3.2211 \(\int \frac{x^7}{(a+b x+c x^2)^4} \, dx\)

Optimal. Leaf size=291 \[ \frac{x^2 \left (b x \left (140 a^2 c^2-32 a b^2 c+3 b^4\right )+3 a \left (64 a^2 c^2-10 a b^2 c+b^4\right )\right )}{6 c^2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac{b x \left (38 a^2 c^2-11 a b^2 c+b^4\right )}{c^3 \left (b^2-4 a c\right )^3}+\frac{b \left (70 a^2 b^2 c^2-140 a^3 c^3-14 a b^4 c+b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{7/2}}+\frac{x^6 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{x^4 \left (b x \left (b^2-14 a c\right )+a \left (b^2-24 a c\right )\right )}{6 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac{\log \left (a+b x+c x^2\right )}{2 c^4} \]

[Out]

-((b*(b^4 - 11*a*b^2*c + 38*a^2*c^2)*x)/(c^3*(b^2 - 4*a*c)^3)) + (x^6*(2*a + b*x))/(3*(b^2 - 4*a*c)*(a + b*x +
c*x^2)^3) + (x^4*(a*(b^2 - 24*a*c) + b*(b^2 - 14*a*c)*x))/(6*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^2) + (x^2*(3
*a*(b^4 - 10*a*b^2*c + 64*a^2*c^2) + b*(3*b^4 - 32*a*b^2*c + 140*a^2*c^2)*x))/(6*c^2*(b^2 - 4*a*c)^3*(a + b*x
+ c*x^2)) + (b*(b^6 - 14*a*b^4*c + 70*a^2*b^2*c^2 - 140*a^3*c^3)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^4*
(b^2 - 4*a*c)^(7/2)) + Log[a + b*x + c*x^2]/(2*c^4)

________________________________________________________________________________________

Rubi [A]  time = 0.50454, antiderivative size = 291, 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, 773, 634, 618, 206, 628} \[ \frac{x^2 \left (b x \left (140 a^2 c^2-32 a b^2 c+3 b^4\right )+3 a \left (64 a^2 c^2-10 a b^2 c+b^4\right )\right )}{6 c^2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac{b x \left (38 a^2 c^2-11 a b^2 c+b^4\right )}{c^3 \left (b^2-4 a c\right )^3}+\frac{b \left (70 a^2 b^2 c^2-140 a^3 c^3-14 a b^4 c+b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{7/2}}+\frac{x^6 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{x^4 \left (b x \left (b^2-14 a c\right )+a \left (b^2-24 a c\right )\right )}{6 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac{\log \left (a+b x+c x^2\right )}{2 c^4} \]

Antiderivative was successfully verified.

[In]

Int[x^7/(a + b*x + c*x^2)^4,x]

[Out]

-((b*(b^4 - 11*a*b^2*c + 38*a^2*c^2)*x)/(c^3*(b^2 - 4*a*c)^3)) + (x^6*(2*a + b*x))/(3*(b^2 - 4*a*c)*(a + b*x +
c*x^2)^3) + (x^4*(a*(b^2 - 24*a*c) + b*(b^2 - 14*a*c)*x))/(6*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^2) + (x^2*(3
*a*(b^4 - 10*a*b^2*c + 64*a^2*c^2) + b*(3*b^4 - 32*a*b^2*c + 140*a^2*c^2)*x))/(6*c^2*(b^2 - 4*a*c)^3*(a + b*x
+ c*x^2)) + (b*(b^6 - 14*a*b^4*c + 70*a^2*b^2*c^2 - 140*a^3*c^3)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^4*
(b^2 - 4*a*c)^(7/2)) + 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 773

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*g*x)/
c, x] + Dist[1/c, Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*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]

