Optimal. Leaf size=47 \[ \frac{2 \sqrt{b x^2+c x^4}}{c^2 x}-\frac{x^3}{c \sqrt{b x^2+c x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.067865, antiderivative size = 47, 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 = {2015, 1588} \[ \frac{2 \sqrt{b x^2+c x^4}}{c^2 x}-\frac{x^3}{c \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2015
Rule 1588
Rubi steps
\begin{align*} \int \frac{x^6}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=-\frac{x^3}{c \sqrt{b x^2+c x^4}}+\frac{2 \int \frac{x^2}{\sqrt{b x^2+c x^4}} \, dx}{c}\\ &=-\frac{x^3}{c \sqrt{b x^2+c x^4}}+\frac{2 \sqrt{b x^2+c x^4}}{c^2 x}\\ \end{align*}
Mathematica [A] time = 0.0152588, size = 29, normalized size = 0.62 \[ \frac{x \left (2 b+c x^2\right )}{c^2 \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 37, normalized size = 0.8 \begin{align*}{\frac{ \left ( c{x}^{2}+b \right ) \left ( c{x}^{2}+2\,b \right ){x}^{3}}{{c}^{2}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.99026, size = 30, normalized size = 0.64 \begin{align*} \frac{c x^{2} + 2 \, b}{\sqrt{c x^{2} + b} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.24328, size = 74, normalized size = 1.57 \begin{align*} \frac{\sqrt{c x^{4} + b x^{2}}{\left (c x^{2} + 2 \, b\right )}}{c^{3} x^{3} + b 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 \frac{x^{6}}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.16965, size = 70, normalized size = 1.49 \begin{align*} -\frac{2 \, \sqrt{b}}{{\left ({\left (\sqrt{c + \frac{b}{x^{2}}} - \frac{\sqrt{b}}{x}\right )}^{2} - c\right )} c} + \frac{b}{\sqrt{c + \frac{b}{x^{2}}} c^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]