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

Optimal. Leaf size=92 \[ \frac{(a B+3 A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{3/2}}+\frac{x (a B+3 A b)}{8 a^2 b \left (a+b x^2\right )}+\frac{x (A b-a B)}{4 a b \left (a+b x^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.0908262, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{(a B+3 A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{3/2}}+\frac{x (a B+3 A b)}{8 a^2 b \left (a+b x^2\right )}+\frac{x (A b-a B)}{4 a b \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.7736, size = 78, normalized size = 0.85 \[ \frac{x \left (A b - B a\right )}{4 a b \left (a + b x^{2}\right )^{2}} + \frac{x \left (3 A b + B a\right )}{8 a^{2} b \left (a + b x^{2}\right )} + \frac{\left (3 A b + B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.106749, size = 84, normalized size = 0.91 \[ \frac{(a B+3 A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{3/2}}+\frac{x \left (a^2 (-B)+a b \left (5 A+B x^2\right )+3 A b^2 x^2\right )}{8 a^2 b \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 90, normalized size = 1. \[{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{2}} \left ({\frac{ \left ( 3\,Ab+Ba \right ){x}^{3}}{8\,{a}^{2}}}+{\frac{ \left ( 5\,Ab-Ba \right ) x}{8\,ab}} \right ) }+{\frac{3\,A}{8\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{B}{8\,ab}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/(b*x^2+a)^3,x)

[Out]

(1/8*(3*A*b+B*a)/a^2*x^3+1/8*(5*A*b-B*a)/a/b*x)/(b*x^2+a)^2+3/8/a^2/(a*b)^(1/2)*
arctan(x*b/(a*b)^(1/2))*A+1/8/a/b/(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)/(b*x^2 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242552, size = 1, normalized size = 0.01 \[ \left [\frac{{\left ({\left (B a b^{2} + 3 \, A b^{3}\right )} x^{4} + B a^{3} + 3 \, A a^{2} b + 2 \,{\left (B a^{2} b + 3 \, A a b^{2}\right )} x^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left ({\left (B a b + 3 \, A b^{2}\right )} x^{3} -{\left (B a^{2} - 5 \, A a b\right )} x\right )} \sqrt{-a b}}{16 \,{\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )} \sqrt{-a b}}, \frac{{\left ({\left (B a b^{2} + 3 \, A b^{3}\right )} x^{4} + B a^{3} + 3 \, A a^{2} b + 2 \,{\left (B a^{2} b + 3 \, A a b^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left ({\left (B a b + 3 \, A b^{2}\right )} x^{3} -{\left (B a^{2} - 5 \, A a b\right )} x\right )} \sqrt{a b}}{8 \,{\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.82895, size = 150, normalized size = 1.63 \[ - \frac{\sqrt{- \frac{1}{a^{5} b^{3}}} \left (3 A b + B a\right ) \log{\left (- a^{3} b \sqrt{- \frac{1}{a^{5} b^{3}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{a^{5} b^{3}}} \left (3 A b + B a\right ) \log{\left (a^{3} b \sqrt{- \frac{1}{a^{5} b^{3}}} + x \right )}}{16} + \frac{x^{3} \left (3 A b^{2} + B a b\right ) + x \left (5 A a b - B a^{2}\right )}{8 a^{4} b + 16 a^{3} b^{2} x^{2} + 8 a^{2} b^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.22289, size = 105, normalized size = 1.14 \[ \frac{{\left (B a + 3 \, A b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{2} b} + \frac{B a b x^{3} + 3 \, A b^{2} x^{3} - B a^{2} x + 5 \, A a b x}{8 \,{\left (b x^{2} + a\right )}^{2} a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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