Optimal. Leaf size=169 \[ -\frac {512 b^5 \left (a x+b x^{2/3}\right )^{5/2}}{15015 a^6 x^{5/3}}+\frac {256 b^4 \left (a x+b x^{2/3}\right )^{5/2}}{3003 a^5 x^{4/3}}-\frac {64 b^3 \left (a x+b x^{2/3}\right )^{5/2}}{429 a^4 x}+\frac {32 b^2 \left (a x+b x^{2/3}\right )^{5/2}}{143 a^3 x^{2/3}}-\frac {4 b \left (a x+b x^{2/3}\right )^{5/2}}{13 a^2 \sqrt [3]{x}}+\frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{5 a} \]
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Rubi [A] time = 0.25, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2002, 2016, 2014} \[ -\frac {512 b^5 \left (a x+b x^{2/3}\right )^{5/2}}{15015 a^6 x^{5/3}}+\frac {256 b^4 \left (a x+b x^{2/3}\right )^{5/2}}{3003 a^5 x^{4/3}}-\frac {64 b^3 \left (a x+b x^{2/3}\right )^{5/2}}{429 a^4 x}+\frac {32 b^2 \left (a x+b x^{2/3}\right )^{5/2}}{143 a^3 x^{2/3}}-\frac {4 b \left (a x+b x^{2/3}\right )^{5/2}}{13 a^2 \sqrt [3]{x}}+\frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{5 a} \]
Antiderivative was successfully verified.
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Rule 2002
Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int \left (b x^{2/3}+a x\right )^{3/2} \, dx &=\frac {2 \left (b x^{2/3}+a x\right )^{5/2}}{5 a}-\frac {(2 b) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{\sqrt [3]{x}} \, dx}{3 a}\\ &=\frac {2 \left (b x^{2/3}+a x\right )^{5/2}}{5 a}-\frac {4 b \left (b x^{2/3}+a x\right )^{5/2}}{13 a^2 \sqrt [3]{x}}+\frac {\left (16 b^2\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^{2/3}} \, dx}{39 a^2}\\ &=\frac {2 \left (b x^{2/3}+a x\right )^{5/2}}{5 a}+\frac {32 b^2 \left (b x^{2/3}+a x\right )^{5/2}}{143 a^3 x^{2/3}}-\frac {4 b \left (b x^{2/3}+a x\right )^{5/2}}{13 a^2 \sqrt [3]{x}}-\frac {\left (32 b^3\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x} \, dx}{143 a^3}\\ &=\frac {2 \left (b x^{2/3}+a x\right )^{5/2}}{5 a}-\frac {64 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{429 a^4 x}+\frac {32 b^2 \left (b x^{2/3}+a x\right )^{5/2}}{143 a^3 x^{2/3}}-\frac {4 b \left (b x^{2/3}+a x\right )^{5/2}}{13 a^2 \sqrt [3]{x}}+\frac {\left (128 b^4\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^{4/3}} \, dx}{1287 a^4}\\ &=\frac {2 \left (b x^{2/3}+a x\right )^{5/2}}{5 a}+\frac {256 b^4 \left (b x^{2/3}+a x\right )^{5/2}}{3003 a^5 x^{4/3}}-\frac {64 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{429 a^4 x}+\frac {32 b^2 \left (b x^{2/3}+a x\right )^{5/2}}{143 a^3 x^{2/3}}-\frac {4 b \left (b x^{2/3}+a x\right )^{5/2}}{13 a^2 \sqrt [3]{x}}-\frac {\left (256 b^5\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^{5/3}} \, dx}{9009 a^5}\\ &=\frac {2 \left (b x^{2/3}+a x\right )^{5/2}}{5 a}-\frac {512 b^5 \left (b x^{2/3}+a x\right )^{5/2}}{15015 a^6 x^{5/3}}+\frac {256 b^4 \left (b x^{2/3}+a x\right )^{5/2}}{3003 a^5 x^{4/3}}-\frac {64 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{429 a^4 x}+\frac {32 b^2 \left (b x^{2/3}+a x\right )^{5/2}}{143 a^3 x^{2/3}}-\frac {4 b \left (b x^{2/3}+a x\right )^{5/2}}{13 a^2 \sqrt [3]{x}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 98, normalized size = 0.58 \[ \frac {2 \left (a \sqrt [3]{x}+b\right )^2 \sqrt {a x+b x^{2/3}} \left (3003 a^5 x^{5/3}-2310 a^4 b x^{4/3}+1680 a^3 b^2 x-1120 a^2 b^3 x^{2/3}+640 a b^4 \sqrt [3]{x}-256 b^5\right )}{15015 a^6 \sqrt [3]{x}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 434, normalized size = 2.57 \[ \frac {2}{3003} \, b {\left (\frac {256 \, b^{\frac {13}{2}}}{a^{6}} + \frac {\frac {13 \, {\left (63 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} - 385 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b + 990 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{2} - 1386 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{3} + 1155 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{4} - 693 \, \sqrt {a x^{\frac {1}{3}} + b} b^{5}\right )} b}{a^{5}} + \frac {3 \, {\left (231 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} - 1638 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b + 5005 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{2} - 8580 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{3} + 9009 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{4} - 6006 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{5} + 3003 \, \sqrt {a x^{\frac {1}{3}} + b} b^{6}\right )}}{a^{5}}}{a}\right )} - \frac {2}{15015} \, a {\left (\frac {1024 \, b^{\frac {15}{2}}}{a^{7}} - \frac {\frac {15 \, {\left (231 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} - 1638 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b + 5005 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{2} - 8580 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{3} + 9009 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{4} - 6006 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{5} + 3003 \, \sqrt {a x^{\frac {1}{3}} + b} b^{6}\right )} b}{a^{6}} + \frac {7 \, {\left (429 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} - 3465 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b + 12285 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{2} - 25025 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{3} + 32175 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{4} - 27027 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{5} + 15015 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{6} - 6435 \, \sqrt {a x^{\frac {1}{3}} + b} b^{7}\right )}}{a^{6}}}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 79, normalized size = 0.47 \[ \frac {2 \left (a x +b \,x^{\frac {2}{3}}\right )^{\frac {3}{2}} \left (a \,x^{\frac {1}{3}}+b \right ) \left (3003 a^{5} x^{\frac {5}{3}}-2310 a^{4} b \,x^{\frac {4}{3}}+1680 a^{3} b^{2} x -1120 a^{2} b^{3} x^{\frac {2}{3}}+640 a \,b^{4} x^{\frac {1}{3}}-256 b^{5}\right )}{15015 a^{6} x} \]
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 \[ \int {\left (a x + b x^{\frac {2}{3}}\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.14, size = 40, normalized size = 0.24 \[ \frac {x\,{\left (a\,x+b\,x^{2/3}\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},6;\ 7;\ -\frac {a\,x^{1/3}}{b}\right )}{2\,{\left (\frac {a\,x^{1/3}}{b}+1\right )}^{3/2}} \]
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 \[ \int \left (a x + b x^{\frac {2}{3}}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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