Optimal. Leaf size=40 \[ \frac{\left (a-b x^4\right )^{3/2}}{6 b^2}-\frac{a \sqrt{a-b x^4}}{2 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0236956, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {266, 43} \[ \frac{\left (a-b x^4\right )^{3/2}}{6 b^2}-\frac{a \sqrt{a-b x^4}}{2 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^7}{\sqrt{a-b x^4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x}{\sqrt{a-b x}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{a}{b \sqrt{a-b x}}-\frac{\sqrt{a-b x}}{b}\right ) \, dx,x,x^4\right )\\ &=-\frac{a \sqrt{a-b x^4}}{2 b^2}+\frac{\left (a-b x^4\right )^{3/2}}{6 b^2}\\ \end{align*}
Mathematica [A] time = 0.0136454, size = 28, normalized size = 0.7 \[ -\frac{\sqrt{a-b x^4} \left (2 a+b x^4\right )}{6 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 25, normalized size = 0.6 \begin{align*} -{\frac{b{x}^{4}+2\,a}{6\,{b}^{2}}\sqrt{-b{x}^{4}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.961852, size = 43, normalized size = 1.08 \begin{align*} \frac{{\left (-b x^{4} + a\right )}^{\frac{3}{2}}}{6 \, b^{2}} - \frac{\sqrt{-b x^{4} + a} a}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.43101, size = 55, normalized size = 1.38 \begin{align*} -\frac{{\left (b x^{4} + 2 \, a\right )} \sqrt{-b x^{4} + a}}{6 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.425, size = 44, normalized size = 1.1 \begin{align*} \begin{cases} - \frac{a \sqrt{a - b x^{4}}}{3 b^{2}} - \frac{x^{4} \sqrt{a - b x^{4}}}{6 b} & \text{for}\: b \neq 0 \\\frac{x^{8}}{8 \sqrt{a}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.10942, size = 39, normalized size = 0.98 \begin{align*} \frac{{\left (-b x^{4} + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{-b x^{4} + a} a}{6 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]