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

Optimal. Leaf size=55 \[ -\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} b^{3/2}}-\frac{c^2}{a x}+\frac{d^2 x}{b} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.11726, antiderivative size = 55, 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 \[ -\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} b^{3/2}}-\frac{c^2}{a x}+\frac{d^2 x}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*Integral(1/b, x) - c**2/(a*x) - (a*d - b*c)**2*atan(sqrt(b)*x/sqrt(a))/(a**
(3/2)*b**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0937019, size = 55, normalized size = 1. \[ -\frac{(a d-b c)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} b^{3/2}}-\frac{c^2}{a x}+\frac{d^2 x}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 85, normalized size = 1.6 \[{\frac{{d}^{2}x}{b}}-{\frac{a{d}^{2}}{b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+2\,{\frac{cd}{\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }-{\frac{b{c}^{2}}{a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{{c}^{2}}{ax}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.24246, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x \log \left (-\frac{2 \, a b x -{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (a d^{2} x^{2} - b c^{2}\right )} \sqrt{-a b}}{2 \, \sqrt{-a b} a b x}, -\frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x \arctan \left (\frac{\sqrt{a b} x}{a}\right ) -{\left (a d^{2} x^{2} - b c^{2}\right )} \sqrt{a b}}{\sqrt{a b} a b x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 2.76457, size = 165, normalized size = 3. \[ \frac{\sqrt{- \frac{1}{a^{3} b^{3}}} \left (a d - b c\right )^{2} \log{\left (- \frac{a^{2} b \sqrt{- \frac{1}{a^{3} b^{3}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} - \frac{\sqrt{- \frac{1}{a^{3} b^{3}}} \left (a d - b c\right )^{2} \log{\left (\frac{a^{2} b \sqrt{- \frac{1}{a^{3} b^{3}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac{d^{2} x}{b} - \frac{c^{2}}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.223369, size = 85, normalized size = 1.55 \[ \frac{d^{2} x}{b} - \frac{c^{2}}{a x} - \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d^2*x/b - c^2/(a*x) - (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*arctan(b*x/sqrt(a*b))/(sqr
t(a*b)*a*b)