3.270 \(\int \frac{c+d x^2}{x^4 \left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=90 \[ \frac{\sqrt{b} (5 b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2}}+\frac{b x (b c-a d)}{2 a^3 \left (a+b x^2\right )}+\frac{2 b c-a d}{a^3 x}-\frac{c}{3 a^2 x^3} \]

[Out]

-c/(3*a^2*x^3) + (2*b*c - a*d)/(a^3*x) + (b*(b*c - a*d)*x)/(2*a^3*(a + b*x^2)) +
(Sqrt[b]*(5*b*c - 3*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(7/2))

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Rubi [A]  time = 0.261874, antiderivative size = 90, 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 \[ \frac{\sqrt{b} (5 b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2}}+\frac{b x (b c-a d)}{2 a^3 \left (a+b x^2\right )}+\frac{2 b c-a d}{a^3 x}-\frac{c}{3 a^2 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)/(x^4*(a + b*x^2)^2),x]

[Out]

-c/(3*a^2*x^3) + (2*b*c - a*d)/(a^3*x) + (b*(b*c - a*d)*x)/(2*a^3*(a + b*x^2)) +
(Sqrt[b]*(5*b*c - 3*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(7/2))

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Rubi in Sympy [A]  time = 40.5816, size = 82, normalized size = 0.91 \[ - \frac{c}{3 a^{2} x^{3}} - \frac{b x \left (a d - b c\right )}{2 a^{3} \left (a + b x^{2}\right )} - \frac{a d - 2 b c}{a^{3} x} - \frac{\sqrt{b} \left (3 a d - 5 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)/x**4/(b*x**2+a)**2,x)

[Out]

-c/(3*a**2*x**3) - b*x*(a*d - b*c)/(2*a**3*(a + b*x**2)) - (a*d - 2*b*c)/(a**3*x
) - sqrt(b)*(3*a*d - 5*b*c)*atan(sqrt(b)*x/sqrt(a))/(2*a**(7/2))

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Mathematica [A]  time = 0.117096, size = 90, normalized size = 1. \[ -\frac{\sqrt{b} (3 a d-5 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2}}-\frac{b x (a d-b c)}{2 a^3 \left (a+b x^2\right )}+\frac{2 b c-a d}{a^3 x}-\frac{c}{3 a^2 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)/(x^4*(a + b*x^2)^2),x]

[Out]

-c/(3*a^2*x^3) + (2*b*c - a*d)/(a^3*x) - (b*(-(b*c) + a*d)*x)/(2*a^3*(a + b*x^2)
) - (Sqrt[b]*(-5*b*c + 3*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(7/2))

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Maple [A]  time = 0.016, size = 110, normalized size = 1.2 \[ -{\frac{c}{3\,{a}^{2}{x}^{3}}}-{\frac{d}{{a}^{2}x}}+2\,{\frac{bc}{{a}^{3}x}}-{\frac{bxd}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{x{b}^{2}c}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,bd}{2\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{b}^{2}c}{2\,{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)/x^4/(b*x^2+a)^2,x)

[Out]

-1/3*c/a^2/x^3-1/a^2/x*d+2/a^3/x*b*c-1/2/a^2*b*x/(b*x^2+a)*d+1/2/a^3*b^2*x/(b*x^
2+a)*c-3/2/a^2*b/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*d+5/2/a^3*b^2/(a*b)^(1/2)*a
rctan(x*b/(a*b)^(1/2))*c

