\(\int \frac {(d+e x)^2}{(a+b x+c x^2)^5} \, dx\) [2224]

Optimal result

Integrand size = 20, antiderivative size = 330 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^5} \, dx=-\frac {(d+e x) (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}-\frac {4 b^2 d e+12 a c d e-7 b \left (c d^2+a e^2\right )-\left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) x}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac {5 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (b+2 c x)}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac {5 c \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (b+2 c x)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac {10 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}} \]

[Out]

-1/4*(e*x+d)*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^4+1/6*(-4*b^2*d*e-12*a*c*d*e+7*b*(a*e^2+c*d
^2)+(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))*x)/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^3-5/12*(14*c^2*d^2+3*b^2*e^2-2*c
*e*(-a*e+7*b*d))*(2*c*x+b)/(-4*a*c+b^2)^3/(c*x^2+b*x+a)^2+5/2*c*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))*(2*c
*x+b)/(-4*a*c+b^2)^4/(c*x^2+b*x+a)-10*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))*arctanh((2*c*x+b)/(-4*a*c+
b^2)^(1/2))/(-4*a*c+b^2)^(9/2)

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {752, 652, 628, 632, 212} \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^5} \, dx=-\frac {10 c^2 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{\left (b^2-4 a c\right )^{9/2}}+\frac {5 c (b+2 c x) \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac {5 (b+2 c x) \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}-\frac {-x \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )-7 b \left (a e^2+c d^2\right )+12 a c d e+4 b^2 d e}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac {(d+e x) (-2 a e+x (2 c d-b e)+b d)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4} \]

[In]

Int[(d + e*x)^2/(a + b*x + c*x^2)^5,x]

[Out]

-1/4*((d + e*x)*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)^4) - (4*b^2*d*e + 12*a*c*d*e
- 7*b*(c*d^2 + a*e^2) - (14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*x)/(6*(b^2 - 4*a*c)^2*(a + b*x + c*x^2
)^3) - (5*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(b + 2*c*x))/(12*(b^2 - 4*a*c)^3*(a + b*x + c*x^2)^2)
+ (5*c*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(b + 2*c*x))/(2*(b^2 - 4*a*c)^4*(a + b*x + c*x^2)) - (1
0*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(9/
2)

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 628

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1
)*(b^2 - 4*a*c))), x] - Dist[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

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 652

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

Rule 752

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

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

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^5} \, dx=\frac {1}{12} \left (\frac {\left (14 c^2 d^2+3 b^2 e^2+2 c e (-7 b d+a e)\right ) (b+2 c x)}{c \left (b^2-4 a c\right )^2 (a+x (b+c x))^3}-\frac {5 \left (14 c^2 d^2+3 b^2 e^2+2 c e (-7 b d+a e)\right ) (b+2 c x)}{\left (b^2-4 a c\right )^3 (a+x (b+c x))^2}+\frac {30 c \left (14 c^2 d^2+3 b^2 e^2+2 c e (-7 b d+a e)\right ) (b+2 c x)}{\left (b^2-4 a c\right )^4 (a+x (b+c x))}+\frac {3 \left (a b e^2+2 c^2 d^2 x+b^2 e^2 x+b c d (d-2 e x)-2 a c e (2 d+e x)\right )}{c \left (-b^2+4 a c\right ) (a+x (b+c x))^4}+\frac {120 c^2 \left (14 c^2 d^2+3 b^2 e^2+2 c e (-7 b d+a e)\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{9/2}}\right ) \]

[In]

Integrate[(d + e*x)^2/(a + b*x + c*x^2)^5,x]

[Out]

(((14*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-7*b*d + a*e))*(b + 2*c*x))/(c*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^3) - (5*(1
4*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-7*b*d + a*e))*(b + 2*c*x))/((b^2 - 4*a*c)^3*(a + x*(b + c*x))^2) + (30*c*(14*c
^2*d^2 + 3*b^2*e^2 + 2*c*e*(-7*b*d + a*e))*(b + 2*c*x))/((b^2 - 4*a*c)^4*(a + x*(b + c*x))) + (3*(a*b*e^2 + 2*
c^2*d^2*x + b^2*e^2*x + b*c*d*(d - 2*e*x) - 2*a*c*e*(2*d + e*x)))/(c*(-b^2 + 4*a*c)*(a + x*(b + c*x))^4) + (12
0*c^2*(14*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-7*b*d + a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(
9/2))/12

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1016\) vs. \(2(317)=634\).

Time = 20.81 (sec) , antiderivative size = 1017, normalized size of antiderivative = 3.08

method	result	size
default	\(\text {Expression too large to display}\)	\(1017\)
risch	\(\text {Expression too large to display}\)	\(1875\)

[In]

int((e*x+d)^2/(c*x^2+b*x+a)^5,x,method=_RETURNVERBOSE)

[Out]

(5*c^5*(2*a*c*e^2+3*b^2*e^2-14*b*c*d*e+14*c^2*d^2)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)
*x^7+35/2*c^4*(2*a*c*e^2+3*b^2*e^2-14*b*c*d*e+14*c^2*d^2)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6
*c+b^8)*b*x^6+5/3*c^3*(11*a*c+13*b^2)*(2*a*c*e^2+3*b^2*e^2-14*b*c*d*e+14*c^2*d^2)/(256*a^4*c^4-256*a^3*b^2*c^3
+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^5+25/12*b*(22*a*c+5*b^2)*c^2*(2*a*c*e^2+3*b^2*e^2-14*b*c*d*e+14*c^2*d^2)/(25
6*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^4+1/3*(73*a^2*c^2+101*a*b^2*c+3*b^4)*c*(2*a*c*e^2+3
*b^2*e^2-14*b*c*d*e+14*c^2*d^2)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^3+1/6*b*(219*a^2
*c^2+28*a*b^2*c-b^4)*(2*a*c*e^2+3*b^2*e^2-14*b*c*d*e+14*c^2*d^2)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-1
6*a*b^6*c+b^8)*x^2-1/3*(30*a^4*c^3*e^2-279*a^3*b^2*c^2*e^2+558*a^3*b*c^3*d*e-558*a^3*c^4*d^2-28*a^2*b^4*c*e^2+
348*a^2*b^3*c^2*d*e-348*a^2*b^2*c^3*d^2+a*b^6*e^2-38*a*b^5*c*d*e+38*a*b^4*c^2*d^2+2*b^7*d*e-2*b^6*c*d^2)/(256*
a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x+1/12*(324*a^4*b*c^2*e^2-768*a^4*c^3*d*e+28*a^3*b^3*c*
e^2-348*a^3*b^2*c^2*d*e+1116*a^3*b*c^3*d^2-a^2*b^5*e^2+38*a^2*b^4*c*d*e-326*a^2*b^3*c^2*d^2-2*a*b^6*d*e+50*a*b
^5*c*d^2-3*b^7*d^2)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8))/(c*x^2+b*x+a)^4+10*c^2*(2*a*c
*e^2+3*b^2*e^2-14*b*c*d*e+14*c^2*d^2)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^
(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2235 vs. \(2 (318) = 636\).

