\(\int \frac {1}{(a+b x+c x^2)^5} \, dx\) [2226]

Optimal result

Integrand size = 12, antiderivative size = 171 \[ \int \frac {1}{\left (a+b x+c x^2\right )^5} \, dx=-\frac {b+2 c x}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {7 c (b+2 c x)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac {35 c^2 (b+2 c x)}{6 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac {35 c^3 (b+2 c x)}{\left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac {140 c^4 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}} \]

[Out]

1/4*(-2*c*x-b)/(-4*a*c+b^2)/(c*x^2+b*x+a)^4+7/6*c*(2*c*x+b)/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^3-35/6*c^2*(2*c*x+b)/
(-4*a*c+b^2)^3/(c*x^2+b*x+a)^2+35*c^3*(2*c*x+b)/(-4*a*c+b^2)^4/(c*x^2+b*x+a)-140*c^4*arctanh((2*c*x+b)/(-4*a*c
+b^2)^(1/2))/(-4*a*c+b^2)^(9/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {628, 632, 212} \[ \int \frac {1}{\left (a+b x+c x^2\right )^5} \, dx=-\frac {140 c^4 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}}+\frac {35 c^3 (b+2 c x)}{\left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac {35 c^2 (b+2 c x)}{6 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac {7 c (b+2 c x)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac {b+2 c x}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4} \]

[In]

Int[(a + b*x + c*x^2)^(-5),x]

[Out]

-1/4*(b + 2*c*x)/((b^2 - 4*a*c)*(a + b*x + c*x^2)^4) + (7*c*(b + 2*c*x))/(6*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^
3) - (35*c^2*(b + 2*c*x))/(6*(b^2 - 4*a*c)^3*(a + b*x + c*x^2)^2) + (35*c^3*(b + 2*c*x))/((b^2 - 4*a*c)^4*(a +
b*x + c*x^2)) - (140*c^4*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]

Rubi steps \begin{align*} \text {integral}& = -\frac {b+2 c x}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}-\frac {(7 c) \int \frac {1}{\left (a+b x+c x^2\right )^4} \, dx}{2 \left (b^2-4 a c\right )} \\ & = -\frac {b+2 c x}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {7 c (b+2 c x)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}+\frac {\left (35 c^2\right ) \int \frac {1}{\left (a+b x+c x^2\right )^3} \, dx}{3 \left (b^2-4 a c\right )^2} \\ & = -\frac {b+2 c x}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {7 c (b+2 c x)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac {35 c^2 (b+2 c x)}{6 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}-\frac {\left (35 c^3\right ) \int \frac {1}{\left (a+b x+c x^2\right )^2} \, dx}{\left (b^2-4 a c\right )^3} \\ & = -\frac {b+2 c x}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {7 c (b+2 c x)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac {35 c^2 (b+2 c x)}{6 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac {35 c^3 (b+2 c x)}{\left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}+\frac {\left (70 c^4\right ) \int \frac {1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^4} \\ & = -\frac {b+2 c x}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {7 c (b+2 c x)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac {35 c^2 (b+2 c x)}{6 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac {35 c^3 (b+2 c x)}{\left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac {\left (140 c^4\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 {b+2 c x}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {7 c (b+2 c x)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac {35 c^2 (b+2 c x)}{6 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac {35 c^3 (b+2 c x)}{\left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac {140 c^4 \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 = 0.10 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\left (a+b x+c x^2\right )^5} \, dx=\frac {-\frac {3 \left (b^2-4 a c\right )^3 (b+2 c x)}{(a+x (b+c x))^4}+\frac {14 c \left (b^2-4 a c\right )^2 (b+2 c x)}{(a+x (b+c x))^3}-\frac {70 c^2 \left (b^2-4 a c\right ) (b+2 c x)}{(a+x (b+c x))^2}+\frac {420 c^3 (b+2 c x)}{a+x (b+c x)}+\frac {1680 c^4 \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}}{12 \left (b^2-4 a c\right )^4} \]

[In]

Integrate[(a + b*x + c*x^2)^(-5),x]

[Out]

((-3*(b^2 - 4*a*c)^3*(b + 2*c*x))/(a + x*(b + c*x))^4 + (14*c*(b^2 - 4*a*c)^2*(b + 2*c*x))/(a + x*(b + c*x))^3
- (70*c^2*(b^2 - 4*a*c)*(b + 2*c*x))/(a + x*(b + c*x))^2 + (420*c^3*(b + 2*c*x))/(a + x*(b + c*x)) + (1680*c^
4*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(12*(b^2 - 4*a*c)^4)

