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

Optimal. Leaf size=77 \[ -\frac{2 x (4 A b-a B)}{3 a^3 \sqrt{a+b x^2}}-\frac{x (4 A b-a B)}{3 a^2 \left (a+b x^2\right )^{3/2}}-\frac{A}{a x \left (a+b x^2\right )^{3/2}} \]

[Out]

-(A/(a*x*(a + b*x^2)^(3/2))) - ((4*A*b - a*B)*x)/(3*a^2*(a + b*x^2)^(3/2)) - (2*
(4*A*b - a*B)*x)/(3*a^3*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.0933518, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{2 x (4 A b-a B)}{3 a^3 \sqrt{a+b x^2}}-\frac{x (4 A b-a B)}{3 a^2 \left (a+b x^2\right )^{3/2}}-\frac{A}{a x \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(x^2*(a + b*x^2)^(5/2)),x]

[Out]

-(A/(a*x*(a + b*x^2)^(3/2))) - ((4*A*b - a*B)*x)/(3*a^2*(a + b*x^2)^(3/2)) - (2*
(4*A*b - a*B)*x)/(3*a^3*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 10.5702, size = 68, normalized size = 0.88 \[ - \frac{A}{a x \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{x \left (4 A b - B a\right )}{3 a^{2} \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{2 x \left (4 A b - B a\right )}{3 a^{3} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/x**2/(b*x**2+a)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0629823, size = 60, normalized size = 0.78 \[ \frac{-3 a^2 \left (A-B x^2\right )+2 a b x^2 \left (B x^2-6 A\right )-8 A b^2 x^4}{3 a^3 x \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(x^2*(a + b*x^2)^(5/2)),x]

[Out]

(-8*A*b^2*x^4 - 3*a^2*(A - B*x^2) + 2*a*b*x^2*(-6*A + B*x^2))/(3*a^3*x*(a + b*x^
2)^(3/2))

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Maple [A]  time = 0.007, size = 59, normalized size = 0.8 \[ -{\frac{8\,A{b}^{2}{x}^{4}-2\,Bab{x}^{4}+12\,aAb{x}^{2}-3\,B{a}^{2}{x}^{2}+3\,A{a}^{2}}{3\,x{a}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^2/(b*x^2+a)^(5/2),x)

[Out]

-1/3*(8*A*b^2*x^4-2*B*a*b*x^4+12*A*a*b*x^2-3*B*a^2*x^2+3*A*a^2)/(b*x^2+a)^(3/2)/
x/a^3

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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)/((b*x^2 + a)^(5/2)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234176, size = 104, normalized size = 1.35 \[ \frac{{\left (2 \,{\left (B a b - 4 \, A b^{2}\right )} x^{4} - 3 \, A a^{2} + 3 \,{\left (B a^{2} - 4 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{3 \,{\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^(5/2)*x^2),x, algorithm="fricas")

[Out]

1/3*(2*(B*a*b - 4*A*b^2)*x^4 - 3*A*a^2 + 3*(B*a^2 - 4*A*a*b)*x^2)*sqrt(b*x^2 + a
)/(a^3*b^2*x^5 + 2*a^4*b*x^3 + a^5*x)

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Sympy [A]  time = 77.4416, size = 265, normalized size = 3.44 \[ A \left (- \frac{3 a^{2} b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{4}} - \frac{12 a b^{\frac{11}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{4}} - \frac{8 b^{\frac{13}{2}} x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{4}}\right ) + B \left (\frac{3 a x}{3 a^{\frac{7}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 3 a^{\frac{5}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{2 b x^{3}}{3 a^{\frac{7}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 3 a^{\frac{5}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)/x**2/(b*x**2+a)**(5/2),x)

[Out]

A*(-3*a**2*b**(9/2)*sqrt(a/(b*x**2) + 1)/(3*a**5*b**4 + 6*a**4*b**5*x**2 + 3*a**
3*b**6*x**4) - 12*a*b**(11/2)*x**2*sqrt(a/(b*x**2) + 1)/(3*a**5*b**4 + 6*a**4*b*
*5*x**2 + 3*a**3*b**6*x**4) - 8*b**(13/2)*x**4*sqrt(a/(b*x**2) + 1)/(3*a**5*b**4
+ 6*a**4*b**5*x**2 + 3*a**3*b**6*x**4)) + B*(3*a*x/(3*a**(7/2)*sqrt(1 + b*x**2/
a) + 3*a**(5/2)*b*x**2*sqrt(1 + b*x**2/a)) + 2*b*x**3/(3*a**(7/2)*sqrt(1 + b*x**
2/a) + 3*a**(5/2)*b*x**2*sqrt(1 + b*x**2/a)))

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GIAC/XCAS [A]  time = 0.231271, size = 136, normalized size = 1.77 \[ \frac{x{\left (\frac{{\left (2 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{2}}{a^{5} b} + \frac{3 \,{\left (B a^{4} b - 2 \, A a^{3} b^{2}\right )}}{a^{5} b}\right )}}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}}} + \frac{2 \, A \sqrt{b}}{{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^(5/2)*x^2),x, algorithm="giac")

[Out]

1/3*x*((2*B*a^3*b^2 - 5*A*a^2*b^3)*x^2/(a^5*b) + 3*(B*a^4*b - 2*A*a^3*b^2)/(a^5*
b))/(b*x^2 + a)^(3/2) + 2*A*sqrt(b)/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)*a^2)