3.19.68 \(\int \frac {x^7}{(a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=294 \[ \frac {3 x^2 \left (16 a^2 c^2-13 a b^2 c+2 b^4\right )}{2 c^3 \left (b^2-4 a c\right )^2}+\frac {3 b \left (-70 a^3 c^3+70 a^2 b^2 c^2-21 a b^4 c+2 b^6\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^5 \left (b^2-4 a c\right )^{5/2}}+\frac {3 \left (2 b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 c^5}-\frac {3 b x \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{c^4 \left (b^2-4 a c\right )^2}-\frac {b x^3 \left (2 b^2-11 a c\right )}{c^2 \left (b^2-4 a c\right )^2}+\frac {x^6 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 x^4 \left (b x \left (b^2-6 a c\right )+a \left (b^2-8 a c\right )\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )} \]

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Rubi [A]  time = 0.42, antiderivative size = 294, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {3 x^2 \left (16 a^2 c^2-13 a b^2 c+2 b^4\right )}{2 c^3 \left (b^2-4 a c\right )^2}+\frac {3 b \left (70 a^2 b^2 c^2-70 a^3 c^3-21 a b^4 c+2 b^6\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^5 \left (b^2-4 a c\right )^{5/2}}-\frac {b x^3 \left (2 b^2-11 a c\right )}{c^2 \left (b^2-4 a c\right )^2}+\frac {3 \left (2 b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 c^5}-\frac {3 b x \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{c^4 \left (b^2-4 a c\right )^2}+\frac {x^6 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 x^4 \left (b x \left (b^2-6 a c\right )+a \left (b^2-8 a c\right )\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7/(a + b*x + c*x^2)^3,x]

[Out]

(-3*b*(2*b^2 - 9*a*c)*(b^2 - 3*a*c)*x)/(c^4*(b^2 - 4*a*c)^2) + (3*(2*b^4 - 13*a*b^2*c + 16*a^2*c^2)*x^2)/(2*c^
3*(b^2 - 4*a*c)^2) - (b*(2*b^2 - 11*a*c)*x^3)/(c^2*(b^2 - 4*a*c)^2) + (x^6*(2*a + b*x))/(2*(b^2 - 4*a*c)*(a +
b*x + c*x^2)^2) + (3*x^4*(a*(b^2 - 8*a*c) + b*(b^2 - 6*a*c)*x))/(2*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) + (3*b
*(2*b^6 - 21*a*b^4*c + 70*a^2*b^2*c^2 - 70*a^3*c^3)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^5*(b^2 - 4*a*c)
^(5/2)) + (3*(2*b^2 - a*c)*Log[a + b*x + c*x^2])/(2*c^5)

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 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 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]

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 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 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 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])

