### 3.2213 $$\int \frac{x^5}{(a+b x+c x^2)^4} \, dx$$

Optimal. Leaf size=145 $-\frac{20 a^2 b \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}-\frac{x^5 (b+2 c x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{5 b x^3 (2 a+b x)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac{5 a b x (2 a+b x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}$

[Out]

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

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Rubi [A]  time = 0.0638838, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {728, 722, 618, 206} $-\frac{20 a^2 b \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}-\frac{x^5 (b+2 c x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{5 b x^3 (2 a+b x)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac{5 a b x (2 a+b x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^5/(a + b*x + c*x^2)^4,x]

[Out]

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

Rule 728

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

Rule 722

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[(2*(2*p + 3)*(c*d
^2 - b*d*e + a*e^2))/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 2)*(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] && EqQ[m +
2*p + 2, 0] && LtQ[p, -1]

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{x^5}{\left (a+b x+c x^2\right )^4} \, dx &=-\frac{x^5 (b+2 c x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{(5 b) \int \frac{x^4}{\left (a+b x+c x^2\right )^3} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac{x^5 (b+2 c x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{5 b x^3 (2 a+b x)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac{(5 a b) \int \frac{x^2}{\left (a+b x+c x^2\right )^2} \, dx}{\left (b^2-4 a c\right )^2}\\ &=-\frac{x^5 (b+2 c x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{5 b x^3 (2 a+b x)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac{5 a b x (2 a+b x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac{\left (10 a^2 b\right ) \int \frac{1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^3}\\ &=-\frac{x^5 (b+2 c x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{5 b x^3 (2 a+b x)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac{5 a b x (2 a+b x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac{\left (20 a^2 b\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 )^3}\\ &=-\frac{x^5 (b+2 c x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{5 b x^3 (2 a+b x)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac{5 a b x (2 a+b x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac{20 a^2 b \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]  time = 0.278009, size = 266, normalized size = 1.83 $\frac{1}{6} \left (-\frac{2 \left (a^2 b c (5 c x-4 b)+2 a^3 c^2+a b^3 (b-5 c x)+b^5 x\right )}{c^4 \left (4 a c-b^2\right ) (a+x (b+c x))^3}+\frac{3 \left (38 a^2 b^2 c^2-20 a^2 b c^3 x-64 a^3 c^3-12 a b^4 c+b^6\right )}{c^3 \left (4 a c-b^2\right )^3 (a+x (b+c x))}+\frac{-61 a^2 b^2 c^2+70 a^2 b c^3 x+48 a^3 c^3-40 a b^3 c^2 x+19 a b^4 c+5 b^5 c x-2 b^6}{c^4 \left (b^2-4 a c\right )^2 (a+x (b+c x))^2}-\frac{120 a^2 b \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{7/2}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^5/(a + b*x + c*x^2)^4,x]

[Out]

((-2*b^6 + 19*a*b^4*c - 61*a^2*b^2*c^2 + 48*a^3*c^3 + 5*b^5*c*x - 40*a*b^3*c^2*x + 70*a^2*b*c^3*x)/(c^4*(b^2 -
4*a*c)^2*(a + x*(b + c*x))^2) + (3*(b^6 - 12*a*b^4*c + 38*a^2*b^2*c^2 - 64*a^3*c^3 - 20*a^2*b*c^3*x))/(c^3*(-
b^2 + 4*a*c)^3*(a + x*(b + c*x))) - (2*(2*a^3*c^2 + b^5*x + a*b^3*(b - 5*c*x) + a^2*b*c*(-4*b + 5*c*x)))/(c^4*
(-b^2 + 4*a*c)*(a + x*(b + c*x))^3) - (120*a^2*b*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2))
/6