Time = 0.37 (sec) , antiderivative size = 4491, normalized size of antiderivative = 13.61 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^5} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^2/(c*x^2+b*x+a)^5,x, algorithm="fricas")

[Out]

[1/12*(60*(14*(b^2*c^7 - 4*a*c^8)*d^2 - 14*(b^3*c^6 - 4*a*b*c^7)*d*e + (3*b^4*c^5 - 10*a*b^2*c^6 - 8*a^2*c^7)*
e^2)*x^7 + 210*(14*(b^3*c^6 - 4*a*b*c^7)*d^2 - 14*(b^4*c^5 - 4*a*b^2*c^6)*d*e + (3*b^5*c^4 - 10*a*b^3*c^5 - 8*
a^2*b*c^6)*e^2)*x^6 + 20*(14*(13*b^4*c^5 - 41*a*b^2*c^6 - 44*a^2*c^7)*d^2 - 14*(13*b^5*c^4 - 41*a*b^3*c^5 - 44
*a^2*b*c^6)*d*e + (39*b^6*c^3 - 97*a*b^4*c^4 - 214*a^2*b^2*c^5 - 88*a^3*c^6)*e^2)*x^5 + 25*(14*(5*b^5*c^4 + 2*
a*b^3*c^5 - 88*a^2*b*c^6)*d^2 - 14*(5*b^6*c^3 + 2*a*b^4*c^4 - 88*a^2*b^2*c^5)*d*e + (15*b^7*c^2 + 16*a*b^5*c^3
- 260*a^2*b^3*c^4 - 176*a^3*b*c^5)*e^2)*x^4 + 4*(14*(3*b^6*c^3 + 89*a*b^4*c^4 - 331*a^2*b^2*c^5 - 292*a^3*c^6
)*d^2 - 14*(3*b^7*c^2 + 89*a*b^5*c^3 - 331*a^2*b^3*c^4 - 292*a^3*b*c^5)*d*e + (9*b^8*c + 273*a*b^6*c^2 - 815*a
^2*b^4*c^3 - 1538*a^3*b^2*c^4 - 584*a^4*c^5)*e^2)*x^3 - (3*b^9 - 62*a*b^7*c + 526*a^2*b^5*c^2 - 2420*a^3*b^3*c
^3 + 4464*a^4*b*c^4)*d^2 - 2*(a*b^8 - 23*a^2*b^6*c + 250*a^3*b^4*c^2 - 312*a^4*b^2*c^3 - 1536*a^5*c^4)*d*e - (
a^2*b^7 - 32*a^3*b^5*c - 212*a^4*b^3*c^2 + 1296*a^5*b*c^3)*e^2 - 2*(14*(b^7*c^2 - 32*a*b^5*c^3 - 107*a^2*b^3*c
^4 + 876*a^3*b*c^5)*d^2 - 14*(b^8*c - 32*a*b^6*c^2 - 107*a^2*b^4*c^3 + 876*a^3*b^2*c^4)*d*e + (3*b^9 - 94*a*b^
7*c - 385*a^2*b^5*c^2 + 2414*a^3*b^3*c^3 + 1752*a^4*b*c^4)*e^2)*x^2 + 60*(14*a^4*c^4*d^2 - 14*a^4*b*c^3*d*e +
(14*c^8*d^2 - 14*b*c^7*d*e + (3*b^2*c^6 + 2*a*c^7)*e^2)*x^8 + 4*(14*b*c^7*d^2 - 14*b^2*c^6*d*e + (3*b^3*c^5 +
2*a*b*c^6)*e^2)*x^7 + 2*(14*(3*b^2*c^6 + 2*a*c^7)*d^2 - 14*(3*b^3*c^5 + 2*a*b*c^6)*d*e + (9*b^4*c^4 + 12*a*b^2
*c^5 + 4*a^2*c^6)*e^2)*x^6 + 4*(14*(b^3*c^5 + 3*a*b*c^6)*d^2 - 14*(b^4*c^4 + 3*a*b^2*c^5)*d*e + (3*b^5*c^3 + 1
1*a*b^3*c^4 + 6*a^2*b*c^5)*e^2)*x^5 + (14*(b^4*c^4 + 12*a*b^2*c^5 + 6*a^2*c^6)*d^2 - 14*(b^5*c^3 + 12*a*b^3*c^
4 + 6*a^2*b*c^5)*d*e + (3*b^6*c^2 + 38*a*b^4*c^3 + 42*a^2*b^2*c^4 + 12*a^3*c^5)*e^2)*x^4 + 4*(14*(a*b^3*c^4 +
3*a^2*b*c^5)*d^2 - 14*(a*b^4*c^3 + 3*a^2*b^2*c^4)*d*e + (3*a*b^5*c^2 + 11*a^2*b^3*c^3 + 6*a^3*b*c^4)*e^2)*x^3
+ (3*a^4*b^2*c^2 + 2*a^5*c^3)*e^2 + 2*(14*(3*a^2*b^2*c^4 + 2*a^3*c^5)*d^2 - 14*(3*a^2*b^3*c^3 + 2*a^3*b*c^4)*d
*e + (9*a^2*b^4*c^2 + 12*a^3*b^2*c^3 + 4*a^4*c^4)*e^2)*x^2 + 4*(14*a^3*b*c^4*d^2 - 14*a^3*b^2*c^3*d*e + (3*a^3
