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

Optimal. Leaf size=42 \[ \frac{d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}}+\frac{e \log \left (a+c x^2\right )}{2 c} \]

[Out]

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

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Rubi [A]  time = 0.0460827, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}}+\frac{e \log \left (a+c x^2\right )}{2 c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)/(a + c*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.87532, size = 37, normalized size = 0.88 \[ \frac{e \log{\left (a + c x^{2} \right )}}{2 c} + \frac{d \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{\sqrt{a} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0249424, size = 42, normalized size = 1. \[ \frac{d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}}+\frac{e \log \left (a+c x^2\right )}{2 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 32, normalized size = 0.8 \[{\frac{e\ln \left ( c{x}^{2}+a \right ) }{2\,c}}+{d\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(c*x^2+a),x)

[Out]

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

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.213374, size = 1, normalized size = 0.02 \[ \left [\frac{c d \log \left (\frac{2 \, a c x +{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) + \sqrt{-a c} e \log \left (c x^{2} + a\right )}{2 \, \sqrt{-a c} c}, \frac{2 \, c d \arctan \left (\frac{\sqrt{a c} x}{a}\right ) + \sqrt{a c} e \log \left (c x^{2} + a\right )}{2 \, \sqrt{a c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.740287, size = 124, normalized size = 2.95 \[ \left (\frac{e}{2 c} - \frac{d \sqrt{- a c^{3}}}{2 a c^{2}}\right ) \log{\left (x + \frac{2 a c \left (\frac{e}{2 c} - \frac{d \sqrt{- a c^{3}}}{2 a c^{2}}\right ) - a e}{c d} \right )} + \left (\frac{e}{2 c} + \frac{d \sqrt{- a c^{3}}}{2 a c^{2}}\right ) \log{\left (x + \frac{2 a c \left (\frac{e}{2 c} + \frac{d \sqrt{- a c^{3}}}{2 a c^{2}}\right ) - a e}{c d} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)/(c*x**2+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.213887, size = 43, normalized size = 1.02 \[ \frac{d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{\sqrt{a c}} + \frac{e{\rm ln}\left (c x^{2} + a\right )}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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