\(\int \frac {1}{(d+e x) (a+b x+c x^2)^3} \, dx\) [2207]

Optimal result

Integrand size = 20, antiderivative size = 429 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {3 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2-2 b^2 e^2-c e (3 b d-8 a e)\right )-2 c (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac {\left (12 c^5 d^5-b^5 e^5+10 a b^3 c e^5-30 a^2 b c^2 e^5-10 c^4 d^3 e (3 b d-4 a e)+20 c^3 d e^2 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} \left (c d^2-b d e+a e^2\right )^3}+\frac {e^5 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {e^5 \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3} \]

[Out]

1/2*(-b*c*d+b^2*e-2*a*c*e-c*(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(c*x^2+b*x+a)^2+1/2*(-3*a*c*e*(-b
*e+2*c*d)^2+(2*a*c*e-b^2*e+b*c*d)*(6*c^2*d^2-2*b^2*e^2-c*e*(-8*a*e+3*b*d))+2*c*(-b*e+2*c*d)*(3*c^2*d^2-b^2*e^2
-c*e*(-7*a*e+3*b*d))*x)/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)^2/(c*x^2+b*x+a)-(12*c^5*d^5-b^5*e^5+10*a*b^3*c*e^5-
30*a^2*b*c^2*e^5-10*c^4*d^3*e*(-4*a*e+3*b*d)+20*c^3*d*e^2*(3*a^2*e^2-3*a*b*d*e+b^2*d^2))*arctanh((2*c*x+b)/(-4
*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(5/2)/(a*e^2-b*d*e+c*d^2)^3+e^5*ln(e*x+d)/(a*e^2-b*d*e+c*d^2)^3-1/2*e^5*ln(c*x^2
+b*x+a)/(a*e^2-b*d*e+c*d^2)^3

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {754, 836, 814, 648, 632, 212, 642} \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=-\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (20 c^3 d e^2 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )-30 a^2 b c^2 e^5+10 a b^3 c e^5-10 c^4 d^3 e (3 b d-4 a e)-b^5 e^5+12 c^5 d^5\right )}{\left (b^2-4 a c\right )^{5/2} \left (a e^2-b d e+c d^2\right )^3}-\frac {-2 c x (2 c d-b e) \left (-c e (3 b d-7 a e)-b^2 e^2+3 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-c e (3 b d-8 a e)-2 b^2 e^2+6 c^2 d^2\right )+3 a c e (2 c d-b e)^2}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac {2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}-\frac {e^5 \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}+\frac {e^5 \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^3} \]

[In]

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

[Out]

-1/2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^2)
- (3*a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c*e)*(6*c^2*d^2 - 2*b^2*e^2 - c*e*(3*b*d - 8*a*e)) - 2*c*(2
*c*d - b*e)*(3*c^2*d^2 - b^2*e^2 - c*e*(3*b*d - 7*a*e))*x)/(2*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^2*(a + b
*x + c*x^2)) - ((12*c^5*d^5 - b^5*e^5 + 10*a*b^3*c*e^5 - 30*a^2*b*c^2*e^5 - 10*c^4*d^3*e*(3*b*d - 4*a*e) + 20*
c^3*d*e^2*(b^2*d^2 - 3*a*b*d*e + 3*a^2*e^2))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(5/2)*(c*d
^2 - b*d*e + a*e^2)^3) + (e^5*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^3 - (e^5*Log[a + b*x + c*x^2])/(2*(c*d^2 -
b*d*e + a*e^2)^3)

Rule 212

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

Rule 632

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 642

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 648

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 754

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(b
*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e +
a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), 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[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 814

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]

