Optimal. Leaf size=66 \[ \frac{A \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{3/2}}-\frac{x (A+B x)}{b \sqrt{a+b x^2}}+\frac{2 B \sqrt{a+b x^2}}{b^2} \]
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Rubi [A] time = 0.13668, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{A \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{3/2}}-\frac{x (A+B x)}{b \sqrt{a+b x^2}}+\frac{2 B \sqrt{a+b x^2}}{b^2} \]
Antiderivative was successfully verified.
[In] Int[(x^2*(A + B*x))/(a + b*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 14.7082, size = 65, normalized size = 0.98 \[ \frac{A \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{b^{\frac{3}{2}}} + \frac{2 B \sqrt{a + b x^{2}}}{b^{2}} - \frac{x \left (2 A + 2 B x\right )}{2 b \sqrt{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(B*x+A)/(b*x**2+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.105024, size = 60, normalized size = 0.91 \[ \frac{A \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{b^{3/2}}+\frac{2 a B+b x (B x-A)}{b^2 \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^2*(A + B*x))/(a + b*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.009, size = 72, normalized size = 1.1 \[ -{\frac{Ax}{b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{A\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{B{x}^{2}}{b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+2\,{\frac{Ba}{{b}^{2}\sqrt{b{x}^{2}+a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(B*x+A)/(b*x^2+a)^(3/2),x)
[Out]
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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((B*x + A)*x^2/(b*x^2 + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.26142, size = 1, normalized size = 0.02 \[ \left [\frac{2 \,{\left (B b x^{2} - A b x + 2 \, B a\right )} \sqrt{b x^{2} + a} \sqrt{b} +{\left (A b^{2} x^{2} + A a b\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{2 \,{\left (b^{3} x^{2} + a b^{2}\right )} \sqrt{b}}, \frac{{\left (B b x^{2} - A b x + 2 \, B a\right )} \sqrt{b x^{2} + a} \sqrt{-b} +{\left (A b^{2} x^{2} + A a b\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{{\left (b^{3} x^{2} + a b^{2}\right )} \sqrt{-b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^2/(b*x^2 + a)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.06385, size = 83, normalized size = 1.26 \[ A \left (\frac{\operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{b^{\frac{3}{2}}} - \frac{x}{\sqrt{a} b \sqrt{1 + \frac{b x^{2}}{a}}}\right ) + B \left (\begin{cases} \frac{2 a}{b^{2} \sqrt{a + b x^{2}}} + \frac{x^{2}}{b \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(B*x+A)/(b*x**2+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.220276, size = 78, normalized size = 1.18 \[ \frac{{\left (\frac{B x}{b} - \frac{A}{b}\right )} x + \frac{2 \, B a}{b^{2}}}{\sqrt{b x^{2} + a}} - \frac{A{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^2/(b*x^2 + a)^(3/2),x, algorithm="giac")
[Out]