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Maple [B]  time = 0.163, size = 486, normalized size = 3.4 \begin{align*}{\frac{1}{ \left ( c{x}^{2}+bx+a \right ) ^{3}} \left ( -10\,{\frac{{a}^{2}b{c}^{2}{x}^{5}}{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}-{\frac{ \left ( 64\,{a}^{3}{c}^{3}+2\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ){x}^{4}}{ \left ( 128\,{a}^{3}{c}^{3}-96\,{a}^{2}{b}^{2}{c}^{2}+24\,a{b}^{4}c-2\,{b}^{6} \right ) c}}-{\frac{b \left ( 224\,{a}^{3}{c}^{3}+62\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ){x}^{3}}{6\,{c}^{2} \left ( 64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ) }}-{\frac{a \left ( 64\,{a}^{3}{c}^{3}+32\,{a}^{2}{b}^{2}{c}^{2}+17\,a{b}^{4}c-{b}^{6} \right ){x}^{2}}{2\,{c}^{2} \left ( 64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ) }}-{\frac{b{a}^{2} \left ( 44\,{a}^{2}{c}^{2}+18\,ac{b}^{2}-{b}^{4} \right ) x}{2\,{c}^{2} \left ( 64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ) }}-{\frac{{a}^{3} \left ( 64\,{a}^{2}{c}^{2}+18\,ac{b}^{2}-{b}^{4} \right ) }{6\,{c}^{2} \left ( 64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ) }} \right ) }-20\,{\frac{b{a}^{2}}{ \left ( 64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ) \sqrt{4\,ac-{b}^{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(x^5/(c*x^2+b*x+a)^4,x)

[Out]

(-10*b*a^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*c^2*x^5-1/2*(64*a^3*c^3+2*a^2*b^2*c^2+12*a*b^4*c-b^6)/(6
4*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/c*x^4-1/6*b*(224*a^3*c^3+62*a^2*b^2*c^2+12*a*b^4*c-b^6)/c^2/(64*a^3*c
^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^3-1/2*a*(64*a^3*c^3+32*a^2*b^2*c^2+17*a*b^4*c-b^6)/c^2/(64*a^3*c^3-48*a^2*
b^2*c^2+12*a*b^4*c-b^6)*x^2-1/2*a^2*b*(44*a^2*c^2+18*a*b^2*c-b^4)/c^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^
6)*x-1/6*a^3*(64*a^2*c^2+18*a*b^2*c-b^4)/c^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6))/(c*x^2+b*x+a)^3-20*b*
a^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/(4*a*c-b^2)^(1/2)*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(x^5/(c*x^2+b*x+a)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.21565, size = 3379, normalized size = 23.3 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(c*x^2+b*x+a)^4,x, algorithm="fricas")

[Out]

