3.2200 \(\int \frac{x^6}{(a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=238 \[ \frac{3 x \left (10 a^2 c^2-7 a b^2 c+b^4\right )}{c^3 \left (b^2-4 a c\right )^2}-\frac{3 \left (30 a^2 b^2 c^2-20 a^3 c^3-10 a b^4 c+b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{5/2}}-\frac{3 b x^2 \left (b^2-6 a c\right )}{2 c^2 \left (b^2-4 a c\right )^2}+\frac{x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{x^3 \left (b x \left (b^2-7 a c\right )+a \left (b^2-10 a c\right )\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{3 b \log \left (a+b x+c x^2\right )}{2 c^4} \]

[Out]

(3*(b^4 - 7*a*b^2*c + 10*a^2*c^2)*x)/(c^3*(b^2 - 4*a*c)^2) - (3*b*(b^2 - 6*a*c)*x^2)/(2*c^2*(b^2 - 4*a*c)^2) +
(x^5*(2*a + b*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (x^3*(a*(b^2 - 10*a*c) + b*(b^2 - 7*a*c)*x))/(c*(b^
2 - 4*a*c)^2*(a + b*x + c*x^2)) - (3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*ArcTanh[(b + 2*c*x)/Sqrt
[b^2 - 4*a*c]])/(c^4*(b^2 - 4*a*c)^(5/2)) - (3*b*Log[a + b*x + c*x^2])/(2*c^4)

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Rubi [A]  time = 0.287418, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {738, 818, 800, 634, 618, 206, 628} \[ \frac{3 x \left (10 a^2 c^2-7 a b^2 c+b^4\right )}{c^3 \left (b^2-4 a c\right )^2}-\frac{3 \left (30 a^2 b^2 c^2-20 a^3 c^3-10 a b^4 c+b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{5/2}}-\frac{3 b x^2 \left (b^2-6 a c\right )}{2 c^2 \left (b^2-4 a c\right )^2}+\frac{x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{x^3 \left (b x \left (b^2-7 a c\right )+a \left (b^2-10 a c\right )\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{3 b \log \left (a+b x+c x^2\right )}{2 c^4} \]

Antiderivative was successfully verified.

[In]

Int[x^6/(a + b*x + c*x^2)^3,x]

[Out]

(3*(b^4 - 7*a*b^2*c + 10*a^2*c^2)*x)/(c^3*(b^2 - 4*a*c)^2) - (3*b*(b^2 - 6*a*c)*x^2)/(2*c^2*(b^2 - 4*a*c)^2) +
(x^5*(2*a + b*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (x^3*(a*(b^2 - 10*a*c) + b*(b^2 - 7*a*c)*x))/(c*(b^
2 - 4*a*c)^2*(a + b*x + c*x^2)) - (3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*ArcTanh[(b + 2*c*x)/Sqrt
[b^2 - 4*a*c]])/(c^4*(b^2 - 4*a*c)^(5/2)) - (3*b*Log[a + b*x + c*x^2])/(2*c^4)

Rule 738

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]

