Optimal. Leaf size=44 \[ -\frac{A b-a B}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac{B}{b^2 \sqrt{a+b x^2}} \]
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Rubi [A] time = 0.101192, antiderivative size = 44, 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 \[ -\frac{A b-a B}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac{B}{b^2 \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Int[(x*(A + B*x^2))/(a + b*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 13.1774, size = 37, normalized size = 0.84 \[ - \frac{B}{b^{2} \sqrt{a + b x^{2}}} - \frac{A b - B a}{3 b^{2} \left (a + b x^{2}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(B*x**2+A)/(b*x**2+a)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0264469, size = 34, normalized size = 0.77 \[ \frac{-2 a B-A b-3 b B x^2}{3 b^2 \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(A + B*x^2))/(a + b*x^2)^(5/2),x]
[Out]
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Maple [A] time = 0.006, size = 30, normalized size = 0.7 \[ -{\frac{3\,bB{x}^{2}+Ab+2\,Ba}{3\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(B*x^2+A)/(b*x^2+a)^(5/2),x)
[Out]
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Maxima [A] time = 1.35378, size = 68, normalized size = 1.55 \[ -\frac{B x^{2}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}} b} - \frac{2 \, B a}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b^{2}} - \frac{A}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x/(b*x^2 + a)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.213887, size = 70, normalized size = 1.59 \[ -\frac{{\left (3 \, B b x^{2} + 2 \, B a + A b\right )} \sqrt{b x^{2} + a}}{3 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x/(b*x^2 + a)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.85138, size = 143, normalized size = 3.25 \[ \begin{cases} - \frac{A b}{3 a b^{2} \sqrt{a + b x^{2}} + 3 b^{3} x^{2} \sqrt{a + b x^{2}}} - \frac{2 B a}{3 a b^{2} \sqrt{a + b x^{2}} + 3 b^{3} x^{2} \sqrt{a + b x^{2}}} - \frac{3 B b x^{2}}{3 a b^{2} \sqrt{a + b x^{2}} + 3 b^{3} x^{2} \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{2}}{2} + \frac{B x^{4}}{4}}{a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(B*x**2+A)/(b*x**2+a)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.256997, size = 43, normalized size = 0.98 \[ -\frac{3 \,{\left (b x^{2} + a\right )} B - B a + A b}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x/(b*x^2 + a)^(5/2),x, algorithm="giac")
[Out]