Optimal. Leaf size=80 \[ \frac{a^2 \left (a+b x^2\right )^{9/2}}{3 b^4}-\frac{a^3 \left (a+b x^2\right )^{7/2}}{7 b^4}+\frac{\left (a+b x^2\right )^{13/2}}{13 b^4}-\frac{3 a \left (a+b x^2\right )^{11/2}}{11 b^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0467618, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{a^2 \left (a+b x^2\right )^{9/2}}{3 b^4}-\frac{a^3 \left (a+b x^2\right )^{7/2}}{7 b^4}+\frac{\left (a+b x^2\right )^{13/2}}{13 b^4}-\frac{3 a \left (a+b x^2\right )^{11/2}}{11 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^7 \left (a+b x^2\right )^{5/2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^3 (a+b x)^{5/2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^{5/2}}{b^3}+\frac{3 a^2 (a+b x)^{7/2}}{b^3}-\frac{3 a (a+b x)^{9/2}}{b^3}+\frac{(a+b x)^{11/2}}{b^3}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^3 \left (a+b x^2\right )^{7/2}}{7 b^4}+\frac{a^2 \left (a+b x^2\right )^{9/2}}{3 b^4}-\frac{3 a \left (a+b x^2\right )^{11/2}}{11 b^4}+\frac{\left (a+b x^2\right )^{13/2}}{13 b^4}\\ \end{align*}
Mathematica [A] time = 0.0267787, size = 50, normalized size = 0.62 \[ \frac{\left (a+b x^2\right )^{7/2} \left (56 a^2 b x^2-16 a^3-126 a b^2 x^4+231 b^3 x^6\right )}{3003 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 47, normalized size = 0.6 \begin{align*} -{\frac{-231\,{b}^{3}{x}^{6}+126\,a{b}^{2}{x}^{4}-56\,{a}^{2}b{x}^{2}+16\,{a}^{3}}{3003\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{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 = 1.54613, size = 180, normalized size = 2.25 \begin{align*} \frac{{\left (231 \, b^{6} x^{12} + 567 \, a b^{5} x^{10} + 371 \, a^{2} b^{4} x^{8} + 5 \, a^{3} b^{3} x^{6} - 6 \, a^{4} b^{2} x^{4} + 8 \, a^{5} b x^{2} - 16 \, a^{6}\right )} \sqrt{b x^{2} + a}}{3003 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 8.7502, size = 158, normalized size = 1.98 \begin{align*} \begin{cases} - \frac{16 a^{6} \sqrt{a + b x^{2}}}{3003 b^{4}} + \frac{8 a^{5} x^{2} \sqrt{a + b x^{2}}}{3003 b^{3}} - \frac{2 a^{4} x^{4} \sqrt{a + b x^{2}}}{1001 b^{2}} + \frac{5 a^{3} x^{6} \sqrt{a + b x^{2}}}{3003 b} + \frac{53 a^{2} x^{8} \sqrt{a + b x^{2}}}{429} + \frac{27 a b x^{10} \sqrt{a + b x^{2}}}{143} + \frac{b^{2} x^{12} \sqrt{a + b x^{2}}}{13} & \text{for}\: b \neq 0 \\\frac{a^{\frac{5}{2}} x^{8}}{8} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.62763, size = 301, normalized size = 3.76 \begin{align*} \frac{\frac{143 \,{\left (35 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3}\right )} a^{2}}{b^{3}} + \frac{26 \,{\left (315 \,{\left (b x^{2} + a\right )}^{\frac{11}{2}} - 1540 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} a + 2970 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a^{2} - 2772 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{3} + 1155 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{4}\right )} a}{b^{3}} + \frac{5 \,{\left (693 \,{\left (b x^{2} + a\right )}^{\frac{13}{2}} - 4095 \,{\left (b x^{2} + a\right )}^{\frac{11}{2}} a + 10010 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} a^{2} - 12870 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a^{3} + 9009 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{4} - 3003 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{5}\right )}}{b^{3}}}{45045 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]