Rule 836

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rubi steps \begin{align*} \text {integral}& = -\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {\int \frac {6 c^2 d^2-2 b^2 e^2-c e (3 b d-8 a e)+3 c e (2 c d-b e) x}{(d+e x) \left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {3 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2-2 b^2 e^2-c e (3 b d-8 a e)\right )-2 c (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac {\int \frac {2 \left (6 c^4 d^4+b^4 e^4-c^3 d^2 e (9 b d-14 a e)+b^2 c e^3 (b d-8 a e)+c^2 e^2 \left (b^2 d^2-7 a b d e+16 a^2 e^2\right )\right )+2 c e (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x}{(d+e x) \left (a+b x+c x^2\right )} \, dx}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2} \\ & = -\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {3 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2-2 b^2 e^2-c e (3 b d-8 a e)\right )-2 c (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac {\int \left (\frac {2 \left (b^2-4 a c\right )^2 e^6}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac {2 \left (6 c^5 d^5-b^5 e^5+9 a b^3 c e^5-23 a^2 b c^2 e^5-5 c^4 d^3 e (3 b d-4 a e)+10 c^3 d e^2 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )-c \left (b^2-4 a c\right )^2 e^5 x\right )}{\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}\right ) \, dx}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2} \\ & = -\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {3 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2-2 b^2 e^2-c e (3 b d-8 a e)\right )-2 c (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac {e^5 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac {\int \frac {6 c^5 d^5-b^5 e^5+9 a b^3 c e^5-23 a^2 b c^2 e^5-5 c^4 d^3 e (3 b d-4 a e)+10 c^3 d e^2 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )-c \left (b^2-4 a c\right )^2 e^5 x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3} \\ & = -\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {3 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2-2 b^2 e^2-c e (3 b d-8 a e)\right )-2 c (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac {e^5 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {e^5 \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 (12 c^5 d^5-b^5 e^5+10 a b^3 c e^5-30 a^2 b c^2 e^5-10 c^4 d^3 e (3 b d-4 a e)+20 c^3 d e^2 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3} \\ & = -\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {3 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2-2 b^2 e^2-c e (3 b d-8 a e)\right )-2 c (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac {e^5 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {e^5 \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3}-\frac {\left (12 c^5 d^5-b^5 e^5+10 a b^3 c e^5-30 a^2 b c^2 e^5-10 c^4 d^3 e (3 b d-4 a e)+20 c^3 d e^2 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3} \\ & = -\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {3 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2-2 b^2 e^2-c e (3 b d-8 a e)\right )-2 c (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac {\left (12 c^5 d^5-b^5 e^5+10 a b^3 c e^5-30 a^2 b c^2 e^5-10 c^4 d^3 e (3 b d-4 a e)+20 c^3 d e^2 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} \left (c d^2-b d e+a e^2\right )^3}+\frac {e^5 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {e^5 \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=\frac {1}{2} \left (\frac {-b^2 e+2 c (a e+c d x)+b c (d-e x)}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) (a+x (b+c x))^2}+\frac {2 b^4 e^3+b^3 c e^2 (d+2 e x)+4 c^2 \left (4 a^2 e^3+3 c^2 d^3 x+7 a c d e^2 x\right )+2 b c^2 \left (3 c d^2 (d-3 e x)+7 a e^2 (d-e x)\right )+b^2 c e \left (-15 a e^2+c d (-9 d+2 e x)\right )}{\left (b^2-4 a c\right )^2 \left (c d^2+e (-b d+a e)\right )^2 (a+x (b+c x))}+\frac {2 \left (-12 c^5 d^5+b^5 e^5-10 a b^3 c e^5+30 a^2 b c^2 e^5+10 c^4 d^3 e (3 b d-4 a e)-20 c^3 d e^2 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2} \left (-c d^2+e (b d-a e)\right )^3}+\frac {2 e^5 \log (d+e x)}{\left (c d^2+e (-b d+a e)\right )^3}-\frac {e^5 \log (a+x (b+c x))}{\left (c d^2+e (-b d+a e)\right )^3}\right ) \]

[In]

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

[Out]

