Integrand size = 74, antiderivative size = 39 \[ \int \frac {x \left (a+b x^2\right ) \left (a e+3 b e x^2\right )}{c-a^4 d x^4-4 a^3 b d x^6-6 a^2 b^2 d x^8-4 a b^3 d x^{10}-b^4 d x^{12}} \, dx=\frac {e \text {arctanh}\left (\frac {\sqrt {d} x^2 \left (a+b x^2\right )^2}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d}} \] Output:
1/2*e*arctanh(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 = 103, normalized size of antiderivative = 2.64 \[ \int \frac {x \left (a+b x^2\right ) \left (a e+3 b e x^2\right )}{c-a^4 d x^4-4 a^3 b d x^6-6 a^2 b^2 d x^8-4 a b^3 d x^{10}-b^4 d x^{12}} \, dx=-\frac {e \text {RootSum}\left [c-a^4 d \text {$\#$1}^2-4 a^3 b d \text {$\#$1}^3-6 a^2 b^2 d \text {$\#$1}^4-4 a b^3 d \text {$\#$1}^5-b^4 d \text {$\#$1}^6\&,\frac {\log \left (x^2-\text {$\#$1}\right )}{a^2 \text {$\#$1}+2 a b \text {$\#$1}^2+b^2 \text {$\#$1}^3}\&\right ]}{4 d} \] 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:
-1/4*(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) & ])/d
Time = 1.29 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {7239, 27, 7261, 221}
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 |
\(\displaystyle \int \frac {e x \left (a+b x^2\right ) \left (a+3 b x^2\right )}{c-d x^4 \left (a+b x^2\right )^4}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle e \int \frac {x \left (b x^2+a\right ) \left (3 b x^2+a\right )}{c-d x^4 \left (b x^2+a\right )^4}dx\) |
\(\Big \downarrow \) 7261 |
\(\displaystyle \frac {1}{2} e \int \frac {1}{c-d x^4 \left (b x^2+a\right )^4}d\left (x^2 \left (b x^2+a\right )^2\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {e \text {arctanh}\left (\frac {\sqrt {d} x^2 \left (a+b x^2\right )^2}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d}}\) |
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*ArcTanh[(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)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[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 = 0.08 (sec) , antiderivative size = 136, normalized size of antiderivative = 3.49
method | result | size |
default | \(\frac {e \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{4} d \,\textit {\_Z}^{6}+4 d a \,b^{3} \textit {\_Z}^{5}+6 a^{2} d \,b^{2} \textit {\_Z}^{4}+4 a^{3} d b \,\textit {\_Z}^{3}+a^{4} d \,\textit {\_Z}^{2}-c \right )}{\sum }\frac {\left (-3 \textit {\_R}^{2} b^{2}-4 \textit {\_R} a b -a^{2}\right ) \ln \left (x^{2}-\textit {\_R} \right )}{3 b^{4} \textit {\_R}^{5}+10 a \,b^{3} \textit {\_R}^{4}+12 a^{2} b^{2} \textit {\_R}^{3}+6 a^{3} b \,\textit {\_R}^{2}+a^{4} \textit {\_R}}\right )}{4 d}\) | \(136\) |
risch | \(\frac {e \ln \left (\left (-4 a^{3} b^{2} d -27 \sqrt {c d}\, b^{3}\right ) x^{6}+\left (-8 a^{4} b d -54 \sqrt {c d}\, a \,b^{2}\right ) x^{4}+\left (-4 d \,a^{5}-27 \sqrt {c d}\, a^{2} b \right ) x^{2}-4 \sqrt {c d}\, a^{3}-27 b c \right )}{4 \sqrt {c d}}-\frac {e \ln \left (\left (4 a^{3} b^{2} d -27 \sqrt {c d}\, b^{3}\right ) x^{6}+\left (8 a^{4} b d -54 \sqrt {c d}\, a \,b^{2}\right ) x^{4}+\left (4 d \,a^{5}-27 \sqrt {c d}\, a^{2} b \right ) x^{2}-4 \sqrt {c d}\, a^{3}+27 b c \right )}{4 \sqrt {c d}}\) | \(188\) |
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 = 0.11 (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. 97 vs. \(2 (36) = 72\).