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^7}{\left (a+b x+c x^2\right )^4} \, dx &=\frac{x^6 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac{\int \frac{x^5 (12 a+b x)}{\left (a+b x+c x^2\right )^3} \, dx}{3 \left (b^2-4 a c\right )}\\ &=\frac{x^6 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{x^4 \left (a \left (b^2-24 a c\right )+b \left (b^2-14 a c\right ) x\right )}{6 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac{\int \frac{x^3 \left (4 a \left (b^2-24 a c\right )+b \left (3 b^2-22 a c\right ) x\right )}{\left (a+b x+c x^2\right )^2} \, dx}{6 c \left (b^2-4 a c\right )^2}\\ &=\frac{x^6 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{x^4 \left (a \left (b^2-24 a c\right )+b \left (b^2-14 a c\right ) x\right )}{6 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac{x^2 \left (3 a \left (b^4-10 a b^2 c+64 a^2 c^2\right )+b \left (3 b^4-32 a b^2 c+140 a^2 c^2\right ) x\right )}{6 c^2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac{\int \frac{x \left (6 a \left (b^4-10 a b^2 c+64 a^2 c^2\right )+6 b \left (b^4-11 a b^2 c+38 a^2 c^2\right ) x\right )}{a+b x+c x^2} \, dx}{6 c^2 \left (b^2-4 a c\right )^3}\\ &=-\frac{b \left (b^4-11 a b^2 c+38 a^2 c^2\right ) x}{c^3 \left (b^2-4 a c\right )^3}+\frac{x^6 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{x^4 \left (a \left (b^2-24 a c\right )+b \left (b^2-14 a c\right ) x\right )}{6 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac{x^2 \left (3 a \left (b^4-10 a b^2 c+64 a^2 c^2\right )+b \left (3 b^4-32 a b^2 c+140 a^2 c^2\right ) x\right )}{6 c^2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac{\int \frac{-6 a b \left (b^4-11 a b^2 c+38 a^2 c^2\right )+\left (-6 b^2 \left (b^4-11 a b^2 c+38 a^2 c^2\right )+6 a c \left (b^4-10 a b^2 c+64 a^2 c^2\right )\right ) x}{a+b x+c x^2} \, dx}{6 c^3 \left (b^2-4 a c\right )^3}\\ &=-\frac{b \left (b^4-11 a b^2 c+38 a^2 c^2\right ) x}{c^3 \left (b^2-4 a c\right )^3}+\frac{x^6 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{x^4 \left (a \left (b^2-24 a c\right )+b \left (b^2-14 a c\right ) x\right )}{6 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac{x^2 \left (3 a \left (b^4-10 a b^2 c+64 a^2 c^2\right )+b \left (3 b^4-32 a b^2 c+140 a^2 c^2\right ) x\right )}{6 c^2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac{\int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 c^4}-\frac{\left (b \left (b^6-14 a b^4 c+70 a^2 b^2 c^2-140 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 )^3}\\ &=-\frac{b \left (b^4-11 a b^2 c+38 a^2 c^2\right ) x}{c^3 \left (b^2-4 a c\right )^3}+\frac{x^6 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{x^4 \left (a \left (b^2-24 a c\right )+b \left (b^2-14 a c\right ) x\right )}{6 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac{x^2 \left (3 a \left (b^4-10 a b^2 c+64 a^2 c^2\right )+b \left (3 b^4-32 a b^2 c+140 a^2 c^2\right ) x\right )}{6 c^2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac{\log \left (a+b x+c x^2\right )}{2 c^4}+\frac{\left (b \left (b^6-14 a b^4 c+70 a^2 b^2 c^2-140 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 )^3}\\ &=-\frac{b \left (b^4-11 a b^2 c+38 a^2 c^2\right ) x}{c^3 \left (b^2-4 a c\right )^3}+\frac{x^6 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{x^4 \left (a \left (b^2-24 a c\right )+b \left (b^2-14 a c\right ) x\right )}{6 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac{x^2 \left (3 a \left (b^4-10 a b^2 c+64 a^2 c^2\right )+b \left (3 b^4-32 a b^2 c+140 a^2 c^2\right ) x\right )}{6 c^2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac{b \left (b^6-14 a b^4 c+70 a^2 b^2 c^2-140 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 )^{7/2}}+\frac{\log \left (a+b x+c x^2\right )}{2 c^4}\\ \end{align*}

