Optimal. Leaf size=76 \[ -\frac{4 x^{9/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac{8 x^{3/2}}{105 b^3 \left (a x+b x^3\right )^{3/2}}-\frac{x^{15/2}}{7 b \left (a x+b x^3\right )^{7/2}} \]
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Rubi [A] time = 0.118776, antiderivative size = 76, normalized size of antiderivative = 1., 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 2015
Rule 2014
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*}
Mathematica [A] time = 0.0256023, 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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Maple [A] time = 0.005, size = 48, normalized size = 0.6 \begin{align*} -{\frac{ \left ( b{x}^{2}+a \right ) \left ( 35\,{x}^{4}{b}^{2}+28\,a{x}^{2}b+8\,{a}^{2} \right ) }{105\,{b}^{3}}{x}^{{\frac{9}{2}}} \left ( b{x}^{3}+ax \right ) ^{-{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{19}{2}}}{{\left (b x^{3} + a x\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42644, size = 184, normalized size = 2.42 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24878, size = 68, normalized size = 0.89 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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