\(\int \frac {x^7}{(a x^2+b x^3+c x^4)^2} \, dx\) [20]

Optimal result

Integrand size = 22, antiderivative size = 114 \[ \int \frac {x^7}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=-\frac {b x}{c \left (b^2-4 a c\right )}+\frac {x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {b \left (b^2-6 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac {\log \left (a+b x+c x^2\right )}{2 c^2} \]

[Out]

-b*x/c/(-4*a*c+b^2)+x^2*(b*x+2*a)/(-4*a*c+b^2)/(c*x^2+b*x+a)+b*(-6*a*c+b^2)*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/
2))/c^2/(-4*a*c+b^2)^(3/2)+1/2*ln(c*x^2+b*x+a)/c^2

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1599, 752, 787, 648, 632, 212, 642} \[ \int \frac {x^7}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\frac {b \left (b^2-6 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac {x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {b x}{c \left (b^2-4 a c\right )}+\frac {\log \left (a+b x+c x^2\right )}{2 c^2} \]

[In]

Int[x^7/(a*x^2 + b*x^3 + c*x^4)^2,x]

[Out]

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

Rule 212

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

Rule 632

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 642

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 648

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 752

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 787

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 1599

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(m +
n*p)*(a + b*x^(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, m, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] &
& PosQ[r - p]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3}{\left (a+b x+c x^2\right )^2} \, dx \\ & = \frac {x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \frac {x (4 a+b x)}{a+b x+c x^2} \, dx}{-b^2+4 a c} \\ & = -\frac {b x}{c \left (b^2-4 a c\right )}+\frac {x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {-a b+\left (-b^2+4 a c\right ) x}{a+b x+c x^2} \, dx}{c \left (b^2-4 a c\right )} \\ & = -\frac {b x}{c \left (b^2-4 a c\right )}+\frac {x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^2}-\frac {\left (b \left (b^2-6 a c\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c^2 \left (b^2-4 a c\right )} \\ & = -\frac {b x}{c \left (b^2-4 a c\right )}+\frac {x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\log \left (a+b x+c x^2\right )}{2 c^2}+\frac {\left (b \left (b^2-6 a c\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^2 \left (b^2-4 a c\right )} \\ & = -\frac {b x}{c \left (b^2-4 a c\right )}+\frac {x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac {\log \left (a+b x+c x^2\right )}{2 c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.96 \[ \int \frac {x^7}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\frac {\frac {2 \left (-2 a^2 c+b^3 x+a b (b-3 c x)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac {2 b \left (b^2-6 a c\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+\log (a+x (b+c x))}{2 c^2} \]

[In]

Integrate[x^7/(a*x^2 + b*x^3 + c*x^4)^2,x]

[Out]

((2*(-2*a^2*c + b^3*x + a*b*(b - 3*c*x)))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (2*b*(b^2 - 6*a*c)*ArcTan[(b + 2
*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + Log[a + x*(b + c*x)])/(2*c^2)

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.48

method	result	size
default	\(\frac {\frac {b \left (3 a c -b^{2}\right ) x}{c^{2} \left (4 a c -b^{2}\right )}+\frac {a \left (2 a c -b^{2}\right )}{\left (4 a c -b^{2}\right ) c^{2}}}{c \,x^{2}+b x +a}+\frac {\frac {\left (4 a c -b^{2}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-a b -\frac {\left (4 a c -b^{2}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{c \left (4 a c -b^{2}\right )}\)	\(169\)
risch	\(\frac {\frac {b \left (3 a c -b^{2}\right ) x}{c^{2} \left (4 a c -b^{2}\right )}+\frac {a \left (2 a c -b^{2}\right )}{\left (4 a c -b^{2}\right ) c^{2}}}{c \,x^{2}+b x +a}+\frac {8 \ln \left (-24 a^{2} b \,c^{2}+10 a \,b^{3} c -b^{5}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, c x -\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, b \right ) a^{2}}{\left (4 a c -b^{2}\right )^{2}}-\frac {4 \ln \left (-24 a^{2} b \,c^{2}+10 a \,b^{3} c -b^{5}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, c x -\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, b \right ) a \,b^{2}}{\left (4 a c -b^{2}\right )^{2} c}+\frac {\ln \left (-24 a^{2} b \,c^{2}+10 a \,b^{3} c -b^{5}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, c x -\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, b \right ) b^{4}}{2 \left (4 a c -b^{2}\right )^{2} c^{2}}+\frac {\ln \left (-24 a^{2} b \,c^{2}+10 a \,b^{3} c -b^{5}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, c x -\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, b \right ) \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}}{2 \left (4 a c -b^{2}\right )^{2} c^{2}}+\frac {8 \ln \left (-24 a^{2} b \,c^{2}+10 a \,b^{3} c -b^{5}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, c x +\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, b \right ) a^{2}}{\left (4 a c -b^{2}\right )^{2}}-\frac {4 \ln \left (-24 a^{2} b \,c^{2}+10 a \,b^{3} c -b^{5}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, c x +\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, b \right ) a \,b^{2}}{\left (4 a c -b^{2}\right )^{2} c}+\frac {\ln \left (-24 a^{2} b \,c^{2}+10 a \,b^{3} c -b^{5}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, c x +\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, b \right ) b^{4}}{2 \left (4 a c -b^{2}\right )^{2} c^{2}}-\frac {\ln \left (-24 a^{2} b \,c^{2}+10 a \,b^{3} c -b^{5}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, c x +\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, b \right ) \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}}{2 \left (4 a c -b^{2}\right )^{2} c^{2}}\)	\(979\)

