Optimal. Leaf size=76 \[ \frac{4 x^{11/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac{8 x^{9/2}}{105 a^3 \left (a x+b x^3\right )^{3/2}}+\frac{x^{13/2}}{7 a \left (a x+b x^3\right )^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.112928, 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} \[ \frac{4 x^{11/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac{8 x^{9/2}}{105 a^3 \left (a x+b x^3\right )^{3/2}}+\frac{x^{13/2}}{7 a \left (a x+b x^3\right )^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2015
Rule 2014
Rubi steps
\begin{align*} \int \frac{x^{13/2}}{\left (a x+b x^3\right )^{9/2}} \, dx &=\frac{x^{13/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac{4 \int \frac{x^{11/2}}{\left (a x+b x^3\right )^{7/2}} \, dx}{7 a}\\ &=\frac{x^{13/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac{4 x^{11/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac{8 \int \frac{x^{9/2}}{\left (a x+b x^3\right )^{5/2}} \, dx}{35 a^2}\\ &=\frac{x^{13/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac{4 x^{11/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac{8 x^{9/2}}{105 a^3 \left (a x+b x^3\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.020886, size = 55, normalized size = 0.72 \[ \frac{x^{5/2} \sqrt{x \left (a+b x^2\right )} \left (35 a^2+28 a b x^2+8 b^2 x^4\right )}{105 a^3 \left (a+b x^2\right )^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 48, normalized size = 0.6 \begin{align*}{\frac{ \left ( b{x}^{2}+a \right ) \left ( 8\,{b}^{2}{x}^{4}+28\,ab{x}^{2}+35\,{a}^{2} \right ) }{105\,{a}^{3}}{x}^{{\frac{15}{2}}} \left ( b{x}^{3}+ax \right ) ^{-{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{13}{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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.65983, size = 185, normalized size = 2.43 \begin{align*} \frac{{\left (8 \, b^{2} x^{6} + 28 \, a b x^{4} + 35 \, a^{2} x^{2}\right )} \sqrt{b x^{3} + a x} \sqrt{x}}{105 \,{\left (a^{3} b^{4} x^{8} + 4 \, a^{4} b^{3} x^{6} + 6 \, a^{5} b^{2} x^{4} + 4 \, a^{6} b x^{2} + a^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.27394, size = 58, normalized size = 0.76 \begin{align*} \frac{{\left (4 \, x^{2}{\left (\frac{2 \, b^{2} x^{2}}{a^{3}} + \frac{7 \, b}{a^{2}}\right )} + \frac{35}{a}\right )} x^{3}}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]