Optimal. Leaf size=195 \[ \frac {2048 b^6 \left (a x+b x^{2/3}\right )^{3/2}}{15015 a^7 x}-\frac {1024 b^5 \left (a x+b x^{2/3}\right )^{3/2}}{5005 a^6 x^{2/3}}+\frac {256 b^4 \left (a x+b x^{2/3}\right )^{3/2}}{1001 a^5 \sqrt [3]{x}}-\frac {128 b^3 \left (a x+b x^{2/3}\right )^{3/2}}{429 a^4}+\frac {48 b^2 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{3/2}}{143 a^3}-\frac {24 b x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{65 a^2}+\frac {2 x \left (a x+b x^{2/3}\right )^{3/2}}{5 a} \]
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Rubi [A] time = 0.27, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2016, 2002, 2014} \[ \frac {2048 b^6 \left (a x+b x^{2/3}\right )^{3/2}}{15015 a^7 x}-\frac {1024 b^5 \left (a x+b x^{2/3}\right )^{3/2}}{5005 a^6 x^{2/3}}+\frac {256 b^4 \left (a x+b x^{2/3}\right )^{3/2}}{1001 a^5 \sqrt [3]{x}}-\frac {128 b^3 \left (a x+b x^{2/3}\right )^{3/2}}{429 a^4}+\frac {48 b^2 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{3/2}}{143 a^3}-\frac {24 b x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{65 a^2}+\frac {2 x \left (a x+b x^{2/3}\right )^{3/2}}{5 a} \]
Antiderivative was successfully verified.
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Rule 2002
Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int x \sqrt {b x^{2/3}+a x} \, dx &=\frac {2 x \left (b x^{2/3}+a x\right )^{3/2}}{5 a}-\frac {(4 b) \int x^{2/3} \sqrt {b x^{2/3}+a x} \, dx}{5 a}\\ &=-\frac {24 b x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{65 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{3/2}}{5 a}+\frac {\left (8 b^2\right ) \int \sqrt [3]{x} \sqrt {b x^{2/3}+a x} \, dx}{13 a^2}\\ &=\frac {48 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{143 a^3}-\frac {24 b x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{65 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{3/2}}{5 a}-\frac {\left (64 b^3\right ) \int \sqrt {b x^{2/3}+a x} \, dx}{143 a^3}\\ &=-\frac {128 b^3 \left (b x^{2/3}+a x\right )^{3/2}}{429 a^4}+\frac {48 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{143 a^3}-\frac {24 b x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{65 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{3/2}}{5 a}+\frac {\left (128 b^4\right ) \int \frac {\sqrt {b x^{2/3}+a x}}{\sqrt [3]{x}} \, dx}{429 a^4}\\ &=-\frac {128 b^3 \left (b x^{2/3}+a x\right )^{3/2}}{429 a^4}+\frac {256 b^4 \left (b x^{2/3}+a x\right )^{3/2}}{1001 a^5 \sqrt [3]{x}}+\frac {48 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{143 a^3}-\frac {24 b x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{65 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{3/2}}{5 a}-\frac {\left (512 b^5\right ) \int \frac {\sqrt {b x^{2/3}+a x}}{x^{2/3}} \, dx}{3003 a^5}\\ &=-\frac {128 b^3 \left (b x^{2/3}+a x\right )^{3/2}}{429 a^4}-\frac {1024 b^5 \left (b x^{2/3}+a x\right )^{3/2}}{5005 a^6 x^{2/3}}+\frac {256 b^4 \left (b x^{2/3}+a x\right )^{3/2}}{1001 a^5 \sqrt [3]{x}}+\frac {48 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{143 a^3}-\frac {24 b x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{65 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{3/2}}{5 a}+\frac {\left (1024 b^6\right ) \int \frac {\sqrt {b x^{2/3}+a x}}{x} \, dx}{15015 a^6}\\ &=-\frac {128 b^3 \left (b x^{2/3}+a x\right )^{3/2}}{429 a^4}+\frac {2048 b^6 \left (b x^{2/3}+a x\right )^{3/2}}{15015 a^7 x}-\frac {1024 b^5 \left (b x^{2/3}+a x\right )^{3/2}}{5005 a^6 x^{2/3}}+\frac {256 b^4 \left (b x^{2/3}+a x\right )^{3/2}}{1001 a^5 \sqrt [3]{x}}+\frac {48 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{143 a^3}-\frac {24 b x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{65 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{3/2}}{5 a}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 107, normalized size = 0.55 \[ \frac {2 \left (a \sqrt [3]{x}+b\right ) \sqrt {a x+b x^{2/3}} \left (3003 a^6 x^2-2772 a^5 b x^{5/3}+2520 a^4 b^2 x^{4/3}-2240 a^3 b^3 x+1920 a^2 b^4 x^{2/3}-1536 a b^5 \sqrt [3]{x}+1024 b^6\right )}{15015 a^7 \sqrt [3]{x}} \]
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 [A] time = 0.19, size = 228, normalized size = 1.17 \[ -\frac {2048 \, b^{\frac {15}{2}}}{15015 \, a^{7}} + \frac {2 \, {\left (\frac {15 \, {\left (231 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} - 1638 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b + 5005 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{2} - 8580 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{3} + 9009 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{4} - 6006 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{5} + 3003 \, \sqrt {a x^{\frac {1}{3}} + b} b^{6}\right )} b}{a^{6}} + \frac {7 \, {\left (429 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} - 3465 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b + 12285 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{2} - 25025 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{3} + 32175 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{4} - 27027 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{5} + 15015 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{6} - 6435 \, \sqrt {a x^{\frac {1}{3}} + b} b^{7}\right )}}{a^{6}}\right )}}{15015 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 90, normalized size = 0.46 \[ -\frac {2 \sqrt {a x +b \,x^{\frac {2}{3}}}\, \left (a \,x^{\frac {1}{3}}+b \right ) \left (-3003 a^{6} x^{2}+2772 a^{5} b \,x^{\frac {5}{3}}-2520 a^{4} b^{2} x^{\frac {4}{3}}+2240 a^{3} b^{3} x -1920 a^{2} b^{4} x^{\frac {2}{3}}+1536 a \,b^{5} x^{\frac {1}{3}}-1024 b^{6}\right )}{15015 a^{7} x^{\frac {1}{3}}} \]
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 \[ \int \sqrt {a x + b x^{\frac {2}{3}}} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\sqrt {a\,x+b\,x^{2/3}} \,d x \]
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 \[ \int x \sqrt {a x + b x^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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