Optimal. Leaf size=36 \[ \frac{a}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac{1}{b^2 \sqrt{a+b x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0215647, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{a}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac{1}{b^2 \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a+b x^2\right )^{5/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(a+b x)^{5/2}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^{5/2}}+\frac{1}{b (a+b x)^{3/2}}\right ) \, dx,x,x^2\right )\\ &=\frac{a}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac{1}{b^2 \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [A] time = 0.0109335, size = 28, normalized size = 0.78 \[ \frac{-2 a-3 b x^2}{3 b^2 \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 25, normalized size = 0.7 \begin{align*} -{\frac{3\,b{x}^{2}+2\,a}{3\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 2.36117, size = 45, normalized size = 1.25 \begin{align*} -\frac{x^{2}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}} b} - \frac{2 \, a}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.3013, size = 97, normalized size = 2.69 \begin{align*} -\frac{{\left (3 \, b x^{2} + 2 \, a\right )} \sqrt{b x^{2} + a}}{3 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.01676, size = 92, normalized size = 2.56 \begin{align*} \begin{cases} - \frac{2 a}{3 a b^{2} \sqrt{a + b x^{2}} + 3 b^{3} x^{2} \sqrt{a + b x^{2}}} - \frac{3 b x^{2}}{3 a b^{2} \sqrt{a + b x^{2}} + 3 b^{3} x^{2} \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 2.55065, size = 32, normalized size = 0.89 \begin{align*} -\frac{3 \, b x^{2} + 2 \, a}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]