Optimal. Leaf size=225 \[ -\frac {4096 b^7 \sqrt {a x+b x^{2/3}}}{2145 a^8 \sqrt [3]{x}}+\frac {2048 b^6 \sqrt {a x+b x^{2/3}}}{2145 a^7}-\frac {512 b^5 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{715 a^6}+\frac {256 b^4 x^{2/3} \sqrt {a x+b x^{2/3}}}{429 a^5}-\frac {224 b^3 x \sqrt {a x+b x^{2/3}}}{429 a^4}+\frac {336 b^2 x^{4/3} \sqrt {a x+b x^{2/3}}}{715 a^3}-\frac {28 b x^{5/3} \sqrt {a x+b x^{2/3}}}{65 a^2}+\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a} \]
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Rubi [A] time = 0.35, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2016, 2002, 2014} \[ -\frac {4096 b^7 \sqrt {a x+b x^{2/3}}}{2145 a^8 \sqrt [3]{x}}+\frac {2048 b^6 \sqrt {a x+b x^{2/3}}}{2145 a^7}-\frac {512 b^5 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{715 a^6}+\frac {256 b^4 x^{2/3} \sqrt {a x+b x^{2/3}}}{429 a^5}-\frac {224 b^3 x \sqrt {a x+b x^{2/3}}}{429 a^4}+\frac {336 b^2 x^{4/3} \sqrt {a x+b x^{2/3}}}{715 a^3}-\frac {28 b x^{5/3} \sqrt {a x+b x^{2/3}}}{65 a^2}+\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a} \]
Antiderivative was successfully verified.
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Rule 2002
Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {b x^{2/3}+a x}} \, dx &=\frac {2 x^2 \sqrt {b x^{2/3}+a x}}{5 a}-\frac {(14 b) \int \frac {x^{5/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{15 a}\\ &=-\frac {28 b x^{5/3} \sqrt {b x^{2/3}+a x}}{65 a^2}+\frac {2 x^2 \sqrt {b x^{2/3}+a x}}{5 a}+\frac {\left (56 b^2\right ) \int \frac {x^{4/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{65 a^2}\\ &=\frac {336 b^2 x^{4/3} \sqrt {b x^{2/3}+a x}}{715 a^3}-\frac {28 b x^{5/3} \sqrt {b x^{2/3}+a x}}{65 a^2}+\frac {2 x^2 \sqrt {b x^{2/3}+a x}}{5 a}-\frac {\left (112 b^3\right ) \int \frac {x}{\sqrt {b x^{2/3}+a x}} \, dx}{143 a^3}\\ &=-\frac {224 b^3 x \sqrt {b x^{2/3}+a x}}{429 a^4}+\frac {336 b^2 x^{4/3} \sqrt {b x^{2/3}+a x}}{715 a^3}-\frac {28 b x^{5/3} \sqrt {b x^{2/3}+a x}}{65 a^2}+\frac {2 x^2 \sqrt {b x^{2/3}+a x}}{5 a}+\frac {\left (896 b^4\right ) \int \frac {x^{2/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{1287 a^4}\\ &=\frac {256 b^4 x^{2/3} \sqrt {b x^{2/3}+a x}}{429 a^5}-\frac {224 b^3 x \sqrt {b x^{2/3}+a x}}{429 a^4}+\frac {336 b^2 x^{4/3} \sqrt {b x^{2/3}+a x}}{715 a^3}-\frac {28 b x^{5/3} \sqrt {b x^{2/3}+a x}}{65 a^2}+\frac {2 x^2 \sqrt {b x^{2/3}+a x}}{5 a}-\frac {\left (256 b^5\right ) \int \frac {\sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}} \, dx}{429 a^5}\\ &=-\frac {512 b^5 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{715 a^6}+\frac {256 b^4 x^{2/3} \sqrt {b x^{2/3}+a x}}{429 a^5}-\frac {224 b^3 x \sqrt {b x^{2/3}+a x}}{429 a^4}+\frac {336 b^2 x^{4/3} \sqrt {b x^{2/3}+a x}}{715 a^3}-\frac {28 b x^{5/3} \sqrt {b x^{2/3}+a x}}{65 a^2}+\frac {2 x^2 \sqrt {b x^{2/3}+a x}}{5 a}+\frac {\left (1024 b^6\right ) \int \frac {1}{\sqrt {b x^{2/3}+a x}} \, dx}{2145 a^6}\\ &=\frac {2048 b^6 \sqrt {b x^{2/3}+a x}}{2145 a^7}-\frac {512 b^5 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{715 a^6}+\frac {256 b^4 x^{2/3} \sqrt {b x^{2/3}+a x}}{429 a^5}-\frac {224 b^3 x \sqrt {b x^{2/3}+a x}}{429 a^4}+\frac {336 b^2 x^{4/3} \sqrt {b x^{2/3}+a x}}{715 a^3}-\frac {28 b x^{5/3} \sqrt {b x^{2/3}+a x}}{65 a^2}+\frac {2 x^2 \sqrt {b x^{2/3}+a x}}{5 a}-\frac {\left (2048 b^7\right ) \int \frac {1}{\sqrt [3]{x} \sqrt {b x^{2/3}+a x}} \, dx}{6435 a^7}\\ &=\frac {2048 b^6 \sqrt {b x^{2/3}+a x}}{2145 a^7}-\frac {4096 b^7 \sqrt {b x^{2/3}+a x}}{2145 a^8 \sqrt [3]{x}}-\frac {512 b^5 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{715 a^6}+\frac {256 b^4 x^{2/3} \sqrt {b x^{2/3}+a x}}{429 a^5}-\frac {224 b^3 x \sqrt {b x^{2/3}+a x}}{429 a^4}+\frac {336 b^2 x^{4/3} \sqrt {b x^{2/3}+a x}}{715 a^3}-\frac {28 b x^{5/3} \sqrt {b x^{2/3}+a x}}{65 a^2}+\frac {2 x^2 \sqrt {b x^{2/3}+a x}}{5 a}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 111, normalized size = 0.49 \[ \frac {2 \sqrt {a x+b x^{2/3}} \left (429 a^7 x^{7/3}-462 a^6 b x^2+504 a^5 b^2 x^{5/3}-560 a^4 b^3 x^{4/3}+640 a^3 b^4 x-768 a^2 b^5 x^{2/3}+1024 a b^6 \sqrt [3]{x}-2048 b^7\right )}{2145 a^8 \sqrt [3]{x}} \]
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 = 0.27, size = 122, normalized size = 0.54 \[ \frac {4096 \, b^{\frac {15}{2}}}{2145 \, a^{8}} + \frac {2 \, {\left (429 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} - 3465 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b + 12285 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{2} - 25025 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{3} + 32175 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{4} - 27027 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{5} + 15015 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{6} - 6435 \, \sqrt {a x^{\frac {1}{3}} + b} b^{7}\right )}}{2145 \, a^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 101, normalized size = 0.45 \[ \frac {2 \left (a \,x^{\frac {1}{3}}+b \right ) \left (429 a^{7} x^{\frac {7}{3}}-462 a^{6} b \,x^{2}+504 a^{5} b^{2} x^{\frac {5}{3}}-560 a^{4} b^{3} x^{\frac {4}{3}}+640 a^{3} b^{4} x -768 a^{2} b^{5} x^{\frac {2}{3}}+1024 a \,b^{6} x^{\frac {1}{3}}-2048 b^{7}\right ) x^{\frac {1}{3}}}{2145 \sqrt {a x +b \,x^{\frac {2}{3}}}\, a^{8}} \]
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 \[ \int \frac {x^{2}}{\sqrt {a x + b x^{\frac {2}{3}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2}{\sqrt {a\,x+b\,x^{2/3}}} \,d x \]
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 \[ \int \frac {x^{2}}{\sqrt {a x + b x^{\frac {2}{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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