Optimal. Leaf size=103 \[ -\frac{a^3 B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{3/2}}-\frac{a^2 B x \sqrt{a+b x^2}}{16 b}+\frac{\left (a+b x^2\right )^{5/2} (6 A+5 B x)}{30 b}-\frac{a B x \left (a+b x^2\right )^{3/2}}{24 b} \]
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Rubi [A] time = 0.0332214, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {780, 195, 217, 206} \[ -\frac{a^3 B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{3/2}}-\frac{a^2 B x \sqrt{a+b x^2}}{16 b}+\frac{\left (a+b x^2\right )^{5/2} (6 A+5 B x)}{30 b}-\frac{a B x \left (a+b x^2\right )^{3/2}}{24 b} \]
Antiderivative was successfully verified.
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Rule 780
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x (A+B x) \left (a+b x^2\right )^{3/2} \, dx &=\frac{(6 A+5 B x) \left (a+b x^2\right )^{5/2}}{30 b}-\frac{(a B) \int \left (a+b x^2\right )^{3/2} \, dx}{6 b}\\ &=-\frac{a B x \left (a+b x^2\right )^{3/2}}{24 b}+\frac{(6 A+5 B x) \left (a+b x^2\right )^{5/2}}{30 b}-\frac{\left (a^2 B\right ) \int \sqrt{a+b x^2} \, dx}{8 b}\\ &=-\frac{a^2 B x \sqrt{a+b x^2}}{16 b}-\frac{a B x \left (a+b x^2\right )^{3/2}}{24 b}+\frac{(6 A+5 B x) \left (a+b x^2\right )^{5/2}}{30 b}-\frac{\left (a^3 B\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{16 b}\\ &=-\frac{a^2 B x \sqrt{a+b x^2}}{16 b}-\frac{a B x \left (a+b x^2\right )^{3/2}}{24 b}+\frac{(6 A+5 B x) \left (a+b x^2\right )^{5/2}}{30 b}-\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}\\ &=-\frac{a^2 B x \sqrt{a+b x^2}}{16 b}-\frac{a B x \left (a+b x^2\right )^{3/2}}{24 b}+\frac{(6 A+5 B x) \left (a+b x^2\right )^{5/2}}{30 b}-\frac{a^3 B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.183505, size = 107, normalized size = 1.04 \[ \frac{\sqrt{a+b x^2} \left (\sqrt{b} \left (3 a^2 (16 A+5 B x)+2 a b x^2 (48 A+35 B x)+8 b^2 x^4 (6 A+5 B x)\right )-\frac{15 a^{5/2} B \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{\frac{b x^2}{a}+1}}\right )}{240 b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 94, normalized size = 0.9 \begin{align*}{\frac{Bx}{6\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{Bax}{24\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}Bx}{16\,b}\sqrt{b{x}^{2}+a}}-{\frac{B{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{A}{5\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{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 = 1.61261, size = 500, normalized size = 4.85 \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} + 70 \, B a b^{2} x^{3} + 96 \, A a b^{2} x^{2} + 15 \, B a^{2} b x + 48 \, A a^{2} b\right )} \sqrt{b x^{2} + a}}{480 \, b^{2}}, \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} + 70 \, B a b^{2} x^{3} + 96 \, A a b^{2} x^{2} + 15 \, B a^{2} b x + 48 \, A a^{2} b\right )} \sqrt{b x^{2} + a}}{240 \, b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.0323, size = 223, normalized size = 2.17 \begin{align*} A a \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: b = 0 \\\frac{\left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) + A b \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 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{17 B a^{\frac{3}{2}} x^{3}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{11 B \sqrt{a} b 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{3}{2}}} + \frac{B b^{2} 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 = 1.17139, size = 120, normalized size = 1.17 \begin{align*} \frac{B a^{3} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{16 \, b^{\frac{3}{2}}} + \frac{1}{240} \, \sqrt{b x^{2} + a}{\left (\frac{48 \, A a^{2}}{b} +{\left (\frac{15 \, B a^{2}}{b} + 2 \,{\left (48 \, A a +{\left (35 \, B a + 4 \,{\left (5 \, B b x + 6 \, A b\right )} x\right )} x\right )} x\right )} x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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