Optimal. Leaf size=87 \[ \frac{3 a^2 A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 \sqrt{c}}+\frac{1}{4} A x \left (a+c x^2\right )^{3/2}+\frac{3}{8} a A x \sqrt{a+c x^2}+\frac{B \left (a+c x^2\right )^{5/2}}{5 c} \]
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Rubi [A] time = 0.0257915, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {641, 195, 217, 206} \[ \frac{3 a^2 A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 \sqrt{c}}+\frac{1}{4} A x \left (a+c x^2\right )^{3/2}+\frac{3}{8} a A x \sqrt{a+c x^2}+\frac{B \left (a+c x^2\right )^{5/2}}{5 c} \]
Antiderivative was successfully verified.
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Rule 641
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int (A+B x) \left (a+c x^2\right )^{3/2} \, dx &=\frac{B \left (a+c x^2\right )^{5/2}}{5 c}+A \int \left (a+c x^2\right )^{3/2} \, dx\\ &=\frac{1}{4} A x \left (a+c x^2\right )^{3/2}+\frac{B \left (a+c x^2\right )^{5/2}}{5 c}+\frac{1}{4} (3 a A) \int \sqrt{a+c x^2} \, dx\\ &=\frac{3}{8} a A x \sqrt{a+c x^2}+\frac{1}{4} A x \left (a+c x^2\right )^{3/2}+\frac{B \left (a+c x^2\right )^{5/2}}{5 c}+\frac{1}{8} \left (3 a^2 A\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx\\ &=\frac{3}{8} a A x \sqrt{a+c x^2}+\frac{1}{4} A x \left (a+c x^2\right )^{3/2}+\frac{B \left (a+c x^2\right )^{5/2}}{5 c}+\frac{1}{8} \left (3 a^2 A\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )\\ &=\frac{3}{8} a A x \sqrt{a+c x^2}+\frac{1}{4} A x \left (a+c x^2\right )^{3/2}+\frac{B \left (a+c x^2\right )^{5/2}}{5 c}+\frac{3 a^2 A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0685126, size = 88, normalized size = 1.01 \[ \frac{\sqrt{a+c x^2} \left (8 a^2 B+a c x (25 A+16 B x)+2 c^2 x^3 (5 A+4 B x)\right )+15 a^2 A \sqrt{c} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{40 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 69, normalized size = 0.8 \begin{align*}{\frac{B}{5\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Ax}{4} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,aAx}{8}\sqrt{c{x}^{2}+a}}+{\frac{3\,A{a}^{2}}{8}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}} \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 = 1.87056, size = 425, normalized size = 4.89 \begin{align*} \left [\frac{15 \, A a^{2} \sqrt{c} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \,{\left (8 \, B c^{2} x^{4} + 10 \, A c^{2} x^{3} + 16 \, B a c x^{2} + 25 \, A a c x + 8 \, B a^{2}\right )} \sqrt{c x^{2} + a}}{80 \, c}, -\frac{15 \, A a^{2} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (8 \, B c^{2} x^{4} + 10 \, A c^{2} x^{3} + 16 \, B a c x^{2} + 25 \, A a c x + 8 \, B a^{2}\right )} \sqrt{c x^{2} + a}}{40 \, c}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.90037, size = 219, normalized size = 2.52 \begin{align*} \frac{A a^{\frac{3}{2}} x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{A a^{\frac{3}{2}} x}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 A \sqrt{a} c x^{3}}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 A a^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 \sqrt{c}} + \frac{A c^{2} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} + B a \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + B c \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{a x^{2} \sqrt{a + c x^{2}}}{15 c} + \frac{x^{4} \sqrt{a + c x^{2}}}{5} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12054, size = 103, normalized size = 1.18 \begin{align*} -\frac{3 \, A a^{2} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{8 \, \sqrt{c}} + \frac{1}{40} \, \sqrt{c x^{2} + a}{\left (\frac{8 \, B a^{2}}{c} +{\left (25 \, A a + 2 \,{\left (8 \, B a +{\left (4 \, B c x + 5 \, A c\right )} x\right )} x\right )} x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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