3.559 \(\int \frac{x \left (A+B x^2\right )}{\sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=43 \[ \frac{\sqrt{a+b x^2} (A b-a B)}{b^2}+\frac{B \left (a+b x^2\right )^{3/2}}{3 b^2} \]

[Out]

((A*b - a*B)*Sqrt[a + b*x^2])/b^2 + (B*(a + b*x^2)^(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]

((A*b - a*B)*Sqrt[a + b*x^2])/b^2 + (B*(a + b*x^2)^(3/2))/(3*b^2)

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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]

B*(a + b*x**2)**(3/2)/(3*b**2) + sqrt(a + b*x**2)*(A*b - B*a)/b**2

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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]

(Sqrt[a + b*x^2]*(3*A*b - 2*a*B + b*B*x^2))/(3*b^2)

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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]

1/3*(b*x^2+a)^(1/2)*(B*b*x^2+3*A*b-2*B*a)/b^2

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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]

Exception raised: ValueError

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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]

1/3*(B*b*x^2 - 2*B*a + 3*A*b)*sqrt(b*x^2 + a)/b^2

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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]

Piecewise((A*sqrt(a + b*x**2)/b - 2*B*a*sqrt(a + b*x**2)/(3*b**2) + B*x**2*sqrt(
a + b*x**2)/(3*b), Ne(b, 0)), ((A*x**2/2 + B*x**4/4)/sqrt(a), True))

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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]

1/3*((b*x^2 + a)^(3/2)*B - 3*sqrt(b*x^2 + a)*B*a + 3*sqrt(b*x^2 + a)*A*b)/b^2