3.617 \(\int \frac{1}{(d+e x) (a+b (d+e x)^2+c (d+e x)^4)} \, dx\)

Optimal. Leaf size=94 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{2 a e \sqrt{b^2-4 a c}}-\frac{\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a e}+\frac{\log (d+e x)}{a e} \]

[Out]

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

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Rubi [A]  time = 0.132291, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {1142, 1114, 705, 29, 634, 618, 206, 628} \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{2 a e \sqrt{b^2-4 a c}}-\frac{\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a e}+\frac{\log (d+e x)}{a e} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)),x]

[Out]

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

Rule 1142

Int[(u_)^(m_.)*((a_.) + (b_.)*(v_)^2 + (c_.)*(v_)^4)^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m),
Subst[Int[x^m*(a + b*x^2 + c*x^(2*2))^p, x], x, v], x] /; FreeQ[{a, b, c, m, p}, x] && LinearPairQ[u, v, x]

Rule 1114

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]

Rule 705

Int[1/(((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[e^2/(c*d^2 - b*d*e + a*e^2
), Int[1/(d + e*x), x], x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(c*d - b*e - c*e*x)/(a + b*x + c*x^2), x], x]
/; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{1}{(d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \left (a+b x^2+c x^4\right )} \, dx,x,d+e x\right )}{e}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \left (a+b x+c x^2\right )} \, dx,x,(d+e x)^2\right )}{2 e}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,(d+e x)^2\right )}{2 a e}+\frac{\operatorname{Subst}\left (\int \frac{-b-c x}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{2 a e}\\ &=\frac{\log (d+e x)}{a e}-\frac{\operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{4 a e}-\frac{b \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{4 a e}\\ &=\frac{\log (d+e x)}{a e}-\frac{\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a e}+\frac{b \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c (d+e x)^2\right )}{2 a e}\\ &=\frac{b \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{2 a \sqrt{b^2-4 a c} e}+\frac{\log (d+e x)}{a e}-\frac{\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a e}\\ \end{align*}

Mathematica [A]  time = 0.0802059, size = 128, normalized size = 1.36 \[ \frac{4 \sqrt{b^2-4 a c} \log (d+e x)-\left (\sqrt{b^2-4 a c}+b\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right )+\left (b-\sqrt{b^2-4 a c}\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right )}{4 a e \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)),x]

[Out]

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

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Maple [C]  time = 0.008, size = 184, normalized size = 2. \begin{align*}{\frac{\ln \left ( ex+d \right ) }{ae}}+{\frac{1}{2\,ae}\sum _{{\it \_R}={\it RootOf} \left ( c{e}^{4}{{\it \_Z}}^{4}+4\,cd{e}^{3}{{\it \_Z}}^{3}+ \left ( 6\,c{d}^{2}{e}^{2}+b{e}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,c{d}^{3}e+2\,bde \right ){\it \_Z}+c{d}^{4}+b{d}^{2}+a \right ) }{\frac{ \left ( -c{e}^{3}{{\it \_R}}^{3}-3\,cd{e}^{2}{{\it \_R}}^{2}+e \left ( -3\,c{d}^{2}-b \right ){\it \_R}-c{d}^{3}-bd \right ) \ln \left ( x-{\it \_R} \right ) }{2\,c{e}^{3}{{\it \_R}}^{3}+6\,cd{e}^{2}{{\it \_R}}^{2}+6\,c{d}^{2}e{\it \_R}+2\,c{d}^{3}+be{\it \_R}+bd}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)/(a+b*(e*x+d)^2+c*(e*x+d)^4),x)

[Out]

ln(e*x+d)/a/e+1/2/a/e*sum((-c*e^3*_R^3-3*c*d*e^2*_R^2+e*(-3*c*d^2-b)*_R-c*d^3-b*d)/(2*_R^3*c*e^3+6*_R^2*c*d*e^
2+6*_R*c*d^2*e+2*c*d^3+_R*b*e+b*d)*ln(x-_R),_R=RootOf(c*e^4*_Z^4+4*c*d*e^3*_Z^3+(6*c*d^2*e^2+b*e^2)*_Z^2+(4*c*
d^3*e+2*b*d*e)*_Z+c*d^4+b*d^2+a))