[-1/6*(a^3*b^6 - 22*a^4*b^4*c + 8*a^5*b^2*c^2 + 256*a^6*c^3 - 60*(a^2*b^3*c^4 - 4*a^3*b*c^5)*x^5 + 3*(b^8*c -
16*a*b^6*c^2 + 46*a^2*b^4*c^3 - 56*a^3*b^2*c^4 + 256*a^4*c^5)*x^4 + (b^9 - 16*a*b^7*c - 14*a^2*b^5*c^2 + 24*a^
3*b^3*c^3 + 896*a^4*b*c^4)*x^3 + 3*(a*b^8 - 21*a^2*b^6*c + 36*a^3*b^4*c^2 + 64*a^4*b^2*c^3 + 256*a^5*c^4)*x^2
+ 60*(a^2*b*c^5*x^6 + 3*a^2*b^2*c^4*x^5 + 3*a^4*b^2*c^2*x + a^5*b*c^2 + 3*(a^2*b^3*c^3 + a^3*b*c^4)*x^4 + (a^2
*b^4*c^2 + 6*a^3*b^2*c^3)*x^3 + 3*(a^3*b^3*c^2 + a^4*b*c^3)*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)) + 3*(a^2*b^7 - 22*a^3*b^5*c + 28*a^4*b^3*c^2 +
176*a^5*b*c^3)*x)/(a^3*b^8*c^2 - 16*a^4*b^6*c^3 + 96*a^5*b^4*c^4 - 256*a^6*b^2*c^5 + 256*a^7*c^6 + (b^8*c^5 -
16*a*b^6*c^6 + 96*a^2*b^4*c^7 - 256*a^3*b^2*c^8 + 256*a^4*c^9)*x^6 + 3*(b^9*c^4 - 16*a*b^7*c^5 + 96*a^2*b^5*c
^6 - 256*a^3*b^3*c^7 + 256*a^4*b*c^8)*x^5 + 3*(b^10*c^3 - 15*a*b^8*c^4 + 80*a^2*b^6*c^5 - 160*a^3*b^4*c^6 + 25
6*a^5*c^8)*x^4 + (b^11*c^2 - 10*a*b^9*c^3 + 320*a^3*b^5*c^5 - 1280*a^4*b^3*c^6 + 1536*a^5*b*c^7)*x^3 + 3*(a*b^
10*c^2 - 15*a^2*b^8*c^3 + 80*a^3*b^6*c^4 - 160*a^4*b^4*c^5 + 256*a^6*c^7)*x^2 + 3*(a^2*b^9*c^2 - 16*a^3*b^7*c^
3 + 96*a^4*b^5*c^4 - 256*a^5*b^3*c^5 + 256*a^6*b*c^6)*x), -1/6*(a^3*b^6 - 22*a^4*b^4*c + 8*a^5*b^2*c^2 + 256*a
^6*c^3 - 60*(a^2*b^3*c^4 - 4*a^3*b*c^5)*x^5 + 3*(b^8*c - 16*a*b^6*c^2 + 46*a^2*b^4*c^3 - 56*a^3*b^2*c^4 + 256*
a^4*c^5)*x^4 + (b^9 - 16*a*b^7*c - 14*a^2*b^5*c^2 + 24*a^3*b^3*c^3 + 896*a^4*b*c^4)*x^3 + 3*(a*b^8 - 21*a^2*b^
6*c + 36*a^3*b^4*c^2 + 64*a^4*b^2*c^3 + 256*a^5*c^4)*x^2 + 120*(a^2*b*c^5*x^6 + 3*a^2*b^2*c^4*x^5 + 3*a^4*b^2*
c^2*x + a^5*b*c^2 + 3*(a^2*b^3*c^3 + a^3*b*c^4)*x^4 + (a^2*b^4*c^2 + 6*a^3*b^2*c^3)*x^3 + 3*(a^3*b^3*c^2 + a^4
*b*c^3)*x^2)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 3*(a^2*b^7 - 22*a^3*b^
5*c + 28*a^4*b^3*c^2 + 176*a^5*b*c^3)*x)/(a^3*b^8*c^2 - 16*a^4*b^6*c^3 + 96*a^5*b^4*c^4 - 256*a^6*b^2*c^5 + 25
6*a^7*c^6 + (b^8*c^5 - 16*a*b^6*c^6 + 96*a^2*b^4*c^7 - 256*a^3*b^2*c^8 + 256*a^4*c^9)*x^6 + 3*(b^9*c^4 - 16*a*
b^7*c^5 + 96*a^2*b^5*c^6 - 256*a^3*b^3*c^7 + 256*a^4*b*c^8)*x^5 + 3*(b^10*c^3 - 15*a*b^8*c^4 + 80*a^2*b^6*c^5
- 160*a^3*b^4*c^6 + 256*a^5*c^8)*x^4 + (b^11*c^2 - 10*a*b^9*c^3 + 320*a^3*b^5*c^5 - 1280*a^4*b^3*c^6 + 1536*a^
5*b*c^7)*x^3 + 3*(a*b^10*c^2 - 15*a^2*b^8*c^3 + 80*a^3*b^6*c^4 - 160*a^4*b^4*c^5 + 256*a^6*c^7)*x^2 + 3*(a^2*b
^9*c^2 - 16*a^3*b^7*c^3 + 96*a^4*b^5*c^4 - 256*a^5*b^3*c^5 + 256*a^6*b*c^6)*x)]

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Sympy [B]  time = 3.48953, size = 898, normalized size = 6.19 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(c*x**2+b*x+a)**4,x)

[Out]

