right )^6}{b^3}+\frac {\left (a b+b^2 x\right )^7}{b^4}\right ) \, dx,x,\sqrt [3]{x}\right )}{b^5 \left (a+b \sqrt [3]{x}\right )}\\ &=\frac {a^2 \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}-\frac {6 a \left (a+b \sqrt [3]{x}\right )^6 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}{7 b^3}+\frac {3 \left (a+b \sqrt [3]{x}\right )^7 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}{8 b^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 56, normalized size = 0.41 \[ \frac {\left (a+b \sqrt [3]{x}\right )^5 \sqrt {\left (a+b \sqrt [3]{x}\right )^2} \left (a^2-6 a b \sqrt [3]{x}+21 b^2 x^{2/3}\right )}{56 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 61, normalized size = 0.45 \[ 5 \, a^{2} b^{3} x^{2} + a^{5} x + \frac {3}{8} \, {\left (b^{5} x^{2} + 16 \, a^{3} b^{2} x\right )} x^{\frac {2}{3}} + \frac {15}{28} \, {\left (4 \, a b^{4} x^{2} + 7 \, a^{4} b x\right )} x^{\frac {1}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.53, size = 102, normalized size = 0.74 \[ \frac {3}{8} \, b^{5} x^{\frac {8}{3}} \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right ) + \frac {15}{7} \, a b^{4} x^{\frac {7}{3}} \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right ) + 5 \, a^{2} b^{3} x^{2} \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right ) + 6 \, a^{3} b^{2} x^{\frac {5}{3}} \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right ) + \frac {15}{4} \, a^{4} b x^{\frac {4}{3}} \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right ) + a^{5} x \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 87, normalized size = 0.64 \[ \frac {\sqrt {b^{2} x^{\frac {2}{3}}+2 a b \,x^{\frac {1}{3}}+a^{2}}\, \left (21 b^{5} x^{\frac {8}{3}}+120 a \,b^{4} x^{\frac {7}{3}}+280 a^{2} b^{3} x^{2}+336 a^{3} b^{2} x^{\frac {5}{3}}+210 a^{4} b \,x^{\frac {4}{3}}+56 a^{5} x \right )}{56 b \,x^{\frac {1}{3}}+56 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 114, normalized size = 0.83 \[ \frac {{\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{\frac {5}{2}} a^{2} x^{\frac {1}{3}}}{2 \, b^{2}} + \frac {{\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{\frac {5}{2}} a^{3}}{2 \, b^{3}} + \frac {3 \, {\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{\frac {7}{2}} x^{\frac {1}{3}}}{8 \, b^{2}} - \frac {27 \, {\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{\frac {7}{2}} a}{56 \, b^{3}} \]
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 \[ \int {\left (a^2+b^2\,x^{2/3}+2\,a\,b\,x^{1/3}\right )}^{5/2} \,d x \]
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 \[ \int \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac {2}{3}}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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