3.2189 \(\int \frac {1}{(d+e x)^3 (a+b x+c x^2)} \, dx\)

Optimal. Leaf size=272 \[ -\frac {e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}+\frac {e \log (d+e x) \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )}{\left (a e^2-b d e+c d^2\right )^3}-\frac {(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )^3}-\frac {e (2 c d-b e)}{(d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac {e}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )} \]

[Out]

-1/2*e/(a*e^2-b*d*e+c*d^2)/(e*x+d)^2-e*(-b*e+2*c*d)/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)+e*(3*c^2*d^2+b^2*e^2-c*e*(a*
e+3*b*d))*ln(e*x+d)/(a*e^2-b*d*e+c*d^2)^3-1/2*e*(3*c^2*d^2+b^2*e^2-c*e*(a*e+3*b*d))*ln(c*x^2+b*x+a)/(a*e^2-b*d
*e+c*d^2)^3-(-b*e+2*c*d)*(c^2*d^2+b^2*e^2-c*e*(3*a*e+b*d))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/(a*e^2-b*d*e+
c*d^2)^3/(-4*a*c+b^2)^(1/2)

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Rubi [A]  time = 0.43, antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {709, 800, 634, 618, 206, 628} \[ -\frac {e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}+\frac {e \log (d+e x) \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )}{\left (a e^2-b d e+c d^2\right )^3}-\frac {(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )^3}-\frac {e (2 c d-b e)}{(d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac {e}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^3*(a + b*x + c*x^2)),x]

[Out]

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

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 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 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]

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 709

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

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )} \, dx &=-\frac {e}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\frac {\int \frac {c d-b e-c e x}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx}{c d^2-b d e+a e^2}\\ &=-\frac {e}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\frac {\int \left (-\frac {e^2 (-2 c d+b e)}{\left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\frac {e^2 \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {c^3 d^3-b^3 e^3-3 c^2 d e (b d+a e)+b c e^2 (3 b d+2 a e)-c e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) x}{\left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}\right ) \, dx}{c d^2-b d e+a e^2}\\ &=-\frac {e}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {e (2 c d-b e)}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac {\int \frac {c^3 d^3-b^3 e^3-3 c^2 d e (b d+a e)+b c e^2 (3 b d+2 a e)-c e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) x}{a+b x+c x^2} \, dx}{\left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {e}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {e (2 c d-b e)}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {\left (e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right )\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )^3}+\frac {\left ((2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {e}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {e (2 c d-b e)}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3}-\frac {\left ((2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {e}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {e (2 c d-b e)}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac {(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^3}+\frac {e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3}\\ \end {align*}

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Mathematica [A]  time = 0.31, size = 272, normalized size = 1.00 \[ \frac {e \log (d+e x) \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )}{\left (e (a e-b d)+c d^2\right )^3}+\frac {e \left (c e (a e+3 b d)-b^2 e^2-3 c^2 d^2\right ) \log (a+x (b+c x))}{2 \left (e (a e-b d)+c d^2\right )^3}+\frac {(b e-2 c d) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2} \left (e (b d-a e)-c d^2\right )^3}+\frac {e (b e-2 c d)}{(d+e x) \left (e (a e-b d)+c d^2\right )^2}-\frac {e}{2 (d+e x)^2 \left (e (a e-b d)+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^3*(a + b*x + c*x^2)),x]

[Out]

-1/2*e/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + (e*(-2*c*d + b*e))/((c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x))
+ ((-2*c*d + b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(Sqrt[-b^2 +
4*a*c]*(-(c*d^2) + e*(b*d - a*e))^3) + (e*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e))*Log[d + e*x])/(c*d^2 + e*
(-(b*d) + a*e))^3 + (e*(-3*c^2*d^2 - b^2*e^2 + c*e*(3*b*d + a*e))*Log[a + x*(b + c*x)])/(2*(c*d^2 + e*(-(b*d)
+ a*e))^3)

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fricas [B]  time = 92.57, size = 2793, normalized size = 10.27 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