((-(b^2*e) + 2*c*(a*e + c*d*x) + b*c*(d - e*x))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*(a + x*(b + c*x))^2)
+ (2*b^4*e^3 + b^3*c*e^2*(d + 2*e*x) + 4*c^2*(4*a^2*e^3 + 3*c^2*d^3*x + 7*a*c*d*e^2*x) + 2*b*c^2*(3*c*d^2*(d
- 3*e*x) + 7*a*e^2*(d - e*x)) + b^2*c*e*(-15*a*e^2 + c*d*(-9*d + 2*e*x)))/((b^2 - 4*a*c)^2*(c*d^2 + e*(-(b*d)
+ a*e))^2*(a + x*(b + c*x))) + (2*(-12*c^5*d^5 + b^5*e^5 - 10*a*b^3*c*e^5 + 30*a^2*b*c^2*e^5 + 10*c^4*d^3*e*(3
*b*d - 4*a*e) - 20*c^3*d*e^2*(b^2*d^2 - 3*a*b*d*e + 3*a^2*e^2))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/((-b^2
+ 4*a*c)^(5/2)*(-(c*d^2) + e*(b*d - a*e))^3) + (2*e^5*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^3 - (e^5*Log[a
+ x*(b + c*x)])/(c*d^2 + e*(-(b*d) + a*e))^3)/2

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1101\) vs. \(2(419)=838\).

Time = 13.82 (sec) , antiderivative size = 1102, normalized size of antiderivative = 2.57

method	result	size
default	\(\text {Expression too large to display}\)	\(1102\)
risch	\(\text {Expression too large to display}\)	\(60346\)

[In]

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

[Out]

-1/(a*e^2-b*d*e+c*d^2)^3*((c^2*(7*a^2*b*c*e^5-14*a^2*c^2*d*e^4-a*b^3*e^5-8*a*b^2*c*d*e^4+30*a*b*c^2*d^2*e^3-20
*a*c^3*d^3*e^2+b^4*d*e^4-10*b^2*c^2*d^3*e^2+15*b*c^3*d^4*e-6*c^4*d^5)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3-1/2*c*(16
*a^3*c^2*e^5-29*a^2*b^2*c*e^5+26*a^2*b*c^2*d*e^4+16*a^2*c^3*d^2*e^3+4*a*b^4*e^5+32*a*b^3*c*d*e^4-98*a*b^2*c^2*
d^2*e^3+60*a*b*c^3*d^3*e^2-4*b^5*d*e^4+b^4*c*d^2*e^3+30*b^3*c^2*d^3*e^2-45*b^2*c^3*d^4*e+18*b*c^4*d^5)/(16*a^2
*c^2-8*a*b^2*c+b^4)*x^2+(a^3*b*c^2*e^5-18*a^3*c^3*d*e^4+6*a^2*b^3*c*e^5-10*a^2*b^2*c^2*d*e^4+34*a^2*b*c^3*d^2*
e^3-28*a^2*c^4*d^3*e^2-a*b^5*e^5-6*a*b^4*c*d*e^4+18*a*b^3*c^2*d^2*e^3-26*a*b^2*c^3*d^3*e^2+25*a*b*c^4*d^4*e-10
*a*c^5*d^5+b^6*d*e^4-b^5*c*d^2*e^3-3*b^4*c^2*d^3*e^2+5*b^3*c^3*d^4*e-2*b^2*c^4*d^5)/(16*a^2*c^2-8*a*b^2*c+b^4)
*x-1/2*(24*a^4*c^2*e^5-21*a^3*b^2*c*e^5-14*a^3*b*c^2*d*e^4+32*a^3*c^3*d^2*e^3+3*a^2*b^4*e^5+27*a^2*b^3*c*d*e^4
-50*a^2*b^2*c^2*d^2*e^3+12*a^2*b*c^3*d^3*e^2+8*a^2*c^4*d^4*e-4*a*b^5*d*e^4-a*b^4*c*d^2*e^3+24*a*b^3*c^2*d^3*e^
2-29*a*b^2*c^3*d^4*e+10*a*b*c^4*d^5+b^6*d^2*e^3-3*b^5*c*d^3*e^2+3*b^4*c^2*d^4*e-b^3*c^3*d^5)/(16*a^2*c^2-8*a*b
^2*c+b^4))/(c*x^2+b*x+a)^2+1/(16*a^2*c^2-8*a*b^2*c+b^4)*(1/2*(16*a^2*c^3*e^5-8*a*b^2*c^2*e^5+b^4*c*e^5)/c*ln(c
*x^2+b*x+a)+2*(23*a^2*b*c^2*e^5-30*a^2*c^3*d*e^4-9*a*b^3*c*e^5+30*a*b*c^3*d^2*e^3-20*a*c^4*d^3*e^2+b^5*e^5-10*
b^2*c^3*d^3*e^2+15*b*c^4*d^4*e-6*c^5*d^5-1/2*(16*a^2*c^3*e^5-8*a*b^2*c^2*e^5+b^4*c*e^5)*b/c)/(4*a*c-b^2)^(1/2)
*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))))+e^5*ln(e*x+d)/(a*e^2-b*d*e+c*d^2)^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3778 vs. \(2 (419) = 838\).

