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

Optimal. Leaf size=97 \[ -\frac{3 (5 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2} \sqrt{b}}-\frac{x (7 A b-3 a B)}{8 a^3 \left (a+b x^2\right )}-\frac{A}{a^3 x}-\frac{x (A b-a B)}{4 a^2 \left (a+b x^2\right )^2} \]

[Out]

-(A/(a^3*x)) - ((A*b - a*B)*x)/(4*a^2*(a + b*x^2)^2) - ((7*A*b - 3*a*B)*x)/(8*a^
3*(a + b*x^2)) - (3*(5*A*b - a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/2)*Sqrt[b
])

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Rubi [A]  time = 0.274091, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{3 (5 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2} \sqrt{b}}-\frac{x (7 A b-3 a B)}{8 a^3 \left (a+b x^2\right )}-\frac{A}{a^3 x}-\frac{x (A b-a B)}{4 a^2 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(A/(a^3*x)) - ((A*b - a*B)*x)/(4*a^2*(a + b*x^2)^2) - ((7*A*b - 3*a*B)*x)/(8*a^
3*(a + b*x^2)) - (3*(5*A*b - a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/2)*Sqrt[b
])

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Rubi in Sympy [A]  time = 30.3122, size = 88, normalized size = 0.91 \[ - \frac{A}{a^{3} x} - \frac{x \left (A b - B a\right )}{4 a^{2} \left (a + b x^{2}\right )^{2}} - \frac{x \left (7 A b - 3 B a\right )}{8 a^{3} \left (a + b x^{2}\right )} - \frac{3 \left (5 A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 a^{\frac{7}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/(a**3*x) - x*(A*b - B*a)/(4*a**2*(a + b*x**2)**2) - x*(7*A*b - 3*B*a)/(8*a**3
*(a + b*x**2)) - 3*(5*A*b - B*a)*atan(sqrt(b)*x/sqrt(a))/(8*a**(7/2)*sqrt(b))

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Mathematica [A]  time = 0.0979148, size = 96, normalized size = 0.99 \[ \frac{3 (a B-5 A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2} \sqrt{b}}+\frac{x (3 a B-7 A b)}{8 a^3 \left (a+b x^2\right )}-\frac{A}{a^3 x}+\frac{x (a B-A b)}{4 a^2 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(A/(a^3*x)) + ((-(A*b) + a*B)*x)/(4*a^2*(a + b*x^2)^2) + ((-7*A*b + 3*a*B)*x)/(
8*a^3*(a + b*x^2)) + (3*(-5*A*b + a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/2)*S
qrt[b])

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Maple [A]  time = 0.017, size = 125, normalized size = 1.3 \[ -{\frac{A}{{a}^{3}x}}-{\frac{7\,A{x}^{3}{b}^{2}}{8\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{3\,bB{x}^{3}}{8\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{9\,Axb}{8\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{5\,Bx}{8\,a \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{15\,Ab}{8\,{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,B}{8\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/a^3/x-7/8/a^3/(b*x^2+a)^2*A*x^3*b^2+3/8/a^2/(b*x^2+a)^2*B*x^3*b-9/8/a^2/(b*x^
2+a)^2*A*x*b+5/8/a/(b*x^2+a)^2*B*x-15/8/a^3/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*
A*b+3/8/a^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)/((b*x^2 + a)^3*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240613, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left ({\left (B a b^{2} - 5 \, A b^{3}\right )} x^{5} + 2 \,{\left (B a^{2} b - 5 \, A a b^{2}\right )} x^{3} +{\left (B a^{3} - 5 \, A a^{2} b\right )} x\right )} \log \left (-\frac{2 \, a b x -{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left (3 \,{\left (B a b - 5 \, A b^{2}\right )} x^{4} - 8 \, A a^{2} + 5 \,{\left (B a^{2} - 5 \, A a b\right )} x^{2}\right )} \sqrt{-a b}}{16 \,{\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )} \sqrt{-a b}}, \frac{3 \,{\left ({\left (B a b^{2} - 5 \, A b^{3}\right )} x^{5} + 2 \,{\left (B a^{2} b - 5 \, A a b^{2}\right )} x^{3} +{\left (B a^{3} - 5 \, A a^{2} b\right )} x\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (3 \,{\left (B a b - 5 \, A b^{2}\right )} x^{4} - 8 \, A a^{2} + 5 \,{\left (B a^{2} - 5 \, A a b\right )} x^{2}\right )} \sqrt{a b}}{8 \,{\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.54787, size = 194, normalized size = 2. \[ - \frac{3 \sqrt{- \frac{1}{a^{7} b}} \left (- 5 A b + B a\right ) \log{\left (- \frac{3 a^{4} \sqrt{- \frac{1}{a^{7} b}} \left (- 5 A b + B a\right )}{- 15 A b + 3 B a} + x \right )}}{16} + \frac{3 \sqrt{- \frac{1}{a^{7} b}} \left (- 5 A b + B a\right ) \log{\left (\frac{3 a^{4} \sqrt{- \frac{1}{a^{7} b}} \left (- 5 A b + B a\right )}{- 15 A b + 3 B a} + x \right )}}{16} + \frac{- 8 A a^{2} + x^{4} \left (- 15 A b^{2} + 3 B a b\right ) + x^{2} \left (- 25 A a b + 5 B a^{2}\right )}{8 a^{5} x + 16 a^{4} b x^{3} + 8 a^{3} b^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*sqrt(-1/(a**7*b))*(-5*A*b + B*a)*log(-3*a**4*sqrt(-1/(a**7*b))*(-5*A*b + B*a)
/(-15*A*b + 3*B*a) + x)/16 + 3*sqrt(-1/(a**7*b))*(-5*A*b + B*a)*log(3*a**4*sqrt(
-1/(a**7*b))*(-5*A*b + B*a)/(-15*A*b + 3*B*a) + x)/16 + (-8*A*a**2 + x**4*(-15*A
*b**2 + 3*B*a*b) + x**2*(-25*A*a*b + 5*B*a**2))/(8*a**5*x + 16*a**4*b*x**3 + 8*a
**3*b**2*x**5)

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GIAC/XCAS [A]  time = 0.23221, size = 111, normalized size = 1.14 \[ \frac{3 \,{\left (B a - 5 \, A b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{3}} - \frac{A}{a^{3} x} + \frac{3 \, B a b x^{3} - 7 \, A b^{2} x^{3} + 5 \, B a^{2} x - 9 \, A a b x}{8 \,{\left (b x^{2} + a\right )}^{2} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/8*(B*a - 5*A*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^3) - A/(a^3*x) + 1/8*(3*B*a
*b*x^3 - 7*A*b^2*x^3 + 5*B*a^2*x - 9*A*a*b*x)/((b*x^2 + a)^2*a^3)