Optimal. Leaf size=103 \[ \frac{a^2 \left (a+b x^2\right )^{7/2} (A b-a B)}{7 b^4}+\frac{\left (a+b x^2\right )^{11/2} (A b-3 a B)}{11 b^4}-\frac{a \left (a+b x^2\right )^{9/2} (2 A b-3 a B)}{9 b^4}+\frac{B \left (a+b x^2\right )^{13/2}}{13 b^4} \]
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Rubi [A] time = 0.0763276, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {446, 77} \[ \frac{a^2 \left (a+b x^2\right )^{7/2} (A b-a B)}{7 b^4}+\frac{\left (a+b x^2\right )^{11/2} (A b-3 a B)}{11 b^4}-\frac{a \left (a+b x^2\right )^{9/2} (2 A b-3 a B)}{9 b^4}+\frac{B \left (a+b x^2\right )^{13/2}}{13 b^4} \]
Antiderivative was successfully verified.
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Rule 446
Rule 77
Rubi steps
\begin{align*} \int x^5 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 (a+b x)^{5/2} (A+B x) \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a^2 (-A b+a B) (a+b x)^{5/2}}{b^3}+\frac{a (-2 A b+3 a B) (a+b x)^{7/2}}{b^3}+\frac{(A b-3 a B) (a+b x)^{9/2}}{b^3}+\frac{B (a+b x)^{11/2}}{b^3}\right ) \, dx,x,x^2\right )\\ &=\frac{a^2 (A b-a B) \left (a+b x^2\right )^{7/2}}{7 b^4}-\frac{a (2 A b-3 a B) \left (a+b x^2\right )^{9/2}}{9 b^4}+\frac{(A b-3 a B) \left (a+b x^2\right )^{11/2}}{11 b^4}+\frac{B \left (a+b x^2\right )^{13/2}}{13 b^4}\\ \end{align*}
Mathematica [A] time = 0.0622146, size = 78, normalized size = 0.76 \[ \frac{\left (a+b x^2\right )^{7/2} \left (8 a^2 b \left (13 A+21 B x^2\right )-48 a^3 B-14 a b^2 x^2 \left (26 A+27 B x^2\right )+63 b^3 x^4 \left (13 A+11 B x^2\right )\right )}{9009 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 77, normalized size = 0.8 \begin{align*}{\frac{693\,B{x}^{6}{b}^{3}+819\,A{b}^{3}{x}^{4}-378\,Ba{b}^{2}{x}^{4}-364\,Aa{b}^{2}{x}^{2}+168\,B{a}^{2}b{x}^{2}+104\,A{a}^{2}b-48\,B{a}^{3}}{9009\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A] time = 1.63686, size = 340, normalized size = 3.3 \begin{align*} \frac{{\left (693 \, B b^{6} x^{12} + 63 \,{\left (27 \, B a b^{5} + 13 \, A b^{6}\right )} x^{10} + 7 \,{\left (159 \, B a^{2} b^{4} + 299 \, A a b^{5}\right )} x^{8} - 48 \, B a^{6} + 104 \, A a^{5} b +{\left (15 \, B a^{3} b^{3} + 1469 \, A a^{2} b^{4}\right )} x^{6} - 3 \,{\left (6 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{4} + 4 \,{\left (6 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{9009 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.15945, size = 313, normalized size = 3.04 \begin{align*} \begin{cases} \frac{8 A a^{5} \sqrt{a + b x^{2}}}{693 b^{3}} - \frac{4 A a^{4} x^{2} \sqrt{a + b x^{2}}}{693 b^{2}} + \frac{A a^{3} x^{4} \sqrt{a + b x^{2}}}{231 b} + \frac{113 A a^{2} x^{6} \sqrt{a + b x^{2}}}{693} + \frac{23 A a b x^{8} \sqrt{a + b x^{2}}}{99} + \frac{A b^{2} x^{10} \sqrt{a + b x^{2}}}{11} - \frac{16 B a^{6} \sqrt{a + b x^{2}}}{3003 b^{4}} + \frac{8 B a^{5} x^{2} \sqrt{a + b x^{2}}}{3003 b^{3}} - \frac{2 B a^{4} x^{4} \sqrt{a + b x^{2}}}{1001 b^{2}} + \frac{5 B a^{3} x^{6} \sqrt{a + b x^{2}}}{3003 b} + \frac{53 B a^{2} x^{8} \sqrt{a + b x^{2}}}{429} + \frac{27 B a b x^{10} \sqrt{a + b x^{2}}}{143} + \frac{B b^{2} x^{12} \sqrt{a + b x^{2}}}{13} & \text{for}\: b \neq 0 \\a^{\frac{5}{2}} \left (\frac{A x^{6}}{6} + \frac{B x^{8}}{8}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14551, size = 545, normalized size = 5.29 \begin{align*} \frac{\frac{429 \,{\left (15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2}\right )} A a^{2}}{b^{2}} + \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 )} B a^{2}}{b^{3}} + \frac{286 \,{\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 a}{b^{2}} + \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 )} B a}{b^{3}} + \frac{13 \,{\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^{2}} + \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}{b^{3}}}{45045 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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