Time = 209.03 (sec) , antiderivative size = 7575, normalized size of antiderivative = 17.66 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]

[In]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1461 vs. \(2 (419) = 838\).

Time = 0.29 (sec) , antiderivative size = 1461, normalized size of antiderivative = 3.41 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

e^6*log(abs(e*x + 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) - 1/2*e^5*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) + (12*c^5*d^5 - 30*b*c^4*d^4*e + 20*b^2*c^3*d^3*e^2 + 40*a*c^4*d^3*e^2
- 60*a*b*c^3*d^2*e^3 + 60*a^2*c^3*d*e^4 - b^5*e^5 + 10*a*b^3*c*e^5 - 30*a^2*b*c^2*e^5)*arctan((2*c*x + b)/sqrt
(-b^2 + 4*a*c))/((b^4*c^3*d^6 - 8*a*b^2*c^4*d^6 + 16*a^2*c^5*d^6 - 3*b^5*c^2*d^5*e + 24*a*b^3*c^3*d^5*e - 48*a
^2*b*c^4*d^5*e + 3*b^6*c*d^4*e^2 - 21*a*b^4*c^2*d^4*e^2 + 24*a^2*b^2*c^3*d^4*e^2 + 48*a^3*c^4*d^4*e^2 - b^7*d^
3*e^3 + 2*a*b^5*c*d^3*e^3 + 32*a^2*b^3*c^2*d^3*e^3 - 96*a^3*b*c^3*d^3*e^3 + 3*a*b^6*d^2*e^4 - 21*a^2*b^4*c*d^2
*e^4 + 24*a^3*b^2*c^2*d^2*e^4 + 48*a^4*c^3*d^2*e^4 - 3*a^2*b^5*d*e^5 + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5
+ a^3*b^4*e^6 - 8*a^4*b^2*c*e^6 + 16*a^5*c^2*e^6)*sqrt(-b^2 + 4*a*c)) - 1/2*(b^3*c^3*d^5 - 10*a*b*c^4*d^5 - 3
*b^4*c^2*d^4*e + 29*a*b^2*c^3*d^4*e - 8*a^2*c^4*d^4*e + 3*b^5*c*d^3*e^2 - 24*a*b^3*c^2*d^3*e^2 - 12*a^2*b*c^3*
d^3*e^2 - b^6*d^2*e^3 + a*b^4*c*d^2*e^3 + 50*a^2*b^2*c^2*d^2*e^3 - 32*a^3*c^3*d^2*e^3 + 4*a*b^5*d*e^4 - 27*a^2
*b^3*c*d*e^4 + 14*a^3*b*c^2*d*e^4 - 3*a^2*b^4*e^5 + 21*a^3*b^2*c*e^5 - 24*a^4*c^2*e^5 - 2*(6*c^6*d^5 - 15*b*c^
5*d^4*e + 10*b^2*c^4*d^3*e^2 + 20*a*c^5*d^3*e^2 - 30*a*b*c^4*d^2*e^3 - b^4*c^2*d*e^4 + 8*a*b^2*c^3*d*e^4 + 14*
a^2*c^4*d*e^4 + a*b^3*c^2*e^5 - 7*a^2*b*c^3*e^5)*x^3 - (18*b*c^5*d^5 - 45*b^2*c^4*d^4*e + 30*b^3*c^3*d^3*e^2 +
60*a*b*c^4*d^3*e^2 + b^4*c^2*d^2*e^3 - 98*a*b^2*c^3*d^2*e^3 + 16*a^2*c^4*d^2*e^3 - 4*b^5*c*d*e^4 + 32*a*b^3*c
^2*d*e^4 + 26*a^2*b*c^3*d*e^4 + 4*a*b^4*c*e^5 - 29*a^2*b^2*c^2*e^5 + 16*a^3*c^3*e^5)*x^2 - 2*(2*b^2*c^4*d^5 +
10*a*c^5*d^5 - 5*b^3*c^3*d^4*e - 25*a*b*c^4*d^4*e + 3*b^4*c^2*d^3*e^2 + 26*a*b^2*c^3*d^3*e^2 + 28*a^2*c^4*d^3*
e^2 + b^5*c*d^2*e^3 - 18*a*b^3*c^2*d^2*e^3 - 34*a^2*b*c^3*d^2*e^3 - b^6*d*e^4 + 6*a*b^4*c*d*e^4 + 10*a^2*b^2*c
^2*d*e^4 + 18*a^3*c^3*d*e^4 + a*b^5*e^5 - 6*a^2*b^3*c*e^5 - a^3*b*c^2*e^5)*x)/((c*d^2 - b*d*e + a*e^2)^3*(c*x^
2 + b*x + a)^2*(b^2 - 4*a*c)^2)

