Integrand size = 37, antiderivative size = 172 \[ \int \frac {1}{(c e+d e x)^{13/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{11 d e (c e+d e x)^{11/2}}-\frac {18 \sqrt {1-c^2-2 c d x-d^2 x^2}}{77 d e^3 (c e+d e x)^{7/2}}-\frac {30 \sqrt {1-c^2-2 c d x-d^2 x^2}}{77 d e^5 (c e+d e x)^{3/2}}+\frac {30 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right ),-1\right )}{77 d e^{13/2}} \]
30/77*EllipticF((d*e*x+c*e)^(1/2)/e^(1/2),I)/d/e^(13/2)-2/11*(-d^2*x^2-2*c *d*x-c^2+1)^(1/2)/d/e/(d*e*x+c*e)^(11/2)-18/77*(-d^2*x^2-2*c*d*x-c^2+1)^(1 /2)/d/e^3/(d*e*x+c*e)^(7/2)-30/77*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)/d/e^5/(d* e*x+c*e)^(3/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.23 \[ \int \frac {1}{(c e+d e x)^{13/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {11}{4},\frac {1}{2},-\frac {7}{4},(c+d x)^2\right )}{11 d (e (c+d x))^{13/2}} \]
Time = 0.37 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {1117, 1117, 1117, 1113, 762}
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.
| \(\displaystyle \int \frac {1}{\sqrt {-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{13/2}} \, dx\) |
| \(\Big \downarrow \) 1117 |
| \(\displaystyle \frac {9 \int \frac {1}{(c e+d x e)^{9/2} \sqrt {-c^2-2 d x c-d^2 x^2+1}}dx}{11 e^2}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{11 d e (c e+d e x)^{11/2}}\) |
| \(\Big \downarrow \) 1117 |
| \(\displaystyle \frac {9 \left (\frac {5 \int \frac {1}{(c e+d x e)^{5/2} \sqrt {-c^2-2 d x c-d^2 x^2+1}}dx}{7 e^2}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{7 d e (c e+d e x)^{7/2}}\right )}{11 e^2}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{11 d e (c e+d e x)^{11/2}}\) |
| \(\Big \downarrow \) 1117 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (\frac {\int \frac {1}{\sqrt {c e+d x e} \sqrt {-c^2-2 d x c-d^2 x^2+1}}dx}{3 e^2}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{3 d e (c e+d e x)^{3/2}}\right )}{7 e^2}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{7 d e (c e+d e x)^{7/2}}\right )}{11 e^2}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{11 d e (c e+d e x)^{11/2}}\) |
| \(\Big \downarrow \) 1113 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (\frac {2 \int \frac {1}{\sqrt {1-\frac {(c e+d x e)^2}{e^2}}}d\sqrt {c e+d x e}}{3 d e^3}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{3 d e (c e+d e x)^{3/2}}\right )}{7 e^2}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{7 d e (c e+d e x)^{7/2}}\right )}{11 e^2}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{11 d e (c e+d e x)^{11/2}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (\frac {2 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right ),-1\right )}{3 d e^{5/2}}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{3 d e (c e+d e x)^{3/2}}\right )}{7 e^2}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{7 d e (c e+d e x)^{7/2}}\right )}{11 e^2}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{11 d e (c e+d e x)^{11/2}}\) |
(-2*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(11*d*e*(c*e + d*e*x)^(11/2)) + (9* ((-2*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(7*d*e*(c*e + d*e*x)^(7/2)) + (5*( (-2*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(3*d*e*(c*e + d*e*x)^(3/2)) + (2*El lipticF[ArcSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(3*d*e^(5/2))))/(7*e^2)))/ (11*e^2)
3.15.6.3.1 Defintions of rubi rules used
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/(Sqrt[(d_) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_ Symbol] :> Simp[(4/e)*Sqrt[-c/(b^2 - 4*a*c)] Subst[Int[1/Sqrt[Simp[1 - b^ 2*(x^4/(d^2*(b^2 - 4*a*c))), x]], x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0] && LtQ[c/(b^2 - 4*a*c), 0]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[-2*b*d*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(d^2*(m + 1)*(b^2 - 4*a*c))), x] + Simp[b^2*((m + 2*p + 3)/(d^2*(m + 1)*(b^2 - 4*a* c))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[m, -1] & & (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || IntegerQ[(m + 2*p + 3) /2])
Leaf count of result is larger than twice the leaf count of optimal. \(456\) vs. \(2(146)=292\).
