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

Optimal. Leaf size=59 \[ \frac{\sqrt{b} (b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2}}+\frac{b c-a d}{a^2 x}-\frac{c}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.101354, antiderivative size = 59, 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 \[ \frac{\sqrt{b} (b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2}}+\frac{b c-a d}{a^2 x}-\frac{c}{3 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.0667, size = 51, normalized size = 0.86 \[ - \frac{c}{3 a x^{3}} - \frac{a d - b c}{a^{2} x} - \frac{\sqrt{b} \left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0949335, size = 60, normalized size = 1.02 \[ -\frac{\sqrt{b} (a d-b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2}}+\frac{b c-a d}{a^2 x}-\frac{c}{3 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 72, normalized size = 1.2 \[ -{\frac{c}{3\,a{x}^{3}}}-{\frac{d}{ax}}+{\frac{bc}{{a}^{2}x}}-{\frac{bd}{a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{b}^{2}c}{{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)/x^4/(b*x^2+a),x)

[Out]

-1/3*c/a/x^3-1/a/x*d+1/a^2/x*b*c-b/a/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*d+b^2/a
^2/(a*b)^(1/2)*arctan(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)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236148, size = 1, normalized size = 0.02 \[ \left [-\frac{3 \,{\left (b c - a d\right )} x^{3} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) - 6 \,{\left (b c - a d\right )} x^{2} + 2 \, a c}{6 \, a^{2} x^{3}}, \frac{3 \,{\left (b c - a d\right )} x^{3} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right ) + 3 \,{\left (b c - a d\right )} x^{2} - a c}{3 \, a^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/6*(3*(b*c - a*d)*x^3*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 +
a)) - 6*(b*c - a*d)*x^2 + 2*a*c)/(a^2*x^3), 1/3*(3*(b*c - a*d)*x^3*sqrt(b/a)*arc
tan(b*x/(a*sqrt(b/a))) + 3*(b*c - a*d)*x^2 - a*c)/(a^2*x^3)]

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Sympy [A]  time = 2.2177, size = 129, normalized size = 2.19 \[ \frac{\sqrt{- \frac{b}{a^{5}}} \left (a d - b c\right ) \log{\left (- \frac{a^{3} \sqrt{- \frac{b}{a^{5}}} \left (a d - b c\right )}{a b d - b^{2} c} + x \right )}}{2} - \frac{\sqrt{- \frac{b}{a^{5}}} \left (a d - b c\right ) \log{\left (\frac{a^{3} \sqrt{- \frac{b}{a^{5}}} \left (a d - b c\right )}{a b d - b^{2} c} + x \right )}}{2} - \frac{a c + x^{2} \left (3 a d - 3 b c\right )}{3 a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.220508, size = 77, normalized size = 1.31 \[ \frac{{\left (b^{2} c - a b d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2}} + \frac{3 \, b c x^{2} - 3 \, a d x^{2} - a c}{3 \, a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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