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

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

[Out]

-d/a^2/x^2-3/2*e/a^2/x+1/2*(e*x+d)/a/x^2/(c*x^2+a)-2*c*d*ln(x)/a^3+c*d*ln(c*x^2+a)/a^3-3/2*e*arctan(x*c^(1/2)/
a^(1/2))*c^(1/2)/a^(5/2)

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Rubi [A]  time = 0.08, antiderivative size = 96, normalized size of antiderivative = 1.00, 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 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]

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 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 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])

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*}

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Mathematica [A]  time = 0.10, 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]

-1/2*((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])/a^3

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fricas [A]  time = 1.05, size = 285, normalized size = 2.97 \[ \left [-\frac {6 \, a c e x^{3} + 4 \, a c d x^{2} + 4 \, a^{2} e x + 2 \, a^{2} d - 3 \, {\left (a c e x^{4} + a^{2} e x^{2}\right )} \sqrt {-\frac {c}{a}} \log \left (\frac {c x^{2} - 2 \, a x \sqrt {-\frac {c}{a}} - a}{c x^{2} + a}\right ) - 4 \, {\left (c^{2} d x^{4} + a c d x^{2}\right )} \log \left (c x^{2} + a\right ) + 8 \, {\left (c^{2} d x^{4} + a c d x^{2}\right )} \log \relax (x)}{4 \, {\left (a^{3} c x^{4} + a^{4} x^{2}\right )}}, -\frac {3 \, a c e x^{3} + 2 \, a c d x^{2} + 2 \, a^{2} e x + a^{2} d + 3 \, {\left (a c e x^{4} + a^{2} e x^{2}\right )} \sqrt {\frac {c}{a}} \arctan \left (x \sqrt {\frac {c}{a}}\right ) - 2 \, {\left (c^{2} d x^{4} + a c d x^{2}\right )} \log \left (c x^{2} + a\right ) + 4 \, {\left (c^{2} d x^{4} + a c d x^{2}\right )} \log \relax (x)}{2 \, {\left (a^{3} c x^{4} + a^{4} x^{2}\right )}}\right ] \]

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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giac [A]  time = 0.16, size = 95, normalized size = 0.99 \[ -\frac {3 \, c \arctan \left (\frac {c x}{\sqrt {a c}}\right ) e}{2 \, \sqrt {a c} a^{2}} + \frac {c d \log \left (c x^{2} + a\right )}{a^{3}} - \frac {2 \, c d \log \left ({\left | x \right |}\right )}{a^{3}} - \frac {3 \, a c x^{3} e + 2 \, a c d x^{2} + 2 \, a^{2} x e + a^{2} d}{2 \, {\left (c x^{2} + a\right )} a^{3} x^{2}} \]

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)

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maple [A]  time = 0.06, size = 97, normalized size = 1.01 \[ -\frac {c e x}{2 \left (c \,x^{2}+a \right ) a^{2}}-\frac {3 c e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, a^{2}}-\frac {c d}{2 \left (c \,x^{2}+a \right ) a^{2}}-\frac {2 c d \ln \relax (x )}{a^{3}}+\frac {c d \ln \left (c \,x^{2}+a \right )}{a^{3}}-\frac {e}{a^{2} x}-\frac {d}{2 a^{2} x^{2}} \]

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

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maxima [A]  time = 1.18, size = 88, normalized size = 0.92 \[ -\frac {3 \, c e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a^{2}} - \frac {3 \, c e x^{3} + 2 \, c d x^{2} + 2 \, a e x + a d}{2 \, {\left (a^{2} c x^{4} + a^{3} x^{2}\right )}} + \frac {c d \log \left (c x^{2} + a\right )}{a^{3}} - \frac {2 \, c d \log \relax (x)}{a^{3}} \]

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]

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

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mupad [B]  time = 1.26, size = 186, normalized size = 1.94 \[ \frac {\ln \left (a\,e\,\sqrt {-a^7\,c}+4\,a^4\,c\,d-a^4\,c\,e\,x+4\,c\,d\,x\,\sqrt {-a^7\,c}\right )\,\left (3\,e\,\sqrt {-a^7\,c}+4\,a^3\,c\,d\right )}{4\,a^6}-\frac {\ln \left (a\,e\,\sqrt {-a^7\,c}-4\,a^4\,c\,d+a^4\,c\,e\,x+4\,c\,d\,x\,\sqrt {-a^7\,c}\right )\,\left (3\,e\,\sqrt {-a^7\,c}-4\,a^3\,c\,d\right )}{4\,a^6}-\frac {\frac {d}{2\,a}+\frac {e\,x}{a}+\frac {c\,d\,x^2}{a^2}+\frac {3\,c\,e\,x^3}{2\,a^2}}{c\,x^4+a\,x^2}-\frac {2\,c\,d\,\ln \relax (x)}{a^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)/(x^3*(a + c*x^2)^2),x)

[Out]

(log(a*e*(-a^7*c)^(1/2) + 4*a^4*c*d - a^4*c*e*x + 4*c*d*x*(-a^7*c)^(1/2))*(3*e*(-a^7*c)^(1/2) + 4*a^3*c*d))/(4
*a^6) - (log(a*e*(-a^7*c)^(1/2) - 4*a^4*c*d + a^4*c*e*x + 4*c*d*x*(-a^7*c)^(1/2))*(3*e*(-a^7*c)^(1/2) - 4*a^3*
c*d))/(4*a^6) - (d/(2*a) + (e*x)/a + (c*d*x^2)/a^2 + (3*c*e*x^3)/(2*a^2))/(a*x^2 + c*x^4) - (2*c*d*log(x))/a^3

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sympy [B]  time = 2.04, size = 398, normalized size = 4.15 \[ \left (\frac {c d}{a^{3}} - \frac {3 e \sqrt {- a^{7} c}}{4 a^{6}}\right ) \log {\left (x + \frac {- 64 a^{6} d \left (\frac {c d}{a^{3}} - \frac {3 e \sqrt {- a^{7} c}}{4 a^{6}}\right )^{2} - 12 a^{4} e^{2} \left (\frac {c d}{a^{3}} - \frac {3 e \sqrt {- a^{7} c}}{4 a^{6}}\right ) - 64 a^{3} c d^{2} \left (\frac {c d}{a^{3}} - \frac {3 e \sqrt {- a^{7} c}}{4 a^{6}}\right ) - 24 a c d e^{2} + 128 c^{2} d^{3}}{9 a c e^{3} + 144 c^{2} d^{2} e} \right )} + \left (\frac {c d}{a^{3}} + \frac {3 e \sqrt {- a^{7} c}}{4 a^{6}}\right ) \log {\left (x + \frac {- 64 a^{6} d \left (\frac {c d}{a^{3}} + \frac {3 e \sqrt {- a^{7} c}}{4 a^{6}}\right )^{2} - 12 a^{4} e^{2} \left (\frac {c d}{a^{3}} + \frac {3 e \sqrt {- a^{7} c}}{4 a^{6}}\right ) - 64 a^{3} c d^{2} \left (\frac {c d}{a^{3}} + \frac {3 e \sqrt {- a^{7} c}}{4 a^{6}}\right ) - 24 a c d e^{2} + 128 c^{2} d^{3}}{9 a c e^{3} + 144 c^{2} d^{2} e} \right )} + \frac {- a d - 2 a e x - 2 c d x^{2} - 3 c e x^{3}}{2 a^{3} x^{2} + 2 a^{2} c x^{4}} - \frac {2 c d \log {\relax (x )}}{a^{3}} \]

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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