Optimal. Leaf size=87 \[ \frac{3 a^2 A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 \sqrt{c}}+\frac{1}{4} A x \left (a+c x^2\right )^{3/2}+\frac{3}{8} a A x \sqrt{a+c x^2}+\frac{B \left (a+c x^2\right )^{5/2}}{5 c} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0626748, 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 \[ \frac{3 a^2 A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 \sqrt{c}}+\frac{1}{4} A x \left (a+c x^2\right )^{3/2}+\frac{3}{8} a A x \sqrt{a+c x^2}+\frac{B \left (a+c x^2\right )^{5/2}}{5 c} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*(a + c*x^2)^(3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 8.35005, size = 80, normalized size = 0.92 \[ \frac{3 A a^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{8 \sqrt{c}} + \frac{3 A a x \sqrt{a + c x^{2}}}{8} + \frac{A x \left (a + c x^{2}\right )^{\frac{3}{2}}}{4} + \frac{B \left (a + c x^{2}\right )^{\frac{5}{2}}}{5 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+a)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.084575, 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.
[In] Integrate[(A + B*x)*(a + c*x^2)^(3/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.007, size = 69, normalized size = 0.8 \[{\frac{Ax}{4} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,aAx}{8}\sqrt{c{x}^{2}+a}}+{\frac{3\,A{a}^{2}}{8}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{B}{5\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((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, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.355813, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, A a^{2} c \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) + 2 \,{\left (8 \, B c^{2} x^{4} + 10 \, A c^{2} x^{3} + 16 \, B a c x^{2} + 25 \, A a c x + 8 \, B a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{c}}{80 \, c^{\frac{3}{2}}}, \frac{15 \, A a^{2} c \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) +{\left (8 \, B c^{2} x^{4} + 10 \, A c^{2} x^{3} + 16 \, B a c x^{2} + 25 \, A a c x + 8 \, B a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{-c}}{40 \, \sqrt{-c} c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(3/2)*(B*x + A),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 18.4172, size = 219, normalized size = 2.52 \[ \frac{A a^{\frac{3}{2}} x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{A a^{\frac{3}{2}} x}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 A \sqrt{a} c x^{3}}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 A a^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 \sqrt{c}} + \frac{A c^{2} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} + B a \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + B c \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{a x^{2} \sqrt{a + c x^{2}}}{15 c} + \frac{x^{4} \sqrt{a + c x^{2}}}{5} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+a)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.280137, size = 103, normalized size = 1.18 \[ -\frac{3 \, A a^{2}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{8 \, \sqrt{c}} + \frac{1}{40} \, \sqrt{c x^{2} + a}{\left (\frac{8 \, B a^{2}}{c} +{\left (25 \, A a + 2 \,{\left (8 \, B a +{\left (4 \, B c x + 5 \, A c\right )} x\right )} x\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(3/2)*(B*x + A),x, algorithm="giac")
[Out]