Optimal. Leaf size=76 \[ -\frac {8 x^{3/2}}{105 b^3 \left (a x+b x^3\right )^{3/2}}-\frac {4 x^{9/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac {x^{15/2}}{7 b \left (a x+b x^3\right )^{7/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2015, 2014} \[ -\frac {4 x^{9/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac {8 x^{3/2}}{105 b^3 \left (a x+b x^3\right )^{3/2}}-\frac {x^{15/2}}{7 b \left (a x+b x^3\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2014
Rule 2015
Rubi steps
\begin {align*} \int \frac {x^{19/2}}{\left (a x+b x^3\right )^{9/2}} \, dx &=-\frac {x^{15/2}}{7 b \left (a x+b x^3\right )^{7/2}}+\frac {4 \int \frac {x^{13/2}}{\left (a x+b x^3\right )^{7/2}} \, dx}{7 b}\\ &=-\frac {x^{15/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac {4 x^{9/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}+\frac {8 \int \frac {x^{7/2}}{\left (a x+b x^3\right )^{5/2}} \, dx}{35 b^2}\\ &=-\frac {x^{15/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac {4 x^{9/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac {8 x^{3/2}}{105 b^3 \left (a x+b x^3\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 55, normalized size = 0.72 \[ -\frac {\sqrt {x} \left (8 a^2+28 a b x^2+35 b^2 x^4\right )}{105 b^3 \left (a+b x^2\right )^3 \sqrt {x \left (a+b x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 86, normalized size = 1.13 \[ -\frac {{\left (35 \, b^{2} x^{4} + 28 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt {b x^{3} + a x} \sqrt {x}}{105 \, {\left (b^{7} x^{9} + 4 \, a b^{6} x^{7} + 6 \, a^{2} b^{5} x^{5} + 4 \, a^{3} b^{4} x^{3} + a^{4} b^{3} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 50, normalized size = 0.66 \[ \frac {8}{105 \, a^{\frac {3}{2}} b^{3}} - \frac {35 \, {\left (b x^{2} + a\right )}^{2} - 42 \, {\left (b x^{2} + a\right )} a + 15 \, a^{2}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 48, normalized size = 0.63 \[ -\frac {\left (b \,x^{2}+a \right ) \left (35 b^{2} x^{4}+28 a b \,x^{2}+8 a^{2}\right ) x^{\frac {9}{2}}}{105 \left (b \,x^{3}+a x \right )^{\frac {9}{2}} b^{3}} \]
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 \[ \int \frac {x^{\frac {19}{2}}}{{\left (b x^{3} + a x\right )}^{\frac {9}{2}}}\,{d x} \]
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 \[ \int \frac {x^{19/2}}{{\left (b\,x^3+a\,x\right )}^{9/2}} \,d x \]
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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