Optimal. Leaf size=56 \[ \frac{4 \left (a+b \left (c x^3\right )^{3/2}\right )^{5/2}}{45 b^2 c^3}-\frac{4 a \left (a+b \left (c x^3\right )^{3/2}\right )^{3/2}}{27 b^2 c^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0367072, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {369, 266, 43} \[ \frac{4 \left (a+b \left (c x^3\right )^{3/2}\right )^{5/2}}{45 b^2 c^3}-\frac{4 a \left (a+b \left (c x^3\right )^{3/2}\right )^{3/2}}{27 b^2 c^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 369
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^8 \sqrt{a+b \left (c x^3\right )^{3/2}} \, dx &=\operatorname{Subst}\left (\int x^8 \sqrt{a+b c^{3/2} x^{9/2}} \, dx,\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\operatorname{Subst}\left (\frac{2}{9} \operatorname{Subst}\left (\int x \sqrt{a+b c^{3/2} x} \, dx,x,x^{9/2}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\operatorname{Subst}\left (\frac{2}{9} \operatorname{Subst}\left (\int \left (-\frac{a \sqrt{a+b c^{3/2} x}}{b c^{3/2}}+\frac{\left (a+b c^{3/2} x\right )^{3/2}}{b c^{3/2}}\right ) \, dx,x,x^{9/2}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\frac{4 a \left (a+b \left (c x^3\right )^{3/2}\right )^{3/2}}{27 b^2 c^3}+\frac{4 \left (a+b \left (c x^3\right )^{3/2}\right )^{5/2}}{45 b^2 c^3}\\ \end{align*}
Mathematica [A] time = 0.0700529, size = 43, normalized size = 0.77 \[ \frac{4 \left (a+b \left (c x^3\right )^{3/2}\right )^{3/2} \left (3 b \left (c x^3\right )^{3/2}-2 a\right )}{135 b^2 c^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.057, size = 0, normalized size = 0. \begin{align*} \int{x}^{8}\sqrt{a+b \left ( c{x}^{3} \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.957482, size = 58, normalized size = 1.04 \begin{align*} \frac{4 \,{\left (\frac{3 \,{\left (\left (c x^{3}\right )^{\frac{3}{2}} b + a\right )}^{\frac{5}{2}}}{b^{2}} - \frac{5 \,{\left (\left (c x^{3}\right )^{\frac{3}{2}} b + a\right )}^{\frac{3}{2}} a}{b^{2}}\right )}}{135 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 10.9713, size = 128, normalized size = 2.29 \begin{align*} \frac{4 \,{\left (3 \, b^{2} c^{3} x^{9} + \sqrt{c x^{3}} a b c x^{3} - 2 \, a^{2}\right )} \sqrt{\sqrt{c x^{3}} b c x^{3} + a}}{135 \, b^{2} c^{3}} \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^{8} \sqrt{a + b \left (c x^{3}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1606, size = 115, normalized size = 2.05 \begin{align*} \frac{4 \,{\left (\frac{2 \, \sqrt{a c^{3}} a^{2}}{b^{2} c^{2}} - \frac{5 \,{\left (\sqrt{c x} b c^{4} x^{4} + a c^{3}\right )}^{\frac{3}{2}} a c^{3} - 3 \,{\left (\sqrt{c x} b c^{4} x^{4} + a c^{3}\right )}^{\frac{5}{2}}}{b^{2} c^{8}}\right )}{\left | c \right |}}{135 \, c^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]