Integrand size = 22, antiderivative size = 67 \[ \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\frac {x (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {4 a \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \] Output:
x*(b*x+2*a)/(-4*a*c+b^2)/(c*x^2+b*x+a)+4*a*arctanh((2*c*x+b)/(-4*a*c+b^2)^ (1/2))/(-4*a*c+b^2)^(3/2)
Time = 0.10 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.21 \[ \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\frac {b^2 x+a (b-2 c x)}{c \left (-b^2+4 a c\right ) (a+x (b+c x))}+\frac {4 a \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}} \] Input:
Integrate[x^6/(a*x^2 + b*x^3 + c*x^4)^2,x]
Output:
(b^2*x + a*(b - 2*c*x))/(c*(-b^2 + 4*a*c)*(a + x*(b + c*x))) + (4*a*ArcTan [(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2)
Time = 0.22 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {9, 1153, 1083, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^2} \, dx\) |
\(\Big \downarrow \) 9 |
\(\displaystyle \int \frac {x^2}{\left (a+b x+c x^2\right )^2}dx\) |
\(\Big \downarrow \) 1153 |
\(\displaystyle \frac {x (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {2 a \int \frac {1}{c x^2+b x+a}dx}{b^2-4 a c}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {4 a \int \frac {1}{b^2-(b+2 c x)^2-4 a c}d(b+2 c x)}{b^2-4 a c}+\frac {x (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {4 a \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {x (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
Input:
Int[x^6/(a*x^2 + b*x^3 + c*x^4)^2,x]
Output:
(x*(2*a + b*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)) + (4*a*ArcTanh[(b + 2*c* x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2)
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> 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] - Simp[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] && EqQ[m + 2*p + 2, 0] && LtQ[p, -1]
Time = 0.09 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.45
method | result | size |
default | \(\frac {-\frac {\left (2 a c -b^{2}\right ) x}{c \left (4 a c -b^{2}\right )}+\frac {a b}{c \left (4 a c -b^{2}\right )}}{c \,x^{2}+b x +a}+\frac {4 a \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}\) | \(97\) |
risch | \(\frac {-\frac {\left (2 a c -b^{2}\right ) x}{c \left (4 a c -b^{2}\right )}+\frac {a b}{c \left (4 a c -b^{2}\right )}}{c \,x^{2}+b x +a}+\frac {2 a \ln \left (\left (-8 a \,c^{2}+2 c \,b^{2}\right ) x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}}-4 a b c +b^{3}\right )}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}-\frac {2 a \ln \left (\left (8 a \,c^{2}-2 c \,b^{2}\right ) x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}}+4 a b c -b^{3}\right )}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}\) | \(160\) |
Input:
int(x^6/(c*x^4+b*x^3+a*x^2)^2,x,method=_RETURNVERBOSE)
Output:
(-1/c*(2*a*c-b^2)/(4*a*c-b^2)*x+1/c*a*b/(4*a*c-b^2))/(c*x^2+b*x+a)+4*a/(4* a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (63) = 126\).
Time = 0.08 (sec) , antiderivative size = 387, normalized size of antiderivative = 5.78 \[ \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\left [-\frac {a b^{3} - 4 \, a^{2} b c + 2 \, {\left (a c^{2} x^{2} + a b c x + a^{2} c\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + {\left (b^{4} - 6 \, a b^{2} c + 8 \, a^{2} c^{2}\right )} x}{a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}, -\frac {a b^{3} - 4 \, a^{2} b c - 4 \, {\left (a c^{2} x^{2} + a b c x + a^{2} c\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + {\left (b^{4} - 6 \, a b^{2} c + 8 \, a^{2} c^{2}\right )} x}{a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}\right ] \] Input:
integrate(x^6/(c*x^4+b*x^3+a*x^2)^2,x, algorithm="fricas")
Output:
[-(a*b^3 - 4*a^2*b*c + 2*(a*c^2*x^2 + a*b*c*x + a^2*c)*sqrt(b^2 - 4*a*c)*l og((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)) + (b^4 - 6*a*b^2*c + 8*a^2*c^2)*x)/(a*b^4*c - 8*a^2*b^2*c^ 2 + 16*a^3*c^3 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^2 + (b^5*c - 8*a*b ^3*c^2 + 16*a^2*b*c^3)*x), -(a*b^3 - 4*a^2*b*c - 4*(a*c^2*x^2 + a*b*c*x + a^2*c)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4* a*c)) + (b^4 - 6*a*b^2*c + 8*a^2*c^2)*x)/(a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3 *c^3 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^2 + (b^5*c - 8*a*b^3*c^2 + 1 6*a^2*b*c^3)*x)]
Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (61) = 122\).