[In]

int(x^7/(c*x^4+b*x^3+a*x^2)^2,x,method=_RETURNVERBOSE)

[Out]

(b/c^2*(3*a*c-b^2)/(4*a*c-b^2)*x+a*(2*a*c-b^2)/(4*a*c-b^2)/c^2)/(c*x^2+b*x+a)+1/c/(4*a*c-b^2)*(1/2*(4*a*c-b^2)
/c*ln(c*x^2+b*x+a)+2*(-a*b-1/2*(4*a*c-b^2)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 308 vs. \(2 (108) = 216\).

Time = 0.27 (sec) , antiderivative size = 635, normalized size of antiderivative = 5.57 \[ \int \frac {x^7}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\left [\frac {2 \, a b^{4} - 12 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (a b^{3} - 6 \, a^{2} b c + {\left (b^{3} c - 6 \, a b c^{2}\right )} x^{2} + {\left (b^{4} - 6 \, a b^{2} c\right )} x\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 2 \, {\left (b^{5} - 7 \, a b^{3} c + 12 \, a^{2} b c^{2}\right )} x + {\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{2} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4} + {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} x^{2} + {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} x\right )}}, \frac {2 \, a b^{4} - 12 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + 2 \, {\left (a b^{3} - 6 \, a^{2} b c + {\left (b^{3} c - 6 \, a b c^{2}\right )} x^{2} + {\left (b^{4} - 6 \, a b^{2} c\right )} x\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 2 \, {\left (b^{5} - 7 \, a b^{3} c + 12 \, a^{2} b c^{2}\right )} x + {\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{2} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4} + {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} x^{2} + {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} x\right )}}\right ] \]

[In]

integrate(x^7/(c*x^4+b*x^3+a*x^2)^2,x, algorithm="fricas")

[Out]

[1/2*(2*a*b^4 - 12*a^2*b^2*c + 16*a^3*c^2 + (a*b^3 - 6*a^2*b*c + (b^3*c - 6*a*b*c^2)*x^2 + (b^4 - 6*a*b^2*c)*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*(b^5 - 7*a*b^3*c + 12*a^2*b*c^2)*x + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2 + (b^4*c - 8*a*b^2*c^2 + 16*a^2*c^
3)*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*x)*log(c*x^2 + b*x + a))/(a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4 + (
b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^2 + (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x), 1/2*(2*a*b^4 - 12*a^2*b^2
*c + 16*a^3*c^2 + 2*(a*b^3 - 6*a^2*b*c + (b^3*c - 6*a*b*c^2)*x^2 + (b^4 - 6*a*b^2*c)*x)*sqrt(-b^2 + 4*a*c)*arc
tan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 2*(b^5 - 7*a*b^3*c + 12*a^2*b*c^2)*x + (a*b^4 - 8*a^2*b^2
*c + 16*a^3*c^2 + (b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*x)*log(c*x^2 + b*x
+ a))/(a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^2 + (b^5*c^2 - 8*a*b^3
*c^3 + 16*a^2*b*c^4)*x)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 729 vs. \(2 (104) = 208\).

Time = 0.73 (sec) , antiderivative size = 729, normalized size of antiderivative = 6.39 \[ \int \frac {x^7}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a c - b^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{2 c^{2}}\right ) \log {\left (x + \frac {- 16 a^{2} c^{3} \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a c - b^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{2 c^{2}}\right ) + 8 a^{2} c + 8 a b^{2} c^{2} \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a c - b^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{2 c^{2}}\right ) - a b^{2} - b^{4} c \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a c - b^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{2 c^{2}}\right )}{6 a b c - b^{3}} \right )} + \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a c - b^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{2 c^{2}}\right ) \log {\left (x + \frac {- 16 a^{2} c^{3} \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a c - b^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{2 c^{2}}\right ) + 8 a^{2} c + 8 a b^{2} c^{2} \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a c - b^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{2 c^{2}}\right ) - a b^{2} - b^{4} c \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a c - b^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{2 c^{2}}\right )}{6 a b c - b^{3}} \right )} + \frac {2 a^{2} c - a b^{2} + x \left (3 a b c - b^{3}\right )}{4 a^{2} c^{3} - a b^{2} c^{2} + x^{2} \cdot \left (4 a c^{4} - b^{2} c^{3}\right ) + x \left (4 a b c^{3} - b^{3} c^{2}\right )} \]