*b^3*c^2 + 2*a^4*b*c^3)*e^2)*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)) + 4*(2*(b^8*c - 23*a*b^6*c^2 + 250*a^2*b^4*c^3 - 417*a^3*b^2*c^4 - 1116*a^4*c^5
)*d^2 - 2*(b^9 - 23*a*b^7*c + 250*a^2*b^5*c^2 - 417*a^3*b^3*c^3 - 1116*a^4*b*c^4)*d*e - (a*b^8 - 32*a^2*b^6*c
- 167*a^3*b^4*c^2 + 1146*a^4*b^2*c^3 - 120*a^5*c^4)*e^2)*x)/(a^4*b^10 - 20*a^5*b^8*c + 160*a^6*b^6*c^2 - 640*a
^7*b^4*c^3 + 1280*a^8*b^2*c^4 - 1024*a^9*c^5 + (b^10*c^4 - 20*a*b^8*c^5 + 160*a^2*b^6*c^6 - 640*a^3*b^4*c^7 +
1280*a^4*b^2*c^8 - 1024*a^5*c^9)*x^8 + 4*(b^11*c^3 - 20*a*b^9*c^4 + 160*a^2*b^7*c^5 - 640*a^3*b^5*c^6 + 1280*a
^4*b^3*c^7 - 1024*a^5*b*c^8)*x^7 + 2*(3*b^12*c^2 - 58*a*b^10*c^3 + 440*a^2*b^8*c^4 - 1600*a^3*b^6*c^5 + 2560*a
^4*b^4*c^6 - 512*a^5*b^2*c^7 - 2048*a^6*c^8)*x^6 + 4*(b^13*c - 17*a*b^11*c^2 + 100*a^2*b^9*c^3 - 160*a^3*b^7*c
^4 - 640*a^4*b^5*c^5 + 2816*a^5*b^3*c^6 - 3072*a^6*b*c^7)*x^5 + (b^14 - 8*a*b^12*c - 74*a^2*b^10*c^2 + 1160*a^
3*b^8*c^3 - 5440*a^4*b^6*c^4 + 10496*a^5*b^4*c^5 - 4608*a^6*b^2*c^6 - 6144*a^7*c^7)*x^4 + 4*(a*b^13 - 17*a^2*b
^11*c + 100*a^3*b^9*c^2 - 160*a^4*b^7*c^3 - 640*a^5*b^5*c^4 + 2816*a^6*b^3*c^5 - 3072*a^7*b*c^6)*x^3 + 2*(3*a^
2*b^12 - 58*a^3*b^10*c + 440*a^4*b^8*c^2 - 1600*a^5*b^6*c^3 + 2560*a^6*b^4*c^4 - 512*a^7*b^2*c^5 - 2048*a^8*c^
6)*x^2 + 4*(a^3*b^11 - 20*a^4*b^9*c + 160*a^5*b^7*c^2 - 640*a^6*b^5*c^3 + 1280*a^7*b^3*c^4 - 1024*a^8*b*c^5)*x
), 1/12*(60*(14*(b^2*c^7 - 4*a*c^8)*d^2 - 14*(b^3*c^6 - 4*a*b*c^7)*d*e + (3*b^4*c^5 - 10*a*b^2*c^6 - 8*a^2*c^7
)*e^2)*x^7 + 210*(14*(b^3*c^6 - 4*a*b*c^7)*d^2 - 14*(b^4*c^5 - 4*a*b^2*c^6)*d*e + (3*b^5*c^4 - 10*a*b^3*c^5 -
8*a^2*b*c^6)*e^2)*x^6 + 20*(14*(13*b^4*c^5 - 41*a*b^2*c^6 - 44*a^2*c^7)*d^2 - 14*(13*b^5*c^4 - 41*a*b^3*c^5 -
44*a^2*b*c^6)*d*e + (39*b^6*c^3 - 97*a*b^4*c^4 - 214*a^2*b^2*c^5 - 88*a^3*c^6)*e^2)*x^5 + 25*(14*(5*b^5*c^4 +
2*a*b^3*c^5 - 88*a^2*b*c^6)*d^2 - 14*(5*b^6*c^3 + 2*a*b^4*c^4 - 88*a^2*b^2*c^5)*d*e + (15*b^7*c^2 + 16*a*b^5*c
^3 - 260*a^2*b^3*c^4 - 176*a^3*b*c^5)*e^2)*x^4 + 4*(14*(3*b^6*c^3 + 89*a*b^4*c^4 - 331*a^2*b^2*c^5 - 292*a^3*c
^6)*d^2 - 14*(3*b^7*c^2 + 89*a*b^5*c^3 - 331*a^2*b^3*c^4 - 292*a^3*b*c^5)*d*e + (9*b^8*c + 273*a*b^6*c^2 - 815
*a^2*b^4*c^3 - 1538*a^3*b^2*c^4 - 584*a^4*c^5)*e^2)*x^3 - (3*b^9 - 62*a*b^7*c + 526*a^2*b^5*c^2 - 2420*a^3*b^3
*c^3 + 4464*a^4*b*c^4)*d^2 - 2*(a*b^8 - 23*a^2*b^6*c + 250*a^3*b^4*c^2 - 312*a^4*b^2*c^3 - 1536*a^5*c^4)*d*e -
(a^2*b^7 - 32*a^3*b^5*c - 212*a^4*b^3*c^2 + 1296*a^5*b*c^3)*e^2 - 2*(14*(b^7*c^2 - 32*a*b^5*c^3 - 107*a^2*b^3
*c^4 + 876*a^3*b*c^5)*d^2 - 14*(b^8*c - 32*a*b^6*c^2 - 107*a^2*b^4*c^3 + 876*a^3*b^2*c^4)*d*e + (3*b^9 - 94*a*