[-1/2*(5*(b^2*c^2 - 4*a*c^3)*d^4*e - 8*(b^3*c - 4*a*b*c^2)*d^3*e^2 + 3*(b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^3 -
4*(a*b^3 - 4*a^2*b*c)*d*e^4 + (a^2*b^2 - 4*a^3*c)*e^5 - (2*c^3*d^5 - 3*b*c^2*d^4*e + 3*(b^2*c - 2*a*c^2)*d^3*
e^2 - (b^3 - 3*a*b*c)*d^2*e^3 + (2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*a*b*c)
*e^5)*x^2 + 2*(2*c^3*d^4*e - 3*b*c^2*d^3*e^2 + 3*(b^2*c - 2*a*c^2)*d^2*e^3 - (b^3 - 3*a*b*c)*d*e^4)*x)*sqrt(b^
2 - 4*a*c)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c - sqrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) + 2*(2*(
b^2*c^2 - 4*a*c^3)*d^3*e^2 - 3*(b^3*c - 4*a*b*c^2)*d^2*e^3 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d*e^4 - (a*b^3 - 4*
a^2*b*c)*e^5)*x + (3*(b^2*c^2 - 4*a*c^3)*d^4*e - 3*(b^3*c - 4*a*b*c^2)*d^3*e^2 + (b^4 - 5*a*b^2*c + 4*a^2*c^2)
*d^2*e^3 + (3*(b^2*c^2 - 4*a*c^3)*d^2*e^3 - 3*(b^3*c - 4*a*b*c^2)*d*e^4 + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e^5)*x
^2 + 2*(3*(b^2*c^2 - 4*a*c^3)*d^3*e^2 - 3*(b^3*c - 4*a*b*c^2)*d^2*e^3 + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*d*e^4)*x
)*log(c*x^2 + b*x + a) - 2*(3*(b^2*c^2 - 4*a*c^3)*d^4*e - 3*(b^3*c - 4*a*b*c^2)*d^3*e^2 + (b^4 - 5*a*b^2*c + 4
*a^2*c^2)*d^2*e^3 + (3*(b^2*c^2 - 4*a*c^3)*d^2*e^3 - 3*(b^3*c - 4*a*b*c^2)*d*e^4 + (b^4 - 5*a*b^2*c + 4*a^2*c^
2)*e^5)*x^2 + 2*(3*(b^2*c^2 - 4*a*c^3)*d^3*e^2 - 3*(b^3*c - 4*a*b*c^2)*d^2*e^3 + (b^4 - 5*a*b^2*c + 4*a^2*c^2)
*d*e^4)*x)*log(e*x + d))/((b^2*c^3 - 4*a*c^4)*d^8 - 3*(b^3*c^2 - 4*a*b*c^3)*d^7*e + 3*(b^4*c - 3*a*b^2*c^2 - 4
*a^2*c^3)*d^6*e^2 - (b^5 + 2*a*b^3*c - 24*a^2*b*c^2)*d^5*e^3 + 3*(a*b^4 - 3*a^2*b^2*c - 4*a^3*c^2)*d^4*e^4 - 3
*(a^2*b^3 - 4*a^3*b*c)*d^3*e^5 + (a^3*b^2 - 4*a^4*c)*d^2*e^6 + ((b^2*c^3 - 4*a*c^4)*d^6*e^2 - 3*(b^3*c^2 - 4*a
*b*c^3)*d^5*e^3 + 3*(b^4*c - 3*a*b^2*c^2 - 4*a^2*c^3)*d^4*e^4 - (b^5 + 2*a*b^3*c - 24*a^2*b*c^2)*d^3*e^5 + 3*(
a*b^4 - 3*a^2*b^2*c - 4*a^3*c^2)*d^2*e^6 - 3*(a^2*b^3 - 4*a^3*b*c)*d*e^7 + (a^3*b^2 - 4*a^4*c)*e^8)*x^2 + 2*((
b^2*c^3 - 4*a*c^4)*d^7*e - 3*(b^3*c^2 - 4*a*b*c^3)*d^6*e^2 + 3*(b^4*c - 3*a*b^2*c^2 - 4*a^2*c^3)*d^5*e^3 - (b^
5 + 2*a*b^3*c - 24*a^2*b*c^2)*d^4*e^4 + 3*(a*b^4 - 3*a^2*b^2*c - 4*a^3*c^2)*d^3*e^5 - 3*(a^2*b^3 - 4*a^3*b*c)*
d^2*e^6 + (a^3*b^2 - 4*a^4*c)*d*e^7)*x), -1/2*(5*(b^2*c^2 - 4*a*c^3)*d^4*e - 8*(b^3*c - 4*a*b*c^2)*d^3*e^2 + 3
*(b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^3 - 4*(a*b^3 - 4*a^2*b*c)*d*e^4 + (a^2*b^2 - 4*a^3*c)*e^5 + 2*(2*c^3*d^5
- 3*b*c^2*d^4*e + 3*(b^2*c - 2*a*c^2)*d^3*e^2 - (b^3 - 3*a*b*c)*d^2*e^3 + (2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3
*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*a*b*c)*e^5)*x^2 + 2*(2*c^3*d^4*e - 3*b*c^2*d^3*e^2 + 3*(b^2*c - 2*a*c^2)*d
^2*e^3 - (b^3 - 3*a*b*c)*d*e^4)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) +
2*(2*(b^2*c^2 - 4*a*c^3)*d^3*e^2 - 3*(b^3*c - 4*a*b*c^2)*d^2*e^3 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d*e^4 - (a*b^
3 - 4*a^2*b*c)*e^5)*x + (3*(b^2*c^2 - 4*a*c^3)*d^4*e - 3*(b^3*c - 4*a*b*c^2)*d^3*e^2 + (b^4 - 5*a*b^2*c + 4*a^
2*c^2)*d^2*e^3 + (3*(b^2*c^2 - 4*a*c^3)*d^2*e^3 - 3*(b^3*c - 4*a*b*c^2)*d*e^4 + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*
e^5)*x^2 + 2*(3*(b^2*c^2 - 4*a*c^3)*d^3*e^2 - 3*(b^3*c - 4*a*b*c^2)*d^2*e^3 + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*d*
e^4)*x)*log(c*x^2 + b*x + a) - 2*(3*(b^2*c^2 - 4*a*c^3)*d^4*e - 3*(b^3*c - 4*a*b*c^2)*d^3*e^2 + (b^4 - 5*a*b^2
*c + 4*a^2*c^2)*d^2*e^3 + (3*(b^2*c^2 - 4*a*c^3)*d^2*e^3 - 3*(b^3*c - 4*a*b*c^2)*d*e^4 + (b^4 - 5*a*b^2*c + 4*
a^2*c^2)*e^5)*x^2 + 2*(3*(b^2*c^2 - 4*a*c^3)*d^3*e^2 - 3*(b^3*c - 4*a*b*c^2)*d^2*e^3 + (b^4 - 5*a*b^2*c + 4*a^
2*c^2)*d*e^4)*x)*log(e*x + d))/((b^2*c^3 - 4*a*c^4)*d^8 - 3*(b^3*c^2 - 4*a*b*c^3)*d^7*e + 3*(b^4*c - 3*a*b^2*c
^2 - 4*a^2*c^3)*d^6*e^2 - (b^5 + 2*a*b^3*c - 24*a^2*b*c^2)*d^5*e^3 + 3*(a*b^4 - 3*a^2*b^2*c - 4*a^3*c^2)*d^4*e
^4 - 3*(a^2*b^3 - 4*a^3*b*c)*d^3*e^5 + (a^3*b^2 - 4*a^4*c)*d^2*e^6 + ((b^2*c^3 - 4*a*c^4)*d^6*e^2 - 3*(b^3*c^2
- 4*a*b*c^3)*d^5*e^3 + 3*(b^4*c - 3*a*b^2*c^2 - 4*a^2*c^3)*d^4*e^4 - (b^5 + 2*a*b^3*c - 24*a^2*b*c^2)*d^3*e^5
+ 3*(a*b^4 - 3*a^2*b^2*c - 4*a^3*c^2)*d^2*e^6 - 3*(a^2*b^3 - 4*a^3*b*c)*d*e^7 + (a^3*b^2 - 4*a^4*c)*e^8)*x^2
+ 2*((b^2*c^3 - 4*a*c^4)*d^7*e - 3*(b^3*c^2 - 4*a*b*c^3)*d^6*e^2 + 3*(b^4*c - 3*a*b^2*c^2 - 4*a^2*c^3)*d^5*e^3
- (b^5 + 2*a*b^3*c - 24*a^2*b*c^2)*d^4*e^4 + 3*(a*b^4 - 3*a^2*b^2*c - 4*a^3*c^2)*d^3*e^5 - 3*(a^2*b^3 - 4*a^3
*b*c)*d^2*e^6 + (a^3*b^2 - 4*a^4*c)*d*e^7)*x)]

