Integrand size = 25, antiderivative size = 106 \[ \int x^9 \sqrt {\frac {5-7 x^5}{7+5 x^5}} \, dx=-\frac {27}{350} \sqrt {\frac {5-7 x^5}{7+5 x^5}} \left (7+5 x^5\right )+\frac {1}{250} \sqrt {\frac {5-7 x^5}{7+5 x^5}} \left (7+5 x^5\right )^2+\frac {2257 \arctan \left (\sqrt {\frac {5}{7}} \sqrt {\frac {5-7 x^5}{7+5 x^5}}\right )}{875 \sqrt {35}} \]
2257/30625*arctan(1/7*35^(1/2)*((-7*x^5+5)/(5*x^5+7))^(1/2))*35^(1/2)-27/3 50*(5*x^5+7)*((-7*x^5+5)/(5*x^5+7))^(1/2)+1/250*(5*x^5+7)^2*((-7*x^5+5)/(5 *x^5+7))^(1/2)
Time = 0.65 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00 \[ \int x^9 \sqrt {\frac {5-7 x^5}{7+5 x^5}} \, dx=\frac {\sqrt {\frac {5-7 x^5}{7+5 x^5}} \left (35 \sqrt {5-7 x^5} \left (-602-185 x^5+175 x^{10}\right )+4514 \sqrt {35} \sqrt {7+5 x^5} \arctan \left (\frac {\sqrt {\frac {25}{7}-5 x^5}}{\sqrt {7+5 x^5}}\right )\right )}{61250 \sqrt {5-7 x^5}} \]
(Sqrt[(5 - 7*x^5)/(7 + 5*x^5)]*(35*Sqrt[5 - 7*x^5]*(-602 - 185*x^5 + 175*x ^10) + 4514*Sqrt[35]*Sqrt[7 + 5*x^5]*ArcTan[Sqrt[25/7 - 5*x^5]/Sqrt[7 + 5* x^5]]))/(61250*Sqrt[5 - 7*x^5])
Time = 0.23 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2053, 2052, 360, 27, 298, 216}
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.
| \(\displaystyle \int x^9 \sqrt {\frac {5-7 x^5}{5 x^5+7}} \, dx\) |
| \(\Big \downarrow \) 2053 |
| \(\displaystyle \frac {1}{5} \int x^5 \sqrt {\frac {5-7 x^5}{5 x^5+7}}dx^5\) |
| \(\Big \downarrow \) 2052 |
| \(\displaystyle -\frac {148}{5} \int \frac {x^{10} \left (5-7 x^{10}\right )}{\left (5 x^{10}+7\right )^3}d\sqrt {\frac {5-7 x^5}{5 x^5+7}}\) |
| \(\Big \downarrow \) 360 |
| \(\displaystyle -\frac {148}{5} \left (-\frac {1}{100} \int -\frac {2 \left (37-70 x^{10}\right )}{\left (5 x^{10}+7\right )^2}d\sqrt {\frac {5-7 x^5}{5 x^5+7}}-\frac {37 \sqrt {\frac {5-7 x^5}{5 x^5+7}}}{50 \left (5 x^{10}+7\right )^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {148}{5} \left (\frac {1}{50} \int \frac {37-70 x^{10}}{\left (5 x^{10}+7\right )^2}d\sqrt {\frac {5-7 x^5}{5 x^5+7}}-\frac {37 \sqrt {\frac {5-7 x^5}{5 x^5+7}}}{50 \left (5 x^{10}+7\right )^2}\right )\) |
| \(\Big \downarrow \) 298 |
| \(\displaystyle -\frac {148}{5} \left (\frac {1}{50} \left (\frac {135 \sqrt {\frac {5-7 x^5}{5 x^5+7}}}{14 \left (5 x^{10}+7\right )}-\frac {61}{14} \int \frac {1}{5 x^{10}+7}d\sqrt {\frac {5-7 x^5}{5 x^5+7}}\right )-\frac {37 \sqrt {\frac {5-7 x^5}{5 x^5+7}}}{50 \left (5 x^{10}+7\right )^2}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {148}{5} \left (\frac {1}{50} \left (\frac {135 \sqrt {\frac {5-7 x^5}{5 x^5+7}}}{14 \left (5 x^{10}+7\right )}-\frac {61 \arctan \left (\sqrt {\frac {5}{7}} \sqrt {\frac {5-7 x^5}{5 x^5+7}}\right )}{14 \sqrt {35}}\right )-\frac {37 \sqrt {\frac {5-7 x^5}{5 x^5+7}}}{50 \left (5 x^{10}+7\right )^2}\right )\) |