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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((d*x^2 + c)/((b*x^2 + a)^2*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246263, size = 1, normalized size = 0.01 \[ \left [\frac{6 \,{\left (5 \, b^{2} c - 3 \, a b d\right )} x^{4} - 4 \, a^{2} c + 4 \,{\left (5 \, a b c - 3 \, a^{2} d\right )} x^{2} - 3 \,{\left ({\left (5 \, b^{2} c - 3 \, a b d\right )} x^{5} +{\left (5 \, a b c - 3 \, a^{2} d\right )} x^{3}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{12 \,{\left (a^{3} b x^{5} + a^{4} x^{3}\right )}}, \frac{3 \,{\left (5 \, b^{2} c - 3 \, a b d\right )} x^{4} - 2 \, a^{2} c + 2 \,{\left (5 \, a b c - 3 \, a^{2} d\right )} x^{2} + 3 \,{\left ({\left (5 \, b^{2} c - 3 \, a b d\right )} x^{5} +{\left (5 \, a b c - 3 \, a^{2} d\right )} x^{3}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right )}{6 \,{\left (a^{3} b x^{5} + a^{4} x^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^2*x^4),x, algorithm="fricas")

[Out]

[1/12*(6*(5*b^2*c - 3*a*b*d)*x^4 - 4*a^2*c + 4*(5*a*b*c - 3*a^2*d)*x^2 - 3*((5*b
^2*c - 3*a*b*d)*x^5 + (5*a*b*c - 3*a^2*d)*x^3)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqr
t(-b/a) - a)/(b*x^2 + a)))/(a^3*b*x^5 + a^4*x^3), 1/6*(3*(5*b^2*c - 3*a*b*d)*x^4
- 2*a^2*c + 2*(5*a*b*c - 3*a^2*d)*x^2 + 3*((5*b^2*c - 3*a*b*d)*x^5 + (5*a*b*c -
3*a^2*d)*x^3)*sqrt(b/a)*arctan(b*x/(a*sqrt(b/a))))/(a^3*b*x^5 + a^4*x^3)]

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Sympy [A]  time = 3.15965, size = 184, normalized size = 2.04 \[ \frac{\sqrt{- \frac{b}{a^{7}}} \left (3 a d - 5 b c\right ) \log{\left (- \frac{a^{4} \sqrt{- \frac{b}{a^{7}}} \left (3 a d - 5 b c\right )}{3 a b d - 5 b^{2} c} + x \right )}}{4} - \frac{\sqrt{- \frac{b}{a^{7}}} \left (3 a d - 5 b c\right ) \log{\left (\frac{a^{4} \sqrt{- \frac{b}{a^{7}}} \left (3 a d - 5 b c\right )}{3 a b d - 5 b^{2} c} + x \right )}}{4} - \frac{2 a^{2} c + x^{4} \left (9 a b d - 15 b^{2} c\right ) + x^{2} \left (6 a^{2} d - 10 a b c\right )}{6 a^{4} x^{3} + 6 a^{3} b x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**2+c)/x**4/(b*x**2+a)**2,x)

[Out]

sqrt(-b/a**7)*(3*a*d - 5*b*c)*log(-a**4*sqrt(-b/a**7)*(3*a*d - 5*b*c)/(3*a*b*d -
5*b**2*c) + x)/4 - sqrt(-b/a**7)*(3*a*d - 5*b*c)*log(a**4*sqrt(-b/a**7)*(3*a*d
- 5*b*c)/(3*a*b*d - 5*b**2*c) + x)/4 - (2*a**2*c + x**4*(9*a*b*d - 15*b**2*c) +
x**2*(6*a**2*d - 10*a*b*c))/(6*a**4*x**3 + 6*a**3*b*x**5)

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GIAC/XCAS [A]  time = 0.228804, size = 116, normalized size = 1.29 \[ \frac{{\left (5 \, b^{2} c - 3 \, a b d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{3}} + \frac{b^{2} c x - a b d x}{2 \,{\left (b x^{2} + a\right )} a^{3}} + \frac{6 \, b c x^{2} - 3 \, a d x^{2} - a c}{3 \, a^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^2*x^4),x, algorithm="giac")

[Out]

1/2*(5*b^2*c - 3*a*b*d)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^3) + 1/2*(b^2*c*x - a
*b*d*x)/((b*x^2 + a)*a^3) + 1/3*(6*b*c*x^2 - 3*a*d*x^2 - a*c)/(a^3*x^3)