b^7*c - 385*a^2*b^5*c^2 + 2414*a^3*b^3*c^3 + 1752*a^4*b*c^4)*e^2)*x^2 - 120*(14*a^4*c^4*d^2 - 14*a^4*b*c^3*d*e
+ (14*c^8*d^2 - 14*b*c^7*d*e + (3*b^2*c^6 + 2*a*c^7)*e^2)*x^8 + 4*(14*b*c^7*d^2 - 14*b^2*c^6*d*e + (3*b^3*c^5
+ 2*a*b*c^6)*e^2)*x^7 + 2*(14*(3*b^2*c^6 + 2*a*c^7)*d^2 - 14*(3*b^3*c^5 + 2*a*b*c^6)*d*e + (9*b^4*c^4 + 12*a*
b^2*c^5 + 4*a^2*c^6)*e^2)*x^6 + 4*(14*(b^3*c^5 + 3*a*b*c^6)*d^2 - 14*(b^4*c^4 + 3*a*b^2*c^5)*d*e + (3*b^5*c^3
+ 11*a*b^3*c^4 + 6*a^2*b*c^5)*e^2)*x^5 + (14*(b^4*c^4 + 12*a*b^2*c^5 + 6*a^2*c^6)*d^2 - 14*(b^5*c^3 + 12*a*b^3
*c^4 + 6*a^2*b*c^5)*d*e + (3*b^6*c^2 + 38*a*b^4*c^3 + 42*a^2*b^2*c^4 + 12*a^3*c^5)*e^2)*x^4 + 4*(14*(a*b^3*c^4
+ 3*a^2*b*c^5)*d^2 - 14*(a*b^4*c^3 + 3*a^2*b^2*c^4)*d*e + (3*a*b^5*c^2 + 11*a^2*b^3*c^3 + 6*a^3*b*c^4)*e^2)*x
^3 + (3*a^4*b^2*c^2 + 2*a^5*c^3)*e^2 + 2*(14*(3*a^2*b^2*c^4 + 2*a^3*c^5)*d^2 - 14*(3*a^2*b^3*c^3 + 2*a^3*b*c^4
)*d*e + (9*a^2*b^4*c^2 + 12*a^3*b^2*c^3 + 4*a^4*c^4)*e^2)*x^2 + 4*(14*a^3*b*c^4*d^2 - 14*a^3*b^2*c^3*d*e + (3*
a^3*b^3*c^2 + 2*a^4*b*c^3)*e^2)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) +
4*(2*(b^8*c - 23*a*b^6*c^2 + 250*a^2*b^4*c^3 - 417*a^3*b^2*c^4 - 1116*a^4*c^5)*d^2 - 2*(b^9 - 23*a*b^7*c + 250
*a^2*b^5*c^2 - 417*a^3*b^3*c^3 - 1116*a^4*b*c^4)*d*e - (a*b^8 - 32*a^2*b^6*c - 167*a^3*b^4*c^2 + 1146*a^4*b^2*
c^3 - 120*a^5*c^4)*e^2)*x)/(a^4*b^10 - 20*a^5*b^8*c + 160*a^6*b^6*c^2 - 640*a^7*b^4*c^3 + 1280*a^8*b^2*c^4 - 1
024*a^9*c^5 + (b^10*c^4 - 20*a*b^8*c^5 + 160*a^2*b^6*c^6 - 640*a^3*b^4*c^7 + 1280*a^4*b^2*c^8 - 1024*a^5*c^9)*
x^8 + 4*(b^11*c^3 - 20*a*b^9*c^4 + 160*a^2*b^7*c^5 - 640*a^3*b^5*c^6 + 1280*a^4*b^3*c^7 - 1024*a^5*b*c^8)*x^7
+ 2*(3*b^12*c^2 - 58*a*b^10*c^3 + 440*a^2*b^8*c^4 - 1600*a^3*b^6*c^5 + 2560*a^4*b^4*c^6 - 512*a^5*b^2*c^7 - 20
48*a^6*c^8)*x^6 + 4*(b^13*c - 17*a*b^11*c^2 + 100*a^2*b^9*c^3 - 160*a^3*b^7*c^4 - 640*a^4*b^5*c^5 + 2816*a^5*b
^3*c^6 - 3072*a^6*b*c^7)*x^5 + (b^14 - 8*a*b^12*c - 74*a^2*b^10*c^2 + 1160*a^3*b^8*c^3 - 5440*a^4*b^6*c^4 + 10
496*a^5*b^4*c^5 - 4608*a^6*b^2*c^6 - 6144*a^7*c^7)*x^4 + 4*(a*b^13 - 17*a^2*b^11*c + 100*a^3*b^9*c^2 - 160*a^4
*b^7*c^3 - 640*a^5*b^5*c^4 + 2816*a^6*b^3*c^5 - 3072*a^7*b*c^6)*x^3 + 2*(3*a^2*b^12 - 58*a^3*b^10*c + 440*a^4*
b^8*c^2 - 1600*a^5*b^6*c^3 + 2560*a^6*b^4*c^4 - 512*a^7*b^2*c^5 - 2048*a^8*c^6)*x^2 + 4*(a^3*b^11 - 20*a^4*b^9
*c + 160*a^5*b^7*c^2 - 640*a^6*b^5*c^3 + 1280*a^7*b^3*c^4 - 1024*a^8*b*c^5)*x)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2407 vs. \(2 (333) = 666\).