Rule 818

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 634

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

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])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{x^6}{\left (a+b x+c x^2\right )^3} \, dx &=\frac{x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{\int \frac{x^4 (10 a+2 b x)}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )}\\ &=\frac{x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{x^3 \left (a \left (b^2-10 a c\right )+b \left (b^2-7 a c\right ) x\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{\int \frac{x^2 \left (6 a \left (b^2-10 a c\right )+6 b \left (b^2-6 a c\right ) x\right )}{a+b x+c x^2} \, dx}{2 c \left (b^2-4 a c\right )^2}\\ &=\frac{x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{x^3 \left (a \left (b^2-10 a c\right )+b \left (b^2-7 a c\right ) x\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{\int \left (-\frac{6 \left (b^4-7 a b^2 c+10 a^2 c^2\right )}{c^2}+\frac{6 b \left (b^2-6 a c\right ) x}{c}+\frac{6 \left (a \left (b^4-7 a b^2 c+10 a^2 c^2\right )+b \left (b^2-4 a c\right )^2 x\right )}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx}{2 c \left (b^2-4 a c\right )^2}\\ &=\frac{3 \left (b^4-7 a b^2 c+10 a^2 c^2\right ) x}{c^3 \left (b^2-4 a c\right )^2}-\frac{3 b \left (b^2-6 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )^2}+\frac{x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{x^3 \left (a \left (b^2-10 a c\right )+b \left (b^2-7 a c\right ) x\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{3 \int \frac{a \left (b^4-7 a b^2 c+10 a^2 c^2\right )+b \left (b^2-4 a c\right )^2 x}{a+b x+c x^2} \, dx}{c^3 \left (b^2-4 a c\right )^2}\\ &=\frac{3 \left (b^4-7 a b^2 c+10 a^2 c^2\right ) x}{c^3 \left (b^2-4 a c\right )^2}-\frac{3 b \left (b^2-6 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )^2}+\frac{x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{x^3 \left (a \left (b^2-10 a c\right )+b \left (b^2-7 a c\right ) x\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{(3 b) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 c^4}+\frac{\left (3 \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 c^4 \left (b^2-4 a c\right )^2}\\ &=\frac{3 \left (b^4-7 a b^2 c+10 a^2 c^2\right ) x}{c^3 \left (b^2-4 a c\right )^2}-\frac{3 b \left (b^2-6 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )^2}+\frac{x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{x^3 \left (a \left (b^2-10 a c\right )+b \left (b^2-7 a c\right ) x\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{3 b \log \left (a+b x+c x^2\right )}{2 c^4}-\frac{\left (3 \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^4 \left (b^2-4 a c\right )^2}\\ &=\frac{3 \left (b^4-7 a b^2 c+10 a^2 c^2\right ) x}{c^3 \left (b^2-4 a c\right )^2}-\frac{3 b \left (b^2-6 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )^2}+\frac{x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{x^3 \left (a \left (b^2-10 a c\right )+b \left (b^2-7 a c\right ) x\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{3 \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{5/2}}-\frac{3 b \log \left (a+b x+c x^2\right )}{2 c^4}\\ \end{align*}

Mathematica [A]  time = 0.38183, size = 260, normalized size = 1.09 \[ \frac{\frac{a^2 b^2 c (5 b-9 c x)+a^3 c^2 (2 c x-5 b)-a b^4 (b-6 c x)+b^6 (-x)}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}+\frac{-102 a^2 b^2 c^3 x+61 a^2 b^3 c^2-78 a^3 b c^3+36 a^3 c^4 x+48 a b^4 c^2 x-14 a b^5 c-6 b^6 c x+b^7}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac{6 c \left (30 a^2 b^2 c^2-20 a^3 c^3-10 a b^4 c+b^6\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}-3 b c \log (a+x (b+c x))+2 c^2 x}{2 c^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/(a + b*x + c*x^2)^3,x]

[Out]

(2*c^2*x + (b^7 - 14*a*b^5*c + 61*a^2*b^3*c^2 - 78*a^3*b*c^3 - 6*b^6*c*x + 48*a*b^4*c^2*x - 102*a^2*b^2*c^3*x
+ 36*a^3*c^4*x)/((b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (-(b^6*x) + a^2*b^2*c*(5*b - 9*c*x) - a*b^4*(b - 6*c*x)
+ a^3*c^2*(-5*b + 2*c*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) + (6*c*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a
^3*c^3)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2) - 3*b*c*Log[a + x*(b + c*x)])/(2*c^5)

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Maple [B]  time = 0.181, size = 1040, normalized size = 4.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(c*x^2+b*x+a)^3,x)