Rubi steps

\begin {align*} \int \frac {x^7}{\left (a+b x+c x^2\right )^3} \, dx &=\frac {x^6 (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^5 (12 a+3 b x)}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )}\\ &=\frac {x^6 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 x^4 \left (a \left (b^2-8 a c\right )+b \left (b^2-6 a c\right ) x\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\int \frac {x^3 \left (12 a \left (b^2-8 a c\right )+6 b \left (2 b^2-11 a c\right ) x\right )}{a+b x+c x^2} \, dx}{2 c \left (b^2-4 a c\right )^2}\\ &=\frac {x^6 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 x^4 \left (a \left (b^2-8 a c\right )+b \left (b^2-6 a c\right ) x\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\int \left (\frac {6 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{c^3}-\frac {6 \left (2 b^4-13 a b^2 c+16 a^2 c^2\right ) x}{c^2}+\frac {6 b \left (2 b^2-11 a c\right ) x^2}{c}-\frac {6 \left (a b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )+\left (b^2-4 a c\right )^2 \left (2 b^2-a c\right ) x\right )}{c^3 \left (a+b x+c x^2\right )}\right ) \, dx}{2 c \left (b^2-4 a c\right )^2}\\ &=-\frac {3 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) x}{c^4 \left (b^2-4 a c\right )^2}+\frac {3 \left (2 b^4-13 a b^2 c+16 a^2 c^2\right ) x^2}{2 c^3 \left (b^2-4 a c\right )^2}-\frac {b \left (2 b^2-11 a c\right ) x^3}{c^2 \left (b^2-4 a c\right )^2}+\frac {x^6 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 x^4 \left (a \left (b^2-8 a c\right )+b \left (b^2-6 a c\right ) x\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {3 \int \frac {a b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )+\left (b^2-4 a c\right )^2 \left (2 b^2-a c\right ) x}{a+b x+c x^2} \, dx}{c^4 \left (b^2-4 a c\right )^2}\\ &=-\frac {3 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) x}{c^4 \left (b^2-4 a c\right )^2}+\frac {3 \left (2 b^4-13 a b^2 c+16 a^2 c^2\right ) x^2}{2 c^3 \left (b^2-4 a c\right )^2}-\frac {b \left (2 b^2-11 a c\right ) x^3}{c^2 \left (b^2-4 a c\right )^2}+\frac {x^6 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 x^4 \left (a \left (b^2-8 a c\right )+b \left (b^2-6 a c\right ) x\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\left (3 \left (2 b^2-a c\right )\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^5}-\frac {\left (3 b \left (2 b^6-21 a b^4 c+70 a^2 b^2 c^2-70 a^3 c^3\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c^5 \left (b^2-4 a c\right )^2}\\ &=-\frac {3 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) x}{c^4 \left (b^2-4 a c\right )^2}+\frac {3 \left (2 b^4-13 a b^2 c+16 a^2 c^2\right ) x^2}{2 c^3 \left (b^2-4 a c\right )^2}-\frac {b \left (2 b^2-11 a c\right ) x^3}{c^2 \left (b^2-4 a c\right )^2}+\frac {x^6 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 x^4 \left (a \left (b^2-8 a c\right )+b \left (b^2-6 a c\right ) x\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {3 \left (2 b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 c^5}+\frac {\left (3 b \left (2 b^6-21 a b^4 c+70 a^2 b^2 c^2-70 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^5 \left (b^2-4 a c\right )^2}\\ &=-\frac {3 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) x}{c^4 \left (b^2-4 a c\right )^2}+\frac {3 \left (2 b^4-13 a b^2 c+16 a^2 c^2\right ) x^2}{2 c^3 \left (b^2-4 a c\right )^2}-\frac {b \left (2 b^2-11 a c\right ) x^3}{c^2 \left (b^2-4 a c\right )^2}+\frac {x^6 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 x^4 \left (a \left (b^2-8 a c\right )+b \left (b^2-6 a c\right ) x\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {3 b \left (2 b^6-21 a b^4 c+70 a^2 b^2 c^2-70 a^3 c^3\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^5 \left (b^2-4 a c\right )^{5/2}}+\frac {3 \left (2 b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 c^5}\\ \end {align*}

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Mathematica [A]  time = 0.42, size = 299, normalized size = 1.02 \begin {gather*} \frac {\frac {6 b c \left (70 a^3 c^3-70 a^2 b^2 c^2+21 a b^4 c-2 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}}+\frac {-2 a^4 c^3+a^3 b c^2 (9 b-7 c x)+2 a^2 b^3 c (7 c x-3 b)+a b^5 (b-7 c x)+b^7 x}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}-\frac {48 a^4 c^4-153 a^3 b^2 c^3+126 a^3 b c^4 x+88 a^2 b^4 c^2-182 a^2 b^3 c^3 x-17 a b^6 c+70 a b^5 c^2 x+b^8-8 b^7 c x}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}-3 c \left (a c-2 b^2\right ) \log (a+x (b+c x))-6 b c^2 x+c^3 x^2}{2 c^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a + b*x + c*x^2)^3,x]

[Out]

(-6*b*c^2*x + c^3*x^2 - (b^8 - 17*a*b^6*c + 88*a^2*b^4*c^2 - 153*a^3*b^2*c^3 + 48*a^4*c^4 - 8*b^7*c*x + 70*a*b
^5*c^2*x - 182*a^2*b^3*c^3*x + 126*a^3*b*c^4*x)/((b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (-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))^2) +
(6*b*c*(-2*b^6 + 21*a*b^4*c - 70*a^2*b^2*c^2 + 70*a^3*c^3)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a
*c)^(5/2) - 3*c*(-2*b^2 + a*c)*Log[a + x*(b + c*x)])/(2*c^6)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^7}{\left (a+b x+c x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^7/(a + b*x + c*x^2)^3,x]

