Optimal. Leaf size=101 \[ \frac {16 x^{3/2}}{35 a^4 \sqrt {a x+b x^3}}+\frac {8 x^{5/2}}{35 a^3 \left (a x+b x^3\right )^{3/2}}+\frac {6 x^{7/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {x^{9/2}}{7 a \left (a x+b x^3\right )^{7/2}} \]
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Rubi [A] time = 0.16, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2015, 2014} \[ \frac {6 x^{7/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {8 x^{5/2}}{35 a^3 \left (a x+b x^3\right )^{3/2}}+\frac {16 x^{3/2}}{35 a^4 \sqrt {a x+b x^3}}+\frac {x^{9/2}}{7 a \left (a x+b x^3\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2014
Rule 2015
Rubi steps
\begin {align*} \int \frac {x^{9/2}}{\left (a x+b x^3\right )^{9/2}} \, dx &=\frac {x^{9/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {6 \int \frac {x^{7/2}}{\left (a x+b x^3\right )^{7/2}} \, dx}{7 a}\\ &=\frac {x^{9/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {6 x^{7/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {24 \int \frac {x^{5/2}}{\left (a x+b x^3\right )^{5/2}} \, dx}{35 a^2}\\ &=\frac {x^{9/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {6 x^{7/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {8 x^{5/2}}{35 a^3 \left (a x+b x^3\right )^{3/2}}+\frac {16 \int \frac {x^{3/2}}{\left (a x+b x^3\right )^{3/2}} \, dx}{35 a^3}\\ &=\frac {x^{9/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {6 x^{7/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {8 x^{5/2}}{35 a^3 \left (a x+b x^3\right )^{3/2}}+\frac {16 x^{3/2}}{35 a^4 \sqrt {a x+b x^3}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 66, normalized size = 0.65 \[ \frac {\sqrt {x} \sqrt {x \left (a+b x^2\right )} \left (35 a^3+70 a^2 b x^2+56 a b^2 x^4+16 b^3 x^6\right )}{35 a^4 \left (a+b x^2\right )^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 95, normalized size = 0.94 \[ \frac {{\left (16 \, b^{3} x^{6} + 56 \, a b^{2} x^{4} + 70 \, a^{2} b x^{2} + 35 \, a^{3}\right )} \sqrt {b x^{3} + a x} \sqrt {x}}{35 \, {\left (a^{4} b^{4} x^{8} + 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 55, normalized size = 0.54 \[ \frac {{\left (2 \, {\left (4 \, x^{2} {\left (\frac {2 \, b^{3} x^{2}}{a^{4}} + \frac {7 \, b^{2}}{a^{3}}\right )} + \frac {35 \, b}{a^{2}}\right )} x^{2} + \frac {35}{a}\right )} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 59, normalized size = 0.58 \[ \frac {\left (b \,x^{2}+a \right ) \left (16 b^{3} x^{6}+56 a \,b^{2} x^{4}+70 a^{2} b \,x^{2}+35 a^{3}\right ) x^{\frac {11}{2}}}{35 \left (b \,x^{3}+a x \right )^{\frac {9}{2}} a^{4}} \]
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 \[ \int \frac {x^{\frac {9}{2}}}{{\left (b x^{3} + a x\right )}^{\frac {9}{2}}}\,{d x} \]
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 \[ \int \frac {x^{9/2}}{{\left (b\,x^3+a\,x\right )}^{9/2}} \,d x \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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