Maple [A] (verified)

Time = 17.32 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.24

method	result	size
default	\(\frac {2 c x +b}{4 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{4}}+\frac {7 c \left (\frac {2 c x +b}{3 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{3}}+\frac {10 c \left (\frac {2 c x +b}{2 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{2}}+\frac {3 c \left (\frac {2 c x +b}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )}+\frac {4 c \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}\right )}{4 a c -b^{2}}\right )}{3 \left (4 a c -b^{2}\right )}\right )}{2 \left (4 a c -b^{2}\right )}\)	\(212\)
risch	\(\frac {\frac {70 c^{7} x^{7}}{256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}}+\frac {245 c^{6} b \,x^{6}}{256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}}+\frac {70 c^{5} \left (11 a c +13 b^{2}\right ) x^{5}}{3 \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right )}+\frac {175 b \left (22 a c +5 b^{2}\right ) c^{4} x^{4}}{6 \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right )}+\frac {14 \left (73 a^{2} c^{2}+101 a \,b^{2} c +3 b^{4}\right ) c^{3} x^{3}}{3 \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right )}+\frac {7 b \,c^{2} \left (219 a^{2} c^{2}+28 a \,b^{2} c -b^{4}\right ) x^{2}}{3 \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right )}+\frac {2 c \left (279 c^{3} a^{3}+174 a^{2} b^{2} c^{2}-19 a \,b^{4} c +b^{6}\right ) x}{3 \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right )}+\frac {\left (1116 c^{3} a^{3}-326 a^{2} b^{2} c^{2}+50 a \,b^{4} c -3 b^{6}\right ) b}{3072 a^{4} c^{4}-3072 a^{3} b^{2} c^{3}+1152 a^{2} b^{4} c^{2}-192 a \,b^{6} c +12 b^{8}}}{\left (c \,x^{2}+b x +a \right )^{4}}-\frac {70 c^{4} \ln \left (\left (512 a^{4} c^{5}-512 a^{3} b^{2} c^{4}+192 a^{2} b^{4} c^{3}-32 c^{2} b^{6} a +2 b^{8} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {9}{2}}+256 a^{4} b \,c^{4}-256 a^{3} b^{3} c^{3}+96 a^{2} b^{5} c^{2}-16 a \,b^{7} c +b^{9}\right )}{\left (-4 a c +b^{2}\right )^{\frac {9}{2}}}+\frac {70 c^{4} \ln \left (\left (-512 a^{4} c^{5}+512 a^{3} b^{2} c^{4}-192 a^{2} b^{4} c^{3}+32 c^{2} b^{6} a -2 b^{8} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {9}{2}}-256 a^{4} b \,c^{4}+256 a^{3} b^{3} c^{3}-96 a^{2} b^{5} c^{2}+16 a \,b^{7} c -b^{9}\right )}{\left (-4 a c +b^{2}\right )^{\frac {9}{2}}}\)	\(776\)

[In]

int(1/(c*x^2+b*x+a)^5,x,method=_RETURNVERBOSE)

[Out]

1/4*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^4+7/2*c/(4*a*c-b^2)*(1/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^3+10/3*c/
(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. 1098 vs. \(2 (161) = 322\).

Time = 0.39 (sec) , antiderivative size = 2217, normalized size of antiderivative = 12.96 \[ \int \frac {1}{\left (a+b x+c x^2\right )^5} \, dx=\text {Too large to display} \]

[In]

integrate(1/(c*x^2+b*x+a)^5,x, algorithm="fricas")

[Out]