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giac [B]  time = 0.16, size = 593, normalized size = 2.18 \[ -\frac {{\left (3 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3} - a c e^{3}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )}} + \frac {{\left (3 \, c^{2} d^{2} e^{2} - 3 \, b c d e^{3} + b^{2} e^{4} - a c e^{4}\right )} \log \left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e - 3 \, b c^{2} d^{5} e^{2} + 3 \, b^{2} c d^{4} e^{3} + 3 \, a c^{2} d^{4} e^{3} - b^{3} d^{3} e^{4} - 6 \, a b c d^{3} e^{4} + 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} c d^{2} e^{5} - 3 \, a^{2} b d e^{6} + a^{3} e^{7}} + \frac {{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - 6 \, a c^{2} d e^{2} - b^{3} e^{3} + 3 \, a b c e^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {5 \, c^{2} d^{4} e - 8 \, b c d^{3} e^{2} + 3 \, b^{2} d^{2} e^{3} + 6 \, a c d^{2} e^{3} - 4 \, a b d e^{4} + a^{2} e^{5} + 2 \, {\left (2 \, c^{2} d^{3} e^{2} - 3 \, b c d^{2} e^{3} + b^{2} d e^{4} + 2 \, a c d e^{4} - a b e^{5}\right )} x}{2 \, {\left (c d^{2} - b d e + a e^{2}\right )}^{3} {\left (x e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

