Integrand size = 68, antiderivative size = 41 \[ \int \frac {x^2 \sqrt {a+b x^2} \left (a e+2 b e x^2\right )}{c-a^3 d x^6-3 a^2 b d x^8-3 a b^2 d x^{10}-b^3 d x^{12}} \, dx=\frac {e \text {arctanh}\left (\frac {\sqrt {d} x^3 \left (a+b x^2\right )^{3/2}}{\sqrt {c}}\right )}{3 \sqrt {c} \sqrt {d}} \] Output:
1/3*e*arctanh(d^(1/2)*x^3*(b*x^2+a)^(3/2)/c^(1/2))/c^(1/2)/d^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(45670\) vs. \(2(41)=82\).
Time = 17.41 (sec) , antiderivative size = 45670, normalized size of antiderivative = 1113.90 \[ \int \frac {x^2 \sqrt {a+b x^2} \left (a e+2 b e x^2\right )}{c-a^3 d x^6-3 a^2 b d x^8-3 a b^2 d x^{10}-b^3 d x^{12}} \, dx=\text {Result too large to show} \] Input:
Integrate[(x^2*Sqrt[a + b*x^2]*(a*e + 2*b*e*x^2))/(c - a^3*d*x^6 - 3*a^2*b *d*x^8 - 3*a*b^2*d*x^10 - b^3*d*x^12),x]
Output:
Result too large to show
Time = 2.00 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, 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.
\(\displaystyle \int \frac {x^2 \sqrt {a+b x^2} \left (a e+2 b e x^2\right )}{-a^3 d x^6-3 a^2 b d x^8-3 a b^2 d x^{10}-b^3 d x^{12}+c} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {e x^2 \sqrt {a+b x^2} \left (a+2 b x^2\right )}{c-d x^6 \left (a+b x^2\right )^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle e \int \frac {x^2 \sqrt {b x^2+a} \left (2 b x^2+a\right )}{c-d x^6 \left (b x^2+a\right )^3}dx\) |
\(\Big \downarrow \) 7261 |
\(\displaystyle \frac {1}{3} e \int \frac {1}{c-d x^6 \left (b x^2+a\right )^3}d\left (x^3 \left (b x^2+a\right )^{3/2}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {e \text {arctanh}\left (\frac {\sqrt {d} x^3 \left (a+b x^2\right )^{3/2}}{\sqrt {c}}\right )}{3 \sqrt {c} \sqrt {d}}\) |
Input:
Int[(x^2*Sqrt[a + b*x^2]*(a*e + 2*b*e*x^2))/(c - a^3*d*x^6 - 3*a^2*b*d*x^8 - 3*a*b^2*d*x^10 - b^3*d*x^12),x]
Output:
(e*ArcTanh[(Sqrt[d]*x^3*(a + b*x^2)^(3/2))/Sqrt[c]])/(3*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 = 0.16 (sec) , antiderivative size = 165, normalized size of antiderivative = 4.02
method | result | size |
