3.286 \(\int \frac{d+e x}{x^3 (a+c x^2)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0586117, antiderivative size = 73, 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} \[ \frac{c d \log \left (a+c x^2\right )}{2 a^2}-\frac{c d \log (x)}{a^2}-\frac{\sqrt{c} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{d}{2 a x^2}-\frac{e}{a x} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(x^3*(a + c*x^2)),x]

[Out]

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

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

Mathematica [A]  time = 0.0304453, size = 73, normalized size = 1. \[ \frac{c d \log \left (a+c x^2\right )}{2 a^2}-\frac{c d \log (x)}{a^2}-\frac{\sqrt{c} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{d}{2 a x^2}-\frac{e}{a x} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)/(x^3*(a + c*x^2)),x]

[Out]

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

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Maple [A]  time = 0.007, size = 65, normalized size = 0.9 \begin{align*} -{\frac{d}{2\,a{x}^{2}}}-{\frac{e}{ax}}-{\frac{cd\ln \left ( x \right ) }{{a}^{2}}}+{\frac{cd\ln \left ( c{x}^{2}+a \right ) }{2\,{a}^{2}}}-{\frac{ce}{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^3/(c*x^2+a),x)

[Out]

-1/2*d/a/x^2-e/a/x-c*d*ln(x)/a^2+1/2*c*d*ln(c*x^2+a)/a^2-c/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^3/(c*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.65319, size = 359, normalized size = 4.92 \begin{align*} \left [\frac{a e x^{2} \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{2} - 2 \, a x \sqrt{-\frac{c}{a}} - a}{c x^{2} + a}\right ) + c d x^{2} \log \left (c x^{2} + a\right ) - 2 \, c d x^{2} \log \left (x\right ) - 2 \, a e x - a d}{2 \, a^{2} x^{2}}, -\frac{2 \, a e x^{2} \sqrt{\frac{c}{a}} \arctan \left (x \sqrt{\frac{c}{a}}\right ) - c d x^{2} \log \left (c x^{2} + a\right ) + 2 \, c d x^{2} \log \left (x\right ) + 2 \, a e x + a d}{2 \, a^{2} x^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x^3/(c*x^2+a),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 1.77561, size = 359, normalized size = 4.92 \begin{align*} \left (\frac{c d}{2 a^{2}} - \frac{e \sqrt{- a^{5} c}}{2 a^{4}}\right ) \log{\left (x + \frac{- 12 a^{4} d \left (\frac{c d}{2 a^{2}} - \frac{e \sqrt{- a^{5} c}}{2 a^{4}}\right )^{2} - 2 a^{3} e^{2} \left (\frac{c d}{2 a^{2}} - \frac{e \sqrt{- a^{5} c}}{2 a^{4}}\right ) - 6 a^{2} c d^{2} \left (\frac{c d}{2 a^{2}} - \frac{e \sqrt{- a^{5} c}}{2 a^{4}}\right ) - 2 a c d e^{2} + 6 c^{2} d^{3}}{a c e^{3} + 9 c^{2} d^{2} e} \right )} + \left (\frac{c d}{2 a^{2}} + \frac{e \sqrt{- a^{5} c}}{2 a^{4}}\right ) \log{\left (x + \frac{- 12 a^{4} d \left (\frac{c d}{2 a^{2}} + \frac{e \sqrt{- a^{5} c}}{2 a^{4}}\right )^{2} - 2 a^{3} e^{2} \left (\frac{c d}{2 a^{2}} + \frac{e \sqrt{- a^{5} c}}{2 a^{4}}\right ) - 6 a^{2} c d^{2} \left (\frac{c d}{2 a^{2}} + \frac{e \sqrt{- a^{5} c}}{2 a^{4}}\right ) - 2 a c d e^{2} + 6 c^{2} d^{3}}{a c e^{3} + 9 c^{2} d^{2} e} \right )} - \frac{d + 2 e x}{2 a x^{2}} - \frac{c 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**3/(c*x**2+a),x)

[Out]

(c*d/(2*a**2) - e*sqrt(-a**5*c)/(2*a**4))*log(x + (-12*a**4*d*(c*d/(2*a**2) - e*sqrt(-a**5*c)/(2*a**4))**2 - 2
*a**3*e**2*(c*d/(2*a**2) - e*sqrt(-a**5*c)/(2*a**4)) - 6*a**2*c*d**2*(c*d/(2*a**2) - e*sqrt(-a**5*c)/(2*a**4))
- 2*a*c*d*e**2 + 6*c**2*d**3)/(a*c*e**3 + 9*c**2*d**2*e)) + (c*d/(2*a**2) + e*sqrt(-a**5*c)/(2*a**4))*log(x +
(-12*a**4*d*(c*d/(2*a**2) + e*sqrt(-a**5*c)/(2*a**4))**2 - 2*a**3*e**2*(c*d/(2*a**2) + e*sqrt(-a**5*c)/(2*a**
4)) - 6*a**2*c*d**2*(c*d/(2*a**2) + e*sqrt(-a**5*c)/(2*a**4)) - 2*a*c*d*e**2 + 6*c**2*d**3)/(a*c*e**3 + 9*c**2
*d**2*e)) - (d + 2*e*x)/(2*a*x**2) - c*d*log(x)/a**2

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Giac [A]  time = 1.15504, size = 89, normalized size = 1.22 \begin{align*} -\frac{c \arctan \left (\frac{c x}{\sqrt{a c}}\right ) e}{\sqrt{a c} a} + \frac{c d \log \left (c x^{2} + a\right )}{2 \, a^{2}} - \frac{c d \log \left ({\left | x \right |}\right )}{a^{2}} - \frac{2 \, a x e + a d}{2 \, a^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x^3/(c*x^2+a),x, algorithm="giac")

[Out]

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