Optimal. Leaf size=101 \[ -\frac {16 \sqrt {x}}{35 b^4 \sqrt {a x+b x^3}}-\frac {8 x^{7/2}}{35 b^3 \left (a x+b x^3\right )^{3/2}}-\frac {6 x^{13/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac {x^{19/2}}{7 b \left (a x+b x^3\right )^{7/2}} \]
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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} \[ -\frac {6 x^{13/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac {8 x^{7/2}}{35 b^3 \left (a x+b x^3\right )^{3/2}}-\frac {16 \sqrt {x}}{35 b^4 \sqrt {a x+b x^3}}-\frac {x^{19/2}}{7 b \left (a x+b x^3\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2014
Rule 2015
Rubi steps
\begin {align*} \int \frac {x^{23/2}}{\left (a x+b x^3\right )^{9/2}} \, dx &=-\frac {x^{19/2}}{7 b \left (a x+b x^3\right )^{7/2}}+\frac {6 \int \frac {x^{17/2}}{\left (a x+b x^3\right )^{7/2}} \, dx}{7 b}\\ &=-\frac {x^{19/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac {6 x^{13/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}+\frac {24 \int \frac {x^{11/2}}{\left (a x+b x^3\right )^{5/2}} \, dx}{35 b^2}\\ &=-\frac {x^{19/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac {6 x^{13/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac {8 x^{7/2}}{35 b^3 \left (a x+b x^3\right )^{3/2}}+\frac {16 \int \frac {x^{5/2}}{\left (a x+b x^3\right )^{3/2}} \, dx}{35 b^3}\\ &=-\frac {x^{19/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac {6 x^{13/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac {8 x^{7/2}}{35 b^3 \left (a x+b x^3\right )^{3/2}}-\frac {16 \sqrt {x}}{35 b^4 \sqrt {a x+b x^3}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 66, normalized size = 0.65 \[ -\frac {\sqrt {x} \left (16 a^3+56 a^2 b x^2+70 a b^2 x^4+35 b^3 x^6\right )}{35 b^4 \left (a+b x^2\right )^3 \sqrt {x \left (a+b x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 97, normalized size = 0.96 \[ -\frac {{\left (35 \, b^{3} x^{6} + 70 \, a b^{2} x^{4} + 56 \, a^{2} b x^{2} + 16 \, a^{3}\right )} \sqrt {b x^{3} + a x} \sqrt {x}}{35 \, {\left (b^{8} x^{9} + 4 \, a b^{7} x^{7} + 6 \, a^{2} b^{6} x^{5} + 4 \, a^{3} b^{5} x^{3} + a^{4} b^{4} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 64, normalized size = 0.63 \[ \frac {16}{35 \, \sqrt {a} b^{4}} - \frac {35 \, {\left (b x^{2} + a\right )}^{3} - 35 \, {\left (b x^{2} + a\right )}^{2} a + 21 \, {\left (b x^{2} + a\right )} a^{2} - 5 \, a^{3}}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 59, normalized size = 0.58 \[ -\frac {\left (b \,x^{2}+a \right ) \left (35 b^{3} x^{6}+70 a \,b^{2} x^{4}+56 a^{2} b \,x^{2}+16 a^{3}\right ) x^{\frac {9}{2}}}{35 \left (b \,x^{3}+a x \right )^{\frac {9}{2}} b^{4}} \]
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 \[ \int \frac {x^{\frac {23}{2}}}{{\left (b x^{3} + a x\right )}^{\frac {9}{2}}}\,{d x} \]
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 \[ \int \frac {x^{23/2}}{{\left (b\,x^3+a\,x\right )}^{9/2}} \,d x \]
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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