Mupad [B] (verification not implemented)

Time = 19.55 (sec) , antiderivative size = 3292, normalized size of antiderivative = 7.67 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((3*a*b^4*e^3 - b^5*d*e^2 + 24*a^3*c^2*e^3 - b^3*c^2*d^3 - 21*a^2*b^2*c*e^3 + 8*a^2*c^3*d^2*e + 10*a*b*c^3*d^3
+ 2*b^4*c*d^2*e + 6*a*b^3*c*d*e^2 - 19*a*b^2*c^2*d^2*e + 10*a^2*b*c^2*d*e^2)/(2*(a^2*b^4*e^4 + 16*a^2*c^4*d^4
+ 16*a^4*c^2*e^4 + b^4*c^2*d^4 + b^6*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2*c*e^4 + 32*a^3*c^3*d^2*e^2 - 2*a*b
^5*d*e^3 - 2*b^5*c*d^3*e + 16*a*b^3*c^2*d^3*e - 6*a*b^4*c*d^2*e^2 - 32*a^2*b*c^3*d^3*e + 16*a^2*b^3*c*d*e^3 -
32*a^3*b*c^2*d*e^3)) + (x^3*(6*c^5*d^3 + b^3*c^2*e^3 + b^2*c^3*d*e^2 - 7*a*b*c^3*e^3 + 14*a*c^4*d*e^2 - 9*b*c^
4*d^2*e))/(a^2*b^4*e^4 + 16*a^2*c^4*d^4 + 16*a^4*c^2*e^4 + b^4*c^2*d^4 + b^6*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3
*b^2*c*e^4 + 32*a^3*c^3*d^2*e^2 - 2*a*b^5*d*e^3 - 2*b^5*c*d^3*e + 16*a*b^3*c^2*d^3*e - 6*a*b^4*c*d^2*e^2 - 32*
a^2*b*c^3*d^3*e + 16*a^2*b^3*c*d*e^3 - 32*a^3*b*c^2*d*e^3) + (x*(b^5*e^3 + 10*a*c^4*d^3 + 2*b^2*c^3*d^3 - a^2*
b*c^2*e^3 + 18*a^2*c^3*d*e^2 - 3*b^3*c^2*d^2*e - 6*a*b^3*c*e^3 - 15*a*b*c^3*d^2*e + 9*a*b^2*c^2*d*e^2))/(a^2*b
^4*e^4 + 16*a^2*c^4*d^4 + 16*a^4*c^2*e^4 + b^4*c^2*d^4 + b^6*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2*c*e^4 + 32*
a^3*c^3*d^2*e^2 - 2*a*b^5*d*e^3 - 2*b^5*c*d^3*e + 16*a*b^3*c^2*d^3*e - 6*a*b^4*c*d^2*e^2 - 32*a^2*b*c^3*d^3*e
+ 16*a^2*b^3*c*d*e^3 - 32*a^3*b*c^2*d*e^3) + (x^2*(18*b*c^4*d^3 + 4*b^4*c*e^3 + 16*a^2*c^3*e^3 - 29*a*b^2*c^2*
e^3 - 27*b^2*c^3*d^2*e + 3*b^3*c^2*d*e^2 + 42*a*b*c^3*d*e^2))/(2*(a^2*b^4*e^4 + 16*a^2*c^4*d^4 + 16*a^4*c^2*e^
4 + b^4*c^2*d^4 + b^6*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2*c*e^4 + 32*a^3*c^3*d^2*e^2 - 2*a*b^5*d*e^3 - 2*b^5
