\(\int \frac {d+e x}{(a+b x+c x^2)^4} \, dx\) [2217]

Optimal result

Integrand size = 18, antiderivative size = 173 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^4} \, dx=-\frac {b d-2 a e+(2 c d-b e) x}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {5 (2 c d-b e) (b+2 c x)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {5 c (2 c d-b e) (b+2 c x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {20 c^2 (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {652, 628, 632, 212} \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^4} \, dx=\frac {20 c^2 (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}-\frac {5 c (b+2 c x) (2 c d-b e)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {5 (b+2 c x) (2 c d-b e)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {-2 a e+x (2 c d-b e)+b d}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3} \]

[In]

Int[(d + e*x)/(a + b*x + c*x^2)^4,x]

[Out]

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

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]

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.97 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^4} \, dx=\frac {\frac {2 \left (b^2-4 a c\right )^2 (-b d+2 a e-2 c d x+b e x)}{(a+x (b+c x))^3}-\frac {5 \left (b^2-4 a c\right ) (-2 c d+b e) (b+2 c x)}{(a+x (b+c x))^2}+\frac {30 c (-2 c d+b e) (b+2 c x)}{a+x (b+c x)}+\frac {120 c^2 (-2 c d+b e) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}}{6 \left (b^2-4 a c\right )^3} \]

[In]

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

[Out]

((2*(b^2 - 4*a*c)^2*(-(b*d) + 2*a*e - 2*c*d*x + b*e*x))/(a + x*(b + c*x))^3 - (5*(b^2 - 4*a*c)*(-2*c*d + b*e)*
(b + 2*c*x))/(a + x*(b + c*x))^2 + (30*c*(-2*c*d + b*e)*(b + 2*c*x))/(a + x*(b + c*x)) + (120*c^2*(-2*c*d + b*
e)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(6*(b^2 - 4*a*c)^3)

Maple [A] (verified)

Time = 17.15 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.07

method	result	size
default	\(\frac {b d -2 a e +\left (-b e +2 c d \right ) x}{3 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{3}}+\frac {5 \left (-b e +2 c d \right ) \left (\frac {2 c x +b}{2 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{2}}+\frac {3 c \left (\frac {2 c x +b}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )}+\frac {4 c \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}\right )}{4 a c -b^{2}}\right )}{3 \left (4 a c -b^{2}\right )}\)	\(185\)
risch	\(\frac {-\frac {10 c^{4} \left (b e -2 c d \right ) x^{5}}{64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {25 c^{3} \left (b e -2 c d \right ) b \,x^{4}}{64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {5 \left (16 a c +11 b^{2}\right ) c^{2} \left (b e -2 c d \right ) x^{3}}{3 \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}-\frac {5 b \left (16 a c +b^{2}\right ) c \left (b e -2 c d \right ) x^{2}}{2 \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}-\frac {\left (44 a^{2} b \,c^{2} e -88 a^{2} c^{3} d +18 a \,b^{3} c e -36 a \,b^{2} c^{2} d -b^{5} e +2 b^{4} c d \right ) x}{2 \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}-\frac {64 a^{3} c^{2} e +18 a^{2} b^{2} c e -132 a^{2} b \,c^{2} d -e a \,b^{4}+26 a \,b^{3} c d -2 b^{5} d}{6 \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}}{\left (c \,x^{2}+b x +a \right )^{3}}+\frac {10 c^{2} \ln \left (\left (-128 a^{3} c^{4}+96 a^{2} b^{2} c^{3}-24 c^{2} a \,b^{4}+2 b^{6} c \right ) x -\left (-4 a c +b^{2}\right )^{\frac {7}{2}}-64 a^{3} c^{3} b +48 a^{2} c^{2} b^{3}-12 a \,b^{5} c +b^{7}\right ) b e}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}-\frac {20 c^{3} \ln \left (\left (-128 a^{3} c^{4}+96 a^{2} b^{2} c^{3}-24 c^{2} a \,b^{4}+2 b^{6} c \right ) x -\left (-4 a c +b^{2}\right )^{\frac {7}{2}}-64 a^{3} c^{3} b +48 a^{2} c^{2} b^{3}-12 a \,b^{5} c +b^{7}\right ) d}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}-\frac {10 c^{2} \ln \left (\left (128 a^{3} c^{4}-96 a^{2} b^{2} c^{3}+24 c^{2} a \,b^{4}-2 b^{6} c \right ) x -\left (-4 a c +b^{2}\right )^{\frac {7}{2}}+64 a^{3} c^{3} b -48 a^{2} c^{2} b^{3}+12 a \,b^{5} c -b^{7}\right ) b e}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}+\frac {20 c^{3} \ln \left (\left (128 a^{3} c^{4}-96 a^{2} b^{2} c^{3}+24 c^{2} a \,b^{4}-2 b^{6} c \right ) x -\left (-4 a c +b^{2}\right )^{\frac {7}{2}}+64 a^{3} c^{3} b -48 a^{2} c^{2} b^{3}+12 a \,b^{5} c -b^{7}\right ) d}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}\)	\(803\)

