Optimal. Leaf size=56 \[ \frac {4 b \sqrt {a x^2+b x^5}}{9 a^2 x^{5/2}}-\frac {2 \sqrt {a x^2+b x^5}}{9 a x^{11/2}} \]
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Rubi [A] time = 0.08, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2016, 2014} \[ \frac {4 b \sqrt {a x^2+b x^5}}{9 a^2 x^{5/2}}-\frac {2 \sqrt {a x^2+b x^5}}{9 a x^{11/2}} \]
Antiderivative was successfully verified.
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Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int \frac {1}{x^{9/2} \sqrt {a x^2+b x^5}} \, dx &=-\frac {2 \sqrt {a x^2+b x^5}}{9 a x^{11/2}}-\frac {(2 b) \int \frac {1}{x^{3/2} \sqrt {a x^2+b x^5}} \, dx}{3 a}\\ &=-\frac {2 \sqrt {a x^2+b x^5}}{9 a x^{11/2}}+\frac {4 b \sqrt {a x^2+b x^5}}{9 a^2 x^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 35, normalized size = 0.62 \[ -\frac {2 \left (a-2 b x^3\right ) \sqrt {x^2 \left (a+b x^3\right )}}{9 a^2 x^{11/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 31, normalized size = 0.55 \[ \frac {2 \, \sqrt {b x^{5} + a x^{2}} {\left (2 \, b x^{3} - a\right )}}{9 \, a^{2} x^{\frac {11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 38, normalized size = 0.68 \[ -\frac {2 \, {\left (b + \frac {a}{x^{3}}\right )}^{\frac {3}{2}}}{9 \, a^{2}} + \frac {2 \, \sqrt {b + \frac {a}{x^{3}}} b}{3 \, a^{2}} - \frac {4 \, b^{\frac {3}{2}}}{9 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 37, normalized size = 0.66 \[ -\frac {2 \left (b \,x^{3}+a \right ) \left (-2 b \,x^{3}+a \right )}{9 \sqrt {b \,x^{5}+a \,x^{2}}\, a^{2} x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 38, normalized size = 0.68 \[ \frac {2 \, {\left (2 \, b^{2} x^{7} + a b x^{4} - a^{2} x\right )}}{9 \, \sqrt {b x^{3} + a} a^{2} x^{\frac {11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x^{9/2}\,\sqrt {b\,x^5+a\,x^2}} \,d x \]
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 \[ \int \frac {1}{x^{\frac {9}{2}} \sqrt {x^{2} \left (a + b x^{3}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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