Optimal. Leaf size=68 \[ -\frac {16 b \sqrt {a x+b x^{2/3}}}{a^3 \sqrt [3]{x}}+\frac {8 \sqrt {a x+b x^{2/3}}}{a^2}-\frac {6 x}{a \sqrt {a x+b x^{2/3}}} \]
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Rubi [A] time = 0.08, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2015, 2002, 2014} \begin {gather*} -\frac {16 b \sqrt {a x+b x^{2/3}}}{a^3 \sqrt [3]{x}}+\frac {8 \sqrt {a x+b x^{2/3}}}{a^2}-\frac {6 x}{a \sqrt {a x+b x^{2/3}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2002
Rule 2014
Rule 2015
Rubi steps
\begin {align*} \int \frac {x}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx &=-\frac {6 x}{a \sqrt {b x^{2/3}+a x}}+\frac {4 \int \frac {1}{\sqrt {b x^{2/3}+a x}} \, dx}{a}\\ &=-\frac {6 x}{a \sqrt {b x^{2/3}+a x}}+\frac {8 \sqrt {b x^{2/3}+a x}}{a^2}-\frac {(8 b) \int \frac {1}{\sqrt [3]{x} \sqrt {b x^{2/3}+a x}} \, dx}{3 a^2}\\ &=-\frac {6 x}{a \sqrt {b x^{2/3}+a x}}+\frac {8 \sqrt {b x^{2/3}+a x}}{a^2}-\frac {16 b \sqrt {b x^{2/3}+a x}}{a^3 \sqrt [3]{x}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 60, normalized size = 0.88 \begin {gather*} \frac {2 \left (a^2 x^{2/3}-4 a b \sqrt [3]{x}-8 b^2\right ) \sqrt {a x+b x^{2/3}}}{a^3 \sqrt [3]{x} \left (a \sqrt [3]{x}+b\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.97, size = 54, normalized size = 0.79 \begin {gather*} -\frac {2 \sqrt [3]{x} \left (a^2 \left (-x^{2/3}\right )+4 a b \sqrt [3]{x}+8 b^2\right )}{a^3 \sqrt {x^{2/3} \left (a \sqrt [3]{x}+b\right )}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 60, normalized size = 0.88 \begin {gather*} \frac {16 \, b^{\frac {3}{2}}}{a^{3}} - \frac {6 \, b^{2}}{\sqrt {a x^{\frac {1}{3}} + b} a^{3}} + \frac {2 \, {\left ({\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{6} - 6 \, \sqrt {a x^{\frac {1}{3}} + b} a^{6} b\right )}}{a^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 45, normalized size = 0.66 \begin {gather*} \frac {2 \left (a \,x^{\frac {1}{3}}+b \right ) \left (a^{2} x^{\frac {2}{3}}-4 a b \,x^{\frac {1}{3}}-8 b^{2}\right ) x}{\left (a x +b \,x^{\frac {2}{3}}\right )^{\frac {3}{2}} a^{3}} \end {gather*}
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 \begin {gather*} \int \frac {x}{{\left (a x + b x^{\frac {2}{3}}\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {x}{{\left (a\,x+b\,x^{2/3}\right )}^{3/2}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {x}{\left (a x + b x^{\frac {2}{3}}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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