[In]

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

[Out]

1/3*(b*d-2*a*e+(-b*e+2*c*d)*x)/(4*a*c-b^2)/(c*x^2+b*x+a)^3+5/3*(-b*e+2*c*d)/(4*a*c-b^2)*(1/2*(2*c*x+b)/(4*a*c-
b^2)/(c*x^2+b*x+a)^2+3*c/(4*a*c-b^2)*((2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)+4*c/(4*a*c-b^2)^(3/2)*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. 970 vs. \(2 (165) = 330\).

Time = 0.34 (sec) , antiderivative size = 1960, normalized size of antiderivative = 11.33 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[-1/6*(60*(2*(b^2*c^5 - 4*a*c^6)*d - (b^3*c^4 - 4*a*b*c^5)*e)*x^5 + 150*(2*(b^3*c^4 - 4*a*b*c^5)*d - (b^4*c^3
- 4*a*b^2*c^4)*e)*x^4 + 10*(2*(11*b^4*c^3 - 28*a*b^2*c^4 - 64*a^2*c^5)*d - (11*b^5*c^2 - 28*a*b^3*c^3 - 64*a^2
*b*c^4)*e)*x^3 + 15*(2*(b^5*c^2 + 12*a*b^3*c^3 - 64*a^2*b*c^4)*d - (b^6*c + 12*a*b^4*c^2 - 64*a^2*b^2*c^3)*e)*
x^2 - 60*(2*a^3*c^3*d - a^3*b*c^2*e + (2*c^6*d - b*c^5*e)*x^6 + 3*(2*b*c^5*d - b^2*c^4*e)*x^5 + 3*(2*(b^2*c^4
+ a*c^5)*d - (b^3*c^3 + a*b*c^4)*e)*x^4 + (2*(b^3*c^3 + 6*a*b*c^4)*d - (b^4*c^2 + 6*a*b^2*c^3)*e)*x^3 + 3*(2*(
a*b^2*c^3 + a^2*c^4)*d - (a*b^3*c^2 + a^2*b*c^3)*e)*x^2 + 3*(2*a^2*b*c^3*d - a^2*b^2*c^2*e)*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*(b^7 - 17*a*
b^5*c + 118*a^2*b^3*c^2 - 264*a^3*b*c^3)*d + (a*b^6 - 22*a^2*b^4*c + 8*a^3*b^2*c^2 + 256*a^4*c^3)*e - 3*(2*(b^
6*c - 22*a*b^4*c^2 + 28*a^2*b^2*c^3 + 176*a^3*c^4)*d - (b^7 - 22*a*b^5*c + 28*a^2*b^3*c^2 + 176*a^3*b*c^3)*e)*
x)/(a^3*b^8 - 16*a^4*b^6*c + 96*a^5*b^4*c^2 - 256*a^6*b^2*c^3 + 256*a^7*c^4 + (b^8*c^3 - 16*a*b^6*c^4 + 96*a^2
*b^4*c^5 - 256*a^3*b^2*c^6 + 256*a^4*c^7)*x^6 + 3*(b^9*c^2 - 16*a*b^7*c^3 + 96*a^2*b^5*c^4 - 256*a^3*b^3*c^5 +
256*a^4*b*c^6)*x^5 + 3*(b^10*c - 15*a*b^8*c^2 + 80*a^2*b^6*c^3 - 160*a^3*b^4*c^4 + 256*a^5*c^6)*x^4 + (b^11 -