Mathematica [A]  time = 0.584542, size = 386, normalized size = 1.33 \[ \frac{-\frac{3 c \left (-266 a^2 b^3 c^3 x+191 a^2 b^4 c^2-374 a^3 b^2 c^3+308 a^3 b c^4 x+192 a^4 c^4+70 a b^5 c^2 x-40 a b^6 c-6 b^7 c x+3 b^8\right )}{\left (b^2-4 a c\right )^3 (a+x (b+c x))}+\frac{2 \left (2 a^2 b^3 c (7 c x-3 b)+a^3 b c^2 (9 b-7 c x)-2 a^4 c^3+a b^5 (b-7 c x)+b^7 x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^3}+\frac{259 a^2 b^3 c^3 x-139 a^2 b^4 c^2+233 a^3 b^2 c^3-182 a^3 b c^4 x-72 a^4 c^4-98 a b^5 c^2 x+29 a b^6 c+11 b^7 c x-2 b^8}{\left (b^2-4 a c\right )^2 (a+x (b+c x))^2}+\frac{6 b c^2 \left (70 a^2 b^2 c^2-140 a^3 c^3-14 a b^4 c+b^6\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{7/2}}+3 c^2 \log (a+x (b+c x))}{6 c^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a + b*x + c*x^2)^4,x]

[Out]

((-2*b^8 + 29*a*b^6*c - 139*a^2*b^4*c^2 + 233*a^3*b^2*c^3 - 72*a^4*c^4 + 11*b^7*c*x - 98*a*b^5*c^2*x + 259*a^2
*b^3*c^3*x - 182*a^3*b*c^4*x)/((b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) - (3*c*(3*b^8 - 40*a*b^6*c + 191*a^2*b^4*c
^2 - 374*a^3*b^2*c^3 + 192*a^4*c^4 - 6*b^7*c*x + 70*a*b^5*c^2*x - 266*a^2*b^3*c^3*x + 308*a^3*b*c^4*x))/((b^2
- 4*a*c)^3*(a + x*(b + c*x))) + (2*(-2*a^4*c^3 + b^7*x + a*b^5*(b - 7*c*x) + a^3*b*c^2*(9*b - 7*c*x) + 2*a^2*b
^3*c*(-3*b + 7*c*x)))/((b^2 - 4*a*c)*(a + x*(b + c*x))^3) + (6*b*c^2*(b^6 - 14*a*b^4*c + 70*a^2*b^2*c^2 - 140*
a^3*c^3)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2) + 3*c^2*Log[a + x*(b + c*x)])/(6*c^6)

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Maple [B]  time = 0.168, size = 1013, normalized size = 3.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(c*x^2+b*x+a)^4,x)

[Out]

((154*a^3*c^3-133*a^2*b^2*c^2+35*a*b^4*c-3*b^6)*b/c^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^5+1/2*(192*
a^4*c^4+242*a^3*b^2*c^3-341*a^2*b^4*c^2+100*a*b^6*c-9*b^8)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/c^3*x^4+
1/6*b/c^4*(2272*a^4*c^4-1698*a^3*b^2*c^3+117*a^2*b^4*c^2+76*a*b^6*c-11*b^8)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^
4*c-b^6)*x^3+1/2/c^4*a*(288*a^4*c^4+152*a^3*b^2*c^3-381*a^2*b^4*c^2+119*a*b^6*c-11*b^8)/(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*(428*a^3*c^3-460*a^2*b^2*c^2+126*a*b^4*c-11*b^6)/(64*a^3*c^3-48*a^2*b^2*c^2
+12*a*b^4*c-b^6)/c^4*x+1/6*(352*a^3*c^3-438*a^2*b^2*c^2+124*a*b^4*c-11*b^6)*a^3/c^4/(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+32/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/c*ln(c*x^2+b*x+a)*a^3-24/(64*a
^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/c^2*ln(c*x^2+b*x+a)*a^2*b^2+6/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)
/c^3*ln(c*x^2+b*x+a)*a*b^4-1/2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/c^4*ln(c*x^2+b*x+a)*b^6-140/(64*a^3*
c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^3*b+70/(64*a^3*c^
3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*b^3-14/(64*a^3*
c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/c^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^5+1/(64*a^3*c
^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/c^4/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^7

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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^7/(c*x^2+b*x+a)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.6627, size = 6337, normalized size = 21.78 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(c*x^2+b*x+a)^4,x, algorithm="fricas")

[Out]

