Optimal. Leaf size=126 \[ -\frac{5 a^4 B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{3/2}}-\frac{5 a^3 B x \sqrt{a+b x^2}}{128 b}-\frac{5 a^2 B x \left (a+b x^2\right )^{3/2}}{192 b}+\frac{\left (a+b x^2\right )^{7/2} (8 A+7 B x)}{56 b}-\frac{a B x \left (a+b x^2\right )^{5/2}}{48 b} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.131859, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{5 a^4 B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{3/2}}-\frac{5 a^3 B x \sqrt{a+b x^2}}{128 b}-\frac{5 a^2 B x \left (a+b x^2\right )^{3/2}}{192 b}+\frac{\left (a+b x^2\right )^{7/2} (8 A+7 B x)}{56 b}-\frac{a B x \left (a+b x^2\right )^{5/2}}{48 b} \]
Antiderivative was successfully verified.
[In] Int[x*(A + B*x)*(a + b*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 13.4098, size = 116, normalized size = 0.92 \[ - \frac{5 B a^{4} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{128 b^{\frac{3}{2}}} - \frac{5 B a^{3} x \sqrt{a + b x^{2}}}{128 b} - \frac{5 B a^{2} x \left (a + b x^{2}\right )^{\frac{3}{2}}}{192 b} - \frac{B a x \left (a + b x^{2}\right )^{\frac{5}{2}}}{48 b} + \frac{\left (8 A + 7 B x\right ) \left (a + b x^{2}\right )^{\frac{7}{2}}}{56 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(B*x+A)*(b*x**2+a)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.146143, size = 119, normalized size = 0.94 \[ \frac{\sqrt{b} \sqrt{a+b x^2} \left (3 a^3 (128 A+35 B x)+2 a^2 b x^2 (576 A+413 B x)+8 a b^2 x^4 (144 A+119 B x)+48 b^3 x^6 (8 A+7 B x)\right )-105 a^4 B \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{2688 b^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x*(A + B*x)*(a + b*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.007, size = 113, normalized size = 0.9 \[{\frac{A}{7\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{Bx}{8\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{Bxa}{48\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,Bx{a}^{2}}{192\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{a}^{3}Bx}{128\,b}\sqrt{b{x}^{2}+a}}-{\frac{5\,B{a}^{4}}{128}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(B*x+A)*(b*x^2+a)^(5/2),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((b*x^2 + a)^(5/2)*(B*x + A)*x,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.267743, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, B a^{4} \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right ) + 2 \,{\left (336 \, B b^{3} x^{7} + 384 \, A b^{3} x^{6} + 952 \, B a b^{2} x^{5} + 1152 \, A a b^{2} x^{4} + 826 \, B a^{2} b x^{3} + 1152 \, A a^{2} b x^{2} + 105 \, B a^{3} x + 384 \, A a^{3}\right )} \sqrt{b x^{2} + a} \sqrt{b}}{5376 \, b^{\frac{3}{2}}}, -\frac{105 \, B a^{4} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (336 \, B b^{3} x^{7} + 384 \, A b^{3} x^{6} + 952 \, B a b^{2} x^{5} + 1152 \, A a b^{2} x^{4} + 826 \, B a^{2} b x^{3} + 1152 \, A a^{2} b x^{2} + 105 \, B a^{3} x + 384 \, A a^{3}\right )} \sqrt{b x^{2} + a} \sqrt{-b}}{2688 \, \sqrt{-b} b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(5/2)*(B*x + A)*x,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 30.001, size = 354, normalized size = 2.81 \[ A a^{2} \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: b = 0 \\\frac{\left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) + 2 A a b \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + b x^{2}}}{15 b^{2}} + \frac{a x^{2} \sqrt{a + b x^{2}}}{15 b} + \frac{x^{4} \sqrt{a + b x^{2}}}{5} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + A b^{2} \left (\begin{cases} \frac{8 a^{3} \sqrt{a + b x^{2}}}{105 b^{3}} - \frac{4 a^{2} x^{2} \sqrt{a + b x^{2}}}{105 b^{2}} + \frac{a x^{4} \sqrt{a + b x^{2}}}{35 b} + \frac{x^{6} \sqrt{a + b x^{2}}}{7} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{6}}{6} & \text{otherwise} \end{cases}\right ) + \frac{5 B a^{\frac{7}{2}} x}{128 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{133 B a^{\frac{5}{2}} x^{3}}{384 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{127 B a^{\frac{3}{2}} b x^{5}}{192 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{23 B \sqrt{a} b^{2} x^{7}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{5 B a^{4} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{128 b^{\frac{3}{2}}} + \frac{B b^{3} x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(B*x+A)*(b*x**2+a)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.230988, size = 154, normalized size = 1.22 \[ \frac{5 \, B a^{4}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{128 \, b^{\frac{3}{2}}} + \frac{1}{2688} \,{\left (\frac{384 \, A a^{3}}{b} +{\left (\frac{105 \, B a^{3}}{b} + 2 \,{\left (576 \, A a^{2} +{\left (413 \, B a^{2} + 4 \,{\left (144 \, A a b +{\left (119 \, B a b + 6 \,{\left (7 \, B b^{2} x + 8 \, A b^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{b x^{2} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(5/2)*(B*x + A)*x,x, algorithm="giac")
[Out]