Optimal. Leaf size=127 \[ \frac{a^2 B x \sqrt{a+b x^2}}{16 b^2}+\frac{a^3 B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{5/2}}-\frac{a \left (a+b x^2\right )^{3/2} (16 A+15 B x)}{120 b^2}+\frac{A x^2 \left (a+b x^2\right )^{3/2}}{5 b}+\frac{B x^3 \left (a+b x^2\right )^{3/2}}{6 b} \]
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Rubi [A] time = 0.0819759, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {833, 780, 195, 217, 206} \[ \frac{a^2 B x \sqrt{a+b x^2}}{16 b^2}+\frac{a^3 B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{5/2}}-\frac{a \left (a+b x^2\right )^{3/2} (16 A+15 B x)}{120 b^2}+\frac{A x^2 \left (a+b x^2\right )^{3/2}}{5 b}+\frac{B x^3 \left (a+b x^2\right )^{3/2}}{6 b} \]
Antiderivative was successfully verified.
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Rule 833
Rule 780
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^3 (A+B x) \sqrt{a+b x^2} \, dx &=\frac{B x^3 \left (a+b x^2\right )^{3/2}}{6 b}+\frac{\int x^2 (-3 a B+6 A b x) \sqrt{a+b x^2} \, dx}{6 b}\\ &=\frac{A x^2 \left (a+b x^2\right )^{3/2}}{5 b}+\frac{B x^3 \left (a+b x^2\right )^{3/2}}{6 b}+\frac{\int x (-12 a A b-15 a b B x) \sqrt{a+b x^2} \, dx}{30 b^2}\\ &=\frac{A x^2 \left (a+b x^2\right )^{3/2}}{5 b}+\frac{B x^3 \left (a+b x^2\right )^{3/2}}{6 b}-\frac{a (16 A+15 B x) \left (a+b x^2\right )^{3/2}}{120 b^2}+\frac{\left (a^2 B\right ) \int \sqrt{a+b x^2} \, dx}{8 b^2}\\ &=\frac{a^2 B x \sqrt{a+b x^2}}{16 b^2}+\frac{A x^2 \left (a+b x^2\right )^{3/2}}{5 b}+\frac{B x^3 \left (a+b x^2\right )^{3/2}}{6 b}-\frac{a (16 A+15 B x) \left (a+b x^2\right )^{3/2}}{120 b^2}+\frac{\left (a^3 B\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{16 b^2}\\ &=\frac{a^2 B x \sqrt{a+b x^2}}{16 b^2}+\frac{A x^2 \left (a+b x^2\right )^{3/2}}{5 b}+\frac{B x^3 \left (a+b x^2\right )^{3/2}}{6 b}-\frac{a (16 A+15 B x) \left (a+b x^2\right )^{3/2}}{120 b^2}+\frac{\left (a^3 B\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{16 b^2}\\ &=\frac{a^2 B x \sqrt{a+b x^2}}{16 b^2}+\frac{A x^2 \left (a+b x^2\right )^{3/2}}{5 b}+\frac{B x^3 \left (a+b x^2\right )^{3/2}}{6 b}-\frac{a (16 A+15 B x) \left (a+b x^2\right )^{3/2}}{120 b^2}+\frac{a^3 B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.186739, size = 107, normalized size = 0.84 \[ \frac{\sqrt{a+b x^2} \left (\sqrt{b} \left (-a^2 (32 A+15 B x)+2 a b x^2 (8 A+5 B x)+8 b^2 x^4 (6 A+5 B x)\right )+\frac{15 a^{5/2} B \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{\frac{b x^2}{a}+1}}\right )}{240 b^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 115, normalized size = 0.9 \begin{align*}{\frac{B{x}^{3}}{6\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{Bax}{8\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}Bx}{16\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{B{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}+{\frac{A{x}^{2}}{5\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{2\,Aa}{15\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\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 = 1.57808, size = 501, normalized size = 3.94 \begin{align*} \left [\frac{15 \, B a^{3} \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \,{\left (40 \, B b^{3} x^{5} + 48 \, A b^{3} x^{4} + 10 \, B a b^{2} x^{3} + 16 \, A a b^{2} x^{2} - 15 \, B a^{2} b x - 32 \, A a^{2} b\right )} \sqrt{b x^{2} + a}}{480 \, b^{3}}, -\frac{15 \, B a^{3} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (40 \, B b^{3} x^{5} + 48 \, A b^{3} x^{4} + 10 \, B a b^{2} x^{3} + 16 \, A a b^{2} x^{2} - 15 \, B a^{2} b x - 32 \, A a^{2} b\right )} \sqrt{b x^{2} + a}}{240 \, b^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.52921, size = 192, normalized size = 1.51 \begin{align*} A \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + b x^{2}}}{15 b^{2}} + \frac{a x^{2} \sqrt{a + b x^{2}}}{15 b} + \frac{x^{4} \sqrt{a + b x^{2}}}{5} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) - \frac{B a^{\frac{5}{2}} x}{16 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B a^{\frac{3}{2}} x^{3}}{48 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 B \sqrt{a} x^{5}}{24 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{B a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 b^{\frac{5}{2}}} + \frac{B b x^{7}}{6 \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 = 1.19043, size = 126, normalized size = 0.99 \begin{align*} -\frac{B a^{3} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{16 \, b^{\frac{5}{2}}} + \frac{1}{240} \, \sqrt{b x^{2} + a}{\left ({\left (2 \,{\left ({\left (4 \,{\left (5 \, B x + 6 \, A\right )} x + \frac{5 \, B a}{b}\right )} x + \frac{8 \, A a}{b}\right )} x - \frac{15 \, B a^{2}}{b^{2}}\right )} x - \frac{32 \, A a^{2}}{b^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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