Time = 5.65 (sec) , antiderivative size = 457, normalized size of antiderivative = 2.66
| method | result | size |
| elliptic | \(\frac {\sqrt {-e \left (d x +c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}-1\right )}\, \left (-\frac {2 \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -e \,c^{3}+d e x +c e}}{11 d^{7} e^{7} \left (x +\frac {c}{d}\right )^{6}}-\frac {18 \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -e \,c^{3}+d e x +c e}}{77 d^{5} e^{7} \left (x +\frac {c}{d}\right )^{4}}-\frac {30 \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -e \,c^{3}+d e x +c e}}{77 e^{7} d^{3} \left (x +\frac {c}{d}\right )^{2}}+\frac {30 \left (-\frac {c +1}{d}+\frac {c -1}{d}\right ) \sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}\, F\left (\sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}, \sqrt {\frac {-\frac {c -1}{d}+\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\right )}{77 e^{6} \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -e \,c^{3}+d e x +c e}}\right )}{\sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}\) | \(457\) |
| default | \(\frac {\left (15 \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, F\left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) d^{5} x^{5}+75 \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, F\left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) c \,d^{4} x^{4}+150 \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, F\left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) c^{2} d^{3} x^{3}-30 d^{6} x^{6}+150 \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, F\left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) c^{3} d^{2} x^{2}-180 c \,d^{5} x^{5}+75 \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, F\left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) c^{4} d x -450 c^{2} d^{4} x^{4}+15 \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, F\left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) c^{5}-600 c^{3} d^{3} x^{3}-450 c^{4} d^{2} x^{2}+12 d^{4} x^{4}-180 c^{5} d x +48 c \,d^{3} x^{3}-30 c^{6}+72 c^{2} d^{2} x^{2}+48 c^{3} d x +12 c^{4}+4 d^{2} x^{2}+8 c d x +4 c^{2}+14\right ) \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, \sqrt {e \left (d x +c \right )}}{77 e^{7} \left (d x +c \right )^{6} \left (d^{2} x^{2}+2 c d x +c^{2}-1\right ) d}\) | \(515\) |
(-e*(d*x+c)*(d^2*x^2+2*c*d*x+c^2-1))^(1/2)/(e*(d*x+c))^(1/2)/(-d^2*x^2-2*c *d*x-c^2+1)^(1/2)*(-2/11/d^7/e^7*(-d^3*e*x^3-3*c*d^2*e*x^2-3*c^2*d*e*x-c^3 *e+d*e*x+c*e)^(1/2)/(x+c/d)^6-18/77/d^5/e^7*(-d^3*e*x^3-3*c*d^2*e*x^2-3*c^ 2*d*e*x-c^3*e+d*e*x+c*e)^(1/2)/(x+c/d)^4-30/77/e^7/d^3*(-d^3*e*x^3-3*c*d^2 *e*x^2-3*c^2*d*e*x-c^3*e+d*e*x+c*e)^(1/2)/(x+c/d)^2+30/77/e^6*(-(c+1)/d+(c -1)/d)*((x+(c-1)/d)/(-(c+1)/d+(c-1)/d))^(1/2)*((x+c/d)/(-(c-1)/d+c/d))^(1/ 2)*((x+(c+1)/d)/(-(c-1)/d+(c+1)/d))^(1/2)/(-d^3*e*x^3-3*c*d^2*e*x^2-3*c^2* d*e*x-c^3*e+d*e*x+c*e)^(1/2)*EllipticF(((x+(c-1)/d)/(-(c+1)/d+(c-1)/d))^(1 /2),((-(c-1)/d+(c+1)/d)/(-(c-1)/d+c/d))^(1/2)))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.60 \[ \int \frac {1}{(c e+d e x)^{13/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 \, {\left (15 \, {\left (d^{6} x^{6} + 6 \, c d^{5} x^{5} + 15 \, c^{2} d^{4} x^{4} + 20 \, c^{3} d^{3} x^{3} + 15 \, c^{4} d^{2} x^{2} + 6 \, c^{5} d x + c^{6}\right )} \sqrt {-d^{3} e} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right ) + {\left (15 \, d^{6} x^{4} + 60 \, c d^{5} x^{3} + 9 \, {\left (10 \, c^{2} + 1\right )} d^{4} x^{2} + 6 \, {\left (10 \, c^{3} + 3 \, c\right )} d^{3} x + {\left (15 \, c^{4} + 9 \, c^{2} + 7\right )} d^{2}\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d e x + c e}\right )}}{77 \, {\left (d^{9} e^{7} x^{6} + 6 \, c d^{8} e^{7} x^{5} + 15 \, c^{2} d^{7} e^{7} x^{4} + 20 \, c^{3} d^{6} e^{7} x^{3} + 15 \, c^{4} d^{5} e^{7} x^{2} + 6 \, c^{5} d^{4} e^{7} x + c^{6} d^{3} e^{7}\right )}} \]
-2/77*(15*(d^6*x^6 + 6*c*d^5*x^5 + 15*c^2*d^4*x^4 + 20*c^3*d^3*x^3 + 15*c^ 4*d^2*x^2 + 6*c^5*d*x + c^6)*sqrt(-d^3*e)*weierstrassPInverse(4/d^2, 0, (d *x + c)/d) + (15*d^6*x^4 + 60*c*d^5*x^3 + 9*(10*c^2 + 1)*d^4*x^2 + 6*(10*c ^3 + 3*c)*d^3*x + (15*c^4 + 9*c^2 + 7)*d^2)*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*sqrt(d*e*x + c*e))/(d^9*e^7*x^6 + 6*c*d^8*e^7*x^5 + 15*c^2*d^7*e^7*x^ 4 + 20*c^3*d^6*e^7*x^3 + 15*c^4*d^5*e^7*x^2 + 6*c^5*d^4*e^7*x + c^6*d^3*e^ 7)
\[ \int \frac {1}{(c e+d e x)^{13/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int \frac {1}{\left (e \left (c + d x\right )\right )^{\frac {13}{2}} \sqrt {- \left (c + d x - 1\right ) \left (c + d x + 1\right )}}\, dx \]
\[ \int \frac {1}{(c e+d e x)^{13/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int { \frac {1}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} {\left (d e x + c e\right )}^{\frac {13}{2}}} \,d x } \]
\[ \int \frac {1}{(c e+d e x)^{13/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int { \frac {1}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} {\left (d e x + c e\right )}^{\frac {13}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(c e+d e x)^{13/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int \frac {1}{{\left (c\,e+d\,e\,x\right )}^{13/2}\,\sqrt {-c^2-2\,c\,d\,x-d^2\,x^2+1}} \,d x \]