3.590 \(\int \frac{x \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=44 \[ -\frac{A b-a B}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac{B}{b^2 \sqrt{a+b x^2}} \]

[Out]

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

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

-1/3*(3*B*b*x^2 + 2*B*a + A*b)*sqrt(b*x^2 + a)/(b^4*x^4 + 2*a*b^3*x^2 + a^2*b^2)

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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]

Piecewise((-A*b/(3*a*b**2*sqrt(a + b*x**2) + 3*b**3*x**2*sqrt(a + b*x**2)) - 2*B
*a/(3*a*b**2*sqrt(a + b*x**2) + 3*b**3*x**2*sqrt(a + b*x**2)) - 3*B*b*x**2/(3*a*
b**2*sqrt(a + b*x**2) + 3*b**3*x**2*sqrt(a + b*x**2)), Ne(b, 0)), ((A*x**2/2 + B
*x**4/4)/a**(5/2), True))

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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]

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