### 3.2226 $$\int \frac{1}{(a+b x+c x^2)^5} \, dx$$

Optimal. Leaf size=171 $\frac{35 c^3 (b+2 c x)}{\left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac{35 c^2 (b+2 c x)}{6 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}-\frac{140 c^4 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}}+\frac{7 c (b+2 c x)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac{b+2 c x}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}$

[Out]

-(b + 2*c*x)/(4*(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)

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Rubi [A]  time = 0.0698263, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {614, 618, 206} $\frac{35 c^3 (b+2 c x)}{\left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac{35 c^2 (b+2 c x)}{6 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}-\frac{140 c^4 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}}+\frac{7 c (b+2 c x)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac{b+2 c x}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^(-5),x]

[Out]

-(b + 2*c*x)/(4*(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 614

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 618

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

Rule 206

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

Rubi steps

\begin{align*} \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) \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 ) \operatorname{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]  time = 0.173295, size = 167, normalized size = 0.98 $\frac{-\frac{70 c^2 \left (b^2-4 a c\right ) (b+2 c x)}{(a+x (b+c x))^2}+\frac{1680 c^4 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\frac{14 c \left (b^2-4 a c\right )^2 (b+2 c x)}{(a+x (b+c x))^3}-\frac{3 \left (b^2-4 a c\right )^3 (b+2 c x)}{(a+x (b+c x))^4}+\frac{420 c^3 (b+2 c x)}{a+x (b+c x)}}{12 \left (b^2-4 a c\right )^4}$

Antiderivative was successfully veriﬁed.

[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)

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Maple [A]  time = 0.162, size = 249, normalized size = 1.5 \begin{align*}{\frac{2\,cx+b}{ \left ( 16\,ac-4\,{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) ^{4}}}+{\frac{7\,{c}^{2}x}{3\, \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{3}}}+{\frac{7\,bc}{6\, \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{3}}}+{\frac{35\,{c}^{3}x}{3\, \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+{\frac{35\,b{c}^{2}}{6\, \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+70\,{\frac{{c}^{4}x}{ \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) }}+35\,{\frac{b{c}^{3}}{ \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) }}+140\,{\frac{{c}^{4}}{ \left ( 4\,ac-{b}^{2} \right ) ^{9/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^4+7/3*c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^3*x+7/6*c/(4*a*c-b^2)^2/(c*x^2+b
*x+a)^3*b+35/3*c^3/(4*a*c-b^2)^3/(c*x^2+b*x+a)^2*x+35/6*c^2/(4*a*c-b^2)^3/(c*x^2+b*x+a)^2*b+70*c^4/(4*a*c-b^2)
^4/(c*x^2+b*x+a)*x+35*c^3/(4*a*c-b^2)^4/(c*x^2+b*x+a)*b+140*c^4/(4*a*c-b^2)^(9/2)*arctan((2*c*x+b)/(4*a*c-b^2)
^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.14844, size = 4934, normalized size = 28.85 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)]

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Sympy [B]  time = 4.93017, size = 1153, normalized size = 6.74 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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))

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Giac [B]  time = 1.09955, size = 454, normalized size = 2.65 \begin{align*} \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}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)