[Out]

IntegrateAlgebraic[x^7/(a + b*x + c*x^2)^3, x]

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fricas [B]  time = 0.49, size = 2207, normalized size = 7.51

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

[1/2*(7*a^2*b^8 - 83*a^3*b^6*c + 335*a^4*b^4*c^2 - 500*a^5*b^2*c^3 + 160*a^6*c^4 + (b^6*c^4 - 12*a*b^4*c^5 + 4
8*a^2*b^2*c^6 - 64*a^3*c^7)*x^6 - 4*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*x^5 - (11*b^8*c^2
- 134*a*b^6*c^3 + 552*a^2*b^4*c^4 - 800*a^3*b^2*c^5 + 128*a^4*c^6)*x^4 + 2*(b^9*c - 20*a*b^7*c^2 + 147*a^2*b^
5*c^3 - 475*a^3*b^3*c^4 + 572*a^4*b*c^5)*x^3 + (7*b^10 - 93*a*b^8*c + 451*a^2*b^6*c^2 - 937*a^3*b^4*c^3 + 660*
a^4*b^2*c^4 + 128*a^5*c^5)*x^2 - 3*(2*a^2*b^7 - 21*a^3*b^5*c + 70*a^4*b^3*c^2 - 70*a^5*b*c^3 + (2*b^7*c^2 - 21
*a*b^5*c^3 + 70*a^2*b^3*c^4 - 70*a^3*b*c^5)*x^4 + 2*(2*b^8*c - 21*a*b^6*c^2 + 70*a^2*b^4*c^3 - 70*a^3*b^2*c^4)
*x^3 + (2*b^9 - 17*a*b^7*c + 28*a^2*b^5*c^2 + 70*a^3*b^3*c^3 - 140*a^4*b*c^4)*x^2 + 2*(2*a*b^8 - 21*a^2*b^6*c
+ 70*a^3*b^4*c^2 - 70*a^4*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)) + 2*(7*a*b^9 - 89*a^2*b^7*c + 404*a^3*b^5*c^2 - 761*a^4*b^3*c^3 + 484*a^5
*b*c^4)*x + 3*(2*a^2*b^8 - 25*a^3*b^6*c + 108*a^4*b^4*c^2 - 176*a^5*b^2*c^3 + 64*a^6*c^4 + (2*b^8*c^2 - 25*a*b
^6*c^3 + 108*a^2*b^4*c^4 - 176*a^3*b^2*c^5 + 64*a^4*c^6)*x^4 + 2*(2*b^9*c - 25*a*b^7*c^2 + 108*a^2*b^5*c^3 - 1
76*a^3*b^3*c^4 + 64*a^4*b*c^5)*x^3 + (2*b^10 - 21*a*b^8*c + 58*a^2*b^6*c^2 + 40*a^3*b^4*c^3 - 288*a^4*b^2*c^4
+ 128*a^5*c^5)*x^2 + 2*(2*a*b^9 - 25*a^2*b^7*c + 108*a^3*b^5*c^2 - 176*a^4*b^3*c^3 + 64*a^5*b*c^4)*x)*log(c*x^
2 + b*x + a))/(a^2*b^6*c^5 - 12*a^3*b^4*c^6 + 48*a^4*b^2*c^7 - 64*a^5*c^8 + (b^6*c^7 - 12*a*b^4*c^8 + 48*a^2*b
^2*c^9 - 64*a^3*c^10)*x^4 + 2*(b^7*c^6 - 12*a*b^5*c^7 + 48*a^2*b^3*c^8 - 64*a^3*b*c^9)*x^3 + (b^8*c^5 - 10*a*b
^6*c^6 + 24*a^2*b^4*c^7 + 32*a^3*b^2*c^8 - 128*a^4*c^9)*x^2 + 2*(a*b^7*c^5 - 12*a^2*b^5*c^6 + 48*a^3*b^3*c^7 -
64*a^4*b*c^8)*x), 1/2*(7*a^2*b^8 - 83*a^3*b^6*c + 335*a^4*b^4*c^2 - 500*a^5*b^2*c^3 + 160*a^6*c^4 + (b^6*c^4
- 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*x^6 - 4*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)
*x^5 - (11*b^8*c^2 - 134*a*b^6*c^3 + 552*a^2*b^4*c^4 - 800*a^3*b^2*c^5 + 128*a^4*c^6)*x^4 + 2*(b^9*c - 20*a*b^
7*c^2 + 147*a^2*b^5*c^3 - 475*a^3*b^3*c^4 + 572*a^4*b*c^5)*x^3 + (7*b^10 - 93*a*b^8*c + 451*a^2*b^6*c^2 - 937*
a^3*b^4*c^3 + 660*a^4*b^2*c^4 + 128*a^5*c^5)*x^2 + 6*(2*a^2*b^7 - 21*a^3*b^5*c + 70*a^4*b^3*c^2 - 70*a^5*b*c^3
+ (2*b^7*c^2 - 21*a*b^5*c^3 + 70*a^2*b^3*c^4 - 70*a^3*b*c^5)*x^4 + 2*(2*b^8*c - 21*a*b^6*c^2 + 70*a^2*b^4*c^3
- 70*a^3*b^2*c^4)*x^3 + (2*b^9 - 17*a*b^7*c + 28*a^2*b^5*c^2 + 70*a^3*b^3*c^3 - 140*a^4*b*c^4)*x^2 + 2*(2*a*b
^8 - 21*a^2*b^6*c + 70*a^3*b^4*c^2 - 70*a^4*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)) + 2*(7*a*b^9 - 89*a^2*b^7*c + 404*a^3*b^5*c^2 - 761*a^4*b^3*c^3 + 484*a^5*b*c^4)*x + 3*(2*a
^2*b^8 - 25*a^3*b^6*c + 108*a^4*b^4*c^2 - 176*a^5*b^2*c^3 + 64*a^6*c^4 + (2*b^8*c^2 - 25*a*b^6*c^3 + 108*a^2*b
^4*c^4 - 176*a^3*b^2*c^5 + 64*a^4*c^6)*x^4 + 2*(2*b^9*c - 25*a*b^7*c^2 + 108*a^2*b^5*c^3 - 176*a^3*b^3*c^4 + 6
4*a^4*b*c^5)*x^3 + (2*b^10 - 21*a*b^8*c + 58*a^2*b^6*c^2 + 40*a^3*b^4*c^3 - 288*a^4*b^2*c^4 + 128*a^5*c^5)*x^2
+ 2*(2*a*b^9 - 25*a^2*b^7*c + 108*a^3*b^5*c^2 - 176*a^4*b^3*c^3 + 64*a^5*b*c^4)*x)*log(c*x^2 + b*x + a))/(a^2
*b^6*c^5 - 12*a^3*b^4*c^6 + 48*a^4*b^2*c^7 - 64*a^5*c^8 + (b^6*c^7 - 12*a*b^4*c^8 + 48*a^2*b^2*c^9 - 64*a^3*c^
10)*x^4 + 2*(b^7*c^6 - 12*a*b^5*c^7 + 48*a^2*b^3*c^8 - 64*a^3*b*c^9)*x^3 + (b^8*c^5 - 10*a*b^6*c^6 + 24*a^2*b^
4*c^7 + 32*a^3*b^2*c^8 - 128*a^4*c^9)*x^2 + 2*(a*b^7*c^5 - 12*a^2*b^5*c^6 + 48*a^3*b^3*c^7 - 64*a^4*b*c^8)*x)]

