Integrand size = 72, antiderivative size = 39 \[ \int \frac {x \left (a+b x^2\right ) \left (a e+3 b e x^2\right )}{c+a^4 d x^4+4 a^3 b d x^6+6 a^2 b^2 d x^8+4 a b^3 d x^{10}+b^4 d x^{12}} \, dx=\frac {e \arctan \left (\frac {\sqrt {d} x^2 \left (a+b x^2\right )^2}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d}} \] Output:
1/2*e*arctan(d^(1/2)*x^2*(b*x^2+a)^2/c^(1/2))/c^(1/2)/d^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.11 (sec) , antiderivative size = 101, normalized size of antiderivative = 2.59 \[ \int \frac {x \left (a+b x^2\right ) \left (a e+3 b e x^2\right )}{c+a^4 d x^4+4 a^3 b d x^6+6 a^2 b^2 d x^8+4 a b^3 d x^{10}+b^4 d x^{12}} \, dx=\frac {e \text {RootSum}\left [c+a^4 d \text {$\#$1}^2+4 a^3 b d \text {$\#$1}^3+6 a^2 b^2 d \text {$\#$1}^4+4 a b^3 d \text {$\#$1}^5+b^4 d \text {$\#$1}^6\&,\frac {\log \left (x^2-\text {$\#$1}\right )}{a^2 \text {$\#$1}+2 a b \text {$\#$1}^2+b^2 \text {$\#$1}^3}\&\right ]}{4 d} \] Input:
Integrate[(x*(a + b*x^2)*(a*e + 3*b*e*x^2))/(c + a^4*d*x^4 + 4*a^3*b*d*x^6 + 6*a^2*b^2*d*x^8 + 4*a*b^3*d*x^10 + b^4*d*x^12),x]
Output:
(e*RootSum[c + a^4*d*#1^2 + 4*a^3*b*d*#1^3 + 6*a^2*b^2*d*#1^4 + 4*a*b^3*d* #1^5 + b^4*d*#1^6 & , Log[x^2 - #1]/(a^2*#1 + 2*a*b*#1^2 + b^2*#1^3) & ])/ (4*d)
Time = 1.21 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {7239, 27, 7261, 218}
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.
\(\displaystyle \int \frac {x \left (a+b x^2\right ) \left (a e+3 b e x^2\right )}{a^4 d x^4+4 a^3 b d x^6+6 a^2 b^2 d x^8+4 a b^3 d x^{10}+b^4 d x^{12}+c} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {e x \left (a+b x^2\right ) \left (a+3 b x^2\right )}{d x^4 \left (a+b x^2\right )^4+c}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle e \int \frac {x \left (b x^2+a\right ) \left (3 b x^2+a\right )}{d x^4 \left (b x^2+a\right )^4+c}dx\) |
\(\Big \downarrow \) 7261 |
\(\displaystyle \frac {1}{2} e \int \frac {1}{d x^4 \left (b x^2+a\right )^4+c}d\left (x^2 \left (b x^2+a\right )^2\right )\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {e \arctan \left (\frac {\sqrt {d} x^2 \left (a+b x^2\right )^2}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d}}\) |
Input:
Int[(x*(a + b*x^2)*(a*e + 3*b*e*x^2))/(c + a^4*d*x^4 + 4*a^3*b*d*x^6 + 6*a ^2*b^2*d*x^8 + 4*a*b^3*d*x^10 + b^4*d*x^12),x]
Output:
(e*ArcTan[(Sqrt[d]*x^2*(a + b*x^2)^2)/Sqrt[c]])/(2*Sqrt[c]*Sqrt[d])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)*(v_)^(r_.)*(w_)^(s_.)*((a_) + (b_.)*(v_)^(p_.)*(w_)^(q_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/(p*w*D[v, x] + q*v*D[w, x])]}, Simp[c*(p/ (r + 1)) Subst[Int[(a + b*x^(p/(r + 1)))^m, x], x, v^(r + 1)*w^(s + 1)], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p, q, r, s}, x] && EqQ[p*(s + 1), q*( r + 1)] && NeQ[r, -1] && IntegerQ[p/(r + 1)]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 132, normalized size of antiderivative = 3.38