(-148*((-37*Sqrt[(5 - 7*x^5)/(7 + 5*x^5)])/(50*(7 + 5*x^10)^2) + ((135*Sqr t[(5 - 7*x^5)/(7 + 5*x^5)])/(14*(7 + 5*x^10)) - (61*ArcTan[Sqrt[5/7]*Sqrt[ (5 - 7*x^5)/(7 + 5*x^5)]])/(14*Sqrt[35]))/50))/5
3.3.93.3.1 Defintions of rubi rules used
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[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : > Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1)) Int[(a + b*x^2)^(p + 1)*Expan dToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)] - (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[m/2, 0] & & (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)))/((c_) + (d_.)*(x_)))^(p_), x_S ymbol] :> With[{q = Denominator[p]}, Simp[q*e*(b*c - a*d) Subst[Int[x^(q* (p + 1) - 1)*(((-a)*e + c*x^q)^m/(b*e - d*x^q)^(m + 2)), x], x, (e*((a + b* x)/(c + d*x)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e, m}, x] && FractionQ[p] && IntegerQ[m]
Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.) ))^(p_), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(e*( (a + b*x)/(c + d*x)))^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.78 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.08
| method | result | size |
| trager | \(\frac {\left (5 x^{5}+7\right ) \left (35 x^{5}-86\right ) \sqrt {-\frac {7 x^{5}-5}{5 x^{5}+7}}}{1750}+\frac {2257 \operatorname {RootOf}\left (\textit {\_Z}^{2}+35\right ) \ln \left (175 \sqrt {-\frac {7 x^{5}-5}{5 x^{5}+7}}\, x^{5}+35 \operatorname {RootOf}\left (\textit {\_Z}^{2}+35\right ) x^{5}+245 \sqrt {-\frac {7 x^{5}-5}{5 x^{5}+7}}+12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+35\right )\right )}{61250}\) | \(114\) |
| risch | \(\frac {\left (5 x^{5}+7\right ) \left (35 x^{5}-86\right ) \sqrt {-\frac {7 x^{5}-5}{5 x^{5}+7}}}{1750}+\frac {2257 \operatorname {RootOf}\left (\textit {\_Z}^{2}+35\right ) \ln \left (-35 \operatorname {RootOf}\left (\textit {\_Z}^{2}+35\right ) x^{5}+35 \sqrt {-35 x^{10}-24 x^{5}+35}-12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+35\right )\right ) \sqrt {-\frac {7 x^{5}-5}{5 x^{5}+7}}\, \sqrt {-\left (5 x^{5}+7\right ) \left (7 x^{5}-5\right )}}{61250 \left (7 x^{5}-5\right )}\) | \(130\) |
1/1750*(5*x^5+7)*(35*x^5-86)*(-(7*x^5-5)/(5*x^5+7))^(1/2)+2257/61250*RootO f(_Z^2+35)*ln(175*(-(7*x^5-5)/(5*x^5+7))^(1/2)*x^5+35*RootOf(_Z^2+35)*x^5+ 245*(-(7*x^5-5)/(5*x^5+7))^(1/2)+12*RootOf(_Z^2+35))
