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

Optimal. Leaf size=66 \[ -\frac{a^2}{3 c x^3}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2} \sqrt{d}}-\frac{a (2 b c-a d)}{c^2 x} \]

[Out]

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

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Rubi [A]  time = 0.12916, antiderivative size = 66, 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 \[ -\frac{a^2}{3 c x^3}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2} \sqrt{d}}-\frac{a (2 b c-a d)}{c^2 x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.3941, size = 56, normalized size = 0.85 \[ - \frac{a^{2}}{3 c x^{3}} + \frac{a \left (a d - 2 b c\right )}{c^{2} x} + \frac{\left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{c^{\frac{5}{2}} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0969104, size = 64, normalized size = 0.97 \[ -\frac{a^2}{3 c x^3}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2} \sqrt{d}}+\frac{a (a d-2 b c)}{c^2 x} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235958, size = 1, normalized size = 0.02 \[ \left [\frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{3} \log \left (\frac{2 \, c d x +{\left (d x^{2} - c\right )} \sqrt{-c d}}{d x^{2} + c}\right ) - 2 \,{\left (a^{2} c + 3 \,{\left (2 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt{-c d}}{6 \, \sqrt{-c d} c^{2} x^{3}}, \frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{3} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) -{\left (a^{2} c + 3 \,{\left (2 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt{c d}}{3 \, \sqrt{c d} c^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [A]  time = 3.13898, size = 172, normalized size = 2.61 \[ - \frac{\sqrt{- \frac{1}{c^{5} d}} \left (a d - b c\right )^{2} \log{\left (- \frac{c^{3} \sqrt{- \frac{1}{c^{5} d}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{c^{5} d}} \left (a d - b c\right )^{2} \log{\left (\frac{c^{3} \sqrt{- \frac{1}{c^{5} d}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac{- a^{2} c + x^{2} \left (3 a^{2} d - 6 a b c\right )}{3 c^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.223582, size = 96, normalized size = 1.45 \[ \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{\sqrt{c d} c^{2}} - \frac{6 \, a b c x^{2} - 3 \, a^{2} d x^{2} + a^{2} c}{3 \, c^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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