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

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

[Out]

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

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Rubi [A]  time = 0.0419312, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {801, 635, 205, 260} \[ -\frac{d \log \left (a+c x^2\right )}{2 a}+\frac{e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}}+\frac{d \log (x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(x*(a + c*x^2)),x]

[Out]

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

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{d+e x}{x \left (a+c x^2\right )} \, dx &=\int \left (\frac{d}{a x}+\frac{a e-c d x}{a \left (a+c x^2\right )}\right ) \, dx\\ &=\frac{d \log (x)}{a}+\frac{\int \frac{a e-c d x}{a+c x^2} \, dx}{a}\\ &=\frac{d \log (x)}{a}-\frac{(c d) \int \frac{x}{a+c x^2} \, dx}{a}+e \int \frac{1}{a+c x^2} \, dx\\ &=\frac{e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}}+\frac{d \log (x)}{a}-\frac{d \log \left (a+c x^2\right )}{2 a}\\ \end{align*}

Mathematica [A]  time = 0.0206574, size = 49, normalized size = 1. \[ -\frac{d \log \left (a+c x^2\right )}{2 a}+\frac{e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}}+\frac{d \log (x)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 39, normalized size = 0.8 \begin{align*}{\frac{d\ln \left ( x \right ) }{a}}-{\frac{d\ln \left ( c{x}^{2}+a \right ) }{2\,a}}+{e\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)/x/(c*x^2+a),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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 = 1.55701, size = 269, normalized size = 5.49 \begin{align*} \left [-\frac{c d \log \left (c x^{2} + a\right ) - 2 \, c d \log \left (x\right ) + \sqrt{-a c} e \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right )}{2 \, a c}, -\frac{c d \log \left (c x^{2} + a\right ) - 2 \, c d \log \left (x\right ) - 2 \, \sqrt{a c} e \arctan \left (\frac{\sqrt{a c} x}{a}\right )}{2 \, a c}\right ] \end{align*}

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

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Sympy [B]  time = 1.3596, size = 321, normalized size = 6.55 \begin{align*} \left (- \frac{d}{2 a} - \frac{e \sqrt{- a^{3} c}}{2 a^{2} c}\right ) \log{\left (x + \frac{- 12 a^{2} c d \left (- \frac{d}{2 a} - \frac{e \sqrt{- a^{3} c}}{2 a^{2} c}\right )^{2} + 2 a^{2} e^{2} \left (- \frac{d}{2 a} - \frac{e \sqrt{- a^{3} c}}{2 a^{2} c}\right ) + 6 a c d^{2} \left (- \frac{d}{2 a} - \frac{e \sqrt{- a^{3} c}}{2 a^{2} c}\right ) - 2 a d e^{2} + 6 c d^{3}}{a e^{3} + 9 c d^{2} e} \right )} + \left (- \frac{d}{2 a} + \frac{e \sqrt{- a^{3} c}}{2 a^{2} c}\right ) \log{\left (x + \frac{- 12 a^{2} c d \left (- \frac{d}{2 a} + \frac{e \sqrt{- a^{3} c}}{2 a^{2} c}\right )^{2} + 2 a^{2} e^{2} \left (- \frac{d}{2 a} + \frac{e \sqrt{- a^{3} c}}{2 a^{2} c}\right ) + 6 a c d^{2} \left (- \frac{d}{2 a} + \frac{e \sqrt{- a^{3} c}}{2 a^{2} c}\right ) - 2 a d e^{2} + 6 c d^{3}}{a e^{3} + 9 c d^{2} e} \right )} + \frac{d \log{\left (x \right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-d/(2*a) - e*sqrt(-a**3*c)/(2*a**2*c))*log(x + (-12*a**2*c*d*(-d/(2*a) - e*sqrt(-a**3*c)/(2*a**2*c))**2 + 2*a
**2*e**2*(-d/(2*a) - e*sqrt(-a**3*c)/(2*a**2*c)) + 6*a*c*d**2*(-d/(2*a) - e*sqrt(-a**3*c)/(2*a**2*c)) - 2*a*d*
e**2 + 6*c*d**3)/(a*e**3 + 9*c*d**2*e)) + (-d/(2*a) + e*sqrt(-a**3*c)/(2*a**2*c))*log(x + (-12*a**2*c*d*(-d/(2
*a) + e*sqrt(-a**3*c)/(2*a**2*c))**2 + 2*a**2*e**2*(-d/(2*a) + e*sqrt(-a**3*c)/(2*a**2*c)) + 6*a*c*d**2*(-d/(2
*a) + e*sqrt(-a**3*c)/(2*a**2*c)) - 2*a*d*e**2 + 6*c*d**3)/(a*e**3 + 9*c*d**2*e)) + d*log(x)/a

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Giac [A]  time = 1.11762, size = 54, normalized size = 1.1 \begin{align*} \frac{\arctan \left (\frac{c x}{\sqrt{a c}}\right ) e}{\sqrt{a c}} - \frac{d \log \left (c x^{2} + a\right )}{2 \, a} + \frac{d \log \left ({\left | x \right |}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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