*c*d^3*e + 16*a*b^3*c^2*d^3*e - 6*a*b^4*c*d^2*e^2 - 32*a^2*b*c^3*d^3*e + 16*a^2*b^3*c*d*e^3 - 32*a^3*b*c^2*d*e
^3)))/(x^2*(2*a*c + b^2) + a^2 + c^2*x^4 + 2*a*b*x + 2*b*c*x^3) + (e^5*log(d + e*x))/(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(b^5 + (-(4*a*c - b^2)^5)^(1/2) + 16*a^2*b*c^2 + 32*a^2*c^3*x - 8*a*b^3*c + 2*b^4*c
*x - 16*a*b^2*c^2*x)*((b^10*e^5)/2 - 512*a^5*c^5*e^5 - (b^5*e^5*(-(4*a*c - b^2)^5)^(1/2))/2 + 6*c^5*d^5*(-(4*a
*c - b^2)^5)^(1/2) + 80*a^2*b^6*c^2*e^5 - 320*a^3*b^4*c^3*e^5 + 640*a^4*b^2*c^4*e^5 - 10*a*b^8*c*e^5 + 10*b^2*
c^3*d^3*e^2*(-(4*a*c - b^2)^5)^(1/2) + 5*a*b^3*c*e^5*(-(4*a*c - b^2)^5)^(1/2) - 15*b*c^4*d^4*e*(-(4*a*c - b^2)
^5)^(1/2) - 15*a^2*b*c^2*e^5*(-(4*a*c - b^2)^5)^(1/2) + 20*a*c^4*d^3*e^2*(-(4*a*c - b^2)^5)^(1/2) + 30*a^2*c^3
*d*e^4*(-(4*a*c - b^2)^5)^(1/2) - 30*a*b*c^3*d^2*e^3*(-(4*a*c - b^2)^5)^(1/2)))/(a^3*b^10*e^6 - 1024*a^5*c^8*d
^6 - 1024*a^8*c^5*e^6 + b^10*c^3*d^6 - b^13*d^3*e^3 - 20*a*b^8*c^4*d^6 - 20*a^4*b^8*c*e^6 + 3*a*b^12*d^2*e^4 -
3*a^2*b^11*d*e^5 - 3*b^11*c^2*d^5*e + 3*b^12*c*d^4*e^2 + 160*a^2*b^6*c^5*d^6 - 640*a^3*b^4*c^6*d^6 + 1280*a^4
*b^2*c^7*d^6 + 160*a^5*b^6*c^2*e^6 - 640*a^6*b^4*c^3*e^6 + 1280*a^7*b^2*c^4*e^6 - 3072*a^6*c^7*d^4*e^2 - 3072*
a^7*c^6*d^2*e^4 + 420*a^2*b^8*c^3*d^4*e^2 - 40*a^2*b^9*c^2*d^3*e^3 - 1440*a^3*b^6*c^4*d^4*e^2 - 320*a^3*b^7*c^
3*d^3*e^3 + 420*a^3*b^8*c^2*d^2*e^4 + 1920*a^4*b^4*c^5*d^4*e^2 + 2560*a^4*b^5*c^4*d^3*e^3 - 1440*a^4*b^6*c^3*d
^2*e^4 + 768*a^5*b^2*c^6*d^4*e^2 - 6656*a^5*b^3*c^5*d^3*e^3 + 1920*a^5*b^4*c^4*d^2*e^4 + 768*a^6*b^2*c^5*d^2*e
^4 + 60*a*b^9*c^3*d^5*e + 14*a*b^11*c*d^3*e^3 + 60*a^3*b^9*c*d*e^5 + 3072*a^5*b*c^7*d^5*e + 3072*a^7*b*c^5*d*e
^5 - 57*a*b^10*c^2*d^4*e^2 - 480*a^2*b^7*c^4*d^5*e - 57*a^2*b^10*c*d^2*e^4 + 1920*a^3*b^5*c^5*d^5*e - 3840*a^4
