Integrand size = 66, antiderivative size = 41 \[ \int \frac {x^2 \sqrt {a+b x^2} \left (a e+2 b e x^2\right )}{c+a^3 d x^6+3 a^2 b d x^8+3 a b^2 d x^{10}+b^3 d x^{12}} \, dx=\frac {e \arctan \left (\frac {\sqrt {d} x^3 \left (a+b x^2\right )^{3/2}}{\sqrt {c}}\right )}{3 \sqrt {c} \sqrt {d}} \] Output:
1/3*e*arctan(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. \(45669\) vs. \(2(41)=82\).
Time = 6.12 (sec) , antiderivative size = 45669, normalized size of antiderivative = 1113.88 \[ \int \frac {x^2 \sqrt {a+b x^2} \left (a e+2 b e x^2\right )}{c+a^3 d x^6+3 a^2 b d x^8+3 a b^2 d x^{10}+b^3 d x^{12}} \, dx=\text {Result too large to show} \] 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 = 1.94 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, 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.
| \(\displaystyle \int \frac {x^2 \sqrt {a+b x^2} \left (a e+2 b e x^2\right )}{a^3 d x^6+3 a^2 b d x^8+3 a b^2 d x^{10}+b^3 d x^{12}+c} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e x^2 \sqrt {a+b x^2} \left (a+2 b x^2\right )}{d x^6 \left (a+b x^2\right )^3+c}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle e \int \frac {x^2 \sqrt {b x^2+a} \left (2 b x^2+a\right )}{d \left (b x^2+a\right )^3 x^6+c}dx\) |
| \(\Big \downarrow \) 7261 |
| \(\displaystyle \frac {1}{3} e \int \frac {1}{d \left (b x^2+a\right )^3 x^6+c}d\left (x^3 \left (b x^2+a\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {e \arctan \left (\frac {\sqrt {d} x^3 \left (a+b x^2\right )^{3/2}}{\sqrt {c}}\right )}{3 \sqrt {c} \sqrt {d}}\) |
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*ArcTan[(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)*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 = 0.29 (sec) , antiderivative size = 163, normalized size of antiderivative = 3.98
| method | result | size |
| default | \(-\frac {e \,a^{3} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{12}-6 c b \,\textit {\_Z}^{10}+15 c \,b^{2} \textit {\_Z}^{8}+\left (a^{6} d -20 b^{3} c \right ) \textit {\_Z}^{6}+15 b^{4} c \,\textit {\_Z}^{4}-6 c \,b^{5} \textit {\_Z}^{2}+b^{6} c \right )}{\sum }\frac {\textit {\_R} \left (\textit {\_R}^{2}+b \right ) \left (-\textit {\_R}^{2}+b \right )^{2} \ln \left (\frac {-\textit {\_R} x +\sqrt {b \,x^{2}+a}}{x}\right )}{2 \textit {\_R}^{10} c -10 \textit {\_R}^{8} b c +20 \textit {\_R}^{6} b^{2} c +\left (a^{6} d -20 b^{3} c \right ) \textit {\_R}^{4}+10 \textit {\_R}^{2} b^{4} c -2 b^{5} c}\right )}{6}\) | \(163\) |
| pseudoelliptic | \(-\frac {e \,a^{3} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{12}-6 c b \,\textit {\_Z}^{10}+15 c \,b^{2} \textit {\_Z}^{8}+\left (a^{6} d -20 b^{3} c \right ) \textit {\_Z}^{6}+15 b^{4} c \,\textit {\_Z}^{4}-6 c \,b^{5} \textit {\_Z}^{2}+b^{6} c \right )}{\sum }\frac {\textit {\_R} \left (\textit {\_R}^{2}+b \right ) \left (-\textit {\_R}^{2}+b \right )^{2} \ln \left (\frac {-\textit {\_R} x +\sqrt {b \,x^{2}+a}}{x}\right )}{2 \textit {\_R}^{10} c -10 \textit {\_R}^{8} b c +20 \textit {\_R}^{6} b^{2} c +\left (a^{6} d -20 b^{3} c \right ) \textit {\_R}^{4}+10 \textit {\_R}^{2} b^{4} c -2 b^{5} c}\right )}{6}\) | \(163\) |
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 = 3.63 (sec) , antiderivative size = 463, normalized size of antiderivative = 11.29 \[ \int \frac {x^2 \sqrt {a+b x^2} \left (a e+2 b e x^2\right )}{c+a^3 d x^6+3 a^2 b d x^8+3 a b^2 d x^{10}+b^3 d x^{12}} \, dx=\left [-\frac {\sqrt {-c d} e \log \left (\frac {b^{6} d^{2} x^{24} + 6 \, a b^{5} d^{2} x^{22} + 15 \, a^{2} b^{4} d^{2} x^{20} + 20 \, a^{3} b^{3} d^{2} x^{18} + 15 \, a^{4} b^{2} d^{2} x^{16} + 6 \, a^{5} b d^{2} x^{14} - 18 \, a b^{2} c d x^{10} - 18 \, a^{2} b c d x^{8} + {\left (a^{6} d^{2} - 6 \, b^{3} c d\right )} x^{12} - 6 \, a^{3} c d x^{6} - 4 \, {\left (b^{4} d x^{17} + 4 \, a b^{3} d x^{15} + 6 \, a^{2} b^{2} d x^{13} + 4 \, a^{3} b d x^{11} + a^{4} d x^{9} - b c x^{5} - a c x^{3}\right )} \sqrt {b x^{2} + a} \sqrt {-c d} + c^{2}}{b^{6} d^{2} x^{24} + 6 \, a b^{5} d^{2} x^{22} + 15 \, a^{2} b^{4} d^{2} x^{20} + 20 \, a^{3} b^{3} d^{2} x^{18} + 15 \, a^{4} b^{2} d^{2} x^{16} + 6 \, a^{5} b d^{2} x^{14} + 6 \, a b^{2} c d x^{10} + 6 \, a^{2} b c d x^{8} + {\left (a^{6} d^{2} + 2 \, b^{3} c d\right )} x^{12} + 2 \, a^{3} c d x^{6} + c^{2}}\right )}{12 \, c d}, \frac {\sqrt {c d} e \arctan \left (\frac {{\left (b^{3} d x^{12} + 3 \, a b^{2} d x^{10} + 3 \, a^{2} b d x^{8} + a^{3} d x^{6} - c\right )} \sqrt {b x^{2} + a} \sqrt {c d}}{2 \, {\left (b^{2} c d x^{7} + 2 \, a b c d x^{5} + a^{2} c d x^{3}\right )}}\right )}{6 \, c d}\right ] \] 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)* sqrt(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))/(c + a^3*d*x^6 + 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))/(c + a^3*d*x^6 + 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)