-1/2*(3*c^2*d^2*e - 3*b*c*d*e^2 + b^2*e^3 - a*c*e^3)*log(c*x^2 + b*x + a)/(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d
^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 +
a^3*e^6) + (3*c^2*d^2*e^2 - 3*b*c*d*e^3 + b^2*e^4 - a*c*e^4)*log(abs(x*e + d))/(c^3*d^6*e - 3*b*c^2*d^5*e^2 +
3*b^2*c*d^4*e^3 + 3*a*c^2*d^4*e^3 - b^3*d^3*e^4 - 6*a*b*c*d^3*e^4 + 3*a*b^2*d^2*e^5 + 3*a^2*c*d^2*e^5 - 3*a^2
*b*d*e^6 + a^3*e^7) + (2*c^3*d^3 - 3*b*c^2*d^2*e + 3*b^2*c*d*e^2 - 6*a*c^2*d*e^2 - b^3*e^3 + 3*a*b*c*e^3)*arct
an((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3
- 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6)*sqrt(-b^2 + 4*a*c)) - 1/2*(5
*c^2*d^4*e - 8*b*c*d^3*e^2 + 3*b^2*d^2*e^3 + 6*a*c*d^2*e^3 - 4*a*b*d*e^4 + a^2*e^5 + 2*(2*c^2*d^3*e^2 - 3*b*c*
d^2*e^3 + b^2*d*e^4 + 2*a*c*d*e^4 - a*b*e^5)*x)/((c*d^2 - b*d*e + a*e^2)^3*(x*e + d)^2)

