Optimal. Leaf size=101 \[ -\frac{6 x^{13/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac{8 x^{7/2}}{35 b^3 \left (a x+b x^3\right )^{3/2}}-\frac{16 \sqrt{x}}{35 b^4 \sqrt{a x+b x^3}}-\frac{x^{19/2}}{7 b \left (a x+b x^3\right )^{7/2}} \]
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Rubi [A] time = 0.161451, antiderivative size = 101, normalized size of antiderivative = 1., 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 2015
Rule 2014
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*}
Mathematica [A] time = 0.0302392, size = 66, normalized size = 0.65 \[ -\frac{\sqrt{x} \left (56 a^2 b x^2+16 a^3+70 a b^2 x^4+35 b^3 x^6\right )}{35 b^4 \left (a+b x^2\right )^3 \sqrt{x \left (a+b x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 59, normalized size = 0.6 \begin{align*} -{\frac{ \left ( b{x}^{2}+a \right ) \left ( 35\,{x}^{6}{b}^{3}+70\,a{x}^{4}{b}^{2}+56\,{a}^{2}{x}^{2}b+16\,{a}^{3} \right ) }{35\,{b}^{4}}{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{23}{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 = 1.4131, size = 207, normalized size = 2.05 \begin{align*} -\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 )}} \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 = 1.27062, size = 86, normalized size = 0.85 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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