Time = 6.74 (sec) , antiderivative size = 2407, normalized size of antiderivative = 7.29 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^5} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)**2/(c*x**2+b*x+a)**5,x)

[Out]

-5*c**2*sqrt(-1/(4*a*c - b**2)**9)*(2*a*c*e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c**2*d**2)*log(x + (-5120*a**5*
c**7*sqrt(-1/(4*a*c - b**2)**9)*(2*a*c*e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c**2*d**2) + 6400*a**4*b**2*c**6*s
qrt(-1/(4*a*c - b**2)**9)*(2*a*c*e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c**2*d**2) - 3200*a**3*b**4*c**5*sqrt(-1
/(4*a*c - b**2)**9)*(2*a*c*e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c**2*d**2) + 800*a**2*b**6*c**4*sqrt(-1/(4*a*c
- b**2)**9)*(2*a*c*e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c**2*d**2) - 100*a*b**8*c**3*sqrt(-1/(4*a*c - b**2)**
9)*(2*a*c*e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c**2*d**2) + 10*a*b*c**3*e**2 + 5*b**10*c**2*sqrt(-1/(4*a*c - b
**2)**9)*(2*a*c*e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c**2*d**2) + 15*b**3*c**2*e**2 - 70*b**2*c**3*d*e + 70*b*
c**4*d**2)/(20*a*c**4*e**2 + 30*b**2*c**3*e**2 - 140*b*c**4*d*e + 140*c**5*d**2)) + 5*c**2*sqrt(-1/(4*a*c - b*
*2)**9)*(2*a*c*e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c**2*d**2)*log(x + (5120*a**5*c**7*sqrt(-1/(4*a*c - b**2)*
*9)*(2*a*c*e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c**2*d**2) - 6400*a**4*b**2*c**6*sqrt(-1/(4*a*c - b**2)**9)*(2
*a*c*e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c**2*d**2) + 3200*a**3*b**4*c**5*sqrt(-1/(4*a*c - b**2)**9)*(2*a*c*e
**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c**2*d**2) - 800*a**2*b**6*c**4*sqrt(-1/(4*a*c - b**2)**9)*(2*a*c*e**2 + 3
*b**2*e**2 - 14*b*c*d*e + 14*c**2*d**2) + 100*a*b**8*c**3*sqrt(-1/(4*a*c - b**2)**9)*(2*a*c*e**2 + 3*b**2*e**2
- 14*b*c*d*e + 14*c**2*d**2) + 10*a*b*c**3*e**2 - 5*b**10*c**2*sqrt(-1/(4*a*c - b**2)**9)*(2*a*c*e**2 + 3*b**
2*e**2 - 14*b*c*d*e + 14*c**2*d**2) + 15*b**3*c**2*e**2 - 70*b**2*c**3*d*e + 70*b*c**4*d**2)/(20*a*c**4*e**2 +
30*b**2*c**3*e**2 - 140*b*c**4*d*e + 140*c**5*d**2)) + (324*a**4*b*c**2*e**2 - 768*a**4*c**3*d*e + 28*a**3*b*
*3*c*e**2 - 348*a**3*b**2*c**2*d*e + 1116*a**3*b*c**3*d**2 - a**2*b**5*e**2 + 38*a**2*b**4*c*d*e - 326*a**2*b*
*3*c**2*d**2 - 2*a*b**6*d*e + 50*a*b**5*c*d**2 - 3*b**7*d**2 + x**7*(120*a*c**6*e**2 + 180*b**2*c**5*e**2 - 84
0*b*c**6*d*e + 840*c**7*d**2) + x**6*(420*a*b*c**5*e**2 + 630*b**3*c**4*e**2 - 2940*b**2*c**5*d*e + 2940*b*c**
6*d**2) + x**5*(440*a**2*c**5*e**2 + 1180*a*b**2*c**4*e**2 - 3080*a*b*c**5*d*e + 3080*a*c**6*d**2 + 780*b**4*c
**3*e**2 - 3640*b**3*c**4*d*e + 3640*b**2*c**5*d**2) + x**4*(1100*a**2*b*c**4*e**2 + 1900*a*b**3*c**3*e**2 - 7
700*a*b**2*c**4*d*e + 7700*a*b*c**5*d**2 + 375*b**5*c**2*e**2 - 1750*b**4*c**3*d*e + 1750*b**3*c**4*d**2) + x*
*3*(584*a**3*c**4*e**2 + 1684*a**2*b**2*c**3*e**2 - 4088*a**2*b*c**4*d*e + 4088*a**2*c**5*d**2 + 1236*a*b**4*c
**2*e**2 - 5656*a*b**3*c**3*d*e + 5656*a*b**2*c**4*d**2 + 36*b**6*c*e**2 - 168*b**5*c**2*d*e + 168*b**4*c**3*d
**2) + x**2*(876*a**3*b*c**3*e**2 + 1426*a**2*b**3*c**2*e**2 - 6132*a**2*b**2*c**3*d*e + 6132*a**2*b*c**4*d**2
+ 164*a*b**5*c*e**2 - 784*a*b**4*c**2*d*e + 784*a*b**3*c**3*d**2 - 6*b**7*e**2 + 28*b**6*c*d*e - 28*b**5*c**2
*d**2) + x*(-120*a**4*c**3*e**2 + 1116*a**3*b**2*c**2*e**2 - 2232*a**3*b*c**3*d*e + 2232*a**3*c**4*d**2 + 112*
a**2*b**4*c*e**2 - 1392*a**2*b**3*c**2*d*e + 1392*a**2*b**2*c**3*d**2 - 4*a*b**6*e**2 + 152*a*b**5*c*d*e - 152
*a*b**4*c**2*d**2 - 8*b**7*d*e + 8*b**6*c*d**2))/(3072*a**8*c**4 - 3072*a**7*b**2*c**3 + 1152*a**6*b**4*c**2 -
192*a**5*b**6*c + 12*a**4*b**8 + x**8*(3072*a**4*c**8 - 3072*a**3*b**2*c**7 + 1152*a**2*b**4*c**6 - 192*a*b**
6*c**5 + 12*b**8*c**4) + x**7*(12288*a**4*b*c**7 - 12288*a**3*b**3*c**6 + 4608*a**2*b**5*c**5 - 768*a*b**7*c**
4 + 48*b**9*c**3) + x**6*(12288*a**5*c**7 + 6144*a**4*b**2*c**6 - 13824*a**3*b**4*c**5 + 6144*a**2*b**6*c**4 -
1104*a*b**8*c**3 + 72*b**10*c**2) + x**5*(36864*a**5*b*c**6 - 24576*a**4*b**3*c**5 + 1536*a**3*b**5*c**4 + 23
04*a**2*b**7*c**3 - 624*a*b**9*c**2 + 48*b**11*c) + x**4*(18432*a**6*c**6 + 18432*a**5*b**2*c**5 - 26880*a**4*
b**4*c**4 + 9600*a**3*b**6*c**3 - 1080*a**2*b**8*c**2 - 48*a*b**10*c + 12*b**12) + x**3*(36864*a**6*b*c**5 - 2
4576*a**5*b**3*c**4 + 1536*a**4*b**5*c**3 + 2304*a**3*b**7*c**2 - 624*a**2*b**9*c + 48*a*b**11) + x**2*(12288*
a**7*c**5 + 6144*a**6*b**2*c**4 - 13824*a**5*b**4*c**3 + 6144*a**4*b**6*c**2 - 1104*a**3*b**8*c + 72*a**2*b**1
0) + x*(12288*a**7*b*c**4 - 12288*a**6*b**3*c**3 + 4608*a**5*b**5*c**2 - 768*a**4*b**7*c + 48*a**3*b**9))

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^5} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*x+d)^2/(c*x^2+b*x+a)^5,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. 1013 vs. \(2 (318) = 636\).

