Integrand size = 23, antiderivative size = 67 \[ \int x \sqrt {\frac {5-7 x^2}{7+5 x^2}} \, dx=\frac {\left (7+5 x^2\right ) \sqrt {-7+\frac {74}{7+5 x^2}}}{10 \sqrt {5}}-\frac {37 \arctan \left (\frac {\sqrt {-7+\frac {74}{7+5 x^2}}}{\sqrt {7}}\right )}{5 \sqrt {35}} \] Output:
1/50*(5*x^2+7)*(-7+74/(5*x^2+7))^(1/2)*5^(1/2)-37/175*arctan(1/7*(-7+74/(5 *x^2+7))^(1/2)*7^(1/2))*35^(1/2)
Time = 0.22 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.78 \[ \int x \sqrt {\frac {5-7 x^2}{7+5 x^2}} \, dx=\frac {\sqrt {\frac {5-7 x^2}{7+5 x^2}} \left (35 \sqrt {5-7 x^2} \left (7+5 x^2\right )+148 \sqrt {35} \sqrt {7+5 x^2} \arctan \left (\frac {\sqrt {5} \sqrt {5-7 x^2}}{\sqrt {74}-\sqrt {7} \sqrt {7+5 x^2}}\right )\right )}{350 \sqrt {5-7 x^2}} \] Input:
Integrate[x*Sqrt[(5 - 7*x^2)/(7 + 5*x^2)],x]
Output:
(Sqrt[(5 - 7*x^2)/(7 + 5*x^2)]*(35*Sqrt[5 - 7*x^2]*(7 + 5*x^2) + 148*Sqrt[ 35]*Sqrt[7 + 5*x^2]*ArcTan[(Sqrt[5]*Sqrt[5 - 7*x^2])/(Sqrt[74] - Sqrt[7]*S qrt[7 + 5*x^2])]))/(350*Sqrt[5 - 7*x^2])
Time = 0.37 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.13, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2053, 2051, 252, 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.
\(\displaystyle \int x \sqrt {\frac {5-7 x^2}{5 x^2+7}} \, dx\) |
\(\Big \downarrow \) 2053 |
\(\displaystyle \frac {1}{2} \int \sqrt {\frac {5-7 x^2}{5 x^2+7}}dx^2\) |
\(\Big \downarrow \) 2051 |
\(\displaystyle -74 \int \frac {x^4}{\left (5 x^4+7\right )^2}d\sqrt {\frac {5-7 x^2}{5 x^2+7}}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle -74 \left (\frac {1}{10} \int \frac {1}{5 x^4+7}d\sqrt {\frac {5-7 x^2}{5 x^2+7}}-\frac {\sqrt {\frac {5-7 x^2}{5 x^2+7}}}{10 \left (5 x^4+7\right )}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -74 \left (\frac {\arctan \left (\sqrt {\frac {5}{7}} \sqrt {\frac {5-7 x^2}{5 x^2+7}}\right )}{10 \sqrt {35}}-\frac {\sqrt {\frac {5-7 x^2}{5 x^2+7}}}{10 \left (5 x^4+7\right )}\right )\) |
Input:
Int[x*Sqrt[(5 - 7*x^2)/(7 + 5*x^2)],x]
Output:
-74*(-1/10*Sqrt[(5 - 7*x^2)/(7 + 5*x^2)]/(7 + 5*x^4) + ArcTan[Sqrt[5/7]*Sq rt[(5 - 7*x^2)/(7 + 5*x^2)]]/(10*Sqrt[35]))
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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_ Symbol] :> With[{q = Denominator[p]}, Simp[q*e*((b*c - a*d)/n) Subst[Int[ x^(q*(p + 1) - 1)*(((-a)*e + c*x^q)^(1/n - 1)/(b*e - d*x^q)^(1/n + 1)), x], x, (e*((a + b*x^n)/(c + d*x^n)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e}, x] && FractionQ[p] && IntegerQ[1/n]
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]]
Time = 0.35 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.16
method | result | size |
default | \(\frac {\sqrt {-\frac {7 x^{2}-5}{5 x^{2}+7}}\, \left (5 x^{2}+7\right ) \left (37 \sqrt {35}\, \arcsin \left (\frac {35 x^{2}}{37}+\frac {12}{37}\right )+35 \sqrt {-35 x^{4}-24 x^{2}+35}\right )}{350 \sqrt {-\left (7 x^{2}-5\right ) \left (5 x^{2}+7\right )}}\) | \(78\) |
risch | \(\frac {\left (5 x^{2}+7\right ) \sqrt {-\frac {7 x^{2}-5}{5 x^{2}+7}}}{10}-\frac {37 \sqrt {35}\, \arcsin \left (\frac {35 x^{2}}{37}+\frac {12}{37}\right ) \sqrt {-\frac {7 x^{2}-5}{5 x^{2}+7}}\, \sqrt {-\left (7 x^{2}-5\right ) \left (5 x^{2}+7\right )}}{350 \left (7 x^{2}-5\right )}\) | \(91\) |
