3.574 \(\int \frac{x \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=41 \[ \frac{B \sqrt{a+b x^2}}{b^2}-\frac{A b-a B}{b^2 \sqrt{a+b x^2}} \]

[Out]

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

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

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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]

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

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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]

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

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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]

-(-B*b*x^2+A*b-2*B*a)/(b*x^2+a)^(1/2)/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/(b*x^2 + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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]

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

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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)

[Out]

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

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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]

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