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giac [A]  time = 0.22, size = 332, normalized size = 1.13 \begin {gather*} -\frac {3 \, {\left (2 \, b^{7} - 21 \, a b^{5} c + 70 \, a^{2} b^{3} c^{2} - 70 \, a^{3} b c^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} c^{5} - 8 \, a b^{2} c^{6} + 16 \, a^{2} c^{7}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {3 \, {\left (2 \, b^{2} - a c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{5}} + \frac {c^{3} x^{2} - 6 \, b c^{2} x}{2 \, c^{6}} + \frac {7 \, a^{2} b^{6} - 55 \, a^{3} b^{4} c + 115 \, a^{4} b^{2} c^{2} - 40 \, a^{5} c^{3} + 2 \, {\left (4 \, b^{7} c - 35 \, a b^{5} c^{2} + 91 \, a^{2} b^{3} c^{3} - 63 \, a^{3} b c^{4}\right )} x^{3} + {\left (7 \, b^{8} - 53 \, a b^{6} c + 94 \, a^{2} b^{4} c^{2} + 27 \, a^{3} b^{2} c^{3} - 48 \, a^{4} c^{4}\right )} x^{2} + 2 \, {\left (7 \, a b^{7} - 58 \, a^{2} b^{5} c + 136 \, a^{3} b^{3} c^{2} - 73 \, a^{4} b c^{3}\right )} x}{2 \, {\left (c x^{2} + b x + a\right )}^{2} {\left (b^{2} - 4 \, a c\right )}^{2} c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