[1/6*(11*a^3*b^8 - 168*a^4*b^6*c + 934*a^5*b^4*c^2 - 2104*a^6*b^2*c^3 + 1408*a^7*c^4 + 6*(3*b^9*c^2 - 47*a*b^7
*c^3 + 273*a^2*b^5*c^4 - 686*a^3*b^3*c^5 + 616*a^4*b*c^6)*x^5 + 3*(9*b^10*c - 136*a*b^8*c^2 + 741*a^2*b^6*c^3
- 1606*a^3*b^4*c^4 + 776*a^4*b^2*c^5 + 768*a^5*c^6)*x^4 + (11*b^11 - 120*a*b^9*c + 187*a^2*b^7*c^2 + 2166*a^3*
b^5*c^3 - 9064*a^4*b^3*c^4 + 9088*a^5*b*c^5)*x^3 + 3*(11*a*b^10 - 163*a^2*b^8*c + 857*a^3*b^6*c^2 - 1676*a^4*b
^4*c^3 + 320*a^5*b^2*c^4 + 1152*a^6*c^5)*x^2 + 3*(a^3*b^7 - 14*a^4*b^5*c + 70*a^5*b^3*c^2 - 140*a^6*b*c^3 + (b
^7*c^3 - 14*a*b^5*c^4 + 70*a^2*b^3*c^5 - 140*a^3*b*c^6)*x^6 + 3*(b^8*c^2 - 14*a*b^6*c^3 + 70*a^2*b^4*c^4 - 140
*a^3*b^2*c^5)*x^5 + 3*(b^9*c - 13*a*b^7*c^2 + 56*a^2*b^5*c^3 - 70*a^3*b^3*c^4 - 140*a^4*b*c^5)*x^4 + (b^10 - 8
*a*b^8*c - 14*a^2*b^6*c^2 + 280*a^3*b^4*c^3 - 840*a^4*b^2*c^4)*x^3 + 3*(a*b^9 - 13*a^2*b^7*c + 56*a^3*b^5*c^2
- 70*a^4*b^3*c^3 - 140*a^5*b*c^4)*x^2 + 3*(a^2*b^8 - 14*a^3*b^6*c + 70*a^4*b^4*c^2 - 140*a^5*b^2*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)) + 3*(1
1*a^2*b^9 - 170*a^3*b^7*c + 964*a^4*b^5*c^2 - 2268*a^5*b^3*c^3 + 1712*a^6*b*c^4)*x + 3*(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 + 2
56*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)*log(c*x^2 +
b*x + a))/(a^3*b^8*c^4 - 16*a^4*b^6*c^5 + 96*a^5*b^4*c^6 - 256*a^6*b^2*c^7 + 256*a^7*c^8 + (b^8*c^7 - 16*a*b^
6*c^8 + 96*a^2*b^4*c^9 - 256*a^3*b^2*c^10 + 256*a^4*c^11)*x^6 + 3*(b^9*c^6 - 16*a*b^7*c^7 + 96*a^2*b^5*c^8 - 2
56*a^3*b^3*c^9 + 256*a^4*b*c^10)*x^5 + 3*(b^10*c^5 - 15*a*b^8*c^6 + 80*a^2*b^6*c^7 - 160*a^3*b^4*c^8 + 256*a^5
*c^10)*x^4 + (b^11*c^4 - 10*a*b^9*c^5 + 320*a^3*b^5*c^7 - 1280*a^4*b^3*c^8 + 1536*a^5*b*c^9)*x^3 + 3*(a*b^10*c
^4 - 15*a^2*b^8*c^5 + 80*a^3*b^6*c^6 - 160*a^4*b^4*c^7 + 256*a^6*c^9)*x^2 + 3*(a^2*b^9*c^4 - 16*a^3*b^7*c^5 +
