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

Optimal. Leaf size=127 \[ \frac{(b c-a d) (5 b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} \sqrt{b}}+\frac{c (5 b c-6 a d)}{3 a^3 x}+\frac{x \left (3 a^2 d^2-6 a b c d+5 b^2 c^2\right )}{6 a^3 \left (a+b x^2\right )}-\frac{c^2}{3 a x^3 \left (a+b x^2\right )} \]

[Out]

(c*(5*b*c - 6*a*d))/(3*a^3*x) - c^2/(3*a*x^3*(a + b*x^2)) + ((5*b^2*c^2 - 6*a*b*
c*d + 3*a^2*d^2)*x)/(6*a^3*(a + b*x^2)) + ((b*c - a*d)*(5*b*c - a*d)*ArcTan[(Sqr
t[b]*x)/Sqrt[a]])/(2*a^(7/2)*Sqrt[b])

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Rubi [A]  time = 0.363003, antiderivative size = 125, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{(b c-a d) (5 b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} \sqrt{b}}+\frac{c (5 b c-6 a d)}{3 a^3 x}+\frac{x \left (\frac{b c (5 b c-6 a d)}{a^2}+3 d^2\right )}{6 a \left (a+b x^2\right )}-\frac{c^2}{3 a x^3 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(c*(5*b*c - 6*a*d))/(3*a^3*x) - c^2/(3*a*x^3*(a + b*x^2)) + ((3*d^2 + (b*c*(5*b*
c - 6*a*d))/a^2)*x)/(6*a*(a + b*x^2)) + ((b*c - a*d)*(5*b*c - a*d)*ArcTan[(Sqrt[
b]*x)/Sqrt[a]])/(2*a^(7/2)*Sqrt[b])

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Rubi in Sympy [A]  time = 42.1185, size = 110, normalized size = 0.87 \[ - \frac{c^{2}}{3 a x^{3} \left (a + b x^{2}\right )} - \frac{c \left (6 a d - 5 b c\right )}{3 a^{3} x} + \frac{x \left (3 a^{2} d^{2} - b c \left (6 a d - 5 b c\right )\right )}{6 a^{3} \left (a + b x^{2}\right )} + \frac{\left (a d - 5 b c\right ) \left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{7}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c**2/(3*a*x**3*(a + b*x**2)) - c*(6*a*d - 5*b*c)/(3*a**3*x) + x*(3*a**2*d**2 -
b*c*(6*a*d - 5*b*c))/(6*a**3*(a + b*x**2)) + (a*d - 5*b*c)*(a*d - b*c)*atan(sqrt
(b)*x/sqrt(a))/(2*a**(7/2)*sqrt(b))

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Mathematica [A]  time = 0.113971, size = 107, normalized size = 0.84 \[ \frac{x (a d-b c)^2}{2 a^3 \left (a+b x^2\right )}-\frac{2 c (a d-b c)}{a^3 x}-\frac{c^2}{3 a^2 x^3}+\frac{\left (a^2 d^2-6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} \sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

-c^2/(3*a^2*x^3) - (2*c*(-(b*c) + a*d))/(a^3*x) + ((-(b*c) + a*d)^2*x)/(2*a^3*(a
+ b*x^2)) + ((5*b^2*c^2 - 6*a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*
a^(7/2)*Sqrt[b])

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Maple [A]  time = 0.017, size = 161, normalized size = 1.3 \[ -{\frac{{c}^{2}}{3\,{a}^{2}{x}^{3}}}-2\,{\frac{cd}{{a}^{2}x}}+2\,{\frac{b{c}^{2}}{{a}^{3}x}}+{\frac{x{d}^{2}}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{cxbd}{{a}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{2}{c}^{2}x}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{{d}^{2}}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-3\,{\frac{bdc}{{a}^{2}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+{\frac{5\,{b}^{2}{c}^{2}}{2\,{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*c^2/a^2/x^3-2*c/a^2/x*d+2*c^2/a^3/x*b+1/2/a*x/(b*x^2+a)*d^2-1/a^2*x/(b*x^2+
a)*c*b*d+1/2/a^3*x/(b*x^2+a)*b^2*c^2+1/2/a/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*d
^2-3/a^2/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*c*b*d+5/2/a^3/(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)^2*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245404, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left ({\left (5 \, b^{3} c^{2} - 6 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{5} +{\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{3}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (3 \,{\left (5 \, b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2}\right )} x^{4} - 2 \, a^{2} c^{2} + 2 \,{\left (5 \, a b c^{2} - 6 \, a^{2} c d\right )} x^{2}\right )} \sqrt{-a b}}{12 \,{\left (a^{3} b x^{5} + a^{4} x^{3}\right )} \sqrt{-a b}}, \frac{3 \,{\left ({\left (5 \, b^{3} c^{2} - 6 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{5} +{\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{3}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (3 \,{\left (5 \, b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2}\right )} x^{4} - 2 \, a^{2} c^{2} + 2 \,{\left (5 \, a b c^{2} - 6 \, a^{2} c d\right )} x^{2}\right )} \sqrt{a b}}{6 \,{\left (a^{3} b x^{5} + a^{4} x^{3}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.74846, size = 248, normalized size = 1.95 \[ - \frac{\sqrt{- \frac{1}{a^{7} b}} \left (a d - 5 b c\right ) \left (a d - b c\right ) \log{\left (- \frac{a^{4} \sqrt{- \frac{1}{a^{7} b}} \left (a d - 5 b c\right ) \left (a d - b c\right )}{a^{2} d^{2} - 6 a b c d + 5 b^{2} c^{2}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{7} b}} \left (a d - 5 b c\right ) \left (a d - b c\right ) \log{\left (\frac{a^{4} \sqrt{- \frac{1}{a^{7} b}} \left (a d - 5 b c\right ) \left (a d - b c\right )}{a^{2} d^{2} - 6 a b c d + 5 b^{2} c^{2}} + x \right )}}{4} + \frac{- 2 a^{2} c^{2} + x^{4} \left (3 a^{2} d^{2} - 18 a b c d + 15 b^{2} c^{2}\right ) + x^{2} \left (- 12 a^{2} c d + 10 a b c^{2}\right )}{6 a^{4} x^{3} + 6 a^{3} b x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-1/(a**7*b))*(a*d - 5*b*c)*(a*d - b*c)*log(-a**4*sqrt(-1/(a**7*b))*(a*d -
5*b*c)*(a*d - b*c)/(a**2*d**2 - 6*a*b*c*d + 5*b**2*c**2) + x)/4 + sqrt(-1/(a**7*
b))*(a*d - 5*b*c)*(a*d - b*c)*log(a**4*sqrt(-1/(a**7*b))*(a*d - 5*b*c)*(a*d - b*
c)/(a**2*d**2 - 6*a*b*c*d + 5*b**2*c**2) + x)/4 + (-2*a**2*c**2 + x**4*(3*a**2*d
**2 - 18*a*b*c*d + 15*b**2*c**2) + x**2*(-12*a**2*c*d + 10*a*b*c**2))/(6*a**4*x*
*3 + 6*a**3*b*x**5)

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GIAC/XCAS [A]  time = 0.248913, size = 151, normalized size = 1.19 \[ \frac{{\left (5 \, b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{3}} + \frac{b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{2 \,{\left (b x^{2} + a\right )} a^{3}} + \frac{6 \, b c^{2} x^{2} - 6 \, a c d x^{2} - a c^{2}}{3 \, a^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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