method | result | size |
default | \(\frac {e \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{4} d \,\textit {\_Z}^{6}+4 d a \,b^{3} \textit {\_Z}^{5}+6 a^{2} d \,b^{2} \textit {\_Z}^{4}+4 a^{3} d b \,\textit {\_Z}^{3}+a^{4} d \,\textit {\_Z}^{2}+c \right )}{\sum }\frac {\left (3 \textit {\_R}^{2} b^{2}+4 \textit {\_R} a b +a^{2}\right ) \ln \left (x^{2}-\textit {\_R} \right )}{3 b^{4} \textit {\_R}^{5}+10 a \,b^{3} \textit {\_R}^{4}+12 a^{2} b^{2} \textit {\_R}^{3}+6 a^{3} b \,\textit {\_R}^{2}+a^{4} \textit {\_R}}\right )}{4 d}\) | \(132\) |
risch | \(-\frac {e \ln \left (\left (-4 a^{3} b^{2} d -27 \sqrt {-c d}\, b^{3}\right ) x^{6}+\left (-8 a^{4} b d -54 \sqrt {-c d}\, a \,b^{2}\right ) x^{4}+\left (-4 d \,a^{5}-27 \sqrt {-c d}\, a^{2} b \right ) x^{2}-4 \sqrt {-c d}\, a^{3}+27 b c \right )}{4 \sqrt {-c d}}+\frac {e \ln \left (\left (4 a^{3} b^{2} d -27 \sqrt {-c d}\, b^{3}\right ) x^{6}+\left (8 a^{4} b d -54 \sqrt {-c d}\, a \,b^{2}\right ) x^{4}+\left (4 d \,a^{5}-27 \sqrt {-c d}\, a^{2} b \right ) x^{2}-4 \sqrt {-c d}\, a^{3}-27 b c \right )}{4 \sqrt {-c d}}\) | \(198\) |
Input:
int(x*(b*x^2+a)*(3*b*e*x^2+a*e)/(b^4*d*x^12+4*a*b^3*d*x^10+6*a^2*b^2*d*x^8 +4*a^3*b*d*x^6+a^4*d*x^4+c),x,method=_RETURNVERBOSE)
Output:
1/4*e/d*sum((3*_R^2*b^2+4*_R*a*b+a^2)/(3*_R^5*b^4+10*_R^4*a*b^3+12*_R^3*a^ 2*b^2+6*_R^2*a^3*b+_R*a^4)*ln(x^2-_R),_R=RootOf(_Z^6*b^4*d+4*_Z^5*a*b^3*d+ 6*_Z^4*a^2*b^2*d+4*_Z^3*a^3*b*d+_Z^2*a^4*d+c))
Time = 0.14 (sec) , antiderivative size = 200, normalized size of antiderivative = 5.13 \[ \int \frac {x \left (a+b x^2\right ) \left (a e+3 b e x^2\right )}{c+a^4 d x^4+4 a^3 b d x^6+6 a^2 b^2 d x^8+4 a b^3 d x^{10}+b^4 d x^{12}} \, dx=\left [-\frac {\sqrt {-c d} e \log \left (\frac {b^{4} d x^{12} + 4 \, a b^{3} d x^{10} + 6 \, a^{2} b^{2} d x^{8} + 4 \, a^{3} b d x^{6} + a^{4} d x^{4} - 2 \, {\left (b^{2} x^{6} + 2 \, a b x^{4} + a^{2} x^{2}\right )} \sqrt {-c d} - c}{b^{4} d x^{12} + 4 \, a b^{3} d x^{10} + 6 \, a^{2} b^{2} d x^{8} + 4 \, a^{3} b d x^{6} + a^{4} d x^{4} + c}\right )}{4 \, c d}, -\frac {\sqrt {c d} e \arctan \left (\frac {\sqrt {c d}}{b^{2} d x^{6} + 2 \, a b d x^{4} + a^{2} d x^{2}}\right )}{2 \, c d}\right ] \] Input:
integrate(x*(b*x^2+a)*(3*b*e*x^2+a*e)/(b^4*d*x^12+4*a*b^3*d*x^10+6*a^2*b^2 *d*x^8+4*a^3*b*d*x^6+a^4*d*x^4+c),x, algorithm="fricas")
Output:
[-1/4*sqrt(-c*d)*e*log((b^4*d*x^12 + 4*a*b^3*d*x^10 + 6*a^2*b^2*d*x^8 + 4* a^3*b*d*x^6 + a^4*d*x^4 - 2*(b^2*x^6 + 2*a*b*x^4 + a^2*x^2)*sqrt(-c*d) - c )/(b^4*d*x^12 + 4*a*b^3*d*x^10 + 6*a^2*b^2*d*x^8 + 4*a^3*b*d*x^6 + a^4*d*x ^4 + c))/(c*d), -1/2*sqrt(c*d)*e*arctan(sqrt(c*d)/(b^2*d*x^6 + 2*a*b*d*x^4 + a^2*d*x^2))/(c*d)]
Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (36) = 72\).
