Optimal. Leaf size=87 \[ \frac{a (4 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{3/2}}+\frac{x \sqrt{a+b x^2} (4 A b-a B)}{8 b}+\frac{B x \left (a+b x^2\right )^{3/2}}{4 b} \]
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Rubi [A] time = 0.028411, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {388, 195, 217, 206} \[ \frac{a (4 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{3/2}}+\frac{x \sqrt{a+b x^2} (4 A b-a B)}{8 b}+\frac{B x \left (a+b x^2\right )^{3/2}}{4 b} \]
Antiderivative was successfully verified.
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Rule 388
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \sqrt{a+b x^2} \left (A+B x^2\right ) \, dx &=\frac{B x \left (a+b x^2\right )^{3/2}}{4 b}-\frac{(-4 A b+a B) \int \sqrt{a+b x^2} \, dx}{4 b}\\ &=\frac{(4 A b-a B) x \sqrt{a+b x^2}}{8 b}+\frac{B x \left (a+b x^2\right )^{3/2}}{4 b}+\frac{(a (4 A b-a B)) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{8 b}\\ &=\frac{(4 A b-a B) x \sqrt{a+b x^2}}{8 b}+\frac{B x \left (a+b x^2\right )^{3/2}}{4 b}+\frac{(a (4 A b-a B)) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{8 b}\\ &=\frac{(4 A b-a B) x \sqrt{a+b x^2}}{8 b}+\frac{B x \left (a+b x^2\right )^{3/2}}{4 b}+\frac{a (4 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.147091, size = 85, normalized size = 0.98 \[ \frac{\sqrt{a+b x^2} \left (\sqrt{b} x \left (B \left (a+2 b x^2\right )+4 A b\right )-\frac{\sqrt{a} (a B-4 A b) \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{\frac{b x^2}{a}+1}}\right )}{8 b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 96, normalized size = 1.1 \begin{align*}{\frac{Bx}{4\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{Bax}{8\,b}\sqrt{b{x}^{2}+a}}-{\frac{{a}^{2}B}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{Ax}{2}\sqrt{b{x}^{2}+a}}+{\frac{Aa}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}} \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 = 1.78891, size = 370, normalized size = 4.25 \begin{align*} \left [-\frac{{\left (B a^{2} - 4 \, A a b\right )} \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left (2 \, B b^{2} x^{3} +{\left (B a b + 4 \, A b^{2}\right )} x\right )} \sqrt{b x^{2} + a}}{16 \, b^{2}}, \frac{{\left (B a^{2} - 4 \, A a b\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (2 \, B b^{2} x^{3} +{\left (B a b + 4 \, A b^{2}\right )} x\right )} \sqrt{b x^{2} + a}}{8 \, b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.16794, size = 144, normalized size = 1.66 \begin{align*} \frac{A \sqrt{a} x \sqrt{1 + \frac{b x^{2}}{a}}}{2} + \frac{A a \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 \sqrt{b}} + \frac{B a^{\frac{3}{2}} x}{8 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 B \sqrt{a} x^{3}}{8 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{3}{2}}} + \frac{B b x^{5}}{4 \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 = 1.12908, size = 93, normalized size = 1.07 \begin{align*} \frac{1}{8} \,{\left (2 \, B x^{2} + \frac{B a b + 4 \, A b^{2}}{b^{2}}\right )} \sqrt{b x^{2} + a} x + \frac{{\left (B a^{2} - 4 \, A a b\right )} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{8 \, b^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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