Optimal. Leaf size=52 \[ \frac {2 x^2 \sqrt {a x^2+b x^5}}{9 b}-\frac {4 a \sqrt {a x^2+b x^5}}{9 b^2 x} \]
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Rubi [A] time = 0.06, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2016, 1588} \[ \frac {2 x^2 \sqrt {a x^2+b x^5}}{9 b}-\frac {4 a \sqrt {a x^2+b x^5}}{9 b^2 x} \]
Antiderivative was successfully verified.
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Rule 1588
Rule 2016
Rubi steps
\begin {align*} \int \frac {x^6}{\sqrt {a x^2+b x^5}} \, dx &=\frac {2 x^2 \sqrt {a x^2+b x^5}}{9 b}-\frac {(2 a) \int \frac {x^3}{\sqrt {a x^2+b x^5}} \, dx}{3 b}\\ &=-\frac {4 a \sqrt {a x^2+b x^5}}{9 b^2 x}+\frac {2 x^2 \sqrt {a x^2+b x^5}}{9 b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 34, normalized size = 0.65 \[ \frac {2 \left (b x^3-2 a\right ) \sqrt {x^2 \left (a+b x^3\right )}}{9 b^2 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 30, normalized size = 0.58 \[ \frac {2 \, \sqrt {b x^{5} + a x^{2}} {\left (b x^{3} - 2 \, a\right )}}{9 \, b^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{6}}{\sqrt {b x^{5} + a x^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 37, normalized size = 0.71 \[ -\frac {2 \left (b \,x^{3}+a \right ) \left (-b \,x^{3}+2 a \right ) x}{9 \sqrt {b \,x^{5}+a \,x^{2}}\, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.44, size = 34, normalized size = 0.65 \[ \frac {2 \, {\left (b^{2} x^{6} - a b x^{3} - 2 \, a^{2}\right )}}{9 \, \sqrt {b x^{3} + a} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.33, size = 33, normalized size = 0.63 \[ -\frac {\sqrt {b\,x^5+a\,x^2}\,\left (\frac {4\,a}{9\,b^2}-\frac {2\,x^3}{9\,b}\right )}{x} \]
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 \[ \int \frac {x^{6}}{\sqrt {x^{2} \left (a + b x^{3}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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