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maple [B]  time = 0.08, size = 719, normalized size = 2.64 \[ \frac {3 a b c \,e^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {4 a c -b^{2}}}-\frac {6 a \,c^{2} d \,e^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {4 a c -b^{2}}}-\frac {b^{3} e^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {4 a c -b^{2}}}+\frac {3 b^{2} c d \,e^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {4 a c -b^{2}}}-\frac {3 b \,c^{2} d^{2} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {4 a c -b^{2}}}+\frac {2 c^{3} d^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {4 a c -b^{2}}}-\frac {a c \,e^{3} \ln \left (e x +d \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}+\frac {a c \,e^{3} \ln \left (c \,x^{2}+b x +a \right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}+\frac {b^{2} e^{3} \ln \left (e x +d \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}-\frac {b^{2} e^{3} \ln \left (c \,x^{2}+b x +a \right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}-\frac {3 b c d \,e^{2} \ln \left (e x +d \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}+\frac {3 b c d \,e^{2} \ln \left (c \,x^{2}+b x +a \right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}+\frac {3 c^{2} d^{2} e \ln \left (e x +d \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}-\frac {3 c^{2} d^{2} e \ln \left (c \,x^{2}+b x +a \right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}+\frac {b \,e^{2}}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )}-\frac {2 c d e}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )}-\frac {e}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (e x +d \right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^3/(c*x^2+b*x+a),x)

[Out]

-1/2*e/(a*e^2-b*d*e+c*d^2)/(e*x+d)^2+e^2/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)*b-2*e/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)*c*d
-e^3/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*a*c+e^3/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*b^2-3*e^2/(a*e^2-b*d*e+c*d^2)^3*l
n(e*x+d)*b*c*d+3*e/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*c^2*d^2+1/2/(a*e^2-b*d*e+c*d^2)^3*c*ln(c*x^2+b*x+a)*a*e^3-1
/2/(a*e^2-b*d*e+c*d^2)^3*ln(c*x^2+b*x+a)*b^2*e^3+3/2/(a*e^2-b*d*e+c*d^2)^3*c*ln(c*x^2+b*x+a)*b*d*e^2-3/2/(a*e^
2-b*d*e+c*d^2)^3*c^2*ln(c*x^2+b*x+a)*d^2*e+3/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b
^2)^(1/2))*a*b*c*e^3-6/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c^2*a*d*e^2
-1/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^3*e^3+3/(a*e^2-b*d*e+c*d^2)^3
/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*c*d*e^2-3/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*a
rctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*c^2*d^2*e+2/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a
*c-b^2)^(1/2))*c^3*d^3

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive or negative?

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mupad [B]  time = 7.66, size = 3506, normalized size = 12.89 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d + e*x)^3*(a + b*x + c*x^2)),x)

[Out]