10*a**2*b*sqrt(-1/(4*a*c - b**2)**7)*log(x + (-2560*a**6*b*c**4*sqrt(-1/(4*a*c - b**2)**7) + 2560*a**5*b**3*c*
*3*sqrt(-1/(4*a*c - b**2)**7) - 960*a**4*b**5*c**2*sqrt(-1/(4*a*c - b**2)**7) + 160*a**3*b**7*c*sqrt(-1/(4*a*c
- b**2)**7) - 10*a**2*b**9*sqrt(-1/(4*a*c - b**2)**7) + 10*a**2*b**2)/(20*a**2*b*c)) - 10*a**2*b*sqrt(-1/(4*a
*c - b**2)**7)*log(x + (2560*a**6*b*c**4*sqrt(-1/(4*a*c - b**2)**7) - 2560*a**5*b**3*c**3*sqrt(-1/(4*a*c - b**
2)**7) + 960*a**4*b**5*c**2*sqrt(-1/(4*a*c - b**2)**7) - 160*a**3*b**7*c*sqrt(-1/(4*a*c - b**2)**7) + 10*a**2*
b**9*sqrt(-1/(4*a*c - b**2)**7) + 10*a**2*b**2)/(20*a**2*b*c)) - (64*a**5*c**2 + 18*a**4*b**2*c - a**3*b**4 +
60*a**2*b*c**4*x**5 + x**4*(192*a**3*c**4 + 6*a**2*b**2*c**3 + 36*a*b**4*c**2 - 3*b**6*c) + x**3*(224*a**3*b*c
**3 + 62*a**2*b**3*c**2 + 12*a*b**5*c - b**7) + x**2*(192*a**4*c**3 + 96*a**3*b**2*c**2 + 51*a**2*b**4*c - 3*a
*b**6) + x*(132*a**4*b*c**2 + 54*a**3*b**3*c - 3*a**2*b**5))/(384*a**6*c**5 - 288*a**5*b**2*c**4 + 72*a**4*b**
4*c**3 - 6*a**3*b**6*c**2 + x**6*(384*a**3*c**8 - 288*a**2*b**2*c**7 + 72*a*b**4*c**6 - 6*b**6*c**5) + x**5*(1
152*a**3*b*c**7 - 864*a**2*b**3*c**6 + 216*a*b**5*c**5 - 18*b**7*c**4) + x**4*(1152*a**4*c**7 + 288*a**3*b**2*
c**6 - 648*a**2*b**4*c**5 + 198*a*b**6*c**4 - 18*b**8*c**3) + x**3*(2304*a**4*b*c**6 - 1344*a**3*b**3*c**5 + 1
44*a**2*b**5*c**4 + 36*a*b**7*c**3 - 6*b**9*c**2) + x**2*(1152*a**5*c**6 + 288*a**4*b**2*c**5 - 648*a**3*b**4*
c**4 + 198*a**2*b**6*c**3 - 18*a*b**8*c**2) + x*(1152*a**5*b*c**5 - 864*a**4*b**3*c**4 + 216*a**3*b**5*c**3 -
18*a**2*b**7*c**2))

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Giac [B]  time = 1.12755, size = 440, normalized size = 3.03 \begin{align*} \frac{20 \, a^{2} b \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{60 \, a^{2} b c^{4} x^{5} - 3 \, b^{6} c x^{4} + 36 \, a b^{4} c^{2} x^{4} + 6 \, a^{2} b^{2} c^{3} x^{4} + 192 \, a^{3} c^{4} x^{4} - b^{7} x^{3} + 12 \, a b^{5} c x^{3} + 62 \, a^{2} b^{3} c^{2} x^{3} + 224 \, a^{3} b c^{3} x^{3} - 3 \, a b^{6} x^{2} + 51 \, a^{2} b^{4} c x^{2} + 96 \, a^{3} b^{2} c^{2} x^{2} + 192 \, a^{4} c^{3} x^{2} - 3 \, a^{2} b^{5} x + 54 \, a^{3} b^{3} c x + 132 \, a^{4} b c^{2} x - a^{3} b^{4} + 18 \, a^{4} b^{2} c + 64 \, a^{5} c^{2}}{6 \,{\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )}{\left (c x^{2} + b x + a\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(c*x^2+b*x+a)^4,x, algorithm="giac")

[Out]

20*a^2*b*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*(60*a^2*b*c^4*x^5 - 3*b^6*c*x^4 + 36*a*b^4*c^2*x^4 + 6*a^2*b^2*c^3*x^4 + 192*a^3*c^4*x^4 - b^7*x
^3 + 12*a*b^5*c*x^3 + 62*a^2*b^3*c^2*x^3 + 224*a^3*b*c^3*x^3 - 3*a*b^6*x^2 + 51*a^2*b^4*c*x^2 + 96*a^3*b^2*c^2
*x^2 + 192*a^4*c^3*x^2 - 3*a^2*b^5*x + 54*a^3*b^3*c*x + 132*a^4*b*c^2*x - a^3*b^4 + 18*a^4*b^2*c + 64*a^5*c^2)
/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*(c*x^2 + b*x + a)^3)