96*a^4*b^5*c^6 - 256*a^5*b^3*c^7 + 256*a^6*b*c^8)*x), 1/6*(11*a^3*b^8 - 168*a^4*b^6*c + 934*a^5*b^4*c^2 - 2104
*a^6*b^2*c^3 + 1408*a^7*c^4 + 6*(3*b^9*c^2 - 47*a*b^7*c^3 + 273*a^2*b^5*c^4 - 686*a^3*b^3*c^5 + 616*a^4*b*c^6)
*x^5 + 3*(9*b^10*c - 136*a*b^8*c^2 + 741*a^2*b^6*c^3 - 1606*a^3*b^4*c^4 + 776*a^4*b^2*c^5 + 768*a^5*c^6)*x^4 +
(11*b^11 - 120*a*b^9*c + 187*a^2*b^7*c^2 + 2166*a^3*b^5*c^3 - 9064*a^4*b^3*c^4 + 9088*a^5*b*c^5)*x^3 + 3*(11*
a*b^10 - 163*a^2*b^8*c + 857*a^3*b^6*c^2 - 1676*a^4*b^4*c^3 + 320*a^5*b^2*c^4 + 1152*a^6*c^5)*x^2 + 6*(a^3*b^7
- 14*a^4*b^5*c + 70*a^5*b^3*c^2 - 140*a^6*b*c^3 + (b^7*c^3 - 14*a*b^5*c^4 + 70*a^2*b^3*c^5 - 140*a^3*b*c^6)*x
^6 + 3*(b^8*c^2 - 14*a*b^6*c^3 + 70*a^2*b^4*c^4 - 140*a^3*b^2*c^5)*x^5 + 3*(b^9*c - 13*a*b^7*c^2 + 56*a^2*b^5*
c^3 - 70*a^3*b^3*c^4 - 140*a^4*b*c^5)*x^4 + (b^10 - 8*a*b^8*c - 14*a^2*b^6*c^2 + 280*a^3*b^4*c^3 - 840*a^4*b^2
*c^4)*x^3 + 3*(a*b^9 - 13*a^2*b^7*c + 56*a^3*b^5*c^2 - 70*a^4*b^3*c^3 - 140*a^5*b*c^4)*x^2 + 3*(a^2*b^8 - 14*a
^3*b^6*c + 70*a^4*b^4*c^2 - 140*a^5*b^2*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)) + 3*(11*a^2*b^9 - 170*a^3*b^7*c + 964*a^4*b^5*c^2 - 2268*a^5*b^3*c^3 + 1712*a^6*b*c^4)*x + 3*(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 - 16
0*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)*log(c*x^2 + b*x + a))/(a^3*b^8*c^4 - 16*a^4*b^6*c^5 + 96*a^5*b^4*c^6 - 256*a^6*b^2*c^7 + 256*a^7*c^8 + (
b^8*c^7 - 16*a*b^6*c^8 + 96*a^2*b^4*c^9 - 256*a^3*b^2*c^10 + 256*a^4*c^11)*x^6 + 3*(b^9*c^6 - 16*a*b^7*c^7 + 9
6*a^2*b^5*c^8 - 256*a^3*b^3*c^9 + 256*a^4*b*c^10)*x^5 + 3*(b^10*c^5 - 15*a*b^8*c^6 + 80*a^2*b^6*c^7 - 160*a^3*
b^4*c^8 + 256*a^5*c^10)*x^4 + (b^11*c^4 - 10*a*b^9*c^5 + 320*a^3*b^5*c^7 - 1280*a^4*b^3*c^8 + 1536*a^5*b*c^9)*
x^3 + 3*(a*b^10*c^4 - 15*a^2*b^8*c^5 + 80*a^3*b^6*c^6 - 160*a^4*b^4*c^7 + 256*a^6*c^9)*x^2 + 3*(a^2*b^9*c^4 -
16*a^3*b^7*c^5 + 96*a^4*b^5*c^6 - 256*a^5*b^3*c^7 + 256*a^6*b*c^8)*x)]

