Optimal. Leaf size=72 \[ \frac{3 a^2 \left (a+b x^2\right )^7}{14 b^4}-\frac{a^3 \left (a+b x^2\right )^6}{12 b^4}+\frac{\left (a+b x^2\right )^9}{18 b^4}-\frac{3 a \left (a+b x^2\right )^8}{16 b^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0905814, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{3 a^2 \left (a+b x^2\right )^7}{14 b^4}-\frac{a^3 \left (a+b x^2\right )^6}{12 b^4}+\frac{\left (a+b x^2\right )^9}{18 b^4}-\frac{3 a \left (a+b x^2\right )^8}{16 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^7 \left (a+b x^2\right )^5 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^3 (a+b x)^5 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^5}{b^3}+\frac{3 a^2 (a+b x)^6}{b^3}-\frac{3 a (a+b x)^7}{b^3}+\frac{(a+b x)^8}{b^3}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^3 \left (a+b x^2\right )^6}{12 b^4}+\frac{3 a^2 \left (a+b x^2\right )^7}{14 b^4}-\frac{3 a \left (a+b x^2\right )^8}{16 b^4}+\frac{\left (a+b x^2\right )^9}{18 b^4}\\ \end{align*}
Mathematica [A] time = 0.0021167, size = 69, normalized size = 0.96 \[ \frac{5}{7} a^2 b^3 x^{14}+\frac{5}{6} a^3 b^2 x^{12}+\frac{1}{2} a^4 b x^{10}+\frac{a^5 x^8}{8}+\frac{5}{16} a b^4 x^{16}+\frac{b^5 x^{18}}{18} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.001, size = 58, normalized size = 0.8 \begin{align*}{\frac{{b}^{5}{x}^{18}}{18}}+{\frac{5\,a{b}^{4}{x}^{16}}{16}}+{\frac{5\,{a}^{2}{b}^{3}{x}^{14}}{7}}+{\frac{5\,{a}^{3}{b}^{2}{x}^{12}}{6}}+{\frac{{a}^{4}b{x}^{10}}{2}}+{\frac{{a}^{5}{x}^{8}}{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 2.27767, size = 77, normalized size = 1.07 \begin{align*} \frac{1}{18} \, b^{5} x^{18} + \frac{5}{16} \, a b^{4} x^{16} + \frac{5}{7} \, a^{2} b^{3} x^{14} + \frac{5}{6} \, a^{3} b^{2} x^{12} + \frac{1}{2} \, a^{4} b x^{10} + \frac{1}{8} \, a^{5} x^{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.12387, size = 138, normalized size = 1.92 \begin{align*} \frac{1}{18} x^{18} b^{5} + \frac{5}{16} x^{16} b^{4} a + \frac{5}{7} x^{14} b^{3} a^{2} + \frac{5}{6} x^{12} b^{2} a^{3} + \frac{1}{2} x^{10} b a^{4} + \frac{1}{8} x^{8} a^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.071009, size = 65, normalized size = 0.9 \begin{align*} \frac{a^{5} x^{8}}{8} + \frac{a^{4} b x^{10}}{2} + \frac{5 a^{3} b^{2} x^{12}}{6} + \frac{5 a^{2} b^{3} x^{14}}{7} + \frac{5 a b^{4} x^{16}}{16} + \frac{b^{5} x^{18}}{18} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 2.33403, size = 77, normalized size = 1.07 \begin{align*} \frac{1}{18} \, b^{5} x^{18} + \frac{5}{16} \, a b^{4} x^{16} + \frac{5}{7} \, a^{2} b^{3} x^{14} + \frac{5}{6} \, a^{3} b^{2} x^{12} + \frac{1}{2} \, a^{4} b x^{10} + \frac{1}{8} \, a^{5} x^{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]