10*a*b^9*c + 320*a^3*b^5*c^3 - 1280*a^4*b^3*c^4 + 1536*a^5*b*c^5)*x^3 + 3*(a*b^10 - 15*a^2*b^8*c + 80*a^3*b^6
*c^2 - 160*a^4*b^4*c^3 + 256*a^6*c^5)*x^2 + 3*(a^2*b^9 - 16*a^3*b^7*c + 96*a^4*b^5*c^2 - 256*a^5*b^3*c^3 + 256
*a^6*b*c^4)*x), -1/6*(60*(2*(b^2*c^5 - 4*a*c^6)*d - (b^3*c^4 - 4*a*b*c^5)*e)*x^5 + 150*(2*(b^3*c^4 - 4*a*b*c^5
)*d - (b^4*c^3 - 4*a*b^2*c^4)*e)*x^4 + 10*(2*(11*b^4*c^3 - 28*a*b^2*c^4 - 64*a^2*c^5)*d - (11*b^5*c^2 - 28*a*b
^3*c^3 - 64*a^2*b*c^4)*e)*x^3 + 15*(2*(b^5*c^2 + 12*a*b^3*c^3 - 64*a^2*b*c^4)*d - (b^6*c + 12*a*b^4*c^2 - 64*a
^2*b^2*c^3)*e)*x^2 - 120*(2*a^3*c^3*d - a^3*b*c^2*e + (2*c^6*d - b*c^5*e)*x^6 + 3*(2*b*c^5*d - b^2*c^4*e)*x^5
+ 3*(2*(b^2*c^4 + a*c^5)*d - (b^3*c^3 + a*b*c^4)*e)*x^4 + (2*(b^3*c^3 + 6*a*b*c^4)*d - (b^4*c^2 + 6*a*b^2*c^3)
*e)*x^3 + 3*(2*(a*b^2*c^3 + a^2*c^4)*d - (a*b^3*c^2 + a^2*b*c^3)*e)*x^2 + 3*(2*a^2*b*c^3*d - a^2*b^2*c^2*e)*x)
*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 2*(b^7 - 17*a*b^5*c + 118*a^2*b^3*
c^2 - 264*a^3*b*c^3)*d + (a*b^6 - 22*a^2*b^4*c + 8*a^3*b^2*c^2 + 256*a^4*c^3)*e - 3*(2*(b^6*c - 22*a*b^4*c^2 +
28*a^2*b^2*c^3 + 176*a^3*c^4)*d - (b^7 - 22*a*b^5*c + 28*a^2*b^3*c^2 + 176*a^3*b*c^3)*e)*x)/(a^3*b^8 - 16*a^4
*b^6*c + 96*a^5*b^4*c^2 - 256*a^6*b^2*c^3 + 256*a^7*c^4 + (b^8*c^3 - 16*a*b^6*c^4 + 96*a^2*b^4*c^5 - 256*a^3*b
^2*c^6 + 256*a^4*c^7)*x^6 + 3*(b^9*c^2 - 16*a*b^7*c^3 + 96*a^2*b^5*c^4 - 256*a^3*b^3*c^5 + 256*a^4*b*c^6)*x^5
+ 3*(b^10*c - 15*a*b^8*c^2 + 80*a^2*b^6*c^3 - 160*a^3*b^4*c^4 + 256*a^5*c^6)*x^4 + (b^11 - 10*a*b^9*c + 320*a^
3*b^5*c^3 - 1280*a^4*b^3*c^4 + 1536*a^5*b*c^5)*x^3 + 3*(a*b^10 - 15*a^2*b^8*c + 80*a^3*b^6*c^2 - 160*a^4*b^4*c
^3 + 256*a^6*c^5)*x^2 + 3*(a^2*b^9 - 16*a^3*b^7*c + 96*a^4*b^5*c^2 - 256*a^5*b^3*c^3 + 256*a^6*b*c^4)*x)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1062 vs. \(2 (170) = 340\).