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Sympy [B]  time = 7.28591, size = 2565, normalized size = 8.81 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(c*x**2+b*x+a)**4,x)

[Out]

(-b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3 - 70*a**2*b**2*c**2 + 14*a*b**4*c - b**6)/(2*c**4*(16384*a**7*c**7
- 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c*
*2 + 28*a*b**12*c - b**14)) + 1/(2*c**4))*log(x + (-256*a**4*c**7*(-b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3
- 70*a**2*b**2*c**2 + 14*a*b**4*c - b**6)/(2*c**4*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c*
*5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)) + 1/(2*c**4)) +
128*a**4*c**3 + 256*a**3*b**2*c**6*(-b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3 - 70*a**2*b**2*c**2 + 14*a*b**4
*c - b**6)/(2*c**4*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240
*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)) + 1/(2*c**4)) - 58*a**3*b**2*c**2 - 96*a**2*b**
4*c**5*(-b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3 - 70*a**2*b**2*c**2 + 14*a*b**4*c - b**6)/(2*c**4*(16384*a*
*7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b
**10*c**2 + 28*a*b**12*c - b**14)) + 1/(2*c**4)) + 13*a**2*b**4*c + 16*a*b**6*c**4*(-b*sqrt(-(4*a*c - b**2)**7
)*(140*a**3*c**3 - 70*a**2*b**2*c**2 + 14*a*b**4*c - b**6)/(2*c**4*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 2
1504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14))
+ 1/(2*c**4)) - a*b**6 - b**8*c**3*(-b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3 - 70*a**2*b**2*c**2 + 14*a*b**
4*c - b**6)/(2*c**4*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 224
0*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)) + 1/(2*c**4)))/(140*a**3*b*c**3 - 70*a**2*b**3
*c**2 + 14*a*b**5*c - b**7)) + (b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3 - 70*a**2*b**2*c**2 + 14*a*b**4*c -
b**6)/(2*c**4*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3
*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)) + 1/(2*c**4))*log(x + (-256*a**4*c**7*(b*sqrt(-(4*a*
c - b**2)**7)*(140*a**3*c**3 - 70*a**2*b**2*c**2 + 14*a*b**4*c - b**6)/(2*c**4*(16384*a**7*c**7 - 28672*a**6*b
**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12
*c - b**14)) + 1/(2*c**4)) + 128*a**4*c**3 + 256*a**3*b**2*c**6*(b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3 - 7
0*a**2*b**2*c**2 + 14*a*b**4*c - b**6)/(2*c**4*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5
- 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)) + 1/(2*c**4)) - 58*
a**3*b**2*c**2 - 96*a**2*b**4*c**5*(b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3 - 70*a**2*b**2*c**2 + 14*a*b**4*
c - b**6)/(2*c**4*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*
a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)) + 1/(2*c**4)) + 13*a**2*b**4*c + 16*a*b**6*c**4*
(b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3 - 70*a**2*b**2*c**2 + 14*a*b**4*c - b**6)/(2*c**4*(16384*a**7*c**7
- 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**
2 + 28*a*b**12*c - b**14)) + 1/(2*c**4)) - a*b**6 - b**8*c**3*(b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3 - 70*
a**2*b**2*c**2 + 14*a*b**4*c - b**6)/(2*c**4*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 -
8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)) + 1/(2*c**4)))/(140*a
**3*b*c**3 - 70*a**2*b**3*c**2 + 14*a*b**5*c - b**7)) + (352*a**6*c**3 - 438*a**5*b**2*c**2 + 124*a**4*b**4*c
- 11*a**3*b**6 + x**5*(924*a**3*b*c**5 - 798*a**2*b**3*c**4 + 210*a*b**5*c**3 - 18*b**7*c**2) + x**4*(576*a**4
*c**5 + 726*a**3*b**2*c**4 - 1023*a**2*b**4*c**3 + 300*a*b**6*c**2 - 27*b**8*c) + x**3*(2272*a**4*b*c**4 - 169
8*a**3*b**3*c**3 + 117*a**2*b**5*c**2 + 76*a*b**7*c - 11*b**9) + x**2*(864*a**5*c**4 + 456*a**4*b**2*c**3 - 11
43*a**3*b**4*c**2 + 357*a**2*b**6*c - 33*a*b**8) + x*(1284*a**5*b*c**3 - 1380*a**4*b**3*c**2 + 378*a**3*b**5*c
- 33*a**2*b**7))/(384*a**6*c**7 - 288*a**5*b**2*c**6 + 72*a**4*b**4*c**5 - 6*a**3*b**6*c**4 + x**6*(384*a**3*
c**10 - 288*a**2*b**2*c**9 + 72*a*b**4*c**8 - 6*b**6*c**7) + x**5*(1152*a**3*b*c**9 - 864*a**2*b**3*c**8 + 216
*a*b**5*c**7 - 18*b**7*c**6) + x**4*(1152*a**4*c**9 + 288*a**3*b**2*c**8 - 648*a**2*b**4*c**7 + 198*a*b**6*c**
6 - 18*b**8*c**5) + x**3*(2304*a**4*b*c**8 - 1344*a**3*b**3*c**7 + 144*a**2*b**5*c**6 + 36*a*b**7*c**5 - 6*b**
9*c**4) + x**2*(1152*a**5*c**8 + 288*a**4*b**2*c**7 - 648*a**3*b**4*c**6 + 198*a**2*b**6*c**5 - 18*a*b**8*c**4
) + x*(1152*a**5*b*c**7 - 864*a**4*b**3*c**6 + 216*a**3*b**5*c**5 - 18*a**2*b**7*c**4))

