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

Optimal. Leaf size=96 \[ \frac{c d \log \left (a+c x^2\right )}{a^3}-\frac{2 c d \log (x)}{a^3}-\frac{3 \sqrt{c} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{d}{a^2 x^2}-\frac{3 e}{2 a^2 x}+\frac{d+e x}{2 a x^2 \left (a+c x^2\right )} \]

[Out]

-(d/(a^2*x^2)) - (3*e)/(2*a^2*x) + (d + e*x)/(2*a*x^2*(a + c*x^2)) - (3*Sqrt[c]*e*ArcTan[(Sqrt[c]*x)/Sqrt[a]])
/(2*a^(5/2)) - (2*c*d*Log[x])/a^3 + (c*d*Log[a + c*x^2])/a^3

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Rubi [A]  time = 0.0836267, antiderivative size = 96, 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} \[ \frac{c d \log \left (a+c x^2\right )}{a^3}-\frac{2 c d \log (x)}{a^3}-\frac{3 \sqrt{c} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{d}{a^2 x^2}-\frac{3 e}{2 a^2 x}+\frac{d+e x}{2 a x^2 \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(x^3*(a + c*x^2)^2),x]

[Out]

-(d/(a^2*x^2)) - (3*e)/(2*a^2*x) + (d + e*x)/(2*a*x^2*(a + c*x^2)) - (3*Sqrt[c]*e*ArcTan[(Sqrt[c]*x)/Sqrt[a]])
/(2*a^(5/2)) - (2*c*d*Log[x])/a^3 + (c*d*Log[a + c*x^2])/a^3

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^3 \left (a+c x^2\right )^2} \, dx &=\frac{d+e x}{2 a x^2 \left (a+c x^2\right )}-\frac{\int \frac{-4 a c d-3 a c e x}{x^3 \left (a+c x^2\right )} \, dx}{2 a^2 c}\\ &=\frac{d+e x}{2 a x^2 \left (a+c x^2\right )}-\frac{\int \left (-\frac{4 c d}{x^3}-\frac{3 c e}{x^2}+\frac{4 c^2 d}{a x}+\frac{c^2 (3 a e-4 c d x)}{a \left (a+c x^2\right )}\right ) \, dx}{2 a^2 c}\\ &=-\frac{d}{a^2 x^2}-\frac{3 e}{2 a^2 x}+\frac{d+e x}{2 a x^2 \left (a+c x^2\right )}-\frac{2 c d \log (x)}{a^3}-\frac{c \int \frac{3 a e-4 c d x}{a+c x^2} \, dx}{2 a^3}\\ &=-\frac{d}{a^2 x^2}-\frac{3 e}{2 a^2 x}+\frac{d+e x}{2 a x^2 \left (a+c x^2\right )}-\frac{2 c d \log (x)}{a^3}+\frac{\left (2 c^2 d\right ) \int \frac{x}{a+c x^2} \, dx}{a^3}-\frac{(3 c e) \int \frac{1}{a+c x^2} \, dx}{2 a^2}\\ &=-\frac{d}{a^2 x^2}-\frac{3 e}{2 a^2 x}+\frac{d+e x}{2 a x^2 \left (a+c x^2\right )}-\frac{3 \sqrt{c} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{2 c d \log (x)}{a^3}+\frac{c d \log \left (a+c x^2\right )}{a^3}\\ \end{align*}

Mathematica [A]  time = 0.108234, size = 82, normalized size = 0.85 \[ -\frac{\frac{a c (d+e x)}{a+c x^2}-2 c d \log \left (a+c x^2\right )+3 \sqrt{a} \sqrt{c} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )+\frac{a d}{x^2}+\frac{2 a e}{x}+4 c d \log (x)}{2 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*d)/x^2 + (2*a*e)/x + (a*c*(d + e*x))/(a + c*x^2) + 3*Sqrt[a]*Sqrt[c]*e*ArcTan[(Sqrt[c]*x)/Sqrt[a]] + 4*c*
d*Log[x] - 2*c*d*Log[a + c*x^2])/(2*a^3)

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Maple [A]  time = 0.013, size = 97, normalized size = 1. \begin{align*} -{\frac{d}{2\,{a}^{2}{x}^{2}}}-{\frac{e}{{a}^{2}x}}-2\,{\frac{cd\ln \left ( x \right ) }{{a}^{3}}}-{\frac{cex}{2\,{a}^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{cd}{2\,{a}^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{cd\ln \left ( c{x}^{2}+a \right ) }{{a}^{3}}}-{\frac{3\,ce}{2\,{a}^{2}}\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)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.61931, size = 620, normalized size = 6.46 \begin{align*} \left [-\frac{6 \, a c e x^{3} + 4 \, a c d x^{2} + 4 \, a^{2} e x + 2 \, a^{2} d - 3 \,{\left (a c e x^{4} + a^{2} e x^{2}\right )} \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{2} - 2 \, a x \sqrt{-\frac{c}{a}} - a}{c x^{2} + a}\right ) - 4 \,{\left (c^{2} d x^{4} + a c d x^{2}\right )} \log \left (c x^{2} + a\right ) + 8 \,{\left (c^{2} d x^{4} + a c d x^{2}\right )} \log \left (x\right )}{4 \,{\left (a^{3} c x^{4} + a^{4} x^{2}\right )}}, -\frac{3 \, a c e x^{3} + 2 \, a c d x^{2} + 2 \, a^{2} e x + a^{2} d + 3 \,{\left (a c e x^{4} + a^{2} e x^{2}\right )} \sqrt{\frac{c}{a}} \arctan \left (x \sqrt{\frac{c}{a}}\right ) - 2 \,{\left (c^{2} d x^{4} + a c d x^{2}\right )} \log \left (c x^{2} + a\right ) + 4 \,{\left (c^{2} d x^{4} + a c d x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{3} c x^{4} + a^{4} x^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(6*a*c*e*x^3 + 4*a*c*d*x^2 + 4*a^2*e*x + 2*a^2*d - 3*(a*c*e*x^4 + a^2*e*x^2)*sqrt(-c/a)*log((c*x^2 - 2*a
*x*sqrt(-c/a) - a)/(c*x^2 + a)) - 4*(c^2*d*x^4 + a*c*d*x^2)*log(c*x^2 + a) + 8*(c^2*d*x^4 + a*c*d*x^2)*log(x))
/(a^3*c*x^4 + a^4*x^2), -1/2*(3*a*c*e*x^3 + 2*a*c*d*x^2 + 2*a^2*e*x + a^2*d + 3*(a*c*e*x^4 + a^2*e*x^2)*sqrt(c
/a)*arctan(x*sqrt(c/a)) - 2*(c^2*d*x^4 + a*c*d*x^2)*log(c*x^2 + a) + 4*(c^2*d*x^4 + a*c*d*x^2)*log(x))/(a^3*c*
x^4 + a^4*x^2)]