Time = 0.28 (sec) , antiderivative size = 1013, normalized size of antiderivative = 3.07 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^5} \, dx=\frac {10 \, {\left (14 \, c^{4} d^{2} - 14 \, b c^{3} d e + 3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {840 \, c^{7} d^{2} x^{7} - 840 \, b c^{6} d e x^{7} + 180 \, b^{2} c^{5} e^{2} x^{7} + 120 \, a c^{6} e^{2} x^{7} + 2940 \, b c^{6} d^{2} x^{6} - 2940 \, b^{2} c^{5} d e x^{6} + 630 \, b^{3} c^{4} e^{2} x^{6} + 420 \, a b c^{5} e^{2} x^{6} + 3640 \, b^{2} c^{5} d^{2} x^{5} + 3080 \, a c^{6} d^{2} x^{5} - 3640 \, b^{3} c^{4} d e x^{5} - 3080 \, a b c^{5} d e x^{5} + 780 \, b^{4} c^{3} e^{2} x^{5} + 1180 \, a b^{2} c^{4} e^{2} x^{5} + 440 \, a^{2} c^{5} e^{2} x^{5} + 1750 \, b^{3} c^{4} d^{2} x^{4} + 7700 \, a b c^{5} d^{2} x^{4} - 1750 \, b^{4} c^{3} d e x^{4} - 7700 \, a b^{2} c^{4} d e x^{4} + 375 \, b^{5} c^{2} e^{2} x^{4} + 1900 \, a b^{3} c^{3} e^{2} x^{4} + 1100 \, a^{2} b c^{4} e^{2} x^{4} + 168 \, b^{4} c^{3} d^{2} x^{3} + 5656 \, a b^{2} c^{4} d^{2} x^{3} + 4088 \, a^{2} c^{5} d^{2} x^{3} - 168 \, b^{5} c^{2} d e x^{3} - 5656 \, a b^{3} c^{3} d e x^{3} - 4088 \, a^{2} b c^{4} d e x^{3} + 36 \, b^{6} c e^{2} x^{3} + 1236 \, a b^{4} c^{2} e^{2} x^{3} + 1684 \, a^{2} b^{2} c^{3} e^{2} x^{3} + 584 \, a^{3} c^{4} e^{2} x^{3} - 28 \, b^{5} c^{2} d^{2} x^{2} + 784 \, a b^{3} c^{3} d^{2} x^{2} + 6132 \, a^{2} b c^{4} d^{2} x^{2} + 28 \, b^{6} c d e x^{2} - 784 \, a b^{4} c^{2} d e x^{2} - 6132 \, a^{2} b^{2} c^{3} d e x^{2} - 6 \, b^{7} e^{2} x^{2} + 164 \, a b^{5} c e^{2} x^{2} + 1426 \, a^{2} b^{3} c^{2} e^{2} x^{2} + 876 \, a^{3} b c^{3} e^{2} x^{2} + 8 \, b^{6} c d^{2} x - 152 \, a b^{4} c^{2} d^{2} x + 1392 \, a^{2} b^{2} c^{3} d^{2} x + 2232 \, a^{3} c^{4} d^{2} x - 8 \, b^{7} d e x + 152 \, a b^{5} c d e x - 1392 \, a^{2} b^{3} c^{2} d e x - 2232 \, a^{3} b c^{3} d e x - 4 \, a b^{6} e^{2} x + 112 \, a^{2} b^{4} c e^{2} x + 1116 \, a^{3} b^{2} c^{2} e^{2} x - 120 \, a^{4} c^{3} e^{2} x - 3 \, b^{7} d^{2} + 50 \, a b^{5} c d^{2} - 326 \, a^{2} b^{3} c^{2} d^{2} + 1116 \, a^{3} b c^{3} d^{2} - 2 \, a b^{6} d e + 38 \, a^{2} b^{4} c d e - 348 \, a^{3} b^{2} c^{2} d e - 768 \, a^{4} c^{3} d e - a^{2} b^{5} e^{2} + 28 \, a^{3} b^{3} c e^{2} + 324 \, a^{4} b c^{2} e^{2}}{12 \, {\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} {\left (c x^{2} + b x + a\right )}^{4}} \]

[In]

integrate((e*x+d)^2/(c*x^2+b*x+a)^5,x, algorithm="giac")

[Out]

