Optimal. Leaf size=119 \[ \frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^7}{24 b^3}-\frac{2 a \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^6}{21 b^3}+\frac{a^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{18 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0903906, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1355, 266, 43} \[ \frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^7}{24 b^3}-\frac{2 a \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^6}{21 b^3}+\frac{a^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{18 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1355
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^8 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int x^8 \left (a b+b^2 x^3\right )^5 \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \operatorname{Subst}\left (\int x^2 \left (a b+b^2 x\right )^5 \, dx,x,x^3\right )}{3 b^4 \left (a b+b^2 x^3\right )}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \operatorname{Subst}\left (\int \left (\frac{a^2 \left (a b+b^2 x\right )^5}{b^2}-\frac{2 a \left (a b+b^2 x\right )^6}{b^3}+\frac{\left (a b+b^2 x\right )^7}{b^4}\right ) \, dx,x,x^3\right )}{3 b^4 \left (a b+b^2 x^3\right )}\\ &=\frac{a^2 \left (a+b x^3\right )^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{18 b^3}-\frac{2 a \left (a+b x^3\right )^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{21 b^3}+\frac{\left (a+b x^3\right )^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{24 b^3}\\ \end{align*}
Mathematica [A] time = 0.024054, size = 83, normalized size = 0.7 \[ \frac{x^9 \sqrt{\left (a+b x^3\right )^2} \left (280 a^2 b^3 x^9+336 a^3 b^2 x^6+210 a^4 b x^3+56 a^5+120 a b^4 x^{12}+21 b^5 x^{15}\right )}{504 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.009, size = 80, normalized size = 0.7 \begin{align*}{\frac{{x}^{9} \left ( 21\,{b}^{5}{x}^{15}+120\,a{b}^{4}{x}^{12}+280\,{a}^{2}{b}^{3}{x}^{9}+336\,{a}^{3}{b}^{2}{x}^{6}+210\,{a}^{4}b{x}^{3}+56\,{a}^{5} \right ) }{504\, \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.74264, size = 139, normalized size = 1.17 \begin{align*} \frac{1}{24} \, b^{5} x^{24} + \frac{5}{21} \, a b^{4} x^{21} + \frac{5}{9} \, a^{2} b^{3} x^{18} + \frac{2}{3} \, a^{3} b^{2} x^{15} + \frac{5}{12} \, a^{4} b x^{12} + \frac{1}{9} \, a^{5} x^{9} \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} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.12256, size = 142, normalized size = 1.19 \begin{align*} \frac{1}{24} \, b^{5} x^{24} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{21} \, a b^{4} x^{21} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{9} \, a^{2} b^{3} x^{18} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{2}{3} \, a^{3} b^{2} x^{15} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{12} \, a^{4} b x^{12} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{1}{9} \, a^{5} x^{9} \mathrm{sgn}\left (b x^{3} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]