Optimal. Leaf size=155 \[ \frac{a^2 x \sqrt{a+b x^2} (8 A b-3 a B)}{128 b^2}-\frac{a^3 (8 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{5/2}}+\frac{a x^3 \sqrt{a+b x^2} (8 A b-3 a B)}{64 b}+\frac{x^3 \left (a+b x^2\right )^{3/2} (8 A b-3 a B)}{48 b}+\frac{B x^3 \left (a+b x^2\right )^{5/2}}{8 b} \]
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Rubi [A] time = 0.0673113, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {459, 279, 321, 217, 206} \[ \frac{a^2 x \sqrt{a+b x^2} (8 A b-3 a B)}{128 b^2}-\frac{a^3 (8 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{5/2}}+\frac{a x^3 \sqrt{a+b x^2} (8 A b-3 a B)}{64 b}+\frac{x^3 \left (a+b x^2\right )^{3/2} (8 A b-3 a B)}{48 b}+\frac{B x^3 \left (a+b x^2\right )^{5/2}}{8 b} \]
Antiderivative was successfully verified.
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Rule 459
Rule 279
Rule 321
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^2 \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx &=\frac{B x^3 \left (a+b x^2\right )^{5/2}}{8 b}-\frac{(-8 A b+3 a B) \int x^2 \left (a+b x^2\right )^{3/2} \, dx}{8 b}\\ &=\frac{(8 A b-3 a B) x^3 \left (a+b x^2\right )^{3/2}}{48 b}+\frac{B x^3 \left (a+b x^2\right )^{5/2}}{8 b}+\frac{(a (8 A b-3 a B)) \int x^2 \sqrt{a+b x^2} \, dx}{16 b}\\ &=\frac{a (8 A b-3 a B) x^3 \sqrt{a+b x^2}}{64 b}+\frac{(8 A b-3 a B) x^3 \left (a+b x^2\right )^{3/2}}{48 b}+\frac{B x^3 \left (a+b x^2\right )^{5/2}}{8 b}+\frac{\left (a^2 (8 A b-3 a B)\right ) \int \frac{x^2}{\sqrt{a+b x^2}} \, dx}{64 b}\\ &=\frac{a^2 (8 A b-3 a B) x \sqrt{a+b x^2}}{128 b^2}+\frac{a (8 A b-3 a B) x^3 \sqrt{a+b x^2}}{64 b}+\frac{(8 A b-3 a B) x^3 \left (a+b x^2\right )^{3/2}}{48 b}+\frac{B x^3 \left (a+b x^2\right )^{5/2}}{8 b}-\frac{\left (a^3 (8 A b-3 a B)\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{128 b^2}\\ &=\frac{a^2 (8 A b-3 a B) x \sqrt{a+b x^2}}{128 b^2}+\frac{a (8 A b-3 a B) x^3 \sqrt{a+b x^2}}{64 b}+\frac{(8 A b-3 a B) x^3 \left (a+b x^2\right )^{3/2}}{48 b}+\frac{B x^3 \left (a+b x^2\right )^{5/2}}{8 b}-\frac{\left (a^3 (8 A b-3 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{128 b^2}\\ &=\frac{a^2 (8 A b-3 a B) x \sqrt{a+b x^2}}{128 b^2}+\frac{a (8 A b-3 a B) x^3 \sqrt{a+b x^2}}{64 b}+\frac{(8 A b-3 a B) x^3 \left (a+b x^2\right )^{3/2}}{48 b}+\frac{B x^3 \left (a+b x^2\right )^{5/2}}{8 b}-\frac{a^3 (8 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.249302, size = 130, normalized size = 0.84 \[ \frac{\sqrt{a+b x^2} \left (\sqrt{b} x \left (6 a^2 b \left (4 A+B x^2\right )-9 a^3 B+8 a b^2 x^2 \left (14 A+9 B x^2\right )+16 b^3 x^4 \left (4 A+3 B x^2\right )\right )+\frac{3 a^{5/2} (3 a B-8 A b) \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{\frac{b x^2}{a}+1}}\right )}{384 b^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 177, normalized size = 1.1 \begin{align*}{\frac{B{x}^{3}}{8\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{Bax}{16\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{2}Bx}{64\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,B{a}^{3}x}{128\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,B{a}^{4}}{128}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}+{\frac{Ax}{6\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{aAx}{24\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}Ax}{16\,b}\sqrt{b{x}^{2}+a}}-{\frac{A{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{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 = 1.78934, size = 597, normalized size = 3.85 \begin{align*} \left [-\frac{3 \,{\left (3 \, B a^{4} - 8 \, A a^{3} b\right )} \sqrt{b} \log \left (-2 \, b x^{2} + 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left (48 \, B b^{4} x^{7} + 8 \,{\left (9 \, B a b^{3} + 8 \, A b^{4}\right )} x^{5} + 2 \,{\left (3 \, B a^{2} b^{2} + 56 \, A a b^{3}\right )} x^{3} - 3 \,{\left (3 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x\right )} \sqrt{b x^{2} + a}}{768 \, b^{3}}, -\frac{3 \,{\left (3 \, B a^{4} - 8 \, A a^{3} b\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (48 \, B b^{4} x^{7} + 8 \,{\left (9 \, B a b^{3} + 8 \, A b^{4}\right )} x^{5} + 2 \,{\left (3 \, B a^{2} b^{2} + 56 \, A a b^{3}\right )} x^{3} - 3 \,{\left (3 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x\right )} \sqrt{b x^{2} + a}}{384 \, b^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 22.2434, size = 287, normalized size = 1.85 \begin{align*} \frac{A a^{\frac{5}{2}} x}{16 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{17 A a^{\frac{3}{2}} x^{3}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{11 A \sqrt{a} b x^{5}}{24 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{A a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 b^{\frac{3}{2}}} + \frac{A b^{2} x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{3 B a^{\frac{7}{2}} x}{128 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B a^{\frac{5}{2}} x^{3}}{128 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{13 B a^{\frac{3}{2}} x^{5}}{64 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 B \sqrt{a} b x^{7}}{16 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 B a^{4} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{128 b^{\frac{5}{2}}} + \frac{B b^{2} x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13432, size = 180, normalized size = 1.16 \begin{align*} \frac{1}{384} \,{\left (2 \,{\left (4 \,{\left (6 \, B b x^{2} + \frac{9 \, B a b^{6} + 8 \, A b^{7}}{b^{6}}\right )} x^{2} + \frac{3 \, B a^{2} b^{5} + 56 \, A a b^{6}}{b^{6}}\right )} x^{2} - \frac{3 \,{\left (3 \, B a^{3} b^{4} - 8 \, A a^{2} b^{5}\right )}}{b^{6}}\right )} \sqrt{b x^{2} + a} x - \frac{{\left (3 \, B a^{4} - 8 \, A a^{3} b\right )} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{128 \, b^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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