[-1/12*(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 - 840*(b^2*c^7 - 4*a*c^8)*x^7
- 2940*(b^3*c^6 - 4*a*b*c^7)*x^6 - 280*(13*b^4*c^5 - 41*a*b^2*c^6 - 44*a^2*c^7)*x^5 - 350*(5*b^5*c^4 + 2*a*b^
3*c^5 - 88*a^2*b*c^6)*x^4 - 56*(3*b^6*c^3 + 89*a*b^4*c^4 - 331*a^2*b^2*c^5 - 292*a^3*c^6)*x^3 + 28*(b^7*c^2 -
32*a*b^5*c^3 - 107*a^2*b^3*c^4 + 876*a^3*b*c^5)*x^2 - 840*(c^8*x^8 + 4*b*c^7*x^7 + 4*a^3*b*c^4*x + a^4*c^4 + 2
*(3*b^2*c^6 + 2*a*c^7)*x^6 + 4*(b^3*c^5 + 3*a*b*c^6)*x^5 + (b^4*c^4 + 12*a*b^2*c^5 + 6*a^2*c^6)*x^4 + 4*(a*b^3
*c^4 + 3*a^2*b*c^5)*x^3 + 2*(3*a^2*b^2*c^4 + 2*a^3*c^5)*x^2)*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)) - 8*(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)*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*(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 - 840*(b^2*c^7 - 4*a*c^8)*x^7 - 2940*(b^3*c^6 - 4*a*b*c^
7)*x^6 - 280*(13*b^4*c^5 - 41*a*b^2*c^6 - 44*a^2*c^7)*x^5 - 350*(5*b^5*c^4 + 2*a*b^3*c^5 - 88*a^2*b*c^6)*x^4 -
56*(3*b^6*c^3 + 89*a*b^4*c^4 - 331*a^2*b^2*c^5 - 292*a^3*c^6)*x^3 + 28*(b^7*c^2 - 32*a*b^5*c^3 - 107*a^2*b^3*
c^4 + 876*a^3*b*c^5)*x^2 + 1680*(c^8*x^8 + 4*b*c^7*x^7 + 4*a^3*b*c^4*x + a^4*c^4 + 2*(3*b^2*c^6 + 2*a*c^7)*x^6
+ 4*(b^3*c^5 + 3*a*b*c^6)*x^5 + (b^4*c^4 + 12*a*b^2*c^5 + 6*a^2*c^6)*x^4 + 4*(a*b^3*c^4 + 3*a^2*b*c^5)*x^3 +
2*(3*a^2*b^2*c^4 + 2*a^3*c^5)*x^2)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) -
8*(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)*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)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1153 vs. \(2 (168) = 336\).

Time = 1.53 (sec) , antiderivative size = 1153, normalized size of antiderivative = 6.74 \[ \int \frac {1}{\left (a+b x+c x^2\right )^5} \, dx=- 70 c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} \log {\left (x + \frac {- 71680 a^{5} c^{9} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} + 89600 a^{4} b^{2} c^{8} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} - 44800 a^{3} b^{4} c^{7} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} + 11200 a^{2} b^{6} c^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} - 1400 a b^{8} c^{5} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} + 70 b^{10} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} + 70 b c^{4}}{140 c^{5}} \right )} + 70 c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} \log {\left (x + \frac {71680 a^{5} c^{9} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} - 89600 a^{4} b^{2} c^{8} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} + 44800 a^{3} b^{4} c^{7} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} - 11200 a^{2} b^{6} c^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} + 1400 a b^{8} c^{5} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} - 70 b^{10} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} + 70 b c^{4}}{140 c^{5}} \right )} + \frac {1116 a^{3} b c^{3} - 326 a^{2} b^{3} c^{2} + 50 a b^{5} c - 3 b^{7} + 2940 b c^{6} x^{6} + 840 c^{7} x^{7} + x^{5} \cdot \left (3080 a c^{6} + 3640 b^{2} c^{5}\right ) + x^{4} \cdot \left (7700 a b c^{5} + 1750 b^{3} c^{4}\right ) + x^{3} \cdot \left (4088 a^{2} c^{5} + 5656 a b^{2} c^{4} + 168 b^{4} c^{3}\right ) + x^{2} \cdot \left (6132 a^{2} b c^{4} + 784 a b^{3} c^{3} - 28 b^{5} c^{2}\right ) + x \left (2232 a^{3} c^{4} + 1392 a^{2} b^{2} c^{3} - 152 a b^{4} c^{2} + 8 b^{6} c\right )}{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} \cdot \left (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}\right ) + x^{7} \cdot \left (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}\right ) + x^{6} \cdot \left (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}\right ) + x^{5} \cdot \left (36864 a^{5} b c^{6} - 24576 a^{4} b^{3} c^{5} + 1536 a^{3} b^{5} c^{4} + 2304 a^{2} b^{7} c^{3} - 624 a b^{9} c^{2} + 48 b^{11} c\right ) + x^{4} \cdot \left (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}\right ) + x^{3} \cdot \left (36864 a^{6} b c^{5} - 24576 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}\right ) + x^{2} \cdot \left (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^{10}\right ) + x \left (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}\right )} \]

