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

Optimal. Leaf size=82 \[ -\frac{(b c-a d) (a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} d^{5/2}}+\frac{x (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac{b^2 x}{d^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.221086, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{(b c-a d) (a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} d^{5/2}}+\frac{x (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac{b^2 x}{d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int b^{2}\, dx}{d^{2}} + \frac{x \left (a d - b c\right )^{2}}{2 c d^{2} \left (c + d x^{2}\right )} + \frac{\left (a d - b c\right ) \left (a d + 3 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{2 c^{\frac{3}{2}} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0971772, size = 89, normalized size = 1.09 \[ -\frac{\left (-a^2 d^2-2 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} d^{5/2}}+\frac{x (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac{b^2 x}{d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0., size = 129, normalized size = 1.6 \[{\frac{{b}^{2}x}{{d}^{2}}}+{\frac{{a}^{2}x}{2\,c \left ( d{x}^{2}+c \right ) }}-{\frac{abx}{d \left ( d{x}^{2}+c \right ) }}+{\frac{x{b}^{2}c}{2\,{d}^{2} \left ( d{x}^{2}+c \right ) }}+{\frac{{a}^{2}}{2\,c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{ab}{d}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{3\,{b}^{2}c}{2\,{d}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2/(d*x^2+c)^2,x)

[Out]

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

_______________________________________________________________________________________

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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.239638, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d - a^{2} c d^{2} +{\left (3 \, b^{2} c^{2} d - 2 \, a b c d^{2} - a^{2} d^{3}\right )} x^{2}\right )} \log \left (\frac{2 \, c d x +{\left (d x^{2} - c\right )} \sqrt{-c d}}{d x^{2} + c}\right ) - 2 \,{\left (2 \, b^{2} c d x^{3} +{\left (3 \, b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x\right )} \sqrt{-c d}}{4 \,{\left (c d^{3} x^{2} + c^{2} d^{2}\right )} \sqrt{-c d}}, -\frac{{\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d - a^{2} c d^{2} +{\left (3 \, b^{2} c^{2} d - 2 \, a b c d^{2} - a^{2} d^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) -{\left (2 \, b^{2} c d x^{3} +{\left (3 \, b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x\right )} \sqrt{c d}}{2 \,{\left (c d^{3} x^{2} + c^{2} d^{2}\right )} \sqrt{c d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 3.41042, size = 236, normalized size = 2.88 \[ \frac{b^{2} x}{d^{2}} + \frac{x \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{2 c^{2} d^{2} + 2 c d^{3} x^{2}} - \frac{\sqrt{- \frac{1}{c^{3} d^{5}}} \left (a d - b c\right ) \left (a d + 3 b c\right ) \log{\left (- \frac{c^{2} d^{2} \sqrt{- \frac{1}{c^{3} d^{5}}} \left (a d - b c\right ) \left (a d + 3 b c\right )}{a^{2} d^{2} + 2 a b c d - 3 b^{2} c^{2}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{c^{3} d^{5}}} \left (a d - b c\right ) \left (a d + 3 b c\right ) \log{\left (\frac{c^{2} d^{2} \sqrt{- \frac{1}{c^{3} d^{5}}} \left (a d - b c\right ) \left (a d + 3 b c\right )}{a^{2} d^{2} + 2 a b c d - 3 b^{2} c^{2}} + x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.22602, size = 128, normalized size = 1.56 \[ \frac{b^{2} x}{d^{2}} - \frac{{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{2 \, \sqrt{c d} c d^{2}} + \frac{b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{2 \,{\left (d x^{2} + c\right )} c d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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