[In]

integrate(x**7/(c*x**4+b*x**3+a*x**2)**2,x)

[Out]

(-b*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) +
1/(2*c**2))*log(x + (-16*a**2*c**3*(-b*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(2*c**2*(64*a**3*c**3 - 48*a**
2*b**2*c**2 + 12*a*b**4*c - b**6)) + 1/(2*c**2)) + 8*a**2*c + 8*a*b**2*c**2*(-b*sqrt(-(4*a*c - b**2)**3)*(6*a*
c - b**2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + 1/(2*c**2)) - a*b**2 - b**4*c*(-b
*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + 1/
(2*c**2)))/(6*a*b*c - b**3)) + (b*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2
*c**2 + 12*a*b**4*c - b**6)) + 1/(2*c**2))*log(x + (-16*a**2*c**3*(b*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(
2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + 1/(2*c**2)) + 8*a**2*c + 8*a*b**2*c**2*(b*sq
rt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + 1/(2*
c**2)) - a*b**2 - b**4*c*(b*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2
+ 12*a*b**4*c - b**6)) + 1/(2*c**2)))/(6*a*b*c - b**3)) + (2*a**2*c - a*b**2 + x*(3*a*b*c - b**3))/(4*a**2*c**
3 - a*b**2*c**2 + x**2*(4*a*c**4 - b**2*c**3) + x*(4*a*b*c**3 - b**3*c**2))

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^7}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^7/(c*x^4+b*x^3+a*x^2)^2,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 deta

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.10 \[ \int \frac {x^7}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=-\frac {{\left (b^{3} - 6 \, a b c\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {\log \left (c x^{2} + b x + a\right )}{2 \, c^{2}} + \frac {a b^{2} - 2 \, a^{2} c + {\left (b^{3} - 3 \, a b c\right )} x}{{\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} c^{2}} \]

[In]

integrate(x^7/(c*x^4+b*x^3+a*x^2)^2,x, algorithm="giac")

[Out]

-(b^3 - 6*a*b*c)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2*c^2 - 4*a*c^3)*sqrt(-b^2 + 4*a*c)) + 1/2*log(c*x
^2 + b*x + a)/c^2 + (a*b^2 - 2*a^2*c + (b^3 - 3*a*b*c)*x)/((c*x^2 + b*x + a)*(b^2 - 4*a*c)*c^2)

Mupad [B] (verification not implemented)

Time = 8.81 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.45 \[ \int \frac {x^7}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\frac {\frac {a\,\left (2\,a\,c-b^2\right )}{c^2\,\left (4\,a\,c-b^2\right )}+\frac {b\,x\,\left (3\,a\,c-b^2\right )}{c^2\,\left (4\,a\,c-b^2\right )}}{c\,x^2+b\,x+a}-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}{2\,\left (64\,a^3\,c^5-48\,a^2\,b^2\,c^4+12\,a\,b^4\,c^3-b^6\,c^2\right )}+\frac {b\,\mathrm {atan}\left (\frac {c^2\,{\left (4\,a\,c-b^2\right )}^{5/2}\,\left (\frac {2\,b\,x\,\left (6\,a\,c-b^2\right )}{c\,{\left (4\,a\,c-b^2\right )}^3}+\frac {b^2\,\left (4\,a\,c^2-b^2\,c\right )\,\left (6\,a\,c-b^2\right )}{c^3\,{\left (4\,a\,c-b^2\right )}^4}\right )}{b^3-6\,a\,b\,c}\right )\,\left (6\,a\,c-b^2\right )}{c^2\,{\left (4\,a\,c-b^2\right )}^{3/2}} \]

[In]

int(x^7/(a*x^2 + b*x^3 + c*x^4)^2,x)

[Out]

((a*(2*a*c - b^2))/(c^2*(4*a*c - b^2)) + (b*x*(3*a*c - b^2))/(c^2*(4*a*c - b^2)))/(a + b*x + c*x^2) - (log(a +
b*x + c*x^2)*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c))/(2*(64*a^3*c^5 - b^6*c^2 + 12*a*b^4*c^3 - 48*a
^2*b^2*c^4)) + (b*atan((c^2*(4*a*c - b^2)^(5/2)*((2*b*x*(6*a*c - b^2))/(c*(4*a*c - b^2)^3) + (b^2*(4*a*c^2 - b
^2*c)*(6*a*c - b^2))/(c^3*(4*a*c - b^2)^4)))/(b^3 - 6*a*b*c))*(6*a*c - b^2))/(c^2*(4*a*c - b^2)^(3/2))