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Giac [A]  time = 1.12191, size = 564, normalized size = 1.94 \begin{align*} -\frac{{\left (b^{7} - 14 \, a b^{5} c + 70 \, a^{2} b^{3} c^{2} - 140 \, a^{3} b c^{3}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{6} c^{4} - 12 \, a b^{4} c^{5} + 48 \, a^{2} b^{2} c^{6} - 64 \, a^{3} c^{7}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{\log \left (c x^{2} + b x + a\right )}{2 \, c^{4}} + \frac{11 \, a^{3} b^{6} - 124 \, a^{4} b^{4} c + 438 \, a^{5} b^{2} c^{2} - 352 \, a^{6} c^{3} + 6 \,{\left (3 \, b^{7} c^{2} - 35 \, a b^{5} c^{3} + 133 \, a^{2} b^{3} c^{4} - 154 \, a^{3} b c^{5}\right )} x^{5} + 3 \,{\left (9 \, b^{8} c - 100 \, a b^{6} c^{2} + 341 \, a^{2} b^{4} c^{3} - 242 \, a^{3} b^{2} c^{4} - 192 \, a^{4} c^{5}\right )} x^{4} +{\left (11 \, b^{9} - 76 \, a b^{7} c - 117 \, a^{2} b^{5} c^{2} + 1698 \, a^{3} b^{3} c^{3} - 2272 \, a^{4} b c^{4}\right )} x^{3} + 3 \,{\left (11 \, a b^{8} - 119 \, a^{2} b^{6} c + 381 \, a^{3} b^{4} c^{2} - 152 \, a^{4} b^{2} c^{3} - 288 \, a^{5} c^{4}\right )} x^{2} + 3 \,{\left (11 \, a^{2} b^{7} - 126 \, a^{3} b^{5} c + 460 \, a^{4} b^{3} c^{2} - 428 \, a^{5} b c^{3}\right )} x}{6 \,{\left (c x^{2} + b x + a\right )}^{3}{\left (b^{2} - 4 \, a c\right )}^{3} c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(c*x^2+b*x+a)^4,x, algorithm="giac")

[Out]

-(b^7 - 14*a*b^5*c + 70*a^2*b^3*c^2 - 140*a^3*b*c^3)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^6*c^4 - 12*a*b
^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*sqrt(-b^2 + 4*a*c)) + 1/2*log(c*x^2 + b*x + a)/c^4 + 1/6*(11*a^3*b^6 - 1
24*a^4*b^4*c + 438*a^5*b^2*c^2 - 352*a^6*c^3 + 6*(3*b^7*c^2 - 35*a*b^5*c^3 + 133*a^2*b^3*c^4 - 154*a^3*b*c^5)*
x^5 + 3*(9*b^8*c - 100*a*b^6*c^2 + 341*a^2*b^4*c^3 - 242*a^3*b^2*c^4 - 192*a^4*c^5)*x^4 + (11*b^9 - 76*a*b^7*c
- 117*a^2*b^5*c^2 + 1698*a^3*b^3*c^3 - 2272*a^4*b*c^4)*x^3 + 3*(11*a*b^8 - 119*a^2*b^6*c + 381*a^3*b^4*c^2 -
152*a^4*b^2*c^3 - 288*a^5*c^4)*x^2 + 3*(11*a^2*b^7 - 126*a^3*b^5*c + 460*a^4*b^3*c^2 - 428*a^5*b*c^3)*x)/((c*x
^2 + b*x + a)^3*(b^2 - 4*a*c)^3*c^4)