*b^3*c^6*d^5*e - 480*a^4*b^7*c^2*d*e^5 + 1920*a^5*b^5*c^3*d*e^5 + 6144*a^6*b*c^6*d^3*e^3 - 3840*a^6*b^3*c^4*d*
e^5) + (log(b^5 - (-(4*a*c - b^2)^5)^(1/2) + 16*a^2*b*c^2 + 32*a^2*c^3*x - 8*a*b^3*c + 2*b^4*c*x - 16*a*b^2*c^
2*x)*(512*a^5*c^5*e^5 - (b^10*e^5)/2 - (b^5*e^5*(-(4*a*c - b^2)^5)^(1/2))/2 + 6*c^5*d^5*(-(4*a*c - b^2)^5)^(1/
2) - 80*a^2*b^6*c^2*e^5 + 320*a^3*b^4*c^3*e^5 - 640*a^4*b^2*c^4*e^5 + 10*a*b^8*c*e^5 + 10*b^2*c^3*d^3*e^2*(-(4
*a*c - b^2)^5)^(1/2) + 5*a*b^3*c*e^5*(-(4*a*c - b^2)^5)^(1/2) - 15*b*c^4*d^4*e*(-(4*a*c - b^2)^5)^(1/2) - 15*a
^2*b*c^2*e^5*(-(4*a*c - b^2)^5)^(1/2) + 20*a*c^4*d^3*e^2*(-(4*a*c - b^2)^5)^(1/2) + 30*a^2*c^3*d*e^4*(-(4*a*c
- b^2)^5)^(1/2) - 30*a*b*c^3*d^2*e^3*(-(4*a*c - b^2)^5)^(1/2)))/(a^3*b^10*e^6 - 1024*a^5*c^8*d^6 - 1024*a^8*c^
5*e^6 + b^10*c^3*d^6 - b^13*d^3*e^3 - 20*a*b^8*c^4*d^6 - 20*a^4*b^8*c*e^6 + 3*a*b^12*d^2*e^4 - 3*a^2*b^11*d*e^
5 - 3*b^11*c^2*d^5*e + 3*b^12*c*d^4*e^2 + 160*a^2*b^6*c^5*d^6 - 640*a^3*b^4*c^6*d^6 + 1280*a^4*b^2*c^7*d^6 + 1
60*a^5*b^6*c^2*e^6 - 640*a^6*b^4*c^3*e^6 + 1280*a^7*b^2*c^4*e^6 - 3072*a^6*c^7*d^4*e^2 - 3072*a^7*c^6*d^2*e^4
+ 420*a^2*b^8*c^3*d^4*e^2 - 40*a^2*b^9*c^2*d^3*e^3 - 1440*a^3*b^6*c^4*d^4*e^2 - 320*a^3*b^7*c^3*d^3*e^3 + 420*
a^3*b^8*c^2*d^2*e^4 + 1920*a^4*b^4*c^5*d^4*e^2 + 2560*a^4*b^5*c^4*d^3*e^3 - 1440*a^4*b^6*c^3*d^2*e^4 + 768*a^5
*b^2*c^6*d^4*e^2 - 6656*a^5*b^3*c^5*d^3*e^3 + 1920*a^5*b^4*c^4*d^2*e^4 + 768*a^6*b^2*c^5*d^2*e^4 + 60*a*b^9*c^
3*d^5*e + 14*a*b^11*c*d^3*e^3 + 60*a^3*b^9*c*d*e^5 + 3072*a^5*b*c^7*d^5*e + 3072*a^7*b*c^5*d*e^5 - 57*a*b^10*c
^2*d^4*e^2 - 480*a^2*b^7*c^4*d^5*e - 57*a^2*b^10*c*d^2*e^4 + 1920*a^3*b^5*c^5*d^5*e - 3840*a^4*b^3*c^6*d^5*e -
480*a^4*b^7*c^2*d*e^5 + 1920*a^5*b^5*c^3*d*e^5 + 6144*a^6*b*c^6*d^3*e^3 - 3840*a^6*b^3*c^4*d*e^5)