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 1.99223, size = 1040, normalized size = 11.06 \begin{align*} \left [\frac{\sqrt{b^{2} - 4 \, a c} b \log \left (\frac{2 \, c^{2} e^{4} x^{4} + 8 \, c^{2} d e^{3} x^{3} + 2 \, c^{2} d^{4} + 2 \,{\left (6 \, c^{2} d^{2} + b c\right )} e^{2} x^{2} + 2 \, b c d^{2} + 4 \,{\left (2 \, c^{2} d^{3} + b c d\right )} e x + b^{2} - 2 \, a c +{\left (2 \, c e^{2} x^{2} + 4 \, c d e x + 2 \, c d^{2} + b\right )} \sqrt{b^{2} - 4 \, a c}}{c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} +{\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \,{\left (2 \, c d^{3} + b d\right )} e x + a}\right ) -{\left (b^{2} - 4 \, a c\right )} \log \left (c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} +{\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \,{\left (2 \, c d^{3} + b d\right )} e x + a\right ) + 4 \,{\left (b^{2} - 4 \, a c\right )} \log \left (e x + d\right )}{4 \,{\left (a b^{2} - 4 \, a^{2} c\right )} e}, \frac{2 \, \sqrt{-b^{2} + 4 \, a c} b \arctan \left (-\frac{{\left (2 \, c e^{2} x^{2} + 4 \, c d e x + 2 \, c d^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) -{\left (b^{2} - 4 \, a c\right )} \log \left (c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} +{\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \,{\left (2 \, c d^{3} + b d\right )} e x + a\right ) + 4 \,{\left (b^{2} - 4 \, a c\right )} \log \left (e x + d\right )}{4 \,{\left (a b^{2} - 4 \, a^{2} c\right )} e}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 4.86643, size = 320, normalized size = 3.4 \begin{align*} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{4 a e \left (4 a c - b^{2}\right )} - \frac{1}{4 a e}\right ) \log{\left (\frac{2 d x}{e} + x^{2} + \frac{- 8 a^{2} c e \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{4 a e \left (4 a c - b^{2}\right )} - \frac{1}{4 a e}\right ) + 2 a b^{2} e \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{4 a e \left (4 a c - b^{2}\right )} - \frac{1}{4 a e}\right ) - 2 a c + b^{2} + b c d^{2}}{b c e^{2}} \right )} + \left (\frac{b \sqrt{- 4 a c + b^{2}}}{4 a e \left (4 a c - b^{2}\right )} - \frac{1}{4 a e}\right ) \log{\left (\frac{2 d x}{e} + x^{2} + \frac{- 8 a^{2} c e \left (\frac{b \sqrt{- 4 a c + b^{2}}}{4 a e \left (4 a c - b^{2}\right )} - \frac{1}{4 a e}\right ) + 2 a b^{2} e \left (\frac{b \sqrt{- 4 a c + b^{2}}}{4 a e \left (4 a c - b^{2}\right )} - \frac{1}{4 a e}\right ) - 2 a c + b^{2} + b c d^{2}}{b c e^{2}} \right )} + \frac{\log{\left (\frac{d}{e} + x \right )}}{a e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(a+b*(e*x+d)**2+c*(e*x+d)**4),x)

[Out]

(-b*sqrt(-4*a*c + b**2)/(4*a*e*(4*a*c - b**2)) - 1/(4*a*e))*log(2*d*x/e + x**2 + (-8*a**2*c*e*(-b*sqrt(-4*a*c
+ b**2)/(4*a*e*(4*a*c - b**2)) - 1/(4*a*e)) + 2*a*b**2*e*(-b*sqrt(-4*a*c + b**2)/(4*a*e*(4*a*c - b**2)) - 1/(4
*a*e)) - 2*a*c + b**2 + b*c*d**2)/(b*c*e**2)) + (b*sqrt(-4*a*c + b**2)/(4*a*e*(4*a*c - b**2)) - 1/(4*a*e))*log
(2*d*x/e + x**2 + (-8*a**2*c*e*(b*sqrt(-4*a*c + b**2)/(4*a*e*(4*a*c - b**2)) - 1/(4*a*e)) + 2*a*b**2*e*(b*sqrt
(-4*a*c + b**2)/(4*a*e*(4*a*c - b**2)) - 1/(4*a*e)) - 2*a*c + b**2 + b*c*d**2)/(b*c*e**2)) + log(d/e + x)/(a*e
)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="giac")

[Out]

Timed out