Time = 1.63 (sec) , antiderivative size = 1062, normalized size of antiderivative = 6.14 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^4} \, dx=10 c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) \log {\left (x + \frac {- 2560 a^{4} c^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) + 2560 a^{3} b^{2} c^{5} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) - 960 a^{2} b^{4} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) + 160 a b^{6} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) - 10 b^{8} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) + 10 b^{2} c^{2} e - 20 b c^{3} d}{20 b c^{3} e - 40 c^{4} d} \right )} - 10 c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) \log {\left (x + \frac {2560 a^{4} c^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) - 2560 a^{3} b^{2} c^{5} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) + 960 a^{2} b^{4} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) - 160 a b^{6} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) + 10 b^{8} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) + 10 b^{2} c^{2} e - 20 b c^{3} d}{20 b c^{3} e - 40 c^{4} d} \right )} + \frac {- 64 a^{3} c^{2} e - 18 a^{2} b^{2} c e + 132 a^{2} b c^{2} d + a b^{4} e - 26 a b^{3} c d + 2 b^{5} d + x^{5} \left (- 60 b c^{4} e + 120 c^{5} d\right ) + x^{4} \left (- 150 b^{2} c^{3} e + 300 b c^{4} d\right ) + x^{3} \left (- 160 a b c^{3} e + 320 a c^{4} d - 110 b^{3} c^{2} e + 220 b^{2} c^{3} d\right ) + x^{2} \left (- 240 a b^{2} c^{2} e + 480 a b c^{3} d - 15 b^{4} c e + 30 b^{3} c^{2} d\right ) + x \left (- 132 a^{2} b c^{2} e + 264 a^{2} c^{3} d - 54 a b^{3} c e + 108 a b^{2} c^{2} d + 3 b^{5} e - 6 b^{4} c d\right )}{384 a^{6} c^{3} - 288 a^{5} b^{2} c^{2} + 72 a^{4} b^{4} c - 6 a^{3} b^{6} + x^{6} \cdot \left (384 a^{3} c^{6} - 288 a^{2} b^{2} c^{5} + 72 a b^{4} c^{4} - 6 b^{6} c^{3}\right ) + x^{5} \cdot \left (1152 a^{3} b c^{5} - 864 a^{2} b^{3} c^{4} + 216 a b^{5} c^{3} - 18 b^{7} c^{2}\right ) + x^{4} \cdot \left (1152 a^{4} c^{5} + 288 a^{3} b^{2} c^{4} - 648 a^{2} b^{4} c^{3} + 198 a b^{6} c^{2} - 18 b^{8} c\right ) + x^{3} \cdot \left (2304 a^{4} b c^{4} - 1344 a^{3} b^{3} c^{3} + 144 a^{2} b^{5} c^{2} + 36 a b^{7} c - 6 b^{9}\right ) + x^{2} \cdot \left (1152 a^{5} c^{4} + 288 a^{4} b^{2} c^{3} - 648 a^{3} b^{4} c^{2} + 198 a^{2} b^{6} c - 18 a b^{8}\right ) + x \left (1152 a^{5} b c^{3} - 864 a^{4} b^{3} c^{2} + 216 a^{3} b^{5} c - 18 a^{2} b^{7}\right )} \]

[In]

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

[Out]