[Out]

x/c^3+18/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*a^3-51/c/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^
3*a^2*b^2+24/c^2/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*a*b^4-3/c^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^
2*c+b^4)*x^3*b^6-21/c/(c*x^2+b*x+a)^2*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*a^3-41/2/c^2/(c*x^2+b*x+a)^2*b^3/(16*a^
2*c^2-8*a*b^2*c+b^4)*x^2*a^2+17/c^3/(c*x^2+b*x+a)^2*b^5/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*a-5/2/c^4/(c*x^2+b*x+a)
^2*b^7/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2+14/c/(c*x^2+b*x+a)^2*a^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x-71/c^2/(c*x^2+b*x+
a)^2*a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x*b^2+38/c^3/(c*x^2+b*x+a)^2*a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x*b^4-5/c^4/(c
*x^2+b*x+a)^2*a/(16*a^2*c^2-8*a*b^2*c+b^4)*x*b^6-29/c^2/(c*x^2+b*x+a)^2*b*a^4/(16*a^2*c^2-8*a*b^2*c+b^4)+18/c^
3/(c*x^2+b*x+a)^2*b^3*a^3/(16*a^2*c^2-8*a*b^2*c+b^4)-5/2/c^4/(c*x^2+b*x+a)^2*b^5*a^2/(16*a^2*c^2-8*a*b^2*c+b^4
)-24/c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*ln(c*x^2+b*x+a)*a^2*b+12/c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*ln(c*x^2+b*x+a)*a*
b^3-3/2/c^4/(16*a^2*c^2-8*a*b^2*c+b^4)*ln(c*x^2+b*x+a)*b^5-60/c/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*a
rctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^3+90/c^2/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4
*a*c-b^2)^(1/2))*a^2*b^2-30/c^3/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2
))*a*b^4+3/c^4/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^6

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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^6/(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.72527, size = 4120, normalized size = 17.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