default | \(-\frac {e \,a^{3} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{12}-6 c b \,\textit {\_Z}^{10}+15 c \,b^{2} \textit {\_Z}^{8}+\left (-a^{6} d -20 b^{3} c \right ) \textit {\_Z}^{6}+15 b^{4} c \,\textit {\_Z}^{4}-6 c \,b^{5} \textit {\_Z}^{2}+b^{6} c \right )}{\sum }\left (-\frac {\textit {\_R} \left (\textit {\_R}^{2}+b \right ) \left (-\textit {\_R}^{2}+b \right )^{2} \ln \left (\frac {-\textit {\_R} x +\sqrt {b \,x^{2}+a}}{x}\right )}{-2 \textit {\_R}^{10} c +10 \textit {\_R}^{8} b c -20 \textit {\_R}^{6} b^{2} c +\left (a^{6} d +20 b^{3} c \right ) \textit {\_R}^{4}-10 \textit {\_R}^{2} b^{4} c +2 b^{5} c}\right )\right )}{6}\) | \(165\) |
pseudoelliptic | \(-\frac {e \,a^{3} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{12}-6 c b \,\textit {\_Z}^{10}+15 c \,b^{2} \textit {\_Z}^{8}+\left (-a^{6} d -20 b^{3} c \right ) \textit {\_Z}^{6}+15 b^{4} c \,\textit {\_Z}^{4}-6 c \,b^{5} \textit {\_Z}^{2}+b^{6} c \right )}{\sum }\left (-\frac {\textit {\_R} \left (\textit {\_R}^{2}+b \right ) \left (-\textit {\_R}^{2}+b \right )^{2} \ln \left (\frac {-\textit {\_R} x +\sqrt {b \,x^{2}+a}}{x}\right )}{-2 \textit {\_R}^{10} c +10 \textit {\_R}^{8} b c -20 \textit {\_R}^{6} b^{2} c +\left (a^{6} d +20 b^{3} c \right ) \textit {\_R}^{4}-10 \textit {\_R}^{2} b^{4} c +2 b^{5} c}\right )\right )}{6}\) | \(165\) |
Input:
int(x^2*(b*x^2+a)^(1/2)*(2*b*e*x^2+a*e)/(-b^3*d*x^12-3*a*b^2*d*x^10-3*a^2* b*d*x^8-a^3*d*x^6+c),x,method=_RETURNVERBOSE)
Output:
-1/6*e*a^3*sum(-_R*(_R^2+b)*(-_R^2+b)^2*ln((-_R*x+(b*x^2+a)^(1/2))/x)/(-2* _R^10*c+10*_R^8*b*c-20*_R^6*b^2*c+(a^6*d+20*b^3*c)*_R^4-10*_R^2*b^4*c+2*b^ 5*c),_R=RootOf(c*_Z^12-6*c*b*_Z^10+15*c*b^2*_Z^8+(-a^6*d-20*b^3*c)*_Z^6+15 *b^4*c*_Z^4-6*c*b^5*_Z^2+b^6*c))
Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (29) = 58\).
Time = 3.91 (sec) , antiderivative size = 459, normalized size of antiderivative = 11.20 \[ \int \frac {x^2 \sqrt {a+b x^2} \left (a e+2 b e x^2\right )}{c-a^3 d x^6-3 a^2 b d x^8-3 a b^2 d x^{10}-b^3 d x^{12}} \, dx=\left [\frac {\sqrt {c d} e \log \left (\frac {b^{6} d^{2} x^{24} + 6 \, a b^{5} d^{2} x^{22} + 15 \, a^{2} b^{4} d^{2} x^{20} + 20 \, a^{3} b^{3} d^{2} x^{18} + 15 \, a^{4} b^{2} d^{2} x^{16} + 6 \, a^{5} b d^{2} x^{14} + 18 \, a b^{2} c d x^{10} + 18 \, a^{2} b c d x^{8} + {\left (a^{6} d^{2} + 6 \, b^{3} c d\right )} x^{12} + 6 \, a^{3} c d x^{6} + 4 \, {\left (b^{4} d x^{17} + 4 \, a b^{3} d x^{15} + 6 \, a^{2} b^{2} d x^{13} + 4 \, a^{3} b d x^{11} + a^{4} d x^{9} + b c x^{5} + a c x^{3}\right )} \sqrt {b x^{2} + a} \sqrt {c d} + c^{2}}{b^{6} d^{2} x^{24} + 6 \, a b^{5} d^{2} x^{22} + 15 \, a^{2} b^{4} d^{2} x^{20} + 20 \, a^{3} b^{3} d^{2} x^{18} + 15 \, a^{4} b^{2} d^{2} x^{16} + 6 \, a^{5} b d^{2} x^{14} - 6 \, a b^{2} c d x^{10} - 6 \, a^{2} b c d x^{8} + {\left (a^{6} d^{2} - 2 \, b^{3} c d\right )} x^{12} - 2 \, a^{3} c d x^{6} + c^{2}}\right )}{12 \, c d}, -\frac {\sqrt {-c d} e \arctan \left (\frac {{\left (b^{3} d x^{12} + 3 \, a b^{2} d x^{10} + 3 \, a^{2} b d x^{8} + a^{3} d x^{6} + c\right )} \sqrt {b x^{2} + a} \sqrt {-c d}}{2 \, {\left (b^{2} c d x^{7} + 2 \, a b c d x^{5} + a^{2} c d x^{3}\right )}}\right )}{6 \, c d}\right ] \] Input:
integrate(x^2*(b*x^2+a)^(1/2)*(2*b*e*x^2+a*e)/(-b^3*d*x^12-3*a*b^2*d*x^10- 3*a^2*b*d*x^8-a^3*d*x^6+c),x, algorithm="fricas")
Output:
[1/12*sqrt(c*d)*e*log((b^6*d^2*x^24 + 6*a*b^5*d^2*x^22 + 15*a^2*b^4*d^2*x^ 20 + 20*a^3*b^3*d^2*x^18 + 15*a^4*b^2*d^2*x^16 + 6*a^5*b*d^2*x^14 + 18*a*b ^2*c*d*x^10 + 18*a^2*b*c*d*x^8 + (a^6*d^2 + 6*b^3*c*d)*x^12 + 6*a^3*c*d*x^ 6 + 4*(b^4*d*x^17 + 4*a*b^3*d*x^15 + 6*a^2*b^2*d*x^13 + 4*a^3*b*d*x^11 + a ^4*d*x^9 + b*c*x^5 + a*c*x^3)*sqrt(b*x^2 + a)*sqrt(c*d) + c^2)/(b^6*d^2*x^ 24 + 6*a*b^5*d^2*x^22 + 15*a^2*b^4*d^2*x^20 + 20*a^3*b^3*d^2*x^18 + 15*a^4 *b^2*d^2*x^16 + 6*a^5*b*d^2*x^14 - 6*a*b^2*c*d*x^10 - 6*a^2*b*c*d*x^8 + (a ^6*d^2 - 2*b^3*c*d)*x^12 - 2*a^3*c*d*x^6 + c^2))/(c*d), -1/6*sqrt(-c*d)*e* arctan(1/2*(b^3*d*x^12 + 3*a*b^2*d*x^10 + 3*a^2*b*d*x^8 + a^3*d*x^6 + c)*s qrt(b*x^2 + a)*sqrt(-c*d)/(b^2*c*d*x^7 + 2*a*b*c*d*x^5 + a^2*c*d*x^3))/(c* d)]
Timed out. \[ \int \frac {x^2 \sqrt {a+b x^2} \left (a e+2 b e x^2\right )}{c-a^3 d x^6-3 a^2 b d x^8-3 a b^2 d x^{10}-b^3 d x^{12}} \, dx=\text {Timed out} \] Input:
integrate(x**2*(b*x**2+a)**(1/2)*(2*b*e*x**2+a*e)/(-b**3*d*x**12-3*a*b**2* d*x**10-3*a**2*b*d*x**8-a**3*d*x**6+c),x)
Output:
Timed out