10*c**2*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*log(x + (-2560*a**4*c**6*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*
c*d) + 2560*a**3*b**2*c**5*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d) - 960*a**2*b**4*c**4*sqrt(-1/(4*a*c - b**2
)**7)*(b*e - 2*c*d) + 160*a*b**6*c**3*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d) - 10*b**8*c**2*sqrt(-1/(4*a*c -
b**2)**7)*(b*e - 2*c*d) + 10*b**2*c**2*e - 20*b*c**3*d)/(20*b*c**3*e - 40*c**4*d)) - 10*c**2*sqrt(-1/(4*a*c -
b**2)**7)*(b*e - 2*c*d)*log(x + (2560*a**4*c**6*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d) - 2560*a**3*b**2*c**
5*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d) + 960*a**2*b**4*c**4*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d) - 160
*a*b**6*c**3*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d) + 10*b**8*c**2*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)
+ 10*b**2*c**2*e - 20*b*c**3*d)/(20*b*c**3*e - 40*c**4*d)) + (-64*a**3*c**2*e - 18*a**2*b**2*c*e + 132*a**2*b*
c**2*d + a*b**4*e - 26*a*b**3*c*d + 2*b**5*d + x**5*(-60*b*c**4*e + 120*c**5*d) + x**4*(-150*b**2*c**3*e + 300
*b*c**4*d) + x**3*(-160*a*b*c**3*e + 320*a*c**4*d - 110*b**3*c**2*e + 220*b**2*c**3*d) + x**2*(-240*a*b**2*c**
2*e + 480*a*b*c**3*d - 15*b**4*c*e + 30*b**3*c**2*d) + x*(-132*a**2*b*c**2*e + 264*a**2*c**3*d - 54*a*b**3*c*e
+ 108*a*b**2*c**2*d + 3*b**5*e - 6*b**4*c*d))/(384*a**6*c**3 - 288*a**5*b**2*c**2 + 72*a**4*b**4*c - 6*a**3*b
**6 + x**6*(384*a**3*c**6 - 288*a**2*b**2*c**5 + 72*a*b**4*c**4 - 6*b**6*c**3) + x**5*(1152*a**3*b*c**5 - 864*
a**2*b**3*c**4 + 216*a*b**5*c**3 - 18*b**7*c**2) + x**4*(1152*a**4*c**5 + 288*a**3*b**2*c**4 - 648*a**2*b**4*c
**3 + 198*a*b**6*c**2 - 18*b**8*c) + x**3*(2304*a**4*b*c**4 - 1344*a**3*b**3*c**3 + 144*a**2*b**5*c**2 + 36*a*
b**7*c - 6*b**9) + x**2*(1152*a**5*c**4 + 288*a**4*b**2*c**3 - 648*a**3*b**4*c**2 + 198*a**2*b**6*c - 18*a*b**
8) + x*(1152*a**5*b*c**3 - 864*a**4*b**3*c**2 + 216*a**3*b**5*c - 18*a**2*b**7))

Maxima [F(-2)]

Exception generated. \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^4} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*x+d)/(c*x^2+b*x+a)^4,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. 365 vs. \(2 (165) = 330\).

Time = 0.30 (sec) , antiderivative size = 365, normalized size of antiderivative = 2.11 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^4} \, dx=-\frac {20 \, {\left (2 \, c^{3} d - b c^{2} e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {120 \, c^{5} d x^{5} - 60 \, b c^{4} e x^{5} + 300 \, b c^{4} d x^{4} - 150 \, b^{2} c^{3} e x^{4} + 220 \, b^{2} c^{3} d x^{3} + 320 \, a c^{4} d x^{3} - 110 \, b^{3} c^{2} e x^{3} - 160 \, a b c^{3} e x^{3} + 30 \, b^{3} c^{2} d x^{2} + 480 \, a b c^{3} d x^{2} - 15 \, b^{4} c e x^{2} - 240 \, a b^{2} c^{2} e x^{2} - 6 \, b^{4} c d x + 108 \, a b^{2} c^{2} d x + 264 \, a^{2} c^{3} d x + 3 \, b^{5} e x - 54 \, a b^{3} c e x - 132 \, a^{2} b c^{2} e x + 2 \, b^{5} d - 26 \, a b^{3} c d + 132 \, a^{2} b c^{2} d + a b^{4} e - 18 \, a^{2} b^{2} c e - 64 \, a^{3} c^{2} e}{6 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} {\left (c x^{2} + b x + a\right )}^{3}} \]

[In]

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

[Out]