Time = 1.25 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.49 \[ \int \frac {x \left (a+b x^2\right ) \left (a e+3 b e x^2\right )}{c-a^4 d x^4-4 a^3 b d x^6-6 a^2 b^2 d x^8-4 a b^3 d x^{10}-b^4 d x^{12}} \, dx=- e \left (\frac {\sqrt {\frac {1}{c d}} \log {\left (\frac {a^{2} x^{2}}{b^{2}} + \frac {2 a x^{4}}{b} + x^{6} - \frac {c \sqrt {\frac {1}{c d}}}{b^{2}} \right )}}{4} - \frac {\sqrt {\frac {1}{c d}} \log {\left (\frac {a^{2} x^{2}}{b^{2}} + \frac {2 a x^{4}}{b} + x^{6} + \frac {c \sqrt {\frac {1}{c d}}}{b^{2}} \right )}}{4}\right ) \] 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*sqrt(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. 95 vs. \(2 (29) = 58\).
Time = 7.77 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.44 \[ \int \frac {x \left (a+b x^2\right ) \left (a e+3 b e x^2\right )}{c-a^4 d x^4-4 a^3 b d x^6-6 a^2 b^2 d x^8-4 a b^3 d x^{10}-b^4 d x^{12}} \, dx=\frac {\sqrt {c d} e \log \left ({\left | b^{2} d x^{6} + 2 \, a b d x^{4} + a^{2} d x^{2} + \sqrt {c d} \right |}\right )}{4 \, c d} - \frac {\sqrt {c d} e \log \left ({\left | b^{2} d x^{6} + 2 \, a b d x^{4} + a^{2} d x^{2} - \sqrt {c d} \right |}\right )}{4 \, c d} \] 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 - sqrt (c*d)))/(c*d)
Time = 23.81 (sec) , antiderivative size = 170, normalized size of antiderivative = 4.36 \[ \int \frac {x \left (a+b x^2\right ) \left (a e+3 b e x^2\right )}{c-a^4 d x^4-4 a^3 b d x^6-6 a^2 b^2 d x^8-4 a b^3 d x^{10}-b^4 d x^{12}} \, dx=-\frac {e\,\mathrm {atan}\left (\frac {a^8\,d^{3/2}\,x^2\,16{}\mathrm {i}+a^7\,b\,d^{3/2}\,x^4\,32{}\mathrm {i}+b^4\,c\,\sqrt {d}\,x^6\,729{}\mathrm {i}+a^6\,b^2\,d^{3/2}\,x^6\,16{}\mathrm {i}+a^3\,b\,c\,\sqrt {d}\,216{}\mathrm {i}+a\,b^3\,c\,\sqrt {d}\,x^4\,1458{}\mathrm {i}+a^2\,b^2\,c\,\sqrt {d}\,x^2\,729{}\mathrm {i}}{729\,b^2\,c^{3/2}+16\,a^6\,\sqrt {c}\,d+216\,a^5\,b\,\sqrt {c}\,d\,x^2+432\,a^4\,b^2\,\sqrt {c}\,d\,x^4+216\,a^3\,b^3\,\sqrt {c}\,d\,x^6}\right )\,1{}\mathrm {i}}{2\,\sqrt {c}\,\sqrt {d}} \] Input:
int(-(x*(a + b*x^2)*(a*e + 3*b*e*x^2))/(a^4*d*x^4 - c + 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^8*d^(3/2)*x^2*16i + a^7*b*d^(3/2)*x^4*32i + b^4*c*d^(1/2)*x^6* 729i + a^6*b^2*d^(3/2)*x^6*16i + a^3*b*c*d^(1/2)*216i + a*b^3*c*d^(1/2)*x^ 4*1458i + a^2*b^2*c*d^(1/2)*x^2*729i)/(729*b^2*c^(3/2) + 16*a^6*c^(1/2)*d + 216*a^5*b*c^(1/2)*d*x^2 + 432*a^4*b^2*c^(1/2)*d*x^4 + 216*a^3*b^3*c^(1/2 )*d*x^6))*1i)/(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)