10*(14*c^4*d^2 - 14*b*c^3*d*e + 3*b^2*c^2*e^2 + 2*a*c^3*e^2)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^8 - 16
*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4)*sqrt(-b^2 + 4*a*c)) + 1/12*(840*c^7*d^2*x^7 - 840*b
*c^6*d*e*x^7 + 180*b^2*c^5*e^2*x^7 + 120*a*c^6*e^2*x^7 + 2940*b*c^6*d^2*x^6 - 2940*b^2*c^5*d*e*x^6 + 630*b^3*c
^4*e^2*x^6 + 420*a*b*c^5*e^2*x^6 + 3640*b^2*c^5*d^2*x^5 + 3080*a*c^6*d^2*x^5 - 3640*b^3*c^4*d*e*x^5 - 3080*a*b
*c^5*d*e*x^5 + 780*b^4*c^3*e^2*x^5 + 1180*a*b^2*c^4*e^2*x^5 + 440*a^2*c^5*e^2*x^5 + 1750*b^3*c^4*d^2*x^4 + 770
0*a*b*c^5*d^2*x^4 - 1750*b^4*c^3*d*e*x^4 - 7700*a*b^2*c^4*d*e*x^4 + 375*b^5*c^2*e^2*x^4 + 1900*a*b^3*c^3*e^2*x
^4 + 1100*a^2*b*c^4*e^2*x^4 + 168*b^4*c^3*d^2*x^3 + 5656*a*b^2*c^4*d^2*x^3 + 4088*a^2*c^5*d^2*x^3 - 168*b^5*c^
2*d*e*x^3 - 5656*a*b^3*c^3*d*e*x^3 - 4088*a^2*b*c^4*d*e*x^3 + 36*b^6*c*e^2*x^3 + 1236*a*b^4*c^2*e^2*x^3 + 1684
*a^2*b^2*c^3*e^2*x^3 + 584*a^3*c^4*e^2*x^3 - 28*b^5*c^2*d^2*x^2 + 784*a*b^3*c^3*d^2*x^2 + 6132*a^2*b*c^4*d^2*x
^2 + 28*b^6*c*d*e*x^2 - 784*a*b^4*c^2*d*e*x^2 - 6132*a^2*b^2*c^3*d*e*x^2 - 6*b^7*e^2*x^2 + 164*a*b^5*c*e^2*x^2
+ 1426*a^2*b^3*c^2*e^2*x^2 + 876*a^3*b*c^3*e^2*x^2 + 8*b^6*c*d^2*x - 152*a*b^4*c^2*d^2*x + 1392*a^2*b^2*c^3*d
^2*x + 2232*a^3*c^4*d^2*x - 8*b^7*d*e*x + 152*a*b^5*c*d*e*x - 1392*a^2*b^3*c^2*d*e*x - 2232*a^3*b*c^3*d*e*x -
4*a*b^6*e^2*x + 112*a^2*b^4*c*e^2*x + 1116*a^3*b^2*c^2*e^2*x - 120*a^4*c^3*e^2*x - 3*b^7*d^2 + 50*a*b^5*c*d^2
- 326*a^2*b^3*c^2*d^2 + 1116*a^3*b*c^3*d^2 - 2*a*b^6*d*e + 38*a^2*b^4*c*d*e - 348*a^3*b^2*c^2*d*e - 768*a^4*c^
3*d*e - a^2*b^5*e^2 + 28*a^3*b^3*c*e^2 + 324*a^4*b*c^2*e^2)/((b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*
c^3 + 256*a^4*c^4)*(c*x^2 + b*x + a)^4)

Mupad [B] (verification not implemented)

Time = 11.05 (sec) , antiderivative size = 1348, normalized size of antiderivative = 4.08 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^5} \, dx=\frac {\frac {x^2\,\left (219\,a^2\,b\,c^2+28\,a\,b^3\,c-b^5\right )\,\left (3\,b^2\,e^2-14\,b\,c\,d\,e+14\,c^2\,d^2+2\,a\,c\,e^2\right )}{6\,\left (256\,a^4\,c^4-256\,a^3\,b^2\,c^3+96\,a^2\,b^4\,c^2-16\,a\,b^6\,c+b^8\right )}-\frac {x\,\left (30\,a^4\,c^3\,e^2-279\,a^3\,b^2\,c^2\,e^2+558\,a^3\,b\,c^3\,d\,e-558\,a^3\,c^4\,d^2-28\,a^2\,b^4\,c\,e^2+348\,a^2\,b^3\,c^2\,d\,e-348\,a^2\,b^2\,c^3\,d^2+a\,b^6\,e^2-38\,a\,b^5\,c\,d\,e+38\,a\,b^4\,c^2\,d^2+2\,b^7\,d\,e-2\,b^6\,c\,d^2\right )}{3\,\left (256\,a^4\,c^4-256\,a^3\,b^2\,c^3+96\,a^2\,b^4\,c^2-16\,a\,b^6\,c+b^8\right )}-\frac {-324\,a^4\,b\,c^2\,e^2+768\,a^4\,c^3\,d\,e-28\,a^3\,b^3\,c\,e^2+348\,a^3\,b^2\,c^2\,d\,e-1116\,a^3\,b\,c^3\,d^2+a^2\,b^5\,e^2-38\,a^2\,b^4\,c\,d\,e+326\,a^2\,b^3\,c^2\,d^2+2\,a\,b^6\,d\,e-50\,a\,b^5\,c\,d^2+3\,b^7\,d^2}{12\,\left (256\,a^4\,c^4-256\,a^3\,b^2\,c^3+96\,a^2\,b^4\,c^2-16\,a\,b^6\,c+b^8\right )}+\frac {5\,c^5\,x^7\,\left (3\,b^2\,e^2-14\,b\,c\,d\,e+14\,c^2\,d^2+2\,a\,c\,e^2\right )}{256\,a^4\,c^4-256\,a^3\,b^2\,c^3+96\,a^2\,b^4\,c^2-16\,a\,b^6\,c+b^8}+\frac {x^3\,\left (73\,a^2\,c^3+101\,a\,b^2\,c^2+3\,b^4\,c\right )\,\left (3\,b^2\,e^2-14\,b\,c\,d\,e+14\,c^2\,d^2+2\,a\,c\,e^2\right )}{3\,\left (256\,a^4\,c^4-256\,a^3\,b^2\,c^3+96\,a^2\,b^4\,c^2-16\,a\,b^6\,c+b^8\right )}+\frac {25\,x^4\,\left (5\,b^3\,c^2+22\,a\,b\,c^3\right )\,\left (3\,b^2\,e^2-14\,b\,c\,d\,e+14\,c^2\,d^2+2\,a\,c\,e^2\right )}{12\,\left (256\,a^4\,c^4-256\,a^3\,b^2\,c^3+96\,a^2\,b^4\,c^2-16\,a\,b^6\,c+b^8\right )}+\frac {35\,b\,c^4\,x^6\,\left (3\,b^2\,e^2-14\,b\,c\,d\,e+14\,c^2\,d^2+2\,a\,c\,e^2\right )}{2\,\left (256\,a^4\,c^4-256\,a^3\,b^2\,c^3+96\,a^2\,b^4\,c^2-16\,a\,b^6\,c+b^8\right )}+\frac {5\,c\,x^5\,\left (13\,b^2\,c^2+11\,a\,c^3\right )\,\left (3\,b^2\,e^2-14\,b\,c\,d\,e+14\,c^2\,d^2+2\,a\,c\,e^2\right )}{3\,\left (256\,a^4\,c^4-256\,a^3\,b^2\,c^3+96\,a^2\,b^4\,c^2-16\,a\,b^6\,c+b^8\right )}}{x^4\,\left (6\,a^2\,c^2+12\,a\,b^2\,c+b^4\right )+a^4+c^4\,x^8+x^2\,\left (4\,c\,a^3+6\,a^2\,b^2\right )+x^6\,\left (6\,b^2\,c^2+4\,a\,c^3\right )+x^3\,\left (12\,c\,a^2\,b+4\,a\,b^3\right )+x^5\,\left (4\,b^3\,c+12\,a\,b\,c^2\right )+4\,b\,c^3\,x^7+4\,a^3\,b\,x}+\frac {10\,c^2\,\mathrm {atan}\left (\frac {\left (\frac {10\,c^3\,x\,\left (3\,b^2\,e^2-14\,b\,c\,d\,e+14\,c^2\,d^2+2\,a\,c\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{9/2}}+\frac {5\,c^2\,\left (3\,b^2\,e^2-14\,b\,c\,d\,e+14\,c^2\,d^2+2\,a\,c\,e^2\right )\,\left (256\,a^4\,b\,c^4-256\,a^3\,b^3\,c^3+96\,a^2\,b^5\,c^2-16\,a\,b^7\,c+b^9\right )}{{\left (4\,a\,c-b^2\right )}^{9/2}\,\left (256\,a^4\,c^4-256\,a^3\,b^2\,c^3+96\,a^2\,b^4\,c^2-16\,a\,b^6\,c+b^8\right )}\right )\,\left (256\,a^4\,c^4-256\,a^3\,b^2\,c^3+96\,a^2\,b^4\,c^2-16\,a\,b^6\,c+b^8\right )}{15\,b^2\,c^2\,e^2-70\,b\,c^3\,d\,e+70\,c^4\,d^2+10\,a\,c^3\,e^2}\right )\,\left (3\,b^2\,e^2-14\,b\,c\,d\,e+14\,c^2\,d^2+2\,a\,c\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{9/2}} \]

