Integrand size = 36, antiderivative size = 57 \[ \int \frac {x^{\frac {1}{2} (-1+n)}}{\sqrt {1-5 x} \sqrt {2 x^n+3 x^{1+n}}} \, dx=\frac {\sqrt {\frac {2}{5}} x^{n/2} \sqrt {2+3 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {5} \sqrt {x}\right ),-\frac {3}{10}\right )}{\sqrt {2 x^n+3 x^{1+n}}} \] Output:
1/5*10^(1/2)*x^(1/2*n)*(2+3*x)^(1/2)*EllipticF(x^(1/2)*5^(1/2),1/10*I*30^( 1/2))/(2*x^n+3*x^(1+n))^(1/2)
Time = 2.37 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.28 \[ \int \frac {x^{\frac {1}{2} (-1+n)}}{\sqrt {1-5 x} \sqrt {2 x^n+3 x^{1+n}}} \, dx=-\frac {2 x^{\frac {1+n}{2}} \sqrt {\frac {2+3 x}{-1+5 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{\sqrt {1-5 x}}\right ),\frac {13}{3}\right )}{\sqrt {3} \sqrt {x^n (2+3 x)} \sqrt {\frac {x}{-1+5 x}}} \] Input:
Integrate[x^((-1 + n)/2)/(Sqrt[1 - 5*x]*Sqrt[2*x^n + 3*x^(1 + n)]),x]
Output:
(-2*x^((1 + n)/2)*Sqrt[(2 + 3*x)/(-1 + 5*x)]*EllipticF[ArcSin[1/Sqrt[1 - 5 *x]], 13/3])/(Sqrt[3]*Sqrt[x^n*(2 + 3*x)]*Sqrt[x/(-1 + 5*x)])
Time = 0.33 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1948, 125}
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.
\(\displaystyle \int \frac {x^{\frac {n-1}{2}}}{\sqrt {1-5 x} \sqrt {3 x^{n+1}+2 x^n}} \, dx\) |
\(\Big \downarrow \) 1948 |
\(\displaystyle \frac {\sqrt {3 x+2} x^{n/2} \int \frac {1}{\sqrt {1-5 x} \sqrt {x} \sqrt {3 x+2}}dx}{\sqrt {3 x^{n+1}+2 x^n}}\) |
\(\Big \downarrow \) 125 |
\(\displaystyle \frac {\sqrt {\frac {2}{5}} \sqrt {3 x+2} x^{n/2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {5} \sqrt {x}\right ),-\frac {3}{10}\right )}{\sqrt {3 x^{n+1}+2 x^n}}\) |
Input:
Int[x^((-1 + n)/2)/(Sqrt[1 - 5*x]*Sqrt[2*x^n + 3*x^(1 + n)]),x]
Output:
(Sqrt[2/5]*x^(n/2)*Sqrt[2 + 3*x]*EllipticF[ArcSin[Sqrt[5]*Sqrt[x]], -3/10] )/Sqrt[2*x^n + 3*x^(1 + n)]
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[(2/(b*Sqrt[e]))*Rt[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]* Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] & & GtQ[e, 0] && (GtQ[-b/d, 0] || LtQ[-b/f, 0])
Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Simp[e^IntPart[m]*(e*x)^FracPart[m]*( (a*x^j + b*x^(j + n))^FracPart[p]/(x^(FracPart[m] + j*FracPart[p])*(a + b*x ^n)^FracPart[p])) Int[x^(m + j*p)*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, j, m, n, p, q}, x] && EqQ[jn, j + n] && !IntegerQ[p] && NeQ[b*c - a*d, 0] && !(EqQ[n, 1] && EqQ[j, 1])