Time = 1.08 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.62 \[ \int \frac {x \left (a+b x^2\right ) \left (a e+3 b e x^2\right )}{c+a^4 d x^4+4 a^3 b d x^6+6 a^2 b^2 d x^8+4 a b^3 d x^{10}+b^4 d x^{12}} \, dx=e \left (- \frac {\sqrt {- \frac {1}{c d}} \log {\left (\frac {a^{2} x^{2}}{b^{2}} + \frac {2 a x^{4}}{b} + x^{6} - \frac {c \sqrt {- \frac {1}{c d}}}{b^{2}} \right )}}{4} + \frac {\sqrt {- \frac {1}{c d}} \log {\left (\frac {a^{2} x^{2}}{b^{2}} + \frac {2 a x^{4}}{b} + x^{6} + \frac {c \sqrt {- \frac {1}{c d}}}{b^{2}} \right )}}{4}\right ) \] Input:
integrate(x*(b*x**2+a)*(3*b*e*x**2+a*e)/(b**4*d*x**12+4*a*b**3*d*x**10+6*a **2*b**2*d*x**8+4*a**3*b*d*x**6+a**4*d*x**4+c),x)
Output:
e*(-sqrt(-1/(c*d))*log(a**2*x**2/b**2 + 2*a*x**4/b + x**6 - c*sqrt(-1/(c*d ))/b**2)/4 + sqrt(-1/(c*d))*log(a**2*x**2/b**2 + 2*a*x**4/b + x**6 + c*sqr t(-1/(c*d))/b**2)/4)
\[ \int \frac {x \left (a+b x^2\right ) \left (a e+3 b e x^2\right )}{c+a^4 d x^4+4 a^3 b d x^6+6 a^2 b^2 d x^8+4 a b^3 d x^{10}+b^4 d x^{12}} \, dx=\int { \frac {{\left (3 \, b e x^{2} + a e\right )} {\left (b x^{2} + a\right )} x}{b^{4} d x^{12} + 4 \, a b^{3} d x^{10} + 6 \, a^{2} b^{2} d x^{8} + 4 \, a^{3} b d x^{6} + a^{4} d x^{4} + c} \,d x } \] Input:
integrate(x*(b*x^2+a)*(3*b*e*x^2+a*e)/(b^4*d*x^12+4*a*b^3*d*x^10+6*a^2*b^2 *d*x^8+4*a^3*b*d*x^6+a^4*d*x^4+c),x, algorithm="maxima")
Output:
integrate((3*b*e*x^2 + a*e)*(b*x^2 + a)*x/(b^4*d*x^12 + 4*a*b^3*d*x^10 + 6 *a^2*b^2*d*x^8 + 4*a^3*b*d*x^6 + a^4*d*x^4 + c), x)
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (29) = 58\).
