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

Optimal. Leaf size=103 \[ -\frac{(b c-a d) (a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} b^{3/2}}-\frac{x \left (\frac{3 b c^2}{a}+\frac{a d^2}{b}-2 c d\right )}{2 a \left (a+b x^2\right )}-\frac{c^2}{a x \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.184359, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{(b c-a d) (a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} b^{3/2}}-\frac{x \left (\frac{3 b c^2}{a}+\frac{a d^2}{b}-2 c d\right )}{2 a \left (a+b x^2\right )}-\frac{c^2}{a x \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.1629, size = 88, normalized size = 0.85 \[ - \frac{c^{2}}{a x \left (a + b x^{2}\right )} - \frac{x \left (a^{2} d^{2} - b c \left (2 a d - 3 b c\right )\right )}{2 a^{2} b \left (a + b x^{2}\right )} + \frac{\left (a d - b c\right ) \left (a d + 3 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{5}{2}} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.111834, size = 91, normalized size = 0.88 \[ -\frac{x (a d-b c)^2}{2 a^2 b \left (a+b x^2\right )}-\frac{c^2}{a^2 x}+\frac{\left (a^2 d^2+2 a b c d-3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 131, normalized size = 1.3 \[ -{\frac{{c}^{2}}{{a}^{2}x}}-{\frac{x{d}^{2}}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{cxd}{a \left ( b{x}^{2}+a \right ) }}-{\frac{bx{c}^{2}}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{{d}^{2}}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{cd}{a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,b{c}^{2}}{2\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238065, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left ({\left (3 \, b^{3} c^{2} - 2 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{3} +{\left (3 \, a b^{2} c^{2} - 2 \, a^{2} b c d - a^{3} d^{2}\right )} x\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (2 \, a b c^{2} +{\left (3 \, b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2}\right )} \sqrt{-a b}}{4 \,{\left (a^{2} b^{2} x^{3} + a^{3} b x\right )} \sqrt{-a b}}, -\frac{{\left ({\left (3 \, b^{3} c^{2} - 2 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{3} +{\left (3 \, a b^{2} c^{2} - 2 \, a^{2} b c d - a^{3} d^{2}\right )} x\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (2 \, a b c^{2} +{\left (3 \, b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2}\right )} \sqrt{a b}}{2 \,{\left (a^{2} b^{2} x^{3} + a^{3} b x\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/4*(((3*b^3*c^2 - 2*a*b^2*c*d - a^2*b*d^2)*x^3 + (3*a*b^2*c^2 - 2*a^2*b*c*d -
a^3*d^2)*x)*log((2*a*b*x + (b*x^2 - a)*sqrt(-a*b))/(b*x^2 + a)) + 2*(2*a*b*c^2
+ (3*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^2)*sqrt(-a*b))/((a^2*b^2*x^3 + a^3*b*x)*sq
rt(-a*b)), -1/2*(((3*b^3*c^2 - 2*a*b^2*c*d - a^2*b*d^2)*x^3 + (3*a*b^2*c^2 - 2*a
^2*b*c*d - a^3*d^2)*x)*arctan(sqrt(a*b)*x/a) + (2*a*b*c^2 + (3*b^2*c^2 - 2*a*b*c
*d + a^2*d^2)*x^2)*sqrt(a*b))/((a^2*b^2*x^3 + a^3*b*x)*sqrt(a*b))]

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Sympy [A]  time = 4.00616, size = 238, normalized size = 2.31 \[ - \frac{\sqrt{- \frac{1}{a^{5} b^{3}}} \left (a d - b c\right ) \left (a d + 3 b c\right ) \log{\left (- \frac{a^{3} b \sqrt{- \frac{1}{a^{5} b^{3}}} \left (a d - b c\right ) \left (a d + 3 b c\right )}{a^{2} d^{2} + 2 a b c d - 3 b^{2} c^{2}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{5} b^{3}}} \left (a d - b c\right ) \left (a d + 3 b c\right ) \log{\left (\frac{a^{3} b \sqrt{- \frac{1}{a^{5} b^{3}}} \left (a d - b c\right ) \left (a d + 3 b c\right )}{a^{2} d^{2} + 2 a b c d - 3 b^{2} c^{2}} + x \right )}}{4} - \frac{2 a b c^{2} + x^{2} \left (a^{2} d^{2} - 2 a b c d + 3 b^{2} c^{2}\right )}{2 a^{3} b x + 2 a^{2} b^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-1/(a**5*b**3))*(a*d - b*c)*(a*d + 3*b*c)*log(-a**3*b*sqrt(-1/(a**5*b**3))
*(a*d - b*c)*(a*d + 3*b*c)/(a**2*d**2 + 2*a*b*c*d - 3*b**2*c**2) + x)/4 + sqrt(-
1/(a**5*b**3))*(a*d - b*c)*(a*d + 3*b*c)*log(a**3*b*sqrt(-1/(a**5*b**3))*(a*d -
b*c)*(a*d + 3*b*c)/(a**2*d**2 + 2*a*b*c*d - 3*b**2*c**2) + x)/4 - (2*a*b*c**2 +
x**2*(a**2*d**2 - 2*a*b*c*d + 3*b**2*c**2))/(2*a**3*b*x + 2*a**2*b**2*x**3)

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GIAC/XCAS [A]  time = 0.242098, size = 139, normalized size = 1.35 \[ -\frac{{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{2} b} - \frac{3 \, b^{2} c^{2} x^{2} - 2 \, a b c d x^{2} + a^{2} d^{2} x^{2} + 2 \, a b c^{2}}{2 \,{\left (b x^{3} + a x\right )} a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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