Optimal. Leaf size=101 \[ \frac{6 x^{7/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac{8 x^{5/2}}{35 a^3 \left (a x+b x^3\right )^{3/2}}+\frac{16 x^{3/2}}{35 a^4 \sqrt{a x+b x^3}}+\frac{x^{9/2}}{7 a \left (a x+b x^3\right )^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.157272, 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} \[ \frac{6 x^{7/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac{8 x^{5/2}}{35 a^3 \left (a x+b x^3\right )^{3/2}}+\frac{16 x^{3/2}}{35 a^4 \sqrt{a x+b x^3}}+\frac{x^{9/2}}{7 a \left (a x+b x^3\right )^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2015
Rule 2014
Rubi steps
\begin{align*} \int \frac{x^{9/2}}{\left (a x+b x^3\right )^{9/2}} \, dx &=\frac{x^{9/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac{6 \int \frac{x^{7/2}}{\left (a x+b x^3\right )^{7/2}} \, dx}{7 a}\\ &=\frac{x^{9/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac{6 x^{7/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac{24 \int \frac{x^{5/2}}{\left (a x+b x^3\right )^{5/2}} \, dx}{35 a^2}\\ &=\frac{x^{9/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac{6 x^{7/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac{8 x^{5/2}}{35 a^3 \left (a x+b x^3\right )^{3/2}}+\frac{16 \int \frac{x^{3/2}}{\left (a x+b x^3\right )^{3/2}} \, dx}{35 a^3}\\ &=\frac{x^{9/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac{6 x^{7/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac{8 x^{5/2}}{35 a^3 \left (a x+b x^3\right )^{3/2}}+\frac{16 x^{3/2}}{35 a^4 \sqrt{a x+b x^3}}\\ \end{align*}
Mathematica [A] time = 0.0231574, size = 66, normalized size = 0.65 \[ \frac{\sqrt{x} \sqrt{x \left (a+b x^2\right )} \left (70 a^2 b x^2+35 a^3+56 a b^2 x^4+16 b^3 x^6\right )}{35 a^4 \left (a+b x^2\right )^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 59, normalized size = 0.6 \begin{align*}{\frac{ \left ( b{x}^{2}+a \right ) \left ( 16\,{b}^{3}{x}^{6}+56\,{b}^{2}{x}^{4}a+70\,b{x}^{2}{a}^{2}+35\,{a}^{3} \right ) }{35\,{a}^{4}}{x}^{{\frac{11}{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{9}{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 = 1.50133, size = 203, normalized size = 2.01 \begin{align*} \frac{{\left (16 \, b^{3} x^{6} + 56 \, a b^{2} x^{4} + 70 \, a^{2} b x^{2} + 35 \, a^{3}\right )} \sqrt{b x^{3} + a x} \sqrt{x}}{35 \,{\left (a^{4} b^{4} x^{8} + 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8}\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 = 1.31053, size = 74, normalized size = 0.73 \begin{align*} \frac{{\left (2 \,{\left (4 \, x^{2}{\left (\frac{2 \, b^{3} x^{2}}{a^{4}} + \frac{7 \, b^{2}}{a^{3}}\right )} + \frac{35 \, b}{a^{2}}\right )} x^{2} + \frac{35}{a}\right )} x}{35 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]