(log((27*d*e^6*(b^2 - 4*a*c)^(7/2))/16 + (9*e^7*x*(b^2 - 4*a*c)^(7/2))/16 - 8*a*b^6*e^7 + 4*b*c^6*d^7 - 8*b^7*
e^7*x + 8*c^7*d^7*x + 4*c^6*d^7*(b^2 - 4*a*c)^(1/2) + 72*a^4*c^3*e^7 + (57*b^2*e^7*x*(b^2 - 4*a*c)^(5/2))/16 +
(51*b^4*e^7*x*(b^2 - 4*a*c)^(3/2))/16 + (11*b^6*e^7*x*(b^2 - 4*a*c)^(1/2))/16 + 60*a^2*b^4*c*e^7 + 8*b^2*c^5*
d^6*e + 4*b^6*c*d^2*e^5 + (75*c^2*d^3*e^4*(b^2 - 4*a*c)^(5/2))/4 + 25*c^4*d^5*e^2*(b^2 - 4*a*c)^(3/2) - 132*a^
3*b^2*c^2*e^7 + 408*a^2*c^5*d^4*e^3 - 456*a^3*c^4*d^2*e^5 - 20*b^3*c^4*d^5*e^2 + 28*b^4*c^3*d^4*e^3 - 16*b^5*c
^2*d^3*e^4 + (9*a*b*e^7*(b^2 - 4*a*c)^(5/2))/4 - 88*a*c^6*d^6*e + (9*a*b^3*e^7*(b^2 - 4*a*c)^(3/2))/2 + (5*a*b
^5*e^7*(b^2 - 4*a*c)^(1/2))/4 + (111*b^2*d*e^6*(b^2 - 4*a*c)^(5/2))/16 - (79*b^4*d*e^6*(b^2 - 4*a*c)^(3/2))/16
- (59*b^6*d*e^6*(b^2 - 4*a*c)^(1/2))/16 + 40*a*b^5*c*d*e^6 + (23*b^2*c^2*d^3*e^4*(b^2 - 4*a*c)^(3/2))/2 - 45*
b^2*c^4*d^5*e^2*(b^2 - 4*a*c)^(1/2) + 65*b^3*c^3*d^4*e^3*(b^2 - 4*a*c)^(1/2) - (185*b^4*c^2*d^3*e^4*(b^2 - 4*a
*c)^(1/2))/4 + 64*a*b^5*c*e^7*x - 28*b*c^6*d^6*e*x + 48*b^6*c*d*e^6*x + 504*a^2*b^2*c^3*d^2*e^5 - 21*b*c*d^2*e
^5*(b^2 - 4*a*c)^(5/2) + 8*b*c^5*d^6*e*(b^2 - 4*a*c)^(1/2) + 44*c^6*d^6*e*x*(b^2 - 4*a*c)^(1/2) + 164*a*b*c^5*
d^5*e^2 + 348*a^3*b*c^3*d*e^6 + 108*a^3*b*c^3*e^7*x - 200*a*c^6*d^5*e^2*x - 216*a^3*c^4*d*e^6*x - 37*b*c^3*d^4
*e^3*(b^2 - 4*a*c)^(3/2) + 7*b^3*c*d^2*e^5*(b^2 - 4*a*c)^(3/2) + 18*b^5*c*d^2*e^5*(b^2 - 4*a*c)^(1/2) + (57*c^
2*d^2*e^5*x*(b^2 - 4*a*c)^(5/2))/4 + 51*c^4*d^4*e^3*x*(b^2 - 4*a*c)^(3/2) - 284*a*b^2*c^4*d^4*e^3 + 228*a*b^3*
c^3*d^3*e^4 - 124*a*b^4*c^2*d^2*e^5 - 516*a^2*b*c^4*d^3*e^4 - 240*a^2*b^3*c^2*d*e^6 - 156*a^2*b^3*c^2*e^7*x +
600*a^2*c^5*d^3*e^4*x + 92*b^2*c^5*d^5*e^2*x - 160*b^3*c^4*d^4*e^3*x + 180*b^4*c^3*d^3*e^4*x - 124*b^5*c^2*d^2
*e^5*x - 102*b*c^3*d^3*e^4*x*(b^2 - 4*a*c)^(3/2) - 132*b*c^5*d^5*e^2*x*(b^2 - 4*a*c)^(1/2) - 800*a*b^2*c^4*d^3
*e^4*x + 700*a*b^3*c^3*d^2*e^5*x - 900*a^2*b*c^4*d^2*e^5*x + 612*a^2*b^2*c^3*d*e^6*x - (57*b*c*d*e^6*x*(b^2 -
4*a*c)^(5/2))/4 + (153*b^2*c^2*d^2*e^5*x*(b^2 - 4*a*c)^(3/2))/2 + 165*b^2*c^4*d^4*e^3*x*(b^2 - 4*a*c)^(1/2) -