[In]

integrate(1/(c*x**2+b*x+a)**5,x)

[Out]

-70*c**4*sqrt(-1/(4*a*c - b**2)**9)*log(x + (-71680*a**5*c**9*sqrt(-1/(4*a*c - b**2)**9) + 89600*a**4*b**2*c**
8*sqrt(-1/(4*a*c - b**2)**9) - 44800*a**3*b**4*c**7*sqrt(-1/(4*a*c - b**2)**9) + 11200*a**2*b**6*c**6*sqrt(-1/
(4*a*c - b**2)**9) - 1400*a*b**8*c**5*sqrt(-1/(4*a*c - b**2)**9) + 70*b**10*c**4*sqrt(-1/(4*a*c - b**2)**9) +
70*b*c**4)/(140*c**5)) + 70*c**4*sqrt(-1/(4*a*c - b**2)**9)*log(x + (71680*a**5*c**9*sqrt(-1/(4*a*c - b**2)**9
) - 89600*a**4*b**2*c**8*sqrt(-1/(4*a*c - b**2)**9) + 44800*a**3*b**4*c**7*sqrt(-1/(4*a*c - b**2)**9) - 11200*
a**2*b**6*c**6*sqrt(-1/(4*a*c - b**2)**9) + 1400*a*b**8*c**5*sqrt(-1/(4*a*c - b**2)**9) - 70*b**10*c**4*sqrt(-
1/(4*a*c - b**2)**9) + 70*b*c**4)/(140*c**5)) + (1116*a**3*b*c**3 - 326*a**2*b**3*c**2 + 50*a*b**5*c - 3*b**7
+ 2940*b*c**6*x**6 + 840*c**7*x**7 + x**5*(3080*a*c**6 + 3640*b**2*c**5) + x**4*(7700*a*b*c**5 + 1750*b**3*c**
4) + x**3*(4088*a**2*c**5 + 5656*a*b**2*c**4 + 168*b**4*c**3) + x**2*(6132*a**2*b*c**4 + 784*a*b**3*c**3 - 28*
b**5*c**2) + x*(2232*a**3*c**4 + 1392*a**2*b**2*c**3 - 152*a*b**4*c**2 + 8*b**6*c))/(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 + 4
608*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 + 2304*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 - 24576*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**10) + x*(12288*a**7*b*c**4 - 12288*a**6*b**3*c**3 + 4608*a**5*b**5*c**2 - 76
8*a**4*b**7*c + 48*a**3*b**9))

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\left (a+b x+c x^2\right )^5} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(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. 336 vs. \(2 (161) = 322\).

Time = 0.27 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.96 \[ \int \frac {1}{\left (a+b x+c x^2\right )^5} \, dx=\frac {140 \, c^{4} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {840 \, c^{7} x^{7} + 2940 \, b c^{6} x^{6} + 3640 \, b^{2} c^{5} x^{5} + 3080 \, a c^{6} x^{5} + 1750 \, b^{3} c^{4} x^{4} + 7700 \, a b c^{5} x^{4} + 168 \, b^{4} c^{3} x^{3} + 5656 \, a b^{2} c^{4} x^{3} + 4088 \, a^{2} c^{5} x^{3} - 28 \, b^{5} c^{2} x^{2} + 784 \, a b^{3} c^{3} x^{2} + 6132 \, a^{2} b c^{4} x^{2} + 8 \, b^{6} c x - 152 \, a b^{4} c^{2} x + 1392 \, a^{2} b^{2} c^{3} x + 2232 \, a^{3} c^{4} x - 3 \, b^{7} + 50 \, a b^{5} c - 326 \, a^{2} b^{3} c^{2} + 1116 \, a^{3} b c^{3}}{12 \, {\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} {\left (c x^{2} + b x + a\right )}^{4}} \]

[In]

integrate(1/(c*x^2+b*x+a)^5,x, algorithm="giac")

[Out]

