Optimal. Leaf size=41 \[ \frac{B \sqrt{a+b x^2}}{b^2}-\frac{A b-a B}{b^2 \sqrt{a+b x^2}} \]
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Rubi [A] time = 0.0977305, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{B \sqrt{a+b x^2}}{b^2}-\frac{A b-a B}{b^2 \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Int[(x*(A + B*x^2))/(a + b*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 12.9567, size = 34, normalized size = 0.83 \[ \frac{B \sqrt{a + b x^{2}}}{b^{2}} - \frac{A b - B a}{b^{2} \sqrt{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(B*x**2+A)/(b*x**2+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0242189, size = 30, normalized size = 0.73 \[ \frac{2 a B-A b+b B x^2}{b^2 \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(A + B*x^2))/(a + b*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.006, size = 30, normalized size = 0.7 \[ -{\frac{-bB{x}^{2}+Ab-2\,Ba}{{b}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(B*x^2+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^2 + A)*x/(b*x^2 + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.229803, size = 54, normalized size = 1.32 \[ \frac{{\left (B b x^{2} + 2 \, B a - A b\right )} \sqrt{b x^{2} + a}}{b^{3} x^{2} + a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x/(b*x^2 + a)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.2509, size = 66, normalized size = 1.61 \[ \begin{cases} - \frac{A}{b \sqrt{a + b x^{2}}} + \frac{2 B a}{b^{2} \sqrt{a + b x^{2}}} + \frac{B x^{2}}{b \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{2}}{2} + \frac{B x^{4}}{4}}{a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(B*x**2+A)/(b*x**2+a)**(3/2),x)
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GIAC/XCAS [A] time = 0.227732, size = 46, normalized size = 1.12 \[ \frac{\sqrt{b x^{2} + a} B + \frac{B a - A b}{\sqrt{b x^{2} + a}}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x/(b*x^2 + a)^(3/2),x, algorithm="giac")
[Out]