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

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.222816, 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 \[ \frac{(b c-a d) (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{5/2}}+\frac{x (b c-a d)^2}{2 a b^2 \left (a+b x^2\right )}+\frac{d^2 x}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ d^{2} \int \frac{1}{b^{2}}\, dx + \frac{x \left (a d - b c\right )^{2}}{2 a b^{2} \left (a + b x^{2}\right )} - \frac{\left (a d - b c\right ) \left (3 a d + b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.101111, size = 88, normalized size = 1.07 \[ \frac{\left (-3 a^2 d^2+2 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{5/2}}+\frac{x (b c-a d)^2}{2 a b^2 \left (a+b x^2\right )}+\frac{d^2 x}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 129, normalized size = 1.6 \[{\frac{{d}^{2}x}{{b}^{2}}}+{\frac{ax{d}^{2}}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{cxd}{b \left ( b{x}^{2}+a \right ) }}+{\frac{x{c}^{2}}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{3\,a{d}^{2}}{2\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{cd}{b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{c}^{2}}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)^2/(b*x^2+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.211786, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (a b^{2} c^{2} + 2 \, a^{2} b c d - 3 \, a^{3} d^{2} +{\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{2}\right )} \log \left (-\frac{2 \, a b x -{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left (2 \, a b d^{2} x^{3} +{\left (b^{2} c^{2} - 2 \, a b c d + 3 \, a^{2} d^{2}\right )} x\right )} \sqrt{-a b}}{4 \,{\left (a b^{3} x^{2} + a^{2} b^{2}\right )} \sqrt{-a b}}, \frac{{\left (a b^{2} c^{2} + 2 \, a^{2} b c d - 3 \, a^{3} d^{2} +{\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (2 \, a b d^{2} x^{3} +{\left (b^{2} c^{2} - 2 \, a b c d + 3 \, a^{2} d^{2}\right )} x\right )} \sqrt{a b}}{2 \,{\left (a b^{3} x^{2} + a^{2} b^{2}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 3.68655, size = 236, normalized size = 2.88 \[ \frac{x \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{2 a^{2} b^{2} + 2 a b^{3} x^{2}} + \frac{\sqrt{- \frac{1}{a^{3} b^{5}}} \left (a d - b c\right ) \left (3 a d + b c\right ) \log{\left (- \frac{a^{2} b^{2} \sqrt{- \frac{1}{a^{3} b^{5}}} \left (a d - b c\right ) \left (3 a d + b c\right )}{3 a^{2} d^{2} - 2 a b c d - b^{2} c^{2}} + x \right )}}{4} - \frac{\sqrt{- \frac{1}{a^{3} b^{5}}} \left (a d - b c\right ) \left (3 a d + b c\right ) \log{\left (\frac{a^{2} b^{2} \sqrt{- \frac{1}{a^{3} b^{5}}} \left (a d - b c\right ) \left (3 a d + b c\right )}{3 a^{2} d^{2} - 2 a b c d - b^{2} c^{2}} + x \right )}}{4} + \frac{d^{2} x}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.233258, size = 127, normalized size = 1.55 \[ \frac{d^{2} x}{b^{2}} + \frac{{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a b^{2}} + \frac{b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{2 \,{\left (b x^{2} + a\right )} a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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