Optimal. Leaf size=150 \[ \frac{3 a^4 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{128 c^{5/2}}+\frac{3 a^3 B x \sqrt{a+c x^2}}{128 c^2}+\frac{a^2 B x \left (a+c x^2\right )^{3/2}}{64 c^2}-\frac{a \left (a+c x^2\right )^{5/2} (32 A+35 B x)}{560 c^2}+\frac{A x^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac{B x^3 \left (a+c x^2\right )^{5/2}}{8 c} \]
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Rubi [A] time = 0.279377, 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 \[ \frac{3 a^4 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{128 c^{5/2}}+\frac{3 a^3 B x \sqrt{a+c x^2}}{128 c^2}+\frac{a^2 B x \left (a+c x^2\right )^{3/2}}{64 c^2}-\frac{a \left (a+c x^2\right )^{5/2} (32 A+35 B x)}{560 c^2}+\frac{A x^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac{B x^3 \left (a+c x^2\right )^{5/2}}{8 c} \]
Antiderivative was successfully verified.
[In] Int[x^3*(A + B*x)*(a + c*x^2)^(3/2),x]
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Rubi in Sympy [A] time = 29.5437, size = 139, normalized size = 0.93 \[ \frac{A x^{2} \left (a + c x^{2}\right )^{\frac{5}{2}}}{7 c} + \frac{3 B a^{4} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{128 c^{\frac{5}{2}}} + \frac{3 B a^{3} x \sqrt{a + c x^{2}}}{128 c^{2}} + \frac{B a^{2} x \left (a + c x^{2}\right )^{\frac{3}{2}}}{64 c^{2}} + \frac{B x^{3} \left (a + c x^{2}\right )^{\frac{5}{2}}}{8 c} - \frac{a \left (96 A + 105 B x\right ) \left (a + c x^{2}\right )^{\frac{5}{2}}}{1680 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(B*x+A)*(c*x**2+a)**(3/2),x)
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Mathematica [A] time = 0.134418, size = 119, normalized size = 0.79 \[ \frac{105 a^4 B \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )+\sqrt{c} \sqrt{a+c x^2} \left (-a^3 (256 A+105 B x)+2 a^2 c x^2 (64 A+35 B x)+8 a c^2 x^4 (128 A+105 B x)+80 c^3 x^6 (8 A+7 B x)\right )}{4480 c^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^3*(A + B*x)*(a + c*x^2)^(3/2),x]
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Maple [A] time = 0.01, size = 134, normalized size = 0.9 \[{\frac{A{x}^{2}}{7\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{2\,aA}{35\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{B{x}^{3}}{8\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{aBx}{16\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{2}Bx}{64\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{a}^{3}Bx}{128\,{c}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{3\,B{a}^{4}}{128}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(B*x+A)*(c*x^2+a)^(3/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(3/2)*(B*x + A)*x^3,x, algorithm="maxima")
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Fricas [A] time = 0.382751, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, B a^{4} \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) + 2 \,{\left (560 \, B c^{3} x^{7} + 640 \, A c^{3} x^{6} + 840 \, B a c^{2} x^{5} + 1024 \, A a c^{2} x^{4} + 70 \, B a^{2} c x^{3} + 128 \, A a^{2} c x^{2} - 105 \, B a^{3} x - 256 \, A a^{3}\right )} \sqrt{c x^{2} + a} \sqrt{c}}{8960 \, c^{\frac{5}{2}}}, \frac{105 \, B a^{4} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) +{\left (560 \, B c^{3} x^{7} + 640 \, A c^{3} x^{6} + 840 \, B a c^{2} x^{5} + 1024 \, A a c^{2} x^{4} + 70 \, B a^{2} c x^{3} + 128 \, A a^{2} c x^{2} - 105 \, B a^{3} x - 256 \, A a^{3}\right )} \sqrt{c x^{2} + a} \sqrt{-c}}{4480 \, \sqrt{-c} c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(3/2)*(B*x + A)*x^3,x, algorithm="fricas")
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Sympy [A] time = 48.9797, size = 318, normalized size = 2.12 \[ A a \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{a x^{2} \sqrt{a + c x^{2}}}{15 c} + \frac{x^{4} \sqrt{a + c x^{2}}}{5} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + A c \left (\begin{cases} \frac{8 a^{3} \sqrt{a + c x^{2}}}{105 c^{3}} - \frac{4 a^{2} x^{2} \sqrt{a + c x^{2}}}{105 c^{2}} + \frac{a x^{4} \sqrt{a + c x^{2}}}{35 c} + \frac{x^{6} \sqrt{a + c x^{2}}}{7} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{6}}{6} & \text{otherwise} \end{cases}\right ) - \frac{3 B a^{\frac{7}{2}} x}{128 c^{2} \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{B a^{\frac{5}{2}} x^{3}}{128 c \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{13 B a^{\frac{3}{2}} x^{5}}{64 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{5 B \sqrt{a} c x^{7}}{16 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 B a^{4} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{128 c^{\frac{5}{2}}} + \frac{B c^{2} x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(B*x+A)*(c*x**2+a)**(3/2),x)
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GIAC/XCAS [A] time = 0.275331, size = 155, normalized size = 1.03 \[ -\frac{3 \, B a^{4}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{128 \, c^{\frac{5}{2}}} - \frac{1}{4480} \, \sqrt{c x^{2} + a}{\left (\frac{256 \, A a^{3}}{c^{2}} +{\left (\frac{105 \, B a^{3}}{c^{2}} - 2 \,{\left (\frac{64 \, A a^{2}}{c} +{\left (\frac{35 \, B a^{2}}{c} + 4 \,{\left (128 \, A a + 5 \,{\left (21 \, B a + 2 \,{\left (7 \, B c x + 8 \, A c\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(3/2)*(B*x + A)*x^3,x, algorithm="giac")
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