trager | \(7 \left (\frac {x^{2}}{14}+\frac {1}{10}\right ) \sqrt {-\frac {7 x^{2}-5}{5 x^{2}+7}}+\frac {37 \operatorname {RootOf}\left (\textit {\_Z}^{2}+35\right ) \ln \left (-35 \operatorname {RootOf}\left (\textit {\_Z}^{2}+35\right ) x^{2}+175 \sqrt {-\frac {7 x^{2}-5}{5 x^{2}+7}}\, x^{2}-12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+35\right )+245 \sqrt {-\frac {7 x^{2}-5}{5 x^{2}+7}}\right )}{350}\) | \(107\) |
Input:
int(x*((-7*x^2+5)/(5*x^2+7))^(1/2),x,method=_RETURNVERBOSE)
Output:
1/350*(-(7*x^2-5)/(5*x^2+7))^(1/2)*(5*x^2+7)*(37*35^(1/2)*arcsin(35/37*x^2 +12/37)+35*(-35*x^4-24*x^2+35)^(1/2))/(-(7*x^2-5)*(5*x^2+7))^(1/2)
Time = 0.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.13 \[ \int x \sqrt {\frac {5-7 x^2}{7+5 x^2}} \, dx=\frac {1}{10} \, {\left (5 \, x^{2} + 7\right )} \sqrt {-\frac {7 \, x^{2} - 5}{5 \, x^{2} + 7}} - \frac {37}{350} \, \sqrt {35} \arctan \left (\frac {\sqrt {35} {\left (5 \, x^{2} + 7\right )} \sqrt {-\frac {7 \, x^{2} - 5}{5 \, x^{2} + 7}}}{35 \, x^{2} + 12}\right ) \] Input:
integrate(x*((-7*x^2+5)/(5*x^2+7))^(1/2),x, algorithm="fricas")
Output:
1/10*(5*x^2 + 7)*sqrt(-(7*x^2 - 5)/(5*x^2 + 7)) - 37/350*sqrt(35)*arctan(s qrt(35)*(5*x^2 + 7)*sqrt(-(7*x^2 - 5)/(5*x^2 + 7))/(35*x^2 + 12))
\[ \int x \sqrt {\frac {5-7 x^2}{7+5 x^2}} \, dx=\int x \sqrt {- \frac {7 x^{2} - 5}{5 x^{2} + 7}}\, dx \] Input:
integrate(x*((-7*x**2+5)/(5*x**2+7))**(1/2),x)
Output:
Integral(x*sqrt(-(7*x**2 - 5)/(5*x**2 + 7)), x)
Time = 0.12 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.13 \[ \int x \sqrt {\frac {5-7 x^2}{7+5 x^2}} \, dx=-\frac {37}{175} \, \sqrt {35} \arctan \left (\frac {1}{7} \, \sqrt {35} \sqrt {-\frac {7 \, x^{2} - 5}{5 \, x^{2} + 7}}\right ) - \frac {37 \, \sqrt {-\frac {7 \, x^{2} - 5}{5 \, x^{2} + 7}}}{5 \, {\left (\frac {5 \, {\left (7 \, x^{2} - 5\right )}}{5 \, x^{2} + 7} - 7\right )}} \] Input:
integrate(x*((-7*x^2+5)/(5*x^2+7))^(1/2),x, algorithm="maxima")
Output:
-37/175*sqrt(35)*arctan(1/7*sqrt(35)*sqrt(-(7*x^2 - 5)/(5*x^2 + 7))) - 37/ 5*sqrt(-(7*x^2 - 5)/(5*x^2 + 7))/(5*(7*x^2 - 5)/(5*x^2 + 7) - 7)
Time = 0.13 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.45 \[ \int x \sqrt {\frac {5-7 x^2}{7+5 x^2}} \, dx=\frac {37}{350} \, \sqrt {35} \arcsin \left (\frac {35}{37} \, x^{2} + \frac {12}{37}\right ) + \frac {1}{10} \, \sqrt {-35 \, x^{4} - 24 \, x^{2} + 35} \] Input:
integrate(x*((-7*x^2+5)/(5*x^2+7))^(1/2),x, algorithm="giac")
Output:
37/350*sqrt(35)*arcsin(35/37*x^2 + 12/37) + 1/10*sqrt(-35*x^4 - 24*x^2 + 3 5)
Time = 0.22 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.31 \[ \int x \sqrt {\frac {5-7 x^2}{7+5 x^2}} \, dx=-\frac {37\,\sqrt {35}\,\mathrm {atan}\left (\frac {\sqrt {5}\,\sqrt {7}\,\sqrt {-\frac {7\,x^2-5}{5\,x^2+7}}}{7}\right )}{175}-\frac {37\,\sqrt {5}\,\sqrt {7}\,\sqrt {35}\,\sqrt {-\frac {7\,x^2-5}{5\,x^2+7}}}{1225\,\left (\frac {5\,x^2-\frac {25}{7}}{5\,x^2+7}-1\right )} \] Input:
int(x*(-(7*x^2 - 5)/(5*x^2 + 7))^(1/2),x)
Output:
- (37*35^(1/2)*atan((5^(1/2)*7^(1/2)*(-(7*x^2 - 5)/(5*x^2 + 7))^(1/2))/7)) /175 - (37*5^(1/2)*7^(1/2)*35^(1/2)*(-(7*x^2 - 5)/(5*x^2 + 7))^(1/2))/(122 5*((5*x^2 - 25/7)/(5*x^2 + 7) - 1))
Time = 0.39 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.78 \[ \int x \sqrt {\frac {5-7 x^2}{7+5 x^2}} \, dx=-\frac {37 \sqrt {35}\, \mathit {atan} \left (\frac {\sqrt {5 x^{2}+7}\, \sqrt {-7 x^{2}+5}\, \sqrt {35}}{35 x^{2}-25}\right )}{175}+\frac {\sqrt {5 x^{2}+7}\, \sqrt {-7 x^{2}+5}}{10} \] Input:
int(x*((-7*x^2+5)/(5*x^2+7))^(1/2),x)
Output:
( - 74*sqrt(35)*atan((sqrt(5*x**2 + 7)*sqrt( - 7*x**2 + 5)*sqrt(35))/(35*x **2 - 25)) + 35*sqrt(5*x**2 + 7)*sqrt( - 7*x**2 + 5))/350