[In]

int((d + e*x)^2/(a + b*x + c*x^2)^5,x)

[Out]

((x^2*(219*a^2*b*c^2 - b^5 + 28*a*b^3*c)*(3*b^2*e^2 + 14*c^2*d^2 + 2*a*c*e^2 - 14*b*c*d*e))/(6*(b^8 + 256*a^4*
c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)) - (x*(a*b^6*e^2 - 2*b^6*c*d^2 - 558*a^3*c^4*d^2 + 30*a^4
*c^3*e^2 + 2*b^7*d*e + 38*a*b^4*c^2*d^2 - 28*a^2*b^4*c*e^2 - 348*a^2*b^2*c^3*d^2 - 279*a^3*b^2*c^2*e^2 + 558*a
^3*b*c^3*d*e + 348*a^2*b^3*c^2*d*e - 38*a*b^5*c*d*e))/(3*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3
- 16*a*b^6*c)) - (3*b^7*d^2 + a^2*b^5*e^2 - 1116*a^3*b*c^3*d^2 - 28*a^3*b^3*c*e^2 - 324*a^4*b*c^2*e^2 + 2*a*b
^6*d*e + 326*a^2*b^3*c^2*d^2 - 50*a*b^5*c*d^2 + 768*a^4*c^3*d*e - 38*a^2*b^4*c*d*e + 348*a^3*b^2*c^2*d*e)/(12*
(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)) + (5*c^5*x^7*(3*b^2*e^2 + 14*c^2*d^2 + 2*
a*c*e^2 - 14*b*c*d*e))/(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c) + (x^3*(3*b^4*c + 7
3*a^2*c^3 + 101*a*b^2*c^2)*(3*b^2*e^2 + 14*c^2*d^2 + 2*a*c*e^2 - 14*b*c*d*e))/(3*(b^8 + 256*a^4*c^4 + 96*a^2*b
^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)) + (25*x^4*(5*b^3*c^2 + 22*a*b*c^3)*(3*b^2*e^2 + 14*c^2*d^2 + 2*a*c*e^2
- 14*b*c*d*e))/(12*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)) + (35*b*c^4*x^6*(3*b^
2*e^2 + 14*c^2*d^2 + 2*a*c*e^2 - 14*b*c*d*e))/(2*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*
b^6*c)) + (5*c*x^5*(11*a*c^3 + 13*b^2*c^2)*(3*b^2*e^2 + 14*c^2*d^2 + 2*a*c*e^2 - 14*b*c*d*e))/(3*(b^8 + 256*a^
4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)))/(x^4*(b^4 + 6*a^2*c^2 + 12*a*b^2*c) + a^4 + c^4*x^8 +
x^2*(4*a^3*c + 6*a^2*b^2) + x^6*(4*a*c^3 + 6*b^2*c^2) + x^3*(4*a*b^3 + 12*a^2*b*c) + x^5*(4*b^3*c + 12*a*b*c^
2) + 4*b*c^3*x^7 + 4*a^3*b*x) + (10*c^2*atan((((10*c^3*x*(3*b^2*e^2 + 14*c^2*d^2 + 2*a*c*e^2 - 14*b*c*d*e))/(4
*a*c - b^2)^(9/2) + (5*c^2*(3*b^2*e^2 + 14*c^2*d^2 + 2*a*c*e^2 - 14*b*c*d*e)*(b^9 + 256*a^4*b*c^4 + 96*a^2*b^5
*c^2 - 256*a^3*b^3*c^3 - 16*a*b^7*c))/((4*a*c - b^2)^(9/2)*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c
^3 - 16*a*b^6*c)))*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c))/(70*c^4*d^2 + 10*a*c^3
*e^2 + 15*b^2*c^2*e^2 - 70*b*c^3*d*e))*(3*b^2*e^2 + 14*c^2*d^2 + 2*a*c*e^2 - 14*b*c*d*e))/(4*a*c - b^2)^(9/2)