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

Optimal. Leaf size=89 \[ \frac{(3 a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{3/2} b^{5/2}}+\frac{x (A b-5 a B)}{8 a b^2 \left (a+b x^2\right )}-\frac{x (A b-a B)}{4 b^2 \left (a+b x^2\right )^2} \]

[Out]

-((A*b - a*B)*x)/(4*b^2*(a + b*x^2)^2) + ((A*b - 5*a*B)*x)/(8*a*b^2*(a + b*x^2))
+ ((A*b + 3*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(3/2)*b^(5/2))

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Rubi [A]  time = 0.162397, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{(3 a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{3/2} b^{5/2}}+\frac{x (A b-5 a B)}{8 a b^2 \left (a+b x^2\right )}-\frac{x (A b-a B)}{4 b^2 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^2*(A + B*x^2))/(a + b*x^2)^3,x]

[Out]

-((A*b - a*B)*x)/(4*b^2*(a + b*x^2)^2) + ((A*b - 5*a*B)*x)/(8*a*b^2*(a + b*x^2))
+ ((A*b + 3*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(3/2)*b^(5/2))

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Rubi in Sympy [A]  time = 23.9085, size = 78, normalized size = 0.88 \[ - \frac{x \left (A b - B a\right )}{4 b^{2} \left (a + b x^{2}\right )^{2}} + \frac{x \left (A b - 5 B a\right )}{8 a b^{2} \left (a + b x^{2}\right )} + \frac{\left (A b + 3 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 a^{\frac{3}{2}} b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(B*x**2+A)/(b*x**2+a)**3,x)

[Out]

-x*(A*b - B*a)/(4*b**2*(a + b*x**2)**2) + x*(A*b - 5*B*a)/(8*a*b**2*(a + b*x**2)
) + (A*b + 3*B*a)*atan(sqrt(b)*x/sqrt(a))/(8*a**(3/2)*b**(5/2))

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Mathematica [A]  time = 0.140825, size = 83, normalized size = 0.93 \[ \frac{\frac{(3 a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}+\frac{\sqrt{b} x \left (-3 a^2 B-a b \left (A+5 B x^2\right )+A b^2 x^2\right )}{a \left (a+b x^2\right )^2}}{8 b^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^2*(A + B*x^2))/(a + b*x^2)^3,x]

[Out]

((Sqrt[b]*x*(-3*a^2*B + A*b^2*x^2 - a*b*(A + 5*B*x^2)))/(a*(a + b*x^2)^2) + ((A*
b + 3*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2))/(8*b^(5/2))

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Maple [A]  time = 0.012, size = 89, normalized size = 1. \[{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{2}} \left ({\frac{ \left ( Ab-5\,Ba \right ){x}^{3}}{8\,ab}}-{\frac{ \left ( Ab+3\,Ba \right ) x}{8\,{b}^{2}}} \right ) }+{\frac{A}{8\,ab}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,B}{8\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(B*x^2+A)/(b*x^2+a)^3,x)

[Out]

(1/8*(A*b-5*B*a)/a/b*x^3-1/8*(A*b+3*B*a)/b^2*x)/(b*x^2+a)^2+1/8/b/a/(a*b)^(1/2)*
arctan(x*b/(a*b)^(1/2))*A+3/8/b^2/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*B

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247236, size = 1, normalized size = 0.01 \[ \left [\frac{{\left ({\left (3 \, B a b^{2} + A b^{3}\right )} x^{4} + 3 \, B a^{3} + A a^{2} b + 2 \,{\left (3 \, B a^{2} b + A a b^{2}\right )} x^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left ({\left (5 \, B a b - A b^{2}\right )} x^{3} +{\left (3 \, B a^{2} + A a b\right )} x\right )} \sqrt{-a b}}{16 \,{\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \sqrt{-a b}}, \frac{{\left ({\left (3 \, B a b^{2} + A b^{3}\right )} x^{4} + 3 \, B a^{3} + A a^{2} b + 2 \,{\left (3 \, B a^{2} b + A a b^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) -{\left ({\left (5 \, B a b - A b^{2}\right )} x^{3} +{\left (3 \, B a^{2} + A a b\right )} x\right )} \sqrt{a b}}{8 \,{\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^2/(b*x^2 + a)^3,x, algorithm="fricas")

[Out]

[1/16*(((3*B*a*b^2 + A*b^3)*x^4 + 3*B*a^3 + A*a^2*b + 2*(3*B*a^2*b + A*a*b^2)*x^
2)*log((2*a*b*x + (b*x^2 - a)*sqrt(-a*b))/(b*x^2 + a)) - 2*((5*B*a*b - A*b^2)*x^
3 + (3*B*a^2 + A*a*b)*x)*sqrt(-a*b))/((a*b^4*x^4 + 2*a^2*b^3*x^2 + a^3*b^2)*sqrt
(-a*b)), 1/8*(((3*B*a*b^2 + A*b^3)*x^4 + 3*B*a^3 + A*a^2*b + 2*(3*B*a^2*b + A*a*
b^2)*x^2)*arctan(sqrt(a*b)*x/a) - ((5*B*a*b - A*b^2)*x^3 + (3*B*a^2 + A*a*b)*x)*
sqrt(a*b))/((a*b^4*x^4 + 2*a^2*b^3*x^2 + a^3*b^2)*sqrt(a*b))]

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Sympy [A]  time = 3.54251, size = 153, normalized size = 1.72 \[ - \frac{\sqrt{- \frac{1}{a^{3} b^{5}}} \left (A b + 3 B a\right ) \log{\left (- a^{2} b^{2} \sqrt{- \frac{1}{a^{3} b^{5}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{a^{3} b^{5}}} \left (A b + 3 B a\right ) \log{\left (a^{2} b^{2} \sqrt{- \frac{1}{a^{3} b^{5}}} + x \right )}}{16} - \frac{x^{3} \left (- A b^{2} + 5 B a b\right ) + x \left (A a b + 3 B a^{2}\right )}{8 a^{3} b^{2} + 16 a^{2} b^{3} x^{2} + 8 a b^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(B*x**2+A)/(b*x**2+a)**3,x)

[Out]

-sqrt(-1/(a**3*b**5))*(A*b + 3*B*a)*log(-a**2*b**2*sqrt(-1/(a**3*b**5)) + x)/16
+ sqrt(-1/(a**3*b**5))*(A*b + 3*B*a)*log(a**2*b**2*sqrt(-1/(a**3*b**5)) + x)/16
- (x**3*(-A*b**2 + 5*B*a*b) + x*(A*a*b + 3*B*a**2))/(8*a**3*b**2 + 16*a**2*b**3*
x**2 + 8*a*b**4*x**4)

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GIAC/XCAS [A]  time = 0.224734, size = 105, normalized size = 1.18 \[ \frac{{\left (3 \, B a + A b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a b^{2}} - \frac{5 \, B a b x^{3} - A b^{2} x^{3} + 3 \, B a^{2} x + A a b x}{8 \,{\left (b x^{2} + a\right )}^{2} a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^2/(b*x^2 + a)^3,x, algorithm="giac")

[Out]

1/8*(3*B*a + A*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a*b^2) - 1/8*(5*B*a*b*x^3 - A
*b^2*x^3 + 3*B*a^2*x + A*a*b*x)/((b*x^2 + a)^2*a*b^2)