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

Optimal. Leaf size=57 \[ \frac{d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{c}}-\frac{a e-c d x}{2 a c \left (a+c x^2\right )} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0396792, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{c}}-\frac{a e-c d x}{2 a c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)/(a + c*x^2)^2,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.54688, size = 46, normalized size = 0.81 \[ - \frac{a e - c d x}{2 a c \left (a + c x^{2}\right )} + \frac{d \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0425728, size = 57, normalized size = 1. \[ \frac{d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{c}}+\frac{c d x-a e}{2 a c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 49, normalized size = 0.9 \[{\frac{2\,cdx-2\,ae}{4\,ac \left ( c{x}^{2}+a \right ) }}+{\frac{d}{2\,a}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(c*x^2+a)^2,x)

[Out]

1/4*(2*c*d*x-2*a*e)/a/c/(c*x^2+a)+1/2*d/a/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))

_______________________________________________________________________________________

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((e*x + d)/(c*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.28633, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (c^{2} d x^{2} + a c d\right )} \log \left (\frac{2 \, a c x +{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) + 2 \,{\left (c d x - a e\right )} \sqrt{-a c}}{4 \,{\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt{-a c}}, \frac{{\left (c^{2} d x^{2} + a c d\right )} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) +{\left (c d x - a e\right )} \sqrt{a c}}{2 \,{\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 1.75598, size = 90, normalized size = 1.58 \[ d \left (- \frac{\sqrt{- \frac{1}{a^{3} c}} \log{\left (- a^{2} \sqrt{- \frac{1}{a^{3} c}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{3} c}} \log{\left (a^{2} \sqrt{- \frac{1}{a^{3} c}} + x \right )}}{4}\right ) + \frac{- a e + c d x}{2 a^{2} c + 2 a c^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.268605, size = 65, normalized size = 1.14 \[ \frac{d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} a} + \frac{c d x - a e}{2 \,{\left (c x^{2} + a\right )} a c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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