Time = 8.32 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.54 \[ \int \frac {x \left (a+b x^2\right ) \left (a e+3 b e x^2\right )}{c+a^4 d x^4+4 a^3 b d x^6+6 a^2 b^2 d x^8+4 a b^3 d x^{10}+b^4 d x^{12}} \, dx=\frac {\sqrt {-c d} e \log \left ({\left | b^{2} d x^{6} + 2 \, a b d x^{4} + a^{2} d x^{2} + \sqrt {-c d} \right |}\right )}{4 \, c d} - \frac {\sqrt {-c d} e \log \left ({\left | b^{2} d x^{6} + 2 \, a b d x^{4} + a^{2} d x^{2} - \sqrt {-c d} \right |}\right )}{4 \, c d} \] Input:
integrate(x*(b*x^2+a)*(3*b*e*x^2+a*e)/(b^4*d*x^12+4*a*b^3*d*x^10+6*a^2*b^2 *d*x^8+4*a^3*b*d*x^6+a^4*d*x^4+c),x, algorithm="giac")
Output:
1/4*sqrt(-c*d)*e*log(abs(b^2*d*x^6 + 2*a*b*d*x^4 + a^2*d*x^2 + sqrt(-c*d)) )/(c*d) - 1/4*sqrt(-c*d)*e*log(abs(b^2*d*x^6 + 2*a*b*d*x^4 + a^2*d*x^2 - s qrt(-c*d)))/(c*d)
Time = 22.72 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.15 \[ \int \frac {x \left (a+b x^2\right ) \left (a e+3 b e x^2\right )}{c+a^4 d x^4+4 a^3 b d x^6+6 a^2 b^2 d x^8+4 a b^3 d x^{10}+b^4 d x^{12}} \, dx=\frac {e\,\mathrm {atan}\left (\frac {a^2\,\sqrt {d}\,x^2+b^2\,\sqrt {d}\,x^6+2\,a\,b\,\sqrt {d}\,x^4}{\sqrt {c}}\right )}{2\,\sqrt {c}\,\sqrt {d}} \] Input:
int((x*(a + b*x^2)*(a*e + 3*b*e*x^2))/(c + a^4*d*x^4 + b^4*d*x^12 + 6*a^2* b^2*d*x^8 + 4*a^3*b*d*x^6 + 4*a*b^3*d*x^10),x)
Output:
(e*atan((a^2*d^(1/2)*x^2 + b^2*d^(1/2)*x^6 + 2*a*b*d^(1/2)*x^4)/c^(1/2)))/ (2*c^(1/2)*d^(1/2))
\[ \int \frac {x \left (a+b x^2\right ) \left (a e+3 b e x^2\right )}{c+a^4 d x^4+4 a^3 b d x^6+6 a^2 b^2 d x^8+4 a b^3 d x^{10}+b^4 d x^{12}} \, dx=e \left (3 \left (\int \frac {x^{5}}{b^{4} d \,x^{12}+4 a \,b^{3} d \,x^{10}+6 a^{2} b^{2} d \,x^{8}+4 a^{3} b d \,x^{6}+a^{4} d \,x^{4}+c}d x \right ) b^{2}+4 \left (\int \frac {x^{3}}{b^{4} d \,x^{12}+4 a \,b^{3} d \,x^{10}+6 a^{2} b^{2} d \,x^{8}+4 a^{3} b d \,x^{6}+a^{4} d \,x^{4}+c}d x \right ) a b +\left (\int \frac {x}{b^{4} d \,x^{12}+4 a \,b^{3} d \,x^{10}+6 a^{2} b^{2} d \,x^{8}+4 a^{3} b d \,x^{6}+a^{4} d \,x^{4}+c}d x \right ) a^{2}\right ) \] Input:
int(x*(b*x^2+a)*(3*b*e*x^2+a*e)/(b^4*d*x^12+4*a*b^3*d*x^10+6*a^2*b^2*d*x^8 +4*a^3*b*d*x^6+a^4*d*x^4+c),x)
Output:
e*(3*int(x**5/(a**4*d*x**4 + 4*a**3*b*d*x**6 + 6*a**2*b**2*d*x**8 + 4*a*b* *3*d*x**10 + b**4*d*x**12 + c),x)*b**2 + 4*int(x**3/(a**4*d*x**4 + 4*a**3* b*d*x**6 + 6*a**2*b**2*d*x**8 + 4*a*b**3*d*x**10 + b**4*d*x**12 + c),x)*a* b + int(x/(a**4*d*x**4 + 4*a**3*b*d*x**6 + 6*a**2*b**2*d*x**8 + 4*a*b**3*d *x**10 + b**4*d*x**12 + c),x)*a**2)