Optimal. Leaf size=51 \[ -\frac {2 x^{5/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac {x^{11/2}}{7 b \left (a x+b x^3\right )^{7/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 51, 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 = {2015, 2014} \begin {gather*} -\frac {2 x^{5/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac {x^{11/2}}{7 b \left (a x+b x^3\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2014
Rule 2015
Rubi steps
\begin {align*} \int \frac {x^{15/2}}{\left (a x+b x^3\right )^{9/2}} \, dx &=-\frac {x^{11/2}}{7 b \left (a x+b x^3\right )^{7/2}}+\frac {2 \int \frac {x^{9/2}}{\left (a x+b x^3\right )^{7/2}} \, dx}{7 b}\\ &=-\frac {x^{11/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac {2 x^{5/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 0.86 \begin {gather*} -\frac {\sqrt {x} \left (2 a+7 b x^2\right )}{35 b^2 \left (a+b x^2\right )^3 \sqrt {x \left (a+b x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.63, size = 35, normalized size = 0.69 \begin {gather*} -\frac {x^{7/2} \left (2 a+7 b x^2\right )}{35 b^2 \left (a x+b x^3\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 75, normalized size = 1.47 \begin {gather*} -\frac {\sqrt {b x^{3} + a x} {\left (7 \, b x^{2} + 2 \, a\right )} \sqrt {x}}{35 \, {\left (b^{6} x^{9} + 4 \, a b^{5} x^{7} + 6 \, a^{2} b^{4} x^{5} + 4 \, a^{3} b^{3} x^{3} + a^{4} b^{2} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 33, normalized size = 0.65 \begin {gather*} -\frac {7 \, b x^{2} + 2 \, a}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} + \frac {2}{35 \, a^{\frac {5}{2}} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 37, normalized size = 0.73 \begin {gather*} -\frac {\left (b \,x^{2}+a \right ) \left (7 b \,x^{2}+2 a \right ) x^{\frac {9}{2}}}{35 \left (b \,x^{3}+a x \right )^{\frac {9}{2}} b^{2}} \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^{\frac {15}{2}}}{{\left (b x^{3} + a x\right )}^{\frac {9}{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 = -0.02 \begin {gather*} \int \frac {x^{15/2}}{{\left (b\,x^3+a\,x\right )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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