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

Optimal. Leaf size=59 \[ -\frac{\sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{e \log \left (a+c x^2\right )}{2 a}-\frac{d}{a x}+\frac{e \log (x)}{a} \]

[Out]

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

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Rubi [A]  time = 0.0467175, antiderivative size = 59, 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} \[ -\frac{\sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{e \log \left (a+c x^2\right )}{2 a}-\frac{d}{a x}+\frac{e \log (x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(x^2*(a + c*x^2)),x]

[Out]

-(d/(a*x)) - (Sqrt[c]*d*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(3/2) + (e*Log[x])/a - (e*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^2 \left (a+c x^2\right )} \, dx &=\int \left (\frac{d}{a x^2}+\frac{e}{a x}-\frac{c (d+e x)}{a \left (a+c x^2\right )}\right ) \, dx\\ &=-\frac{d}{a x}+\frac{e \log (x)}{a}-\frac{c \int \frac{d+e x}{a+c x^2} \, dx}{a}\\ &=-\frac{d}{a x}+\frac{e \log (x)}{a}-\frac{(c d) \int \frac{1}{a+c x^2} \, dx}{a}-\frac{(c e) \int \frac{x}{a+c x^2} \, dx}{a}\\ &=-\frac{d}{a x}-\frac{\sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{3/2}}+\frac{e \log (x)}{a}-\frac{e \log \left (a+c x^2\right )}{2 a}\\ \end{align*}

Mathematica [A]  time = 0.0327074, size = 59, normalized size = 1. \[ -\frac{\sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{e \log \left (a+c x^2\right )}{2 a}-\frac{d}{a x}+\frac{e \log (x)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 53, normalized size = 0.9 \begin{align*} -{\frac{d}{ax}}+{\frac{e\ln \left ( x \right ) }{a}}-{\frac{e\ln \left ( c{x}^{2}+a \right ) }{2\,a}}-{\frac{cd}{a}\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^2/(c*x^2+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.6425, size = 289, normalized size = 4.9 \begin{align*} \left [\frac{d x \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{2} - 2 \, a x \sqrt{-\frac{c}{a}} - a}{c x^{2} + a}\right ) - e x \log \left (c x^{2} + a\right ) + 2 \, e x \log \left (x\right ) - 2 \, d}{2 \, a x}, -\frac{2 \, d x \sqrt{\frac{c}{a}} \arctan \left (x \sqrt{\frac{c}{a}}\right ) + e x \log \left (c x^{2} + a\right ) - 2 \, e x \log \left (x\right ) + 2 \, d}{2 \, a x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 1.51491, size = 326, normalized size = 5.53 \begin{align*} \left (- \frac{e}{2 a} - \frac{d \sqrt{- a^{3} c}}{2 a^{3}}\right ) \log{\left (x + \frac{12 a^{4} e \left (- \frac{e}{2 a} - \frac{d \sqrt{- a^{3} c}}{2 a^{3}}\right )^{2} - 6 a^{3} e^{2} \left (- \frac{e}{2 a} - \frac{d \sqrt{- a^{3} c}}{2 a^{3}}\right ) - 2 a^{2} c d^{2} \left (- \frac{e}{2 a} - \frac{d \sqrt{- a^{3} c}}{2 a^{3}}\right ) - 6 a^{2} e^{3} + 2 a c d^{2} e}{9 a c d e^{2} + c^{2} d^{3}} \right )} + \left (- \frac{e}{2 a} + \frac{d \sqrt{- a^{3} c}}{2 a^{3}}\right ) \log{\left (x + \frac{12 a^{4} e \left (- \frac{e}{2 a} + \frac{d \sqrt{- a^{3} c}}{2 a^{3}}\right )^{2} - 6 a^{3} e^{2} \left (- \frac{e}{2 a} + \frac{d \sqrt{- a^{3} c}}{2 a^{3}}\right ) - 2 a^{2} c d^{2} \left (- \frac{e}{2 a} + \frac{d \sqrt{- a^{3} c}}{2 a^{3}}\right ) - 6 a^{2} e^{3} + 2 a c d^{2} e}{9 a c d e^{2} + c^{2} d^{3}} \right )} - \frac{d}{a x} + \frac{e \log{\left (x \right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-e/(2*a) - d*sqrt(-a**3*c)/(2*a**3))*log(x + (12*a**4*e*(-e/(2*a) - d*sqrt(-a**3*c)/(2*a**3))**2 - 6*a**3*e**
2*(-e/(2*a) - d*sqrt(-a**3*c)/(2*a**3)) - 2*a**2*c*d**2*(-e/(2*a) - d*sqrt(-a**3*c)/(2*a**3)) - 6*a**2*e**3 +
2*a*c*d**2*e)/(9*a*c*d*e**2 + c**2*d**3)) + (-e/(2*a) + d*sqrt(-a**3*c)/(2*a**3))*log(x + (12*a**4*e*(-e/(2*a)
+ d*sqrt(-a**3*c)/(2*a**3))**2 - 6*a**3*e**2*(-e/(2*a) + d*sqrt(-a**3*c)/(2*a**3)) - 2*a**2*c*d**2*(-e/(2*a)
+ d*sqrt(-a**3*c)/(2*a**3)) - 6*a**2*e**3 + 2*a*c*d**2*e)/(9*a*c*d*e**2 + c**2*d**3)) - d/(a*x) + e*log(x)/a

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Giac [A]  time = 1.16152, size = 74, normalized size = 1.25 \begin{align*} -\frac{c d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{\sqrt{a c} a} - \frac{e \log \left (c x^{2} + a\right )}{2 \, a} + \frac{e \log \left ({\left | x \right |}\right )}{a} - \frac{d}{a x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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