110*b^3*c^3*d^3*e^4*x*(b^2 - 4*a*c)^(1/2) + (165*b^4*c^2*d^2*e^5*x*(b^2 - 4*a*c)^(1/2))/4 - (51*b^3*c*d*e^6*x*
(b^2 - 4*a*c)^(3/2))/2 - (33*b^5*c*d*e^6*x*(b^2 - 4*a*c)^(1/2))/4 + 500*a*b*c^5*d^4*e^3*x - 328*a*b^4*c^2*d*e^
6*x)*(e^2*((3*c*d*(b^2 - 4*a*c)^(3/2))/2 + 3*b*c*d*(4*a*c - b^2) + (3*b^2*c*d*(b^2 - 4*a*c)^(1/2))/2) - e*(3*c
^2*d^2*(4*a*c - b^2) + 3*b*c^2*d^2*(b^2 - 4*a*c)^(1/2)) - e^3*((3*b*(b^2 - 4*a*c)^(3/2))/4 - (4*a*c - b^2)^2/4
+ (3*b^2*(4*a*c - b^2))/4 + (b^3*(b^2 - 4*a*c)^(1/2))/4) + 2*c^3*d^3*(b^2 - 4*a*c)^(1/2)))/((4*a*c - b^2)*((4
*a*c - b^2)*((3*a*d^2*e^4)/2 - 3*b*d^3*e^3 + (3*c*d^4*e^2)/2) + 2*a^3*e^6 + 2*c^3*d^6 - 5*b^3*d^3*e^3 + (15*a*
b^2*d^2*e^4)/2 + (15*b^2*c*d^4*e^2)/2 - 6*a^2*b*d*e^5 - 6*b*c^2*d^5*e)) - (log(d + e*x)*(e^3*(a*c - b^2) - 3*c
^2*d^2*e + 3*b*c*d*e^2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^
4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3) - (log((27*d*e^6*(b^2 - 4*a*c)^(7/2))/1
6 + (9*e^7*x*(b^2 - 4*a*c)^(7/2))/16 + 8*a*b^6*e^7 - 4*b*c^6*d^7 + 8*b^7*e^7*x - 8*c^7*d^7*x + 4*c^6*d^7*(b^2
- 4*a*c)^(1/2) - 72*a^4*c^3*e^7 + (57*b^2*e^7*x*(b^2 - 4*a*c)^(5/2))/16 + (51*b^4*e^7*x*(b^2 - 4*a*c)^(3/2))/1
6 + (11*b^6*e^7*x*(b^2 - 4*a*c)^(1/2))/16 - 60*a^2*b^4*c*e^7 - 8*b^2*c^5*d^6*e - 4*b^6*c*d^2*e^5 + (75*c^2*d^3
*e^4*(b^2 - 4*a*c)^(5/2))/4 + 25*c^4*d^5*e^2*(b^2 - 4*a*c)^(3/2) + 132*a^3*b^2*c^2*e^7 - 408*a^2*c^5*d^4*e^3 +
456*a^3*c^4*d^2*e^5 + 20*b^3*c^4*d^5*e^2 - 28*b^4*c^3*d^4*e^3 + 16*b^5*c^2*d^3*e^4 + (9*a*b*e^7*(b^2 - 4*a*c)
^(5/2))/4 + 88*a*c^6*d^6*e + (9*a*b^3*e^7*(b^2 - 4*a*c)^(3/2))/2 + (5*a*b^5*e^7*(b^2 - 4*a*c)^(1/2))/4 + (111*
b^2*d*e^6*(b^2 - 4*a*c)^(5/2))/16 - (79*b^4*d*e^6*(b^2 - 4*a*c)^(3/2))/16 - (59*b^6*d*e^6*(b^2 - 4*a*c)^(1/2))
/16 - 40*a*b^5*c*d*e^6 + (23*b^2*c^2*d^3*e^4*(b^2 - 4*a*c)^(3/2))/2 - 45*b^2*c^4*d^5*e^2*(b^2 - 4*a*c)^(1/2) +
65*b^3*c^3*d^4*e^3*(b^2 - 4*a*c)^(1/2) - (185*b^4*c^2*d^3*e^4*(b^2 - 4*a*c)^(1/2))/4 - 64*a*b^5*c*e^7*x + 28*
b*c^6*d^6*e*x - 48*b^6*c*d*e^6*x - 504*a^2*b^2*c^3*d^2*e^5 - 21*b*c*d^2*e^5*(b^2 - 4*a*c)^(5/2) + 8*b*c^5*d^6*
