Optimal. Leaf size=126 \[ -\frac {128 \sqrt {a x+b x^3}}{35 a^5 x^{3/2}}+\frac {64}{35 a^4 \sqrt {x} \sqrt {a x+b x^3}}+\frac {16 \sqrt {x}}{35 a^3 \left (a x+b x^3\right )^{3/2}}+\frac {8 x^{3/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {x^{5/2}}{7 a \left (a x+b x^3\right )^{7/2}} \]
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Rubi [A] time = 0.19, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, 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 {8 x^{3/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {16 \sqrt {x}}{35 a^3 \left (a x+b x^3\right )^{3/2}}+\frac {64}{35 a^4 \sqrt {x} \sqrt {a x+b x^3}}-\frac {128 \sqrt {a x+b x^3}}{35 a^5 x^{3/2}}+\frac {x^{5/2}}{7 a \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^{5/2}}{\left (a x+b x^3\right )^{9/2}} \, dx &=\frac {x^{5/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {8 \int \frac {x^{3/2}}{\left (a x+b x^3\right )^{7/2}} \, dx}{7 a}\\ &=\frac {x^{5/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {8 x^{3/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {48 \int \frac {\sqrt {x}}{\left (a x+b x^3\right )^{5/2}} \, dx}{35 a^2}\\ &=\frac {x^{5/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {8 x^{3/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {16 \sqrt {x}}{35 a^3 \left (a x+b x^3\right )^{3/2}}+\frac {64 \int \frac {1}{\sqrt {x} \left (a x+b x^3\right )^{3/2}} \, dx}{35 a^3}\\ &=\frac {x^{5/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {8 x^{3/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {16 \sqrt {x}}{35 a^3 \left (a x+b x^3\right )^{3/2}}+\frac {64}{35 a^4 \sqrt {x} \sqrt {a x+b x^3}}+\frac {128 \int \frac {1}{x^{3/2} \sqrt {a x+b x^3}} \, dx}{35 a^4}\\ &=\frac {x^{5/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {8 x^{3/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {16 \sqrt {x}}{35 a^3 \left (a x+b x^3\right )^{3/2}}+\frac {64}{35 a^4 \sqrt {x} \sqrt {a x+b x^3}}-\frac {128 \sqrt {a x+b x^3}}{35 a^5 x^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 77, normalized size = 0.61 \begin {gather*} -\frac {\sqrt {x \left (a+b x^2\right )} \left (35 a^4+280 a^3 b x^2+560 a^2 b^2 x^4+448 a b^3 x^6+128 b^4 x^8\right )}{35 a^5 x^{3/2} \left (a+b x^2\right )^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.35, size = 68, normalized size = 0.54 \begin {gather*} -\frac {x^{5/2} \left (35 a^4+280 a^3 b x^2+560 a^2 b^2 x^4+448 a b^3 x^6+128 b^4 x^8\right )}{35 a^5 \left (a x+b x^3\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 110, normalized size = 0.87 \begin {gather*} -\frac {{\left (128 \, b^{4} x^{8} + 448 \, a b^{3} x^{6} + 560 \, a^{2} b^{2} x^{4} + 280 \, a^{3} b x^{2} + 35 \, a^{4}\right )} \sqrt {b x^{3} + a x} \sqrt {x}}{35 \, {\left (a^{5} b^{4} x^{10} + 4 \, a^{6} b^{3} x^{8} + 6 \, a^{7} b^{2} x^{6} + 4 \, a^{8} b x^{4} + a^{9} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 90, normalized size = 0.71 \begin {gather*} -\frac {{\left ({\left (x^{2} {\left (\frac {93 \, b^{4} x^{2}}{a^{5}} + \frac {308 \, b^{3}}{a^{4}}\right )} + \frac {350 \, b^{2}}{a^{3}}\right )} x^{2} + \frac {140 \, b}{a^{2}}\right )} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {2 \, \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )} a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 70, normalized size = 0.56 \begin {gather*} -\frac {\left (b \,x^{2}+a \right ) \left (128 x^{8} b^{4}+448 a \,x^{6} b^{3}+560 a^{2} x^{4} b^{2}+280 a^{3} x^{2} b +35 a^{4}\right ) x^{\frac {7}{2}}}{35 \left (b \,x^{3}+a x \right )^{\frac {9}{2}} a^{5}} \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 {5}{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 = -0.01 \begin {gather*} \int \frac {x^{5/2}}{{\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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{\frac {5}{2}}}{\left (x \left (a + b x^{2}\right )\right )^{\frac {9}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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