Optimal. Leaf size=52 \[ \frac{\left (b x^2+c x^4\right )^{3/2}}{5 c x}-\frac{2 b \left (b x^2+c x^4\right )^{3/2}}{15 c^2 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0486033, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2016, 2000} \[ \frac{\left (b x^2+c x^4\right )^{3/2}}{5 c x}-\frac{2 b \left (b x^2+c x^4\right )^{3/2}}{15 c^2 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2016
Rule 2000
Rubi steps
\begin{align*} \int x^2 \sqrt{b x^2+c x^4} \, dx &=\frac{\left (b x^2+c x^4\right )^{3/2}}{5 c x}-\frac{(2 b) \int \sqrt{b x^2+c x^4} \, dx}{5 c}\\ &=-\frac{2 b \left (b x^2+c x^4\right )^{3/2}}{15 c^2 x^3}+\frac{\left (b x^2+c x^4\right )^{3/2}}{5 c x}\\ \end{align*}
Mathematica [A] time = 0.01787, size = 35, normalized size = 0.67 \[ \frac{\left (x^2 \left (b+c x^2\right )\right )^{3/2} \left (3 c x^2-2 b\right )}{15 c^2 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.047, size = 39, normalized size = 0.8 \begin{align*} -{\frac{ \left ( c{x}^{2}+b \right ) \left ( -3\,c{x}^{2}+2\,b \right ) }{15\,{c}^{2}x}\sqrt{c{x}^{4}+b{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.997549, size = 46, normalized size = 0.88 \begin{align*} \frac{{\left (3 \, c^{2} x^{4} + b c x^{2} - 2 \, b^{2}\right )} \sqrt{c x^{2} + b}}{15 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.5388, size = 86, normalized size = 1.65 \begin{align*} \frac{{\left (3 \, c^{2} x^{4} + b c x^{2} - 2 \, b^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{15 \, c^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sqrt{x^{2} \left (b + c x^{2}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.26036, size = 57, normalized size = 1.1 \begin{align*} \frac{2 \, b^{\frac{5}{2}} \mathrm{sgn}\left (x\right )}{15 \, c^{2}} + \frac{{\left (3 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b\right )} \mathrm{sgn}\left (x\right )}{15 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]