-3*(2*b^7 - 21*a*b^5*c + 70*a^2*b^3*c^2 - 70*a^3*b*c^3)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^4*c^5 - 8*a
*b^2*c^6 + 16*a^2*c^7)*sqrt(-b^2 + 4*a*c)) + 3/2*(2*b^2 - a*c)*log(c*x^2 + b*x + a)/c^5 + 1/2*(c^3*x^2 - 6*b*c
^2*x)/c^6 + 1/2*(7*a^2*b^6 - 55*a^3*b^4*c + 115*a^4*b^2*c^2 - 40*a^5*c^3 + 2*(4*b^7*c - 35*a*b^5*c^2 + 91*a^2*
b^3*c^3 - 63*a^3*b*c^4)*x^3 + (7*b^8 - 53*a*b^6*c + 94*a^2*b^4*c^2 + 27*a^3*b^2*c^3 - 48*a^4*c^4)*x^2 + 2*(7*a
*b^7 - 58*a^2*b^5*c + 136*a^3*b^3*c^2 - 73*a^4*b*c^3)*x)/((c*x^2 + b*x + a)^2*(b^2 - 4*a*c)^2*c^5)

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maple [B]  time = 0.08, size = 1187, normalized size = 4.04

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(c*x^2+b*x+a)^3,x)

[Out]

-73/c^2/(c*x^2+b*x+a)^2*a^4*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x+1/2/c^3*x^2-3/c^4*x*b-20/c^2/(c*x^2+b*x+a)^2*a^5/(1
6*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^3+3/c^5/(16*a^2*c^2-8*a*b^2*c+b^4
)*ln(c*x^2+b*x+a)*b^6+115/2/c^3/(c*x^2+b*x+a)^2*a^4/(16*a^2*c^2-8*a*b^2*c+b^4)*b^2+7/2/c^5/(c*x^2+b*x+a)^2*a^2
/(16*a^2*c^2-8*a*b^2*c+b^4)*b^6+60/c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*ln(c*x^2+b*x+a)*a^2*b^2-51/2/c^4/(16*a^2*c^2
-8*a*b^2*c+b^4)*ln(c*x^2+b*x+a)*a*b^4-24/c/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*a^4+27/2/c^2/(c*x^2+
b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*a^3*b^2+47/c^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*a^2*b^4-
55/2/c^4/(c*x^2+b*x+a)^2*a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*b^4-6/c^5/(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^7-35/c^3/(c*x^2+b*x+a)^2*b^5/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*a+4/c^4/(c*
x^2+b*x+a)^2*b^7/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+7/2/c^5/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b^8+136
/c^3/(c*x^2+b*x+a)^2*a^3*b^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x+7/c^5/(c*x^2+b*x+a)^2*a*b^7/(16*a^2*c^2-8*a*b^2*c+b^
4)*x-58/c^4/(c*x^2+b*x+a)^2*a^2*b^5/(16*a^2*c^2-8*a*b^2*c+b^4)*x+63/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))*a*b^5-210/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^2*b^3+210/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^3*b-63/c/(c*x^2+b*x+a)^2*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*a^3+91/c^2/(c*x^2+b*x+a)^2*b^3/(16*
a^2*c^2-8*a*b^2*c+b^4)*x^3*a^2-53/2/c^4/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*a*b^6

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(c*x^2+b*x+a)^3,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 details)Is 4*a*c-b^2 positive or negative?