\[\int \frac {x^{-\frac {1}{2}+\frac {n}{2}}}{\sqrt {1-5 x}\, \sqrt {2 x^{n}+3 x^{1+n}}}d x\]
Input:
int(x^(-1/2+1/2*n)/(1-5*x)^(1/2)/(2*x^n+3*x^(1+n))^(1/2),x)
Output:
int(x^(-1/2+1/2*n)/(1-5*x)^(1/2)/(2*x^n+3*x^(1+n))^(1/2),x)
Time = 0.08 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.19 \[ \int \frac {x^{\frac {1}{2} (-1+n)}}{\sqrt {1-5 x} \sqrt {2 x^n+3 x^{1+n}}} \, dx=-\frac {2}{15} \, \sqrt {-15} {\rm weierstrassPInverse}\left (\frac {556}{675}, -\frac {10304}{91125}, x + \frac {7}{45}\right ) \] Input:
integrate(x^(-1/2+1/2*n)/(1-5*x)^(1/2)/(2*x^n+3*x^(1+n))^(1/2),x, algorith m="fricas")
Output:
-2/15*sqrt(-15)*weierstrassPInverse(556/675, -10304/91125, x + 7/45)
\[ \int \frac {x^{\frac {1}{2} (-1+n)}}{\sqrt {1-5 x} \sqrt {2 x^n+3 x^{1+n}}} \, dx=\int \frac {x^{\frac {n}{2} - \frac {1}{2}}}{\sqrt {1 - 5 x} \sqrt {2 x^{n} + 3 x^{n + 1}}}\, dx \] Input:
integrate(x**(-1/2+1/2*n)/(1-5*x)**(1/2)/(2*x**n+3*x**(1+n))**(1/2),x)
Output:
Integral(x**(n/2 - 1/2)/(sqrt(1 - 5*x)*sqrt(2*x**n + 3*x**(n + 1))), x)
\[ \int \frac {x^{\frac {1}{2} (-1+n)}}{\sqrt {1-5 x} \sqrt {2 x^n+3 x^{1+n}}} \, dx=\int { \frac {x^{\frac {1}{2} \, n - \frac {1}{2}}}{\sqrt {3 \, x^{n + 1} + 2 \, x^{n}} \sqrt {-5 \, x + 1}} \,d x } \] Input:
integrate(x^(-1/2+1/2*n)/(1-5*x)^(1/2)/(2*x^n+3*x^(1+n))^(1/2),x, algorith m="maxima")
Output:
integrate(x^(1/2*n - 1/2)/(sqrt(3*x^(n + 1) + 2*x^n)*sqrt(-5*x + 1)), x)
\[ \int \frac {x^{\frac {1}{2} (-1+n)}}{\sqrt {1-5 x} \sqrt {2 x^n+3 x^{1+n}}} \, dx=\int { \frac {x^{\frac {1}{2} \, n - \frac {1}{2}}}{\sqrt {3 \, x^{n + 1} + 2 \, x^{n}} \sqrt {-5 \, x + 1}} \,d x } \] Input:
integrate(x^(-1/2+1/2*n)/(1-5*x)^(1/2)/(2*x^n+3*x^(1+n))^(1/2),x, algorith m="giac")
Output:
integrate(x^(1/2*n - 1/2)/(sqrt(3*x^(n + 1) + 2*x^n)*sqrt(-5*x + 1)), x)
Timed out. \[ \int \frac {x^{\frac {1}{2} (-1+n)}}{\sqrt {1-5 x} \sqrt {2 x^n+3 x^{1+n}}} \, dx=\int \frac {x^{\frac {n}{2}-\frac {1}{2}}}{\sqrt {2\,x^n+3\,x^{n+1}}\,\sqrt {1-5\,x}} \,d x \] Input:
int(x^(n/2 - 1/2)/((2*x^n + 3*x^(n + 1))^(1/2)*(1 - 5*x)^(1/2)),x)
Output:
int(x^(n/2 - 1/2)/((2*x^n + 3*x^(n + 1))^(1/2)*(1 - 5*x)^(1/2)), x)
\[ \int \frac {x^{\frac {1}{2} (-1+n)}}{\sqrt {1-5 x} \sqrt {2 x^n+3 x^{1+n}}} \, dx=-\left (\int \frac {\sqrt {x}\, \sqrt {3 x +2}\, \sqrt {-5 x +1}}{15 x^{3}+7 x^{2}-2 x}d x \right ) \] Input:
int(x^(-1/2+1/2*n)/(1-5*x)^(1/2)/(2*x^n+3*x^(1+n))^(1/2),x)
Output:
- int((sqrt(x)*sqrt(3*x + 2)*sqrt( - 5*x + 1))/(15*x**3 + 7*x**2 - 2*x),x )