140*c^4*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*x^7 + 2940*b*c^6*x^6 + 3640*b^2*c^5*x^5 + 3080*a*c^6*x^5 + 1750*b^3*
c^4*x^4 + 7700*a*b*c^5*x^4 + 168*b^4*c^3*x^3 + 5656*a*b^2*c^4*x^3 + 4088*a^2*c^5*x^3 - 28*b^5*c^2*x^2 + 784*a*
b^3*c^3*x^2 + 6132*a^2*b*c^4*x^2 + 8*b^6*c*x - 152*a*b^4*c^2*x + 1392*a^2*b^2*c^3*x + 2232*a^3*c^4*x - 3*b^7 +
50*a*b^5*c - 326*a^2*b^3*c^2 + 1116*a^3*b*c^3)/((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 [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x+c x^2\right )^5} \, dx=\left \{\begin {array}{cl} \frac {70\,c^4\,\ln \left (\frac {\frac {b}{2}-\sqrt {\frac {b^2}{4}-a\,c}+c\,x}{\frac {b}{2}+\sqrt {\frac {b^2}{4}-a\,c}+c\,x}\right )}{{\left (b^2-4\,a\,c\right )}^{9/2}}+\frac {70\,\left (\frac {b}{2}+c\,x\right )\,\left (\frac {c^2}{30\,{\left (4\,a\,c-b^2\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^3}+\frac {c^3}{6\,{\left (4\,a\,c-b^2\right )}^3\,{\left (c\,x^2+b\,x+a\right )}^2}+\frac {c^4}{{\left (4\,a\,c-b^2\right )}^4\,\left (c\,x^2+b\,x+a\right )}+\frac {c}{140\,\left (4\,a\,c-b^2\right )\,{\left (c\,x^2+b\,x+a\right )}^4}\right )}{c} & \text {\ if\ \ }0<b^2-4\,a\,c\\ \frac {70\,\left (\frac {b}{2}+c\,x\right )\,\left (\frac {c^2}{30\,{\left (4\,a\,c-b^2\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^3}+\frac {c^3}{6\,{\left (4\,a\,c-b^2\right )}^3\,{\left (c\,x^2+b\,x+a\right )}^2}+\frac {c^4}{{\left (4\,a\,c-b^2\right )}^4\,\left (c\,x^2+b\,x+a\right )}+\frac {c}{140\,\left (4\,a\,c-b^2\right )\,{\left (c\,x^2+b\,x+a\right )}^4}\right )}{c}+\frac {70\,c^4\,\mathrm {atan}\left (\frac {\frac {b}{2}+c\,x}{\sqrt {a\,c-\frac {b^2}{4}}}\right )}{\sqrt {a\,c-\frac {b^2}{4}}\,{\left (4\,a\,c-b^2\right )}^4} & \text {\ if\ \ }b^2-4\,a\,c<0\\ \int \frac {1}{{\left (c\,x^2+b\,x+a\right )}^5} \,d x & \text {\ if\ \ }b^2-4\,a\,c\notin \mathbb {R}\vee b^2=4\,a\,c \end {array}\right . \]

[In]

int(1/(a + b*x + c*x^2)^5,x)

[Out]

piecewise(0 < - 4*a*c + b^2, (70*c^4*log((b/2 - (- a*c + b^2/4)^(1/2) + c*x)/(b/2 + (- a*c + b^2/4)^(1/2) + c*
x)))/(- 4*a*c + b^2)^(9/2) + (70*(b/2 + c*x)*(c^2/(30*(4*a*c - b^2)^2*(a + b*x + c*x^2)^3) + c^3/(6*(4*a*c - b
^2)^3*(a + b*x + c*x^2)^2) + c^4/((4*a*c - b^2)^4*(a + b*x + c*x^2)) + c/(140*(4*a*c - b^2)*(a + b*x + c*x^2)^
4)))/c, - 4*a*c + b^2 < 0, (70*(b/2 + c*x)*(c^2/(30*(4*a*c - b^2)^2*(a + b*x + c*x^2)^3) + c^3/(6*(4*a*c - b^2
)^3*(a + b*x + c*x^2)^2) + c^4/((4*a*c - b^2)^4*(a + b*x + c*x^2)) + c/(140*(4*a*c - b^2)*(a + b*x + c*x^2)^4)
))/c + (70*c^4*atan((b/2 + c*x)/(a*c - b^2/4)^(1/2)))/((a*c - b^2/4)^(1/2)*(4*a*c - b^2)^4), ~in(- 4*a*c + b^2
, 'real') | b^2 == 4*a*c, int(1/(a + b*x + c*x^2)^5, x))