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mupad [B]  time = 1.47, size = 762, normalized size = 2.59 \begin {gather*} \frac {\frac {a\,\left (-40\,a^4\,c^3+115\,a^3\,b^2\,c^2-55\,a^2\,b^4\,c+7\,a\,b^6\right )}{2\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {b\,x^3\,\left (-63\,a^3\,c^3+91\,a^2\,b^2\,c^2-35\,a\,b^4\,c+4\,b^6\right )}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {x^2\,\left (-48\,a^4\,c^4+27\,a^3\,b^2\,c^3+94\,a^2\,b^4\,c^2-53\,a\,b^6\,c+7\,b^8\right )}{2\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {b\,x\,\left (-73\,a^4\,c^3+136\,a^3\,b^2\,c^2-58\,a^2\,b^4\,c+7\,a\,b^6\right )}{c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{a^2\,c^4+c^6\,x^4+x^2\,\left (b^2\,c^4+2\,a\,c^5\right )+2\,b\,c^5\,x^3+2\,a\,b\,c^4\,x}+\frac {x^2}{2\,c^3}-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (3072\,a^6\,c^6-9984\,a^5\,b^2\,c^5+9600\,a^4\,b^4\,c^4-4320\,a^3\,b^6\,c^3+1020\,a^2\,b^8\,c^2-123\,a\,b^{10}\,c+6\,b^{12}\right )}{2\,\left (1024\,a^5\,c^{10}-1280\,a^4\,b^2\,c^9+640\,a^3\,b^4\,c^8-160\,a^2\,b^6\,c^7+20\,a\,b^8\,c^6-b^{10}\,c^5\right )}-\frac {3\,b\,x}{c^4}-\frac {3\,b\,\mathrm {atan}\left (\frac {\left (\frac {3\,b\,x\,\left (-70\,a^3\,c^3+70\,a^2\,b^2\,c^2-21\,a\,b^4\,c+2\,b^6\right )}{c^4\,{\left (4\,a\,c-b^2\right )}^5}+\frac {3\,b^2\,\left (16\,a^2\,c^6-8\,a\,b^2\,c^5+b^4\,c^4\right )\,\left (-70\,a^3\,c^3+70\,a^2\,b^2\,c^2-21\,a\,b^4\,c+2\,b^6\right )}{2\,c^9\,{\left (4\,a\,c-b^2\right )}^5\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )\,\left (32\,a^2\,c^7\,{\left (4\,a\,c-b^2\right )}^{5/2}+2\,b^4\,c^5\,{\left (4\,a\,c-b^2\right )}^{5/2}-16\,a\,b^2\,c^6\,{\left (4\,a\,c-b^2\right )}^{5/2}\right )}{-210\,a^3\,b\,c^3+210\,a^2\,b^3\,c^2-63\,a\,b^5\,c+6\,b^7}\right )\,\left (-70\,a^3\,c^3+70\,a^2\,b^2\,c^2-21\,a\,b^4\,c+2\,b^6\right )}{c^5\,{\left (4\,a\,c-b^2\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(a + b*x + c*x^2)^3,x)

[Out]

((a*(7*a*b^6 - 40*a^4*c^3 - 55*a^2*b^4*c + 115*a^3*b^2*c^2))/(2*c*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (b*x^3*(4*
b^6 - 63*a^3*c^3 + 91*a^2*b^2*c^2 - 35*a*b^4*c))/(b^4 + 16*a^2*c^2 - 8*a*b^2*c) + (x^2*(7*b^8 - 48*a^4*c^4 + 9
4*a^2*b^4*c^2 + 27*a^3*b^2*c^3 - 53*a*b^6*c))/(2*c*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (b*x*(7*a*b^6 - 73*a^4*c^
3 - 58*a^2*b^4*c + 136*a^3*b^2*c^2))/(c*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(a^2*c^4 + c^6*x^4 + x^2*(2*a*c^5 + b
^2*c^4) + 2*b*c^5*x^3 + 2*a*b*c^4*x) + x^2/(2*c^3) - (log(a + b*x + c*x^2)*(6*b^12 + 3072*a^6*c^6 + 1020*a^2*b
^8*c^2 - 4320*a^3*b^6*c^3 + 9600*a^4*b^4*c^4 - 9984*a^5*b^2*c^5 - 123*a*b^10*c))/(2*(1024*a^5*c^10 - b^10*c^5
+ 20*a*b^8*c^6 - 160*a^2*b^6*c^7 + 640*a^3*b^4*c^8 - 1280*a^4*b^2*c^9)) - (3*b*x)/c^4 - (3*b*atan((((3*b*x*(2*
b^6 - 70*a^3*c^3 + 70*a^2*b^2*c^2 - 21*a*b^4*c))/(c^4*(4*a*c - b^2)^5) + (3*b^2*(16*a^2*c^6 + b^4*c^4 - 8*a*b^
2*c^5)*(2*b^6 - 70*a^3*c^3 + 70*a^2*b^2*c^2 - 21*a*b^4*c))/(2*c^9*(4*a*c - b^2)^5*(b^4 + 16*a^2*c^2 - 8*a*b^2*
c)))*(32*a^2*c^7*(4*a*c - b^2)^(5/2) + 2*b^4*c^5*(4*a*c - b^2)^(5/2) - 16*a*b^2*c^6*(4*a*c - b^2)^(5/2)))/(6*b
^7 - 210*a^3*b*c^3 + 210*a^2*b^3*c^2 - 63*a*b^5*c))*(2*b^6 - 70*a^3*c^3 + 70*a^2*b^2*c^2 - 21*a*b^4*c))/(c^5*(
4*a*c - b^2)^(5/2))