Time = 0.38 (sec) , antiderivative size = 280, normalized size of antiderivative = 4.18 \[ \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=- 2 a \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \log {\left (x + \frac {- 32 a^{3} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + 16 a^{2} b^{2} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} - 2 a b^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + 2 a b}{4 a c} \right )} + 2 a \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \log {\left (x + \frac {32 a^{3} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} - 16 a^{2} b^{2} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + 2 a b^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + 2 a b}{4 a c} \right )} + \frac {a b + x \left (- 2 a c + b^{2}\right )}{4 a^{2} c^{2} - a b^{2} c + x^{2} \cdot \left (4 a c^{3} - b^{2} c^{2}\right ) + x \left (4 a b c^{2} - b^{3} c\right )} \] Input:
integrate(x**6/(c*x**4+b*x**3+a*x**2)**2,x)
Output:
-2*a*sqrt(-1/(4*a*c - b**2)**3)*log(x + (-32*a**3*c**2*sqrt(-1/(4*a*c - b* *2)**3) + 16*a**2*b**2*c*sqrt(-1/(4*a*c - b**2)**3) - 2*a*b**4*sqrt(-1/(4* a*c - b**2)**3) + 2*a*b)/(4*a*c)) + 2*a*sqrt(-1/(4*a*c - b**2)**3)*log(x + (32*a**3*c**2*sqrt(-1/(4*a*c - b**2)**3) - 16*a**2*b**2*c*sqrt(-1/(4*a*c - b**2)**3) + 2*a*b**4*sqrt(-1/(4*a*c - b**2)**3) + 2*a*b)/(4*a*c)) + (a*b + x*(-2*a*c + b**2))/(4*a**2*c**2 - a*b**2*c + x**2*(4*a*c**3 - b**2*c**2 ) + x*(4*a*b*c**2 - b**3*c))
Exception generated. \[ \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^6/(c*x^4+b*x^3+a*x^2)^2,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.31 \[ \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=-\frac {4 \, a \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {b^{2} x - 2 \, a c x + a b}{{\left (b^{2} c - 4 \, a c^{2}\right )} {\left (c x^{2} + b x + a\right )}} \] Input:
integrate(x^6/(c*x^4+b*x^3+a*x^2)^2,x, algorithm="giac")
Output:
-4*a*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2 - 4*a*c)*sqrt(-b^2 + 4*a *c)) - (b^2*x - 2*a*c*x + a*b)/((b^2*c - 4*a*c^2)*(c*x^2 + b*x + a))
Time = 0.11 (sec) , antiderivative size = 135, normalized size of antiderivative = 2.01 \[ \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=-\frac {\frac {x\,\left (2\,a\,c-b^2\right )}{c\,\left (4\,a\,c-b^2\right )}-\frac {a\,b}{c\,\left (4\,a\,c-b^2\right )}}{c\,x^2+b\,x+a}-\frac {4\,a\,\mathrm {atan}\left (\frac {\left (\frac {2\,a\,\left (b^3-4\,a\,b\,c\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {4\,a\,c\,x}{{\left (4\,a\,c-b^2\right )}^{3/2}}\right )\,\left (4\,a\,c-b^2\right )}{2\,a}\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}} \] Input:
int(x^6/(a*x^2 + b*x^3 + c*x^4)^2,x)
Output:
- ((x*(2*a*c - b^2))/(c*(4*a*c - b^2)) - (a*b)/(c*(4*a*c - b^2)))/(a + b*x + c*x^2) - (4*a*atan((((2*a*(b^3 - 4*a*b*c))/(4*a*c - b^2)^(5/2) - (4*a*c *x)/(4*a*c - b^2)^(3/2))*(4*a*c - b^2))/(2*a)))/(4*a*c - b^2)^(3/2)
Time = 0.17 (sec) , antiderivative size = 243, normalized size of antiderivative = 3.63 \[ \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\frac {4 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a^{2} b +4 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{2} x +4 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a b c \,x^{2}+8 a^{3} c -2 a^{2} b^{2}+8 a^{2} c^{2} x^{2}-6 a \,b^{2} c \,x^{2}+b^{4} x^{2}}{b \left (16 a^{2} c^{3} x^{2}-8 a \,b^{2} c^{2} x^{2}+b^{4} c \,x^{2}+16 a^{2} b \,c^{2} x -8 a \,b^{3} c x +b^{5} x +16 a^{3} c^{2}-8 a^{2} b^{2} c +a \,b^{4}\right )} \] Input:
int(x^6/(c*x^4+b*x^3+a*x^2)^2,x)
Output:
(4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b + 4*sqrt (4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**2*x + 4*sqrt(4*a* c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*c*x**2 + 8*a**3*c - 2*a **2*b**2 + 8*a**2*c**2*x**2 - 6*a*b**2*c*x**2 + b**4*x**2)/(b*(16*a**3*c** 2 - 8*a**2*b**2*c + 16*a**2*b*c**2*x + 16*a**2*c**3*x**2 + a*b**4 - 8*a*b* *3*c*x - 8*a*b**2*c**2*x**2 + b**5*x + b**4*c*x**2))