[-1/2*(5*a^2*b^7 - 56*a^3*b^5*c + 202*a^4*b^3*c^2 - 232*a^5*b*c^3 - 2*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5
- 64*a^3*c^6)*x^5 - 4*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^4 + 2*(2*b^8*c - 26*a*b^6*c^
2 + 123*a^2*b^4*c^3 - 254*a^3*b^2*c^4 + 200*a^4*c^5)*x^3 + (5*b^9 - 58*a*b^7*c + 225*a^2*b^5*c^2 - 314*a^3*b^3
*c^3 + 88*a^4*b*c^4)*x^2 + 3*(a^2*b^6 - 10*a^3*b^4*c + 30*a^4*b^2*c^2 - 20*a^5*c^3 + (b^6*c^2 - 10*a*b^4*c^3 +
30*a^2*b^2*c^4 - 20*a^3*c^5)*x^4 + 2*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*x^3 + (b^8 - 8*a*
b^6*c + 10*a^2*b^4*c^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4)*x^2 + 2*(a*b^7 - 10*a^2*b^5*c + 30*a^3*b^3*c^2 - 20*a^4*
b*c^3)*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*(5*a*b^8 - 59*a^2*b^6*c + 235*a^3*b^4*c^2 - 346*a^4*b^2*c^3 + 120*a^5*c^4)*x + 3*(a^2*b^7 - 12*a^
3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3 + (b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^4 + 2*(b^
8*c - 12*a*b^6*c^2 + 48*a^2*b^4*c^3 - 64*a^3*b^2*c^4)*x^3 + (b^9 - 10*a*b^7*c + 24*a^2*b^5*c^2 + 32*a^3*b^3*c^
3 - 128*a^4*b*c^4)*x^2 + 2*(a*b^8 - 12*a^2*b^6*c + 48*a^3*b^4*c^2 - 64*a^4*b^2*c^3)*x)*log(c*x^2 + b*x + a))/(
a^2*b^6*c^4 - 12*a^3*b^4*c^5 + 48*a^4*b^2*c^6 - 64*a^5*c^7 + (b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3
*c^9)*x^4 + 2*(b^7*c^5 - 12*a*b^5*c^6 + 48*a^2*b^3*c^7 - 64*a^3*b*c^8)*x^3 + (b^8*c^4 - 10*a*b^6*c^5 + 24*a^2*
b^4*c^6 + 32*a^3*b^2*c^7 - 128*a^4*c^8)*x^2 + 2*(a*b^7*c^4 - 12*a^2*b^5*c^5 + 48*a^3*b^3*c^6 - 64*a^4*b*c^7)*x
), -1/2*(5*a^2*b^7 - 56*a^3*b^5*c + 202*a^4*b^3*c^2 - 232*a^5*b*c^3 - 2*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c
^5 - 64*a^3*c^6)*x^5 - 4*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^4 + 2*(2*b^8*c - 26*a*b^6*
c^2 + 123*a^2*b^4*c^3 - 254*a^3*b^2*c^4 + 200*a^4*c^5)*x^3 + (5*b^9 - 58*a*b^7*c + 225*a^2*b^5*c^2 - 314*a^3*b
^3*c^3 + 88*a^4*b*c^4)*x^2 + 6*(a^2*b^6 - 10*a^3*b^4*c + 30*a^4*b^2*c^2 - 20*a^5*c^3 + (b^6*c^2 - 10*a*b^4*c^3
+ 30*a^2*b^2*c^4 - 20*a^3*c^5)*x^4 + 2*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*x^3 + (b^8 - 8*
a*b^6*c + 10*a^2*b^4*c^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4)*x^2 + 2*(a*b^7 - 10*a^2*b^5*c + 30*a^3*b^3*c^2 - 20*a^
4*b*c^3)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 2*(5*a*b^8 - 59*a^2*b^6
*c + 235*a^3*b^4*c^2 - 346*a^4*b^2*c^3 + 120*a^5*c^4)*x + 3*(a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*
b*c^3 + (b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^4 + 2*(b^8*c - 12*a*b^6*c^2 + 48*a^2*b^4*c^
3 - 64*a^3*b^2*c^4)*x^3 + (b^9 - 10*a*b^7*c + 24*a^2*b^5*c^2 + 32*a^3*b^3*c^3 - 128*a^4*b*c^4)*x^2 + 2*(a*b^8
- 12*a^2*b^6*c + 48*a^3*b^4*c^2 - 64*a^4*b^2*c^3)*x)*log(c*x^2 + b*x + a))/(a^2*b^6*c^4 - 12*a^3*b^4*c^5 + 48*
a^4*b^2*c^6 - 64*a^5*c^7 + (b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)*x^4 + 2*(b^7*c^5 - 12*a*b^5*
c^6 + 48*a^2*b^3*c^7 - 64*a^3*b*c^8)*x^3 + (b^8*c^4 - 10*a*b^6*c^5 + 24*a^2*b^4*c^6 + 32*a^3*b^2*c^7 - 128*a^4
*c^8)*x^2 + 2*(a*b^7*c^4 - 12*a^2*b^5*c^5 + 48*a^3*b^3*c^6 - 64*a^4*b*c^7)*x)]

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Sympy [B]  time = 4.3829, size = 1714, normalized size = 7.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(c*x**2+b*x+a)**3,x)

[Out]

