Optimal. Leaf size=43 \[ \frac{\sqrt{a+b x^2} (A b-a B)}{b^2}+\frac{B \left (a+b x^2\right )^{3/2}}{3 b^2} \]
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Rubi [A] time = 0.100898, antiderivative size = 43, 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 \[ \frac{\sqrt{a+b x^2} (A b-a B)}{b^2}+\frac{B \left (a+b x^2\right )^{3/2}}{3 b^2} \]
Antiderivative was successfully verified.
[In] Int[(x*(A + B*x^2))/Sqrt[a + b*x^2],x]
[Out]
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Rubi in Sympy [A] time = 13.1456, size = 36, normalized size = 0.84 \[ \frac{B \left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b^{2}} + \frac{\sqrt{a + b x^{2}} \left (A b - B a\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(B*x**2+A)/(b*x**2+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0250252, size = 33, normalized size = 0.77 \[ \frac{\sqrt{a+b x^2} \left (-2 a B+3 A b+b B x^2\right )}{3 b^2} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(A + B*x^2))/Sqrt[a + b*x^2],x]
[Out]
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Maple [A] time = 0.005, size = 30, normalized size = 0.7 \[{\frac{bB{x}^{2}+3\,Ab-2\,Ba}{3\,{b}^{2}}\sqrt{b{x}^{2}+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(B*x^2+A)/(b*x^2+a)^(1/2),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/sqrt(b*x^2 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226943, size = 39, normalized size = 0.91 \[ \frac{{\left (B b x^{2} - 2 \, B a + 3 \, A b\right )} \sqrt{b x^{2} + a}}{3 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x/sqrt(b*x^2 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.99033, size = 70, normalized size = 1.63 \[ \begin{cases} \frac{A \sqrt{a + b x^{2}}}{b} - \frac{2 B a \sqrt{a + b x^{2}}}{3 b^{2}} + \frac{B x^{2} \sqrt{a + b x^{2}}}{3 b} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{2}}{2} + \frac{B x^{4}}{4}}{\sqrt{a}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(B*x**2+A)/(b*x**2+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.234658, size = 58, normalized size = 1.35 \[ \frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}} B - 3 \, \sqrt{b x^{2} + a} B a + 3 \, \sqrt{b x^{2} + a} A b}{3 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x/sqrt(b*x^2 + a),x, algorithm="giac")
[Out]