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

Optimal. Leaf size=87 \[ -\frac{3 \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{e \log \left (a+c x^2\right )}{2 a^2}-\frac{3 d}{2 a^2 x}+\frac{e \log (x)}{a^2}+\frac{d+e x}{2 a x \left (a+c x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0680223, antiderivative size = 87, 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} \[ -\frac{3 \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{e \log \left (a+c x^2\right )}{2 a^2}-\frac{3 d}{2 a^2 x}+\frac{e \log (x)}{a^2}+\frac{d+e x}{2 a x \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(x^2*(a + c*x^2)^2),x]

[Out]

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

Mathematica [A]  time = 0.120757, size = 81, normalized size = 0.93 \[ -\frac{\frac{2 a d-a e x+3 c d x^2}{a x+c x^3}+e \log \left (a+c x^2\right )-2 e \log (x)}{2 a^2}-\frac{3 \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.012, size = 85, normalized size = 1. \begin{align*} -{\frac{d}{{a}^{2}x}}+{\frac{e\ln \left ( x \right ) }{{a}^{2}}}-{\frac{cdx}{2\,{a}^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{e}{2\,a \left ( c{x}^{2}+a \right ) }}-{\frac{e\ln \left ( c{x}^{2}+a \right ) }{2\,{a}^{2}}}-{\frac{3\,cd}{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^2/(c*x^2+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.75394, size = 509, normalized size = 5.85 \begin{align*} \left [-\frac{6 \, c d x^{2} - 2 \, a e x - 3 \,{\left (c d x^{3} + a d x\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 \, a d + 2 \,{\left (c e x^{3} + a e x\right )} \log \left (c x^{2} + a\right ) - 4 \,{\left (c e x^{3} + a e x\right )} \log \left (x\right )}{4 \,{\left (a^{2} c x^{3} + a^{3} x\right )}}, -\frac{3 \, c d x^{2} - a e x + 3 \,{\left (c d x^{3} + a d x\right )} \sqrt{\frac{c}{a}} \arctan \left (x \sqrt{\frac{c}{a}}\right ) + 2 \, a d +{\left (c e x^{3} + a e x\right )} \log \left (c x^{2} + a\right ) - 2 \,{\left (c e x^{3} + a e x\right )} \log \left (x\right )}{2 \,{\left (a^{2} c x^{3} + a^{3} x\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 1.95499, size = 389, normalized size = 4.47 \begin{align*} \left (- \frac{e}{2 a^{2}} - \frac{3 d \sqrt{- a^{5} c}}{4 a^{5}}\right ) \log{\left (x + \frac{32 a^{6} e \left (- \frac{e}{2 a^{2}} - \frac{3 d \sqrt{- a^{5} c}}{4 a^{5}}\right )^{2} - 16 a^{4} e^{2} \left (- \frac{e}{2 a^{2}} - \frac{3 d \sqrt{- a^{5} c}}{4 a^{5}}\right ) - 12 a^{3} c d^{2} \left (- \frac{e}{2 a^{2}} - \frac{3 d \sqrt{- a^{5} c}}{4 a^{5}}\right ) - 16 a^{2} e^{3} + 12 a c d^{2} e}{36 a c d e^{2} + 9 c^{2} d^{3}} \right )} + \left (- \frac{e}{2 a^{2}} + \frac{3 d \sqrt{- a^{5} c}}{4 a^{5}}\right ) \log{\left (x + \frac{32 a^{6} e \left (- \frac{e}{2 a^{2}} + \frac{3 d \sqrt{- a^{5} c}}{4 a^{5}}\right )^{2} - 16 a^{4} e^{2} \left (- \frac{e}{2 a^{2}} + \frac{3 d \sqrt{- a^{5} c}}{4 a^{5}}\right ) - 12 a^{3} c d^{2} \left (- \frac{e}{2 a^{2}} + \frac{3 d \sqrt{- a^{5} c}}{4 a^{5}}\right ) - 16 a^{2} e^{3} + 12 a c d^{2} e}{36 a c d e^{2} + 9 c^{2} d^{3}} \right )} - \frac{2 a d - a e x + 3 c d x^{2}}{2 a^{3} x + 2 a^{2} c x^{3}} + \frac{e \log{\left (x \right )}}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-e/(2*a**2) - 3*d*sqrt(-a**5*c)/(4*a**5))*log(x + (32*a**6*e*(-e/(2*a**2) - 3*d*sqrt(-a**5*c)/(4*a**5))**2 -
16*a**4*e**2*(-e/(2*a**2) - 3*d*sqrt(-a**5*c)/(4*a**5)) - 12*a**3*c*d**2*(-e/(2*a**2) - 3*d*sqrt(-a**5*c)/(4*a
**5)) - 16*a**2*e**3 + 12*a*c*d**2*e)/(36*a*c*d*e**2 + 9*c**2*d**3)) + (-e/(2*a**2) + 3*d*sqrt(-a**5*c)/(4*a**
5))*log(x + (32*a**6*e*(-e/(2*a**2) + 3*d*sqrt(-a**5*c)/(4*a**5))**2 - 16*a**4*e**2*(-e/(2*a**2) + 3*d*sqrt(-a
**5*c)/(4*a**5)) - 12*a**3*c*d**2*(-e/(2*a**2) + 3*d*sqrt(-a**5*c)/(4*a**5)) - 16*a**2*e**3 + 12*a*c*d**2*e)/(
36*a*c*d*e**2 + 9*c**2*d**3)) - (2*a*d - a*e*x + 3*c*d*x**2)/(2*a**3*x + 2*a**2*c*x**3) + e*log(x)/a**2

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Giac [A]  time = 1.11416, size = 108, normalized size = 1.24 \begin{align*} -\frac{3 \, c d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} a^{2}} - \frac{e \log \left (c x^{2} + a\right )}{2 \, a^{2}} + \frac{e \log \left ({\left | x \right |}\right )}{a^{2}} - \frac{3 \, c d x^{2} - a x e + 2 \, a d}{2 \,{\left (c x^{3} + a x\right )} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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