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sympy [B]  time = 5.77, size = 1875, normalized size = 6.38

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(c*x**2+b*x+a)**3,x)

[Out]

-3*b*x/c**4 + (-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*c**5
*(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*c - 2*b**2)/(2*c**5))*log(x + (96*a**4*c**3 - 159*a**3*b**2*c**2 + 64*a**3*c**7*(-3*b*sqrt(-(4*a*c - b**2)**
5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*c**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 6
40*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) - 3*(a*c - 2*b**2)/(2*c**5)) + 57*a**2*b**4*c -
48*a**2*b**2*c**6*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2
*c**5*(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*c - 2*b**2)/(2*c**5)) - 6*a*b**6 + 12*a*b**4*c**5*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**
2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*c**5*(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*c - 2*b**2)/(2*c**5)) - b**6*c**4*(-3*b*sqrt(-(4*a*c - b**2)**5)*
(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*c**5*(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*c - 2*b**2)/(2*c**5)))/(210*a**3*b*c**3 - 2
10*a**2*b**3*c**2 + 63*a*b**5*c - 6*b**7)) + (3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 +
21*a*b**4*c - 2*b**6)/(2*c**5*(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*c - 2*b**2)/(2*c**5))*log(x + (96*a**4*c**3 - 159*a**3*b**2*c**2 + 64*a**3*c**
7*(3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*c**5*(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*c - 2*b**2)
/(2*c**5)) + 57*a**2*b**4*c - 48*a**2*b**2*c**6*(3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**
2 + 21*a*b**4*c - 2*b**6)/(2*c**5*(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*c - 2*b**2)/(2*c**5)) - 6*a*b**6 + 12*a*b**4*c**5*(3*b*sqrt(-(4*a*c - b**2)
**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*c**5*(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*c - 2*b**2)/(2*c**5)) - b**6*c**4*(3*b
*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*c**5*(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*c - 2*b**2)/(2*c*
*5)))/(210*a**3*b*c**3 - 210*a**2*b**3*c**2 + 63*a*b**5*c - 6*b**7)) + (-40*a**5*c**3 + 115*a**4*b**2*c**2 - 5
5*a**3*b**4*c + 7*a**2*b**6 + x**3*(-126*a**3*b*c**4 + 182*a**2*b**3*c**3 - 70*a*b**5*c**2 + 8*b**7*c) + x**2*
(-48*a**4*c**4 + 27*a**3*b**2*c**3 + 94*a**2*b**4*c**2 - 53*a*b**6*c + 7*b**8) + x*(-146*a**4*b*c**3 + 272*a**
3*b**3*c**2 - 116*a**2*b**5*c + 14*a*b**7))/(32*a**4*c**7 - 16*a**3*b**2*c**6 + 2*a**2*b**4*c**5 + x**4*(32*a*
*2*c**9 - 16*a*b**2*c**8 + 2*b**4*c**7) + x**3*(64*a**2*b*c**8 - 32*a*b**3*c**7 + 4*b**5*c**6) + x**2*(64*a**3
*c**8 - 12*a*b**4*c**6 + 2*b**6*c**5) + x*(64*a**3*b*c**7 - 32*a**2*b**3*c**6 + 4*a*b**5*c**5)) + x**2/(2*c**3
)

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