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

Optimal. Leaf size=40 \[ \frac{b x}{d}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} d^{3/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0524135, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{b x}{d}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} d^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)/(c + d*x^2),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.76221, size = 34, normalized size = 0.85 \[ \frac{b x}{d} + \frac{\left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{\sqrt{c} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.038996, size = 40, normalized size = 1. \[ \frac{b x}{d}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} d^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 45, normalized size = 1.1 \[{\frac{bx}{d}}+{a\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{bc}{d}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)/(d*x^2+c),x)

[Out]

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

_______________________________________________________________________________________

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)/(d*x^2 + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.206565, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, \sqrt{-c d} b x -{\left (b c - a d\right )} \log \left (\frac{2 \, c d x +{\left (d x^{2} - c\right )} \sqrt{-c d}}{d x^{2} + c}\right )}{2 \, \sqrt{-c d} d}, \frac{\sqrt{c d} b x -{\left (b c - a d\right )} \arctan \left (\frac{\sqrt{c d} x}{c}\right )}{\sqrt{c d} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 1.56986, size = 82, normalized size = 2.05 \[ \frac{b x}{d} - \frac{\sqrt{- \frac{1}{c d^{3}}} \left (a d - b c\right ) \log{\left (- c d \sqrt{- \frac{1}{c d^{3}}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{c d^{3}}} \left (a d - b c\right ) \log{\left (c d \sqrt{- \frac{1}{c d^{3}}} + x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.230823, size = 46, normalized size = 1.15 \[ \frac{b x}{d} - \frac{{\left (b c - a d\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{\sqrt{c d} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x/d - (b*c - a*d)*arctan(d*x/sqrt(c*d))/(sqrt(c*d)*d)