\[ \int \frac {x^2 \sqrt {a+b x^2} \left (a e+2 b e x^2\right )}{c-a^3 d x^6-3 a^2 b d x^8-3 a b^2 d x^{10}-b^3 d x^{12}} \, dx=\int { -\frac {{\left (2 \, b e x^{2} + a e\right )} \sqrt {b x^{2} + a} x^{2}}{b^{3} d x^{12} + 3 \, a b^{2} d x^{10} + 3 \, a^{2} b d x^{8} + a^{3} d x^{6} - c} \,d x } \] Input:
integrate(x^2*(b*x^2+a)^(1/2)*(2*b*e*x^2+a*e)/(-b^3*d*x^12-3*a*b^2*d*x^10- 3*a^2*b*d*x^8-a^3*d*x^6+c),x, algorithm="maxima")
Output:
-integrate((2*b*e*x^2 + a*e)*sqrt(b*x^2 + a)*x^2/(b^3*d*x^12 + 3*a*b^2*d*x ^10 + 3*a^2*b*d*x^8 + a^3*d*x^6 - c), x)
\[ \int \frac {x^2 \sqrt {a+b x^2} \left (a e+2 b e x^2\right )}{c-a^3 d x^6-3 a^2 b d x^8-3 a b^2 d x^{10}-b^3 d x^{12}} \, dx=\int { -\frac {{\left (2 \, b e x^{2} + a e\right )} \sqrt {b x^{2} + a} x^{2}}{b^{3} d x^{12} + 3 \, a b^{2} d x^{10} + 3 \, a^{2} b d x^{8} + a^{3} d x^{6} - c} \,d x } \] Input:
integrate(x^2*(b*x^2+a)^(1/2)*(2*b*e*x^2+a*e)/(-b^3*d*x^12-3*a*b^2*d*x^10- 3*a^2*b*d*x^8-a^3*d*x^6+c),x, algorithm="giac")
Output:
sage0*x
Timed out. \[ \int \frac {x^2 \sqrt {a+b x^2} \left (a e+2 b e x^2\right )}{c-a^3 d x^6-3 a^2 b d x^8-3 a b^2 d x^{10}-b^3 d x^{12}} \, dx=\int -\frac {x^2\,\sqrt {b\,x^2+a}\,\left (2\,b\,e\,x^2+a\,e\right )}{d\,a^3\,x^6+3\,d\,a^2\,b\,x^8+3\,d\,a\,b^2\,x^{10}+d\,b^3\,x^{12}-c} \,d x \] Input:
int(-(x^2*(a + b*x^2)^(1/2)*(a*e + 2*b*e*x^2))/(a^3*d*x^6 - c + b^3*d*x^12 + 3*a^2*b*d*x^8 + 3*a*b^2*d*x^10),x)
Output:
int(-(x^2*(a + b*x^2)^(1/2)*(a*e + 2*b*e*x^2))/(a^3*d*x^6 - c + b^3*d*x^12 + 3*a^2*b*d*x^8 + 3*a*b^2*d*x^10), x)
\[ \int \frac {x^2 \sqrt {a+b x^2} \left (a e+2 b e x^2\right )}{c-a^3 d x^6-3 a^2 b d x^8-3 a b^2 d x^{10}-b^3 d x^{12}} \, dx=e \left (-2 \left (\int \frac {\sqrt {b \,x^{2}+a}\, x^{4}}{b^{3} d \,x^{12}+3 a \,b^{2} d \,x^{10}+3 a^{2} b d \,x^{8}+a^{3} d \,x^{6}-c}d x \right ) b -\left (\int \frac {\sqrt {b \,x^{2}+a}\, x^{2}}{b^{3} d \,x^{12}+3 a \,b^{2} d \,x^{10}+3 a^{2} b d \,x^{8}+a^{3} d \,x^{6}-c}d x \right ) a \right ) \] Input:
int(x^2*(b*x^2+a)^(1/2)*(2*b*e*x^2+a*e)/(-b^3*d*x^12-3*a*b^2*d*x^10-3*a^2* b*d*x^8-a^3*d*x^6+c),x)
Output:
e*( - 2*int((sqrt(a + b*x**2)*x**4)/(a**3*d*x**6 + 3*a**2*b*d*x**8 + 3*a*b **2*d*x**10 + b**3*d*x**12 - c),x)*b - int((sqrt(a + b*x**2)*x**2)/(a**3*d *x**6 + 3*a**2*b*d*x**8 + 3*a*b**2*d*x**10 + b**3*d*x**12 - c),x)*a)