Time = 0.34 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.77 \[ \int x^9 \sqrt {\frac {5-7 x^5}{7+5 x^5}} \, dx=\frac {1}{1750} \, {\left (175 \, x^{10} - 185 \, x^{5} - 602\right )} \sqrt {-\frac {7 \, x^{5} - 5}{5 \, x^{5} + 7}} + \frac {2257}{61250} \, \sqrt {35} \arctan \left (\frac {\sqrt {35} {\left (35 \, x^{5} + 12\right )} \sqrt {-\frac {7 \, x^{5} - 5}{5 \, x^{5} + 7}}}{35 \, {\left (7 \, x^{5} - 5\right )}}\right ) \]
1/1750*(175*x^10 - 185*x^5 - 602)*sqrt(-(7*x^5 - 5)/(5*x^5 + 7)) + 2257/61 250*sqrt(35)*arctan(1/35*sqrt(35)*(35*x^5 + 12)*sqrt(-(7*x^5 - 5)/(5*x^5 + 7))/(7*x^5 - 5))
Timed out. \[ \int x^9 \sqrt {\frac {5-7 x^5}{7+5 x^5}} \, dx=\text {Timed out} \]
Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.14 \[ \int x^9 \sqrt {\frac {5-7 x^5}{7+5 x^5}} \, dx=\frac {2257}{30625} \, \sqrt {35} \arctan \left (\frac {1}{7} \, \sqrt {35} \sqrt {-\frac {7 \, x^{5} - 5}{5 \, x^{5} + 7}}\right ) - \frac {37 \, {\left (675 \, \left (-\frac {7 \, x^{5} - 5}{5 \, x^{5} + 7}\right )^{\frac {3}{2}} + 427 \, \sqrt {-\frac {7 \, x^{5} - 5}{5 \, x^{5} + 7}}\right )}}{875 \, {\left (\frac {25 \, {\left (7 \, x^{5} - 5\right )}^{2}}{{\left (5 \, x^{5} + 7\right )}^{2}} - \frac {70 \, {\left (7 \, x^{5} - 5\right )}}{5 \, x^{5} + 7} + 49\right )}} \]
2257/30625*sqrt(35)*arctan(1/7*sqrt(35)*sqrt(-(7*x^5 - 5)/(5*x^5 + 7))) - 37/875*(675*(-(7*x^5 - 5)/(5*x^5 + 7))^(3/2) + 427*sqrt(-(7*x^5 - 5)/(5*x^ 5 + 7)))/(25*(7*x^5 - 5)^2/(5*x^5 + 7)^2 - 70*(7*x^5 - 5)/(5*x^5 + 7) + 49 )
Time = 0.32 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.44 \[ \int x^9 \sqrt {\frac {5-7 x^5}{7+5 x^5}} \, dx=\frac {1}{61250} \, {\left (35 \, \sqrt {-35 \, x^{10} - 24 \, x^{5} + 35} {\left (35 \, x^{5} - 86\right )} - 2257 \, \sqrt {35} \arcsin \left (\frac {35}{37} \, x^{5} + \frac {12}{37}\right )\right )} \mathrm {sgn}\left (5 \, x^{5} + 7\right ) \]
1/61250*(35*sqrt(-35*x^10 - 24*x^5 + 35)*(35*x^5 - 86) - 2257*sqrt(35)*arc sin(35/37*x^5 + 12/37))*sgn(5*x^5 + 7)
Time = 0.23 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.26 \[ \int x^9 \sqrt {\frac {5-7 x^5}{7+5 x^5}} \, dx=\frac {2257\,\sqrt {35}\,\mathrm {atan}\left (\frac {\sqrt {5}\,\sqrt {7}\,\sqrt {-\frac {7\,x^5-5}{5\,x^5+7}}}{7}\right )}{30625}-\frac {43\,\sqrt {5}\,\sqrt {7}\,\sqrt {35}\,\sqrt {-\frac {7\,x^5-5}{5\,x^5+7}}}{4375}-\frac {37\,\sqrt {5}\,\sqrt {7}\,\sqrt {35}\,x^5\,\sqrt {-\frac {7\,x^5-5}{5\,x^5+7}}}{12250}+\frac {\sqrt {5}\,\sqrt {7}\,\sqrt {35}\,x^{10}\,\sqrt {-\frac {7\,x^5-5}{5\,x^5+7}}}{350} \]
(2257*35^(1/2)*atan((5^(1/2)*7^(1/2)*(-(7*x^5 - 5)/(5*x^5 + 7))^(1/2))/7)) /30625 - (43*5^(1/2)*7^(1/2)*35^(1/2)*(-(7*x^5 - 5)/(5*x^5 + 7))^(1/2))/43 75 - (37*5^(1/2)*7^(1/2)*35^(1/2)*x^5*(-(7*x^5 - 5)/(5*x^5 + 7))^(1/2))/12 250 + (5^(1/2)*7^(1/2)*35^(1/2)*x^10*(-(7*x^5 - 5)/(5*x^5 + 7))^(1/2))/350