(-3*b/(2*c**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*c**4*(1
024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))*log(x +
(-66*a**3*b*c**2 - 64*a**3*c**6*(-3*b/(2*c**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2
+ 10*a*b**4*c - b**6)/(2*c**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2
+ 20*a*b**8*c - b**10))) + 27*a**2*b**3*c + 48*a**2*b**2*c**5*(-3*b/(2*c**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20
*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*c**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*
b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) - 3*a*b**5 - 12*a*b**4*c**4*(-3*b/(2*c**4) - 3*sqrt(-(
4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*c**4*(1024*a**5*c**5 - 1280*a**4*
b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) + b**6*c**3*(-3*b/(2*c**4) - 3*sq
rt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*c**4*(1024*a**5*c**5 - 1280*
a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))))/(60*a**3*c**3 - 90*a**2*b**
2*c**2 + 30*a*b**4*c - 3*b**6)) + (-3*b/(2*c**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**
2 + 10*a*b**4*c - b**6)/(2*c**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**
2 + 20*a*b**8*c - b**10)))*log(x + (-66*a**3*b*c**2 - 64*a**3*c**6*(-3*b/(2*c**4) + 3*sqrt(-(4*a*c - b**2)**5)
*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*c**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a
**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) + 27*a**2*b**3*c + 48*a**2*b**2*c**5*(-3*b/(2*c**4
) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*c**4*(1024*a**5*c**5
- 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) - 3*a*b**5 - 12*a*b*
*4*c**4*(-3*b/(2*c**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2
*c**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))
+ b**6*c**3*(-3*b/(2*c**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**
6)/(2*c**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**
10))))/(60*a**3*c**3 - 90*a**2*b**2*c**2 + 30*a*b**4*c - 3*b**6)) + (-58*a**4*b*c**2 + 36*a**3*b**3*c - 5*a**2
*b**5 + x**3*(36*a**3*c**4 - 102*a**2*b**2*c**3 + 48*a*b**4*c**2 - 6*b**6*c) + x**2*(-42*a**3*b*c**3 - 41*a**2
*b**3*c**2 + 34*a*b**5*c - 5*b**7) + x*(28*a**4*c**3 - 142*a**3*b**2*c**2 + 76*a**2*b**4*c - 10*a*b**6))/(32*a
**4*c**6 - 16*a**3*b**2*c**5 + 2*a**2*b**4*c**4 + x**4*(32*a**2*c**8 - 16*a*b**2*c**7 + 2*b**4*c**6) + x**3*(6
4*a**2*b*c**7 - 32*a*b**3*c**6 + 4*b**5*c**5) + x**2*(64*a**3*c**7 - 12*a*b**4*c**5 + 2*b**6*c**4) + x*(64*a**
3*b*c**6 - 32*a**2*b**3*c**5 + 4*a*b**5*c**4)) + x/c**3

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Giac [A]  time = 1.12738, size = 381, normalized size = 1.6 \begin{align*} \frac{3 \,{\left (b^{6} - 10 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2} - 20 \, a^{3} c^{3}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{x}{c^{3}} - \frac{3 \, b \log \left (c x^{2} + b x + a\right )}{2 \, c^{4}} - \frac{5 \, a^{2} b^{5} - 36 \, a^{3} b^{3} c + 58 \, a^{4} b c^{2} + 6 \,{\left (b^{6} c - 8 \, a b^{4} c^{2} + 17 \, a^{2} b^{2} c^{3} - 6 \, a^{3} c^{4}\right )} x^{3} +{\left (5 \, b^{7} - 34 \, a b^{5} c + 41 \, a^{2} b^{3} c^{2} + 42 \, a^{3} b c^{3}\right )} x^{2} + 2 \,{\left (5 \, a b^{6} - 38 \, a^{2} b^{4} c + 71 \, a^{3} b^{2} c^{2} - 14 \, a^{4} c^{3}\right )} x}{2 \,{\left (c x^{2} + b x + a\right )}^{2}{\left (b^{2} - 4 \, a c\right )}^{2} c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^4*c^4 - 8*a*b^2*
c^5 + 16*a^2*c^6)*sqrt(-b^2 + 4*a*c)) + x/c^3 - 3/2*b*log(c*x^2 + b*x + a)/c^4 - 1/2*(5*a^2*b^5 - 36*a^3*b^3*c
+ 58*a^4*b*c^2 + 6*(b^6*c - 8*a*b^4*c^2 + 17*a^2*b^2*c^3 - 6*a^3*c^4)*x^3 + (5*b^7 - 34*a*b^5*c + 41*a^2*b^3*
c^2 + 42*a^3*b*c^3)*x^2 + 2*(5*a*b^6 - 38*a^2*b^4*c + 71*a^3*b^2*c^2 - 14*a^4*c^3)*x)/((c*x^2 + b*x + a)^2*(b^
2 - 4*a*c)^2*c^4)