e*(b^2 - 4*a*c)^(1/2) + 44*c^6*d^6*e*x*(b^2 - 4*a*c)^(1/2) - 164*a*b*c^5*d^5*e^2 - 348*a^3*b*c^3*d*e^6 - 108*a
^3*b*c^3*e^7*x + 200*a*c^6*d^5*e^2*x + 216*a^3*c^4*d*e^6*x - 37*b*c^3*d^4*e^3*(b^2 - 4*a*c)^(3/2) + 7*b^3*c*d^
2*e^5*(b^2 - 4*a*c)^(3/2) + 18*b^5*c*d^2*e^5*(b^2 - 4*a*c)^(1/2) + (57*c^2*d^2*e^5*x*(b^2 - 4*a*c)^(5/2))/4 +
51*c^4*d^4*e^3*x*(b^2 - 4*a*c)^(3/2) + 284*a*b^2*c^4*d^4*e^3 - 228*a*b^3*c^3*d^3*e^4 + 124*a*b^4*c^2*d^2*e^5 +
516*a^2*b*c^4*d^3*e^4 + 240*a^2*b^3*c^2*d*e^6 + 156*a^2*b^3*c^2*e^7*x - 600*a^2*c^5*d^3*e^4*x - 92*b^2*c^5*d^
5*e^2*x + 160*b^3*c^4*d^4*e^3*x - 180*b^4*c^3*d^3*e^4*x + 124*b^5*c^2*d^2*e^5*x - 102*b*c^3*d^3*e^4*x*(b^2 - 4
*a*c)^(3/2) - 132*b*c^5*d^5*e^2*x*(b^2 - 4*a*c)^(1/2) + 800*a*b^2*c^4*d^3*e^4*x - 700*a*b^3*c^3*d^2*e^5*x + 90
0*a^2*b*c^4*d^2*e^5*x - 612*a^2*b^2*c^3*d*e^6*x - (57*b*c*d*e^6*x*(b^2 - 4*a*c)^(5/2))/4 + (153*b^2*c^2*d^2*e^
5*x*(b^2 - 4*a*c)^(3/2))/2 + 165*b^2*c^4*d^4*e^3*x*(b^2 - 4*a*c)^(1/2) - 110*b^3*c^3*d^3*e^4*x*(b^2 - 4*a*c)^(
1/2) + (165*b^4*c^2*d^2*e^5*x*(b^2 - 4*a*c)^(1/2))/4 - (51*b^3*c*d*e^6*x*(b^2 - 4*a*c)^(3/2))/2 - (33*b^5*c*d*
e^6*x*(b^2 - 4*a*c)^(1/2))/4 - 500*a*b*c^5*d^4*e^3*x + 328*a*b^4*c^2*d*e^6*x)*(e^2*((3*c*d*(b^2 - 4*a*c)^(3/2)
)/2 - 3*b*c*d*(4*a*c - b^2) + (3*b^2*c*d*(b^2 - 4*a*c)^(1/2))/2) + e*(3*c^2*d^2*(4*a*c - b^2) - 3*b*c^2*d^2*(b
^2 - 4*a*c)^(1/2)) - e^3*((4*a*c - b^2)^2/4 + (3*b*(b^2 - 4*a*c)^(3/2))/4 - (3*b^2*(4*a*c - b^2))/4 + (b^3*(b^
2 - 4*a*c)^(1/2))/4) + 2*c^3*d^3*(b^2 - 4*a*c)^(1/2)))/((4*a*c - b^2)*((4*a*c - b^2)*((3*a*d^2*e^4)/2 - 3*b*d^
3*e^3 + (3*c*d^4*e^2)/2) + 2*a^3*e^6 + 2*c^3*d^6 - 5*b^3*d^3*e^3 + (15*a*b^2*d^2*e^4)/2 + (15*b^2*c*d^4*e^2)/2
- 6*a^2*b*d*e^5 - 6*b*c^2*d^5*e)) - ((a*e^3 - 3*b*d*e^2 + 5*c*d^2*e)/(2*(a^2*e^4 + c^2*d^4 + b^2*d^2*e^2 - 2*
a*b*d*e^3 - 2*b*c*d^3*e + 2*a*c*d^2*e^2)) - (x*(b*e^3 - 2*c*d*e^2))/(a^2*e^4 + c^2*d^4 + b^2*d^2*e^2 - 2*a*b*d
*e^3 - 2*b*c*d^3*e + 2*a*c*d^2*e^2))/(d^2 + e^2*x^2 + 2*d*e*x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**3/(c*x**2+b*x+a),x)

[Out]

Timed out

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