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

Optimal. Leaf size=73 \[ -\frac{d \log \left (a+c x^2\right )}{2 a^2}+\frac{e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{c}}+\frac{d \log (x)}{a^2}+\frac{d+e x}{2 a \left (a+c x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0619373, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {823, 801, 635, 205, 260} \[ -\frac{d \log \left (a+c x^2\right )}{2 a^2}+\frac{e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{c}}+\frac{d \log (x)}{a^2}+\frac{d+e x}{2 a \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(x*(a + c*x^2)^2),x]

[Out]

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

Rule 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

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

Mathematica [A]  time = 0.059459, size = 65, normalized size = 0.89 \[ \frac{\frac{a (d+e x)}{a+c x^2}-d \log \left (a+c x^2\right )+\frac{\sqrt{a} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{c}}+2 d \log (x)}{2 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.011, size = 74, normalized size = 1. \begin{align*}{\frac{d\ln \left ( x \right ) }{{a}^{2}}}+{\frac{ex}{2\,a \left ( c{x}^{2}+a \right ) }}+{\frac{d}{2\,a \left ( c{x}^{2}+a \right ) }}-{\frac{d\ln \left ( c{x}^{2}+a \right ) }{2\,{a}^{2}}}+{\frac{e}{2\,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/(c*x^2+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.66515, size = 487, normalized size = 6.67 \begin{align*} \left [\frac{2 \, a c e x + 2 \, a c d -{\left (c e x^{2} + a e\right )} \sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right ) - 2 \,{\left (c^{2} d x^{2} + a c d\right )} \log \left (c x^{2} + a\right ) + 4 \,{\left (c^{2} d x^{2} + a c d\right )} \log \left (x\right )}{4 \,{\left (a^{2} c^{2} x^{2} + a^{3} c\right )}}, \frac{a c e x + a c d +{\left (c e x^{2} + a e\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (c^{2} d x^{2} + a c d\right )} \log \left (c x^{2} + a\right ) + 2 \,{\left (c^{2} d x^{2} + a c d\right )} \log \left (x\right )}{2 \,{\left (a^{2} c^{2} x^{2} + a^{3} c\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(2*a*c*e*x + 2*a*c*d - (c*e*x^2 + a*e)*sqrt(-a*c)*log((c*x^2 - 2*sqrt(-a*c)*x - a)/(c*x^2 + a)) - 2*(c^2*
d*x^2 + a*c*d)*log(c*x^2 + a) + 4*(c^2*d*x^2 + a*c*d)*log(x))/(a^2*c^2*x^2 + a^3*c), 1/2*(a*c*e*x + a*c*d + (c
*e*x^2 + a*e)*sqrt(a*c)*arctan(sqrt(a*c)*x/a) - (c^2*d*x^2 + a*c*d)*log(c*x^2 + a) + 2*(c^2*d*x^2 + a*c*d)*log
(x))/(a^2*c^2*x^2 + a^3*c)]

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Sympy [B]  time = 1.76729, size = 359, normalized size = 4.92 \begin{align*} \left (- \frac{d}{2 a^{2}} - \frac{e \sqrt{- a^{5} c}}{4 a^{4} c}\right ) \log{\left (x + \frac{- 96 a^{4} c d \left (- \frac{d}{2 a^{2}} - \frac{e \sqrt{- a^{5} c}}{4 a^{4} c}\right )^{2} + 4 a^{3} e^{2} \left (- \frac{d}{2 a^{2}} - \frac{e \sqrt{- a^{5} c}}{4 a^{4} c}\right ) + 48 a^{2} c d^{2} \left (- \frac{d}{2 a^{2}} - \frac{e \sqrt{- a^{5} c}}{4 a^{4} c}\right ) - 4 a d e^{2} + 48 c d^{3}}{a e^{3} + 36 c d^{2} e} \right )} + \left (- \frac{d}{2 a^{2}} + \frac{e \sqrt{- a^{5} c}}{4 a^{4} c}\right ) \log{\left (x + \frac{- 96 a^{4} c d \left (- \frac{d}{2 a^{2}} + \frac{e \sqrt{- a^{5} c}}{4 a^{4} c}\right )^{2} + 4 a^{3} e^{2} \left (- \frac{d}{2 a^{2}} + \frac{e \sqrt{- a^{5} c}}{4 a^{4} c}\right ) + 48 a^{2} c d^{2} \left (- \frac{d}{2 a^{2}} + \frac{e \sqrt{- a^{5} c}}{4 a^{4} c}\right ) - 4 a d e^{2} + 48 c d^{3}}{a e^{3} + 36 c d^{2} e} \right )} + \frac{d + e x}{2 a^{2} + 2 a c x^{2}} + \frac{d \log{\left (x \right )}}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-d/(2*a**2) - e*sqrt(-a**5*c)/(4*a**4*c))*log(x + (-96*a**4*c*d*(-d/(2*a**2) - e*sqrt(-a**5*c)/(4*a**4*c))**2
+ 4*a**3*e**2*(-d/(2*a**2) - e*sqrt(-a**5*c)/(4*a**4*c)) + 48*a**2*c*d**2*(-d/(2*a**2) - e*sqrt(-a**5*c)/(4*a
**4*c)) - 4*a*d*e**2 + 48*c*d**3)/(a*e**3 + 36*c*d**2*e)) + (-d/(2*a**2) + e*sqrt(-a**5*c)/(4*a**4*c))*log(x +
(-96*a**4*c*d*(-d/(2*a**2) + e*sqrt(-a**5*c)/(4*a**4*c))**2 + 4*a**3*e**2*(-d/(2*a**2) + e*sqrt(-a**5*c)/(4*a
**4*c)) + 48*a**2*c*d**2*(-d/(2*a**2) + e*sqrt(-a**5*c)/(4*a**4*c)) - 4*a*d*e**2 + 48*c*d**3)/(a*e**3 + 36*c*d
**2*e)) + (d + e*x)/(2*a**2 + 2*a*c*x**2) + d*log(x)/a**2

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Giac [A]  time = 1.13862, size = 90, normalized size = 1.23 \begin{align*} \frac{\arctan \left (\frac{c x}{\sqrt{a c}}\right ) e}{2 \, \sqrt{a c} a} - \frac{d \log \left (c x^{2} + a\right )}{2 \, a^{2}} + \frac{d \log \left ({\left | x \right |}\right )}{a^{2}} + \frac{a x e + a d}{2 \,{\left (c x^{2} + a\right )} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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