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

Optimal. Leaf size=49 \[ -\frac{\sqrt{a} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{c^{3/2}}+\frac{d \log \left (a+c x^2\right )}{2 c}+\frac{e x}{c} \]

[Out]

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

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Rubi [A]  time = 0.0671577, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\sqrt{a} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{c^{3/2}}+\frac{d \log \left (a+c x^2\right )}{2 c}+\frac{e x}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.037829, size = 49, normalized size = 1. \[ -\frac{\sqrt{a} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{c^{3/2}}+\frac{d \log \left (a+c x^2\right )}{2 c}+\frac{e x}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 43, normalized size = 0.9 \[{\frac{ex}{c}}+{\frac{d\ln \left ( c{x}^{2}+a \right ) }{2\,c}}-{\frac{ae}{c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(e*x+d)/(c*x^2+a),x)

[Out]

e*x/c+1/2*d*ln(c*x^2+a)/c-1/c*a*e/(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)*x/(c*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.293521, size = 1, normalized size = 0.02 \[ \left [\frac{e \sqrt{-\frac{a}{c}} \log \left (\frac{c x^{2} - 2 \, c x \sqrt{-\frac{a}{c}} - a}{c x^{2} + a}\right ) + 2 \, e x + d \log \left (c x^{2} + a\right )}{2 \, c}, -\frac{2 \, e \sqrt{\frac{a}{c}} \arctan \left (\frac{x}{\sqrt{\frac{a}{c}}}\right ) - 2 \, e x - d \log \left (c x^{2} + a\right )}{2 \, c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.6512, size = 112, normalized size = 2.29 \[ \left (\frac{d}{2 c} - \frac{e \sqrt{- a c^{3}}}{2 c^{3}}\right ) \log{\left (x + \frac{- 2 c \left (\frac{d}{2 c} - \frac{e \sqrt{- a c^{3}}}{2 c^{3}}\right ) + d}{e} \right )} + \left (\frac{d}{2 c} + \frac{e \sqrt{- a c^{3}}}{2 c^{3}}\right ) \log{\left (x + \frac{- 2 c \left (\frac{d}{2 c} + \frac{e \sqrt{- a c^{3}}}{2 c^{3}}\right ) + d}{e} \right )} + \frac{e x}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.269151, size = 59, normalized size = 1.2 \[ -\frac{a \arctan \left (\frac{c x}{\sqrt{a c}}\right ) e}{\sqrt{a c} c} + \frac{x e}{c} + \frac{d{\rm ln}\left (c x^{2} + a\right )}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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