-20*(2*c^3*d - b*c^2*e)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^
3)*sqrt(-b^2 + 4*a*c)) - 1/6*(120*c^5*d*x^5 - 60*b*c^4*e*x^5 + 300*b*c^4*d*x^4 - 150*b^2*c^3*e*x^4 + 220*b^2*c
^3*d*x^3 + 320*a*c^4*d*x^3 - 110*b^3*c^2*e*x^3 - 160*a*b*c^3*e*x^3 + 30*b^3*c^2*d*x^2 + 480*a*b*c^3*d*x^2 - 15
*b^4*c*e*x^2 - 240*a*b^2*c^2*e*x^2 - 6*b^4*c*d*x + 108*a*b^2*c^2*d*x + 264*a^2*c^3*d*x + 3*b^5*e*x - 54*a*b^3*
c*e*x - 132*a^2*b*c^2*e*x + 2*b^5*d - 26*a*b^3*c*d + 132*a^2*b*c^2*d + a*b^4*e - 18*a^2*b^2*c*e - 64*a^3*c^2*e
)/((b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*(c*x^2 + b*x + a)^3)

Mupad [B] (verification not implemented)

Time = 10.11 (sec) , antiderivative size = 633, normalized size of antiderivative = 3.66 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^4} \, dx=\frac {\frac {10\,c^4\,x^5\,\left (b\,e-2\,c\,d\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-\frac {-64\,e\,a^3\,c^2-18\,e\,a^2\,b^2\,c+132\,d\,a^2\,b\,c^2+e\,a\,b^4-26\,d\,a\,b^3\,c+2\,d\,b^5}{6\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {x\,\left (b\,e-2\,c\,d\right )\,\left (44\,a^2\,c^2+18\,a\,b^2\,c-b^4\right )}{2\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {5\,c\,x^3\,\left (11\,b^2\,c+16\,a\,c^2\right )\,\left (b\,e-2\,c\,d\right )}{3\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {5\,c\,x^2\,\left (b^3+16\,a\,c\,b\right )\,\left (b\,e-2\,c\,d\right )}{2\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {25\,b\,c^3\,x^4\,\left (b\,e-2\,c\,d\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}}{x^2\,\left (3\,c\,a^2+3\,a\,b^2\right )+x^4\,\left (3\,b^2\,c+3\,a\,c^2\right )+a^3+x^3\,\left (b^3+6\,a\,c\,b\right )+c^3\,x^6+3\,b\,c^2\,x^5+3\,a^2\,b\,x}-\frac {20\,c^2\,\mathrm {atan}\left (\frac {\left (\frac {20\,c^3\,x\,\left (b\,e-2\,c\,d\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}}+\frac {10\,c^2\,\left (b\,e-2\,c\,d\right )\,\left (-64\,a^3\,b\,c^3+48\,a^2\,b^3\,c^2-12\,a\,b^5\,c+b^7\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}{20\,c^3\,d-10\,b\,c^2\,e}\right )\,\left (b\,e-2\,c\,d\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}} \]

[In]

int((d + e*x)/(a + b*x + c*x^2)^4,x)

[Out]

((10*c^4*x^5*(b*e - 2*c*d))/(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c) - (2*b^5*d - 64*a^3*c^2*e + a*b^4
*e - 26*a*b^3*c*d + 132*a^2*b*c^2*d - 18*a^2*b^2*c*e)/(6*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (
x*(b*e - 2*c*d)*(44*a^2*c^2 - b^4 + 18*a*b^2*c))/(2*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (5*c*x
^3*(16*a*c^2 + 11*b^2*c)*(b*e - 2*c*d))/(3*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (5*c*x^2*(b^3 +
16*a*b*c)*(b*e - 2*c*d))/(2*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (25*b*c^3*x^4*(b*e - 2*c*d))/
(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c))/(x^2*(3*a*b^2 + 3*a^2*c) + x^4*(3*a*c^2 + 3*b^2*c) + a^3 + x
^3*(b^3 + 6*a*b*c) + c^3*x^6 + 3*b*c^2*x^5 + 3*a^2*b*x) - (20*c^2*atan((((20*c^3*x*(b*e - 2*c*d))/(4*a*c - b^2
)^(7/2) + (10*c^2*(b*e - 2*c*d)*(b^7 - 64*a^3*b*c^3 + 48*a^2*b^3*c^2 - 12*a*b^5*c))/((4*a*c - b^2)^(7/2)*(b^6
- 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)))*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c))/(20*c^3*d - 10
*b*c^2*e))*(b*e - 2*c*d))/(4*a*c - b^2)^(7/2)