3.287 \(\int \frac{d+e x}{x^4 (a+c x^2)} \, dx\)

Optimal. Leaf size=83 \[ \frac{c^{3/2} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{5/2}}+\frac{c d}{a^2 x}+\frac{c e \log \left (a+c x^2\right )}{2 a^2}-\frac{c e \log (x)}{a^2}-\frac{d}{3 a x^3}-\frac{e}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.0577717, antiderivative size = 83, 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} \[ \frac{c^{3/2} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{5/2}}+\frac{c d}{a^2 x}+\frac{c e \log \left (a+c x^2\right )}{2 a^2}-\frac{c e \log (x)}{a^2}-\frac{d}{3 a x^3}-\frac{e}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(x^4*(a + c*x^2)),x]

[Out]

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

Mathematica [A]  time = 0.0419439, size = 77, normalized size = 0.93 \[ \frac{c^{3/2} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{5/2}}-\frac{-3 c e x^3 \log \left (a+c x^2\right )+2 a d+3 a e x-6 c d x^2+6 c e x^3 \log (x)}{6 a^2 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)/(x^4*(a + c*x^2)),x]

[Out]

(c^(3/2)*d*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(5/2) - (2*a*d + 3*a*e*x - 6*c*d*x^2 + 6*c*e*x^3*Log[x] - 3*c*e*x^3*
Log[a + c*x^2])/(6*a^2*x^3)

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Maple [A]  time = 0.007, size = 75, normalized size = 0.9 \begin{align*} -{\frac{d}{3\,a{x}^{3}}}-{\frac{e}{2\,a{x}^{2}}}+{\frac{cd}{{a}^{2}x}}-{\frac{ec\ln \left ( x \right ) }{{a}^{2}}}+{\frac{ec\ln \left ( c{x}^{2}+a \right ) }{2\,{a}^{2}}}+{\frac{{c}^{2}d}{{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^4/(c*x^2+a),x)

[Out]

-1/3*d/a/x^3-1/2*e/a/x^2+c*d/a^2/x-c*e*ln(x)/a^2+1/2*c*e*ln(c*x^2+a)/a^2+c^2/a^2*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^4/(c*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.62692, size = 404, normalized size = 4.87 \begin{align*} \left [\frac{3 \, c d x^{3} \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{2} + 2 \, a x \sqrt{-\frac{c}{a}} - a}{c x^{2} + a}\right ) + 3 \, c e x^{3} \log \left (c x^{2} + a\right ) - 6 \, c e x^{3} \log \left (x\right ) + 6 \, c d x^{2} - 3 \, a e x - 2 \, a d}{6 \, a^{2} x^{3}}, \frac{6 \, c d x^{3} \sqrt{\frac{c}{a}} \arctan \left (x \sqrt{\frac{c}{a}}\right ) + 3 \, c e x^{3} \log \left (c x^{2} + a\right ) - 6 \, c e x^{3} \log \left (x\right ) + 6 \, c d x^{2} - 3 \, a e x - 2 \, a d}{6 \, a^{2} x^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x^4/(c*x^2+a),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 1.88224, size = 408, normalized size = 4.92 \begin{align*} \left (\frac{c e}{2 a^{2}} - \frac{d \sqrt{- a^{5} c^{3}}}{2 a^{5}}\right ) \log{\left (x + \frac{12 a^{6} e \left (\frac{c e}{2 a^{2}} - \frac{d \sqrt{- a^{5} c^{3}}}{2 a^{5}}\right )^{2} + 6 a^{4} c e^{2} \left (\frac{c e}{2 a^{2}} - \frac{d \sqrt{- a^{5} c^{3}}}{2 a^{5}}\right ) + 2 a^{3} c^{2} d^{2} \left (\frac{c e}{2 a^{2}} - \frac{d \sqrt{- a^{5} c^{3}}}{2 a^{5}}\right ) - 6 a^{2} c^{2} e^{3} + 2 a c^{3} d^{2} e}{9 a c^{3} d e^{2} + c^{4} d^{3}} \right )} + \left (\frac{c e}{2 a^{2}} + \frac{d \sqrt{- a^{5} c^{3}}}{2 a^{5}}\right ) \log{\left (x + \frac{12 a^{6} e \left (\frac{c e}{2 a^{2}} + \frac{d \sqrt{- a^{5} c^{3}}}{2 a^{5}}\right )^{2} + 6 a^{4} c e^{2} \left (\frac{c e}{2 a^{2}} + \frac{d \sqrt{- a^{5} c^{3}}}{2 a^{5}}\right ) + 2 a^{3} c^{2} d^{2} \left (\frac{c e}{2 a^{2}} + \frac{d \sqrt{- a^{5} c^{3}}}{2 a^{5}}\right ) - 6 a^{2} c^{2} e^{3} + 2 a c^{3} d^{2} e}{9 a c^{3} d e^{2} + c^{4} d^{3}} \right )} - \frac{c e \log{\left (x \right )}}{a^{2}} + \frac{- 2 a d - 3 a e x + 6 c d x^{2}}{6 a^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x**4/(c*x**2+a),x)

[Out]

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

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Giac [A]  time = 1.10915, size = 103, normalized size = 1.24 \begin{align*} \frac{c^{2} d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{\sqrt{a c} a^{2}} + \frac{c e \log \left (c x^{2} + a\right )}{2 \, a^{2}} - \frac{c e \log \left ({\left | x \right |}\right )}{a^{2}} + \frac{6 \, c d x^{2} - 3 \, a x e - 2 \, a d}{6 \, a^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x^4/(c*x^2+a),x, algorithm="giac")

[Out]

c^2*d*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^2) + 1/2*c*e*log(c*x^2 + a)/a^2 - c*e*log(abs(x))/a^2 + 1/6*(6*c*d*x^
2 - 3*a*x*e - 2*a*d)/(a^2*x^3)