Optimal. Leaf size=105 \[ -\frac{\sqrt{a+c x^2} (16 a B-9 A c x)}{6 c^3}-\frac{x^3 (A+B x)}{c \sqrt{a+c x^2}}-\frac{3 a A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{5/2}}+\frac{4 B x^2 \sqrt{a+c x^2}}{3 c^2} \]
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Rubi [A] time = 0.0714329, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {819, 833, 780, 217, 206} \[ -\frac{\sqrt{a+c x^2} (16 a B-9 A c x)}{6 c^3}-\frac{x^3 (A+B x)}{c \sqrt{a+c x^2}}-\frac{3 a A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{5/2}}+\frac{4 B x^2 \sqrt{a+c x^2}}{3 c^2} \]
Antiderivative was successfully verified.
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Rule 819
Rule 833
Rule 780
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^4 (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx &=-\frac{x^3 (A+B x)}{c \sqrt{a+c x^2}}+\frac{\int \frac{x^2 (3 a A+4 a B x)}{\sqrt{a+c x^2}} \, dx}{a c}\\ &=-\frac{x^3 (A+B x)}{c \sqrt{a+c x^2}}+\frac{4 B x^2 \sqrt{a+c x^2}}{3 c^2}+\frac{\int \frac{x \left (-8 a^2 B+9 a A c x\right )}{\sqrt{a+c x^2}} \, dx}{3 a c^2}\\ &=-\frac{x^3 (A+B x)}{c \sqrt{a+c x^2}}+\frac{4 B x^2 \sqrt{a+c x^2}}{3 c^2}-\frac{(16 a B-9 A c x) \sqrt{a+c x^2}}{6 c^3}-\frac{(3 a A) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{2 c^2}\\ &=-\frac{x^3 (A+B x)}{c \sqrt{a+c x^2}}+\frac{4 B x^2 \sqrt{a+c x^2}}{3 c^2}-\frac{(16 a B-9 A c x) \sqrt{a+c x^2}}{6 c^3}-\frac{(3 a A) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{2 c^2}\\ &=-\frac{x^3 (A+B x)}{c \sqrt{a+c x^2}}+\frac{4 B x^2 \sqrt{a+c x^2}}{3 c^2}-\frac{(16 a B-9 A c x) \sqrt{a+c x^2}}{6 c^3}-\frac{3 a A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0482126, size = 91, normalized size = 0.87 \[ \frac{-16 a^2 B+a c x (9 A-8 B x)-9 a A \sqrt{c} \sqrt{a+c x^2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )+c^2 x^3 (3 A+2 B x)}{6 c^3 \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 115, normalized size = 1.1 \begin{align*}{\frac{{x}^{4}B}{3\,c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{4\,aB{x}^{2}}{3\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{8\,B{a}^{2}}{3\,{c}^{3}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{A{x}^{3}}{2\,c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{3\,aAx}{2\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{3\,aA}{2}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\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 = 1.51209, size = 495, normalized size = 4.71 \begin{align*} \left [\frac{9 \,{\left (A a c x^{2} + A a^{2}\right )} \sqrt{c} \log \left (-2 \, c x^{2} + 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \,{\left (2 \, B c^{2} x^{4} + 3 \, A c^{2} x^{3} - 8 \, B a c x^{2} + 9 \, A a c x - 16 \, B a^{2}\right )} \sqrt{c x^{2} + a}}{12 \,{\left (c^{4} x^{2} + a c^{3}\right )}}, \frac{9 \,{\left (A a c x^{2} + A a^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) +{\left (2 \, B c^{2} x^{4} + 3 \, A c^{2} x^{3} - 8 \, B a c x^{2} + 9 \, A a c x - 16 \, B a^{2}\right )} \sqrt{c x^{2} + a}}{6 \,{\left (c^{4} x^{2} + a c^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 11.7048, size = 144, normalized size = 1.37 \begin{align*} A \left (\frac{3 \sqrt{a} x}{2 c^{2} \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{3 a \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 c^{\frac{5}{2}}} + \frac{x^{3}}{2 \sqrt{a} c \sqrt{1 + \frac{c x^{2}}{a}}}\right ) + B \left (\begin{cases} - \frac{8 a^{2}}{3 c^{3} \sqrt{a + c x^{2}}} - \frac{4 a x^{2}}{3 c^{2} \sqrt{a + c x^{2}}} + \frac{x^{4}}{3 c \sqrt{a + c x^{2}}} & \text{for}\: c \neq 0 \\\frac{x^{6}}{6 a^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22249, size = 112, normalized size = 1.07 \begin{align*} \frac{{\left ({\left ({\left (\frac{2 \, B x}{c} + \frac{3 \, A}{c}\right )} x - \frac{8 \, B a}{c^{2}}\right )} x + \frac{9 \, A a}{c^{2}}\right )} x - \frac{16 \, B a^{2}}{c^{3}}}{6 \, \sqrt{c x^{2} + a}} + \frac{3 \, A a \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{2 \, c^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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