Optimal. Leaf size=103 \[ -\frac{a^3 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{3/2}}-\frac{a^2 B x \sqrt{a+c x^2}}{16 c}+\frac{\left (a+c x^2\right )^{5/2} (6 A+5 B x)}{30 c}-\frac{a B x \left (a+c x^2\right )^{3/2}}{24 c} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0924524, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{a^3 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{3/2}}-\frac{a^2 B x \sqrt{a+c x^2}}{16 c}+\frac{\left (a+c x^2\right )^{5/2} (6 A+5 B x)}{30 c}-\frac{a B x \left (a+c x^2\right )^{3/2}}{24 c} \]
Antiderivative was successfully verified.
[In] Int[x*(A + B*x)*(a + c*x^2)^(3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 10.2918, size = 90, normalized size = 0.87 \[ - \frac{B a^{3} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{16 c^{\frac{3}{2}}} - \frac{B a^{2} x \sqrt{a + c x^{2}}}{16 c} - \frac{B a x \left (a + c x^{2}\right )^{\frac{3}{2}}}{24 c} + \frac{\left (6 A + 5 B x\right ) \left (a + c x^{2}\right )^{\frac{5}{2}}}{30 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(B*x+A)*(c*x**2+a)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.110718, size = 100, normalized size = 0.97 \[ \frac{\sqrt{c} \sqrt{a+c x^2} \left (3 a^2 (16 A+5 B x)+2 a c x^2 (48 A+35 B x)+8 c^2 x^4 (6 A+5 B x)\right )-15 a^3 B \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{240 c^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x*(A + B*x)*(a + c*x^2)^(3/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.007, size = 94, normalized size = 0.9 \[{\frac{A}{5\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Bx}{6\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{aBx}{24\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}Bx}{16\,c}\sqrt{c{x}^{2}+a}}-{\frac{B{a}^{3}}{16}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(B*x+A)*(c*x^2+a)^(3/2),x)
[Out]
_______________________________________________________________________________________
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,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.393439, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, B a^{3} \log \left (2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) + 2 \,{\left (40 \, B c^{2} x^{5} + 48 \, A c^{2} x^{4} + 70 \, B a c x^{3} + 96 \, A a c x^{2} + 15 \, B a^{2} x + 48 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{c}}{480 \, c^{\frac{3}{2}}}, -\frac{15 \, B a^{3} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (40 \, B c^{2} x^{5} + 48 \, A c^{2} x^{4} + 70 \, B a c x^{3} + 96 \, A a c x^{2} + 15 \, B a^{2} x + 48 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{-c}}{240 \, \sqrt{-c} c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(3/2)*(B*x + A)*x,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 30.6655, size = 223, normalized size = 2.17 \[ A a \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + A c \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{a x^{2} \sqrt{a + c x^{2}}}{15 c} + \frac{x^{4} \sqrt{a + c x^{2}}}{5} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + \frac{B a^{\frac{5}{2}} x}{16 c \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{17 B a^{\frac{3}{2}} x^{3}}{48 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{11 B \sqrt{a} c x^{5}}{24 \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{B a^{3} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{16 c^{\frac{3}{2}}} + \frac{B c^{2} x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(B*x+A)*(c*x**2+a)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.275136, size = 120, normalized size = 1.17 \[ \frac{B a^{3}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{16 \, c^{\frac{3}{2}}} + \frac{1}{240} \, \sqrt{c x^{2} + a}{\left (\frac{48 \, A a^{2}}{c} +{\left (\frac{15 \, B a^{2}}{c} + 2 \,{\left (48 \, A a +{\left (35 \, B a + 4 \,{\left (5 \, B c x + 6 \, A c\right )} x\right )} x\right )} x\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(3/2)*(B*x + A)*x,x, algorithm="giac")
[Out]