Optimal. Leaf size=150 \[ \frac{3 a^3 B x \sqrt{a+c x^2}}{128 c^2}+\frac{a^2 B x \left (a+c x^2\right )^{3/2}}{64 c^2}+\frac{3 a^4 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{128 c^{5/2}}-\frac{a \left (a+c x^2\right )^{5/2} (32 A+35 B x)}{560 c^2}+\frac{A x^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac{B x^3 \left (a+c x^2\right )^{5/2}}{8 c} \]
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Rubi [A] time = 0.0943001, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{3 a^3 B x \sqrt{a+c x^2}}{128 c^2}+\frac{a^2 B x \left (a+c x^2\right )^{3/2}}{64 c^2}+\frac{3 a^4 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{128 c^{5/2}}-\frac{a \left (a+c x^2\right )^{5/2} (32 A+35 B x)}{560 c^2}+\frac{A x^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac{B x^3 \left (a+c x^2\right )^{5/2}}{8 c} \]
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) \left (a+c x^2\right )^{3/2} \, dx &=\frac{B x^3 \left (a+c x^2\right )^{5/2}}{8 c}+\frac{\int x^2 (-3 a B+8 A c x) \left (a+c x^2\right )^{3/2} \, dx}{8 c}\\ &=\frac{A x^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac{B x^3 \left (a+c x^2\right )^{5/2}}{8 c}+\frac{\int x (-16 a A c-21 a B c x) \left (a+c x^2\right )^{3/2} \, dx}{56 c^2}\\ &=\frac{A x^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac{B x^3 \left (a+c x^2\right )^{5/2}}{8 c}-\frac{a (32 A+35 B x) \left (a+c x^2\right )^{5/2}}{560 c^2}+\frac{\left (a^2 B\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{16 c^2}\\ &=\frac{a^2 B x \left (a+c x^2\right )^{3/2}}{64 c^2}+\frac{A x^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac{B x^3 \left (a+c x^2\right )^{5/2}}{8 c}-\frac{a (32 A+35 B x) \left (a+c x^2\right )^{5/2}}{560 c^2}+\frac{\left (3 a^3 B\right ) \int \sqrt{a+c x^2} \, dx}{64 c^2}\\ &=\frac{3 a^3 B x \sqrt{a+c x^2}}{128 c^2}+\frac{a^2 B x \left (a+c x^2\right )^{3/2}}{64 c^2}+\frac{A x^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac{B x^3 \left (a+c x^2\right )^{5/2}}{8 c}-\frac{a (32 A+35 B x) \left (a+c x^2\right )^{5/2}}{560 c^2}+\frac{\left (3 a^4 B\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{128 c^2}\\ &=\frac{3 a^3 B x \sqrt{a+c x^2}}{128 c^2}+\frac{a^2 B x \left (a+c x^2\right )^{3/2}}{64 c^2}+\frac{A x^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac{B x^3 \left (a+c x^2\right )^{5/2}}{8 c}-\frac{a (32 A+35 B x) \left (a+c x^2\right )^{5/2}}{560 c^2}+\frac{\left (3 a^4 B\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{128 c^2}\\ &=\frac{3 a^3 B x \sqrt{a+c x^2}}{128 c^2}+\frac{a^2 B x \left (a+c x^2\right )^{3/2}}{64 c^2}+\frac{A x^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac{B x^3 \left (a+c x^2\right )^{5/2}}{8 c}-\frac{a (32 A+35 B x) \left (a+c x^2\right )^{5/2}}{560 c^2}+\frac{3 a^4 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{128 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.23206, size = 126, normalized size = 0.84 \[ \frac{\sqrt{a+c x^2} \left (\sqrt{c} \left (2 a^2 c x^2 (64 A+35 B x)-a^3 (256 A+105 B x)+8 a c^2 x^4 (128 A+105 B x)+80 c^3 x^6 (8 A+7 B x)\right )+\frac{105 a^{7/2} B \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{\frac{c x^2}{a}+1}}\right )}{4480 c^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 134, normalized size = 0.9 \begin{align*}{\frac{B{x}^{3}}{8\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{aBx}{16\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{2}Bx}{64\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{a}^{3}Bx}{128\,{c}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{3\,B{a}^{4}}{128}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{A{x}^{2}}{7\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{2\,aA}{35\,{c}^{2}} \left ( c{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 = 1.85697, size = 630, normalized size = 4.2 \begin{align*} \left [\frac{105 \, B a^{4} \sqrt{c} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \,{\left (560 \, B c^{4} x^{7} + 640 \, A c^{4} x^{6} + 840 \, B a c^{3} x^{5} + 1024 \, A a c^{3} x^{4} + 70 \, B a^{2} c^{2} x^{3} + 128 \, A a^{2} c^{2} x^{2} - 105 \, B a^{3} c x - 256 \, A a^{3} c\right )} \sqrt{c x^{2} + a}}{8960 \, c^{3}}, -\frac{105 \, B a^{4} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (560 \, B c^{4} x^{7} + 640 \, A c^{4} x^{6} + 840 \, B a c^{3} x^{5} + 1024 \, A a c^{3} x^{4} + 70 \, B a^{2} c^{2} x^{3} + 128 \, A a^{2} c^{2} x^{2} - 105 \, B a^{3} c x - 256 \, A a^{3} c\right )} \sqrt{c x^{2} + a}}{4480 \, c^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 18.9878, size = 318, normalized size = 2.12 \begin{align*} A a \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 ) + A c \left (\begin{cases} \frac{8 a^{3} \sqrt{a + c x^{2}}}{105 c^{3}} - \frac{4 a^{2} x^{2} \sqrt{a + c x^{2}}}{105 c^{2}} + \frac{a x^{4} \sqrt{a + c x^{2}}}{35 c} + \frac{x^{6} \sqrt{a + c x^{2}}}{7} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{6}}{6} & \text{otherwise} \end{cases}\right ) - \frac{3 B a^{\frac{7}{2}} x}{128 c^{2} \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{B a^{\frac{5}{2}} x^{3}}{128 c \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{13 B a^{\frac{3}{2}} x^{5}}{64 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{5 B \sqrt{a} c x^{7}}{16 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 B a^{4} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{128 c^{\frac{5}{2}}} + \frac{B c^{2} x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15344, size = 155, normalized size = 1.03 \begin{align*} -\frac{3 \, B a^{4} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{128 \, c^{\frac{5}{2}}} - \frac{1}{4480} \, \sqrt{c x^{2} + a}{\left (\frac{256 \, A a^{3}}{c^{2}} +{\left (\frac{105 \, B a^{3}}{c^{2}} - 2 \,{\left (\frac{64 \, A a^{2}}{c} +{\left (\frac{35 \, B a^{2}}{c} + 4 \,{\left (128 \, A a + 5 \,{\left (21 \, B a + 2 \,{\left (7 \, B c x + 8 \, A c\right )} x\right )} x\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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