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

Optimal. Leaf size=64 \[ \frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} \sqrt{b}}+\frac{c (b c-2 a d)}{a^2 x}-\frac{c^2}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.133459, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} \sqrt{b}}+\frac{c (b c-2 a d)}{a^2 x}-\frac{c^2}{3 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.0205, size = 56, normalized size = 0.88 \[ - \frac{c^{2}}{3 a x^{3}} - \frac{c \left (2 a d - b c\right )}{a^{2} x} + \frac{\left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{5}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.112273, size = 66, normalized size = 1.03 \[ \frac{(a d-b c)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} \sqrt{b}}-\frac{c (2 a d-b c)}{a^2 x}-\frac{c^2}{3 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 98, normalized size = 1.5 \[ -{\frac{{c}^{2}}{3\,a{x}^{3}}}-2\,{\frac{cd}{ax}}+{\frac{b{c}^{2}}{{a}^{2}x}}+{{d}^{2}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-2\,{\frac{bcd}{a\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+{\frac{{b}^{2}{c}^{2}}{{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241347, size = 1, normalized size = 0.02 \[ \left [\frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{3} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left (a c^{2} - 3 \,{\left (b c^{2} - 2 \, a c d\right )} x^{2}\right )} \sqrt{-a b}}{6 \, \sqrt{-a b} a^{2} x^{3}}, \frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{3} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) -{\left (a c^{2} - 3 \,{\left (b c^{2} - 2 \, a c d\right )} x^{2}\right )} \sqrt{a b}}{3 \, \sqrt{a b} a^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.10921, size = 172, normalized size = 2.69 \[ - \frac{\sqrt{- \frac{1}{a^{5} b}} \left (a d - b c\right )^{2} \log{\left (- \frac{a^{3} \sqrt{- \frac{1}{a^{5} b}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{a^{5} b}} \left (a d - b c\right )^{2} \log{\left (\frac{a^{3} \sqrt{- \frac{1}{a^{5} b}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} - \frac{a c^{2} + x^{2} \left (6 a c d - 3 b c^{2}\right )}{3 a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.224712, size = 97, normalized size = 1.52 \[ \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2}} + \frac{3 \, b c^{2} x^{2} - 6 \, a c d x^{2} - a c^{2}}{3 \, a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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