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Sympy [B]  time = 2.12207, size = 396, normalized size = 4.12 \begin{align*} \left (\frac{c d}{a^{3}} - \frac{3 e \sqrt{- a^{7} c}}{4 a^{6}}\right ) \log{\left (x + \frac{- 64 a^{6} d \left (\frac{c d}{a^{3}} - \frac{3 e \sqrt{- a^{7} c}}{4 a^{6}}\right )^{2} - 12 a^{4} e^{2} \left (\frac{c d}{a^{3}} - \frac{3 e \sqrt{- a^{7} c}}{4 a^{6}}\right ) - 64 a^{3} c d^{2} \left (\frac{c d}{a^{3}} - \frac{3 e \sqrt{- a^{7} c}}{4 a^{6}}\right ) - 24 a c d e^{2} + 128 c^{2} d^{3}}{9 a c e^{3} + 144 c^{2} d^{2} e} \right )} + \left (\frac{c d}{a^{3}} + \frac{3 e \sqrt{- a^{7} c}}{4 a^{6}}\right ) \log{\left (x + \frac{- 64 a^{6} d \left (\frac{c d}{a^{3}} + \frac{3 e \sqrt{- a^{7} c}}{4 a^{6}}\right )^{2} - 12 a^{4} e^{2} \left (\frac{c d}{a^{3}} + \frac{3 e \sqrt{- a^{7} c}}{4 a^{6}}\right ) - 64 a^{3} c d^{2} \left (\frac{c d}{a^{3}} + \frac{3 e \sqrt{- a^{7} c}}{4 a^{6}}\right ) - 24 a c d e^{2} + 128 c^{2} d^{3}}{9 a c e^{3} + 144 c^{2} d^{2} e} \right )} - \frac{a d + 2 a e x + 2 c d x^{2} + 3 c e x^{3}}{2 a^{3} x^{2} + 2 a^{2} c x^{4}} - \frac{2 c d \log{\left (x \right )}}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x**3/(c*x**2+a)**2,x)

[Out]

(c*d/a**3 - 3*e*sqrt(-a**7*c)/(4*a**6))*log(x + (-64*a**6*d*(c*d/a**3 - 3*e*sqrt(-a**7*c)/(4*a**6))**2 - 12*a*
*4*e**2*(c*d/a**3 - 3*e*sqrt(-a**7*c)/(4*a**6)) - 64*a**3*c*d**2*(c*d/a**3 - 3*e*sqrt(-a**7*c)/(4*a**6)) - 24*
a*c*d*e**2 + 128*c**2*d**3)/(9*a*c*e**3 + 144*c**2*d**2*e)) + (c*d/a**3 + 3*e*sqrt(-a**7*c)/(4*a**6))*log(x +
(-64*a**6*d*(c*d/a**3 + 3*e*sqrt(-a**7*c)/(4*a**6))**2 - 12*a**4*e**2*(c*d/a**3 + 3*e*sqrt(-a**7*c)/(4*a**6))
- 64*a**3*c*d**2*(c*d/a**3 + 3*e*sqrt(-a**7*c)/(4*a**6)) - 24*a*c*d*e**2 + 128*c**2*d**3)/(9*a*c*e**3 + 144*c*
*2*d**2*e)) - (a*d + 2*a*e*x + 2*c*d*x**2 + 3*c*e*x**3)/(2*a**3*x**2 + 2*a**2*c*x**4) - 2*c*d*log(x)/a**3

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Giac [A]  time = 1.18226, size = 128, normalized size = 1.33 \begin{align*} -\frac{3 \, c \arctan \left (\frac{c x}{\sqrt{a c}}\right ) e}{2 \, \sqrt{a c} a^{2}} + \frac{c d \log \left (c x^{2} + a\right )}{a^{3}} - \frac{2 \, c d \log \left ({\left | x \right |}\right )}{a^{3}} - \frac{3 \, a c x^{3} e + 2 \, a c d x^{2} + 2 \, a^{2} x e + a^{2} d}{2 \,{\left (c x^{2} + a\right )} a^{3} 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)^2,x, algorithm="giac")

[Out]

-3/2*c*arctan(c*x/sqrt(a*c))*e/(sqrt(a*c)*a^2) + c*d*log(c*x^2 + a)/a^3 - 2*c*d*log(abs(x))/a^3 - 1/2*(3*a*c*x
^3*e + 2*a*c*d*x^2 + 2*a^2*x*e + a^2*d)/((c*x^2 + a)*a^3*x^2)