Integrand size = 24, antiderivative size = 51 \[ \int \frac {\sqrt {d+e x}}{\sqrt {2-3 x} \sqrt {x}} \, dx=\frac {2 \sqrt {d+e x} E\left (\arcsin \left (\sqrt {\frac {3}{2}} \sqrt {x}\right )|-\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {1+\frac {e x}{d}}} \] Output:
2/3*(e*x+d)^(1/2)*EllipticE(1/2*6^(1/2)*x^(1/2),1/3*(-6*e/d)^(1/2))*3^(1/2 )/(1+e*x/d)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(125\) vs. \(2(51)=102\).
Time = 3.20 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.45 \[ \int \frac {\sqrt {d+e x}}{\sqrt {2-3 x} \sqrt {x}} \, dx=\frac {2 \sqrt {x} \left (\frac {3 (d+e x)}{\sqrt {2-3 x}}-\frac {(3 d+2 e) \sqrt {\frac {d+e x}{e (-2+3 x)}} E\left (\arcsin \left (\frac {\sqrt {2+\frac {3 d}{e}}}{\sqrt {2-3 x}}\right )|\frac {2 e}{3 d+2 e}\right )}{\sqrt {2+\frac {3 d}{e}} \sqrt {\frac {x}{-2+3 x}}}\right )}{3 \sqrt {d+e x}} \] Input:
Integrate[Sqrt[d + e*x]/(Sqrt[2 - 3*x]*Sqrt[x]),x]
Output:
(2*Sqrt[x]*((3*(d + e*x))/Sqrt[2 - 3*x] - ((3*d + 2*e)*Sqrt[(d + e*x)/(e*( -2 + 3*x))]*EllipticE[ArcSin[Sqrt[2 + (3*d)/e]/Sqrt[2 - 3*x]], (2*e)/(3*d + 2*e)])/(Sqrt[2 + (3*d)/e]*Sqrt[x/(-2 + 3*x)])))/(3*Sqrt[d + e*x])
Time = 0.16 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {122, 27, 120}
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.
\(\displaystyle \int \frac {\sqrt {d+e x}}{\sqrt {2-3 x} \sqrt {x}} \, dx\) |
\(\Big \downarrow \) 122 |
\(\displaystyle \frac {\sqrt {d+e x} \int \frac {\sqrt {2} \sqrt {\frac {e x}{d}+1}}{\sqrt {2-3 x} \sqrt {x}}dx}{\sqrt {2} \sqrt {\frac {e x}{d}+1}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {d+e x} \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {2-3 x} \sqrt {x}}dx}{\sqrt {\frac {e x}{d}+1}}\) |
\(\Big \downarrow \) 120 |
\(\displaystyle \frac {2 \sqrt {d+e x} E\left (\arcsin \left (\sqrt {\frac {3}{2}} \sqrt {x}\right )|-\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {\frac {e x}{d}+1}}\) |
Input:
Int[Sqrt[d + e*x]/(Sqrt[2 - 3*x]*Sqrt[x]),x]
Output:
(2*Sqrt[d + e*x]*EllipticE[ArcSin[Sqrt[3/2]*Sqrt[x]], (-2*e)/(3*d)])/(Sqrt [3]*Sqrt[1 + (e*x)/d])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] :> Simp[2*(Sqrt[e]/b)*Rt[-b/d, 2]*EllipticE[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] && Gt Q[e, 0] && !LtQ[-b/d, 0]
Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[1 + d*(x/c)]/(Sqrt[c + d*x]*Sqrt[1 + f*(x/e)]) ) Int[Sqrt[1 + f*(x/e)]/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]), x], x] /; FreeQ[{b , c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(211\) vs. \(2(41)=82\).
Time = 0.92 (sec) , antiderivative size = 212, normalized size of antiderivative = 4.16
method | result | size |
default | \(-\frac {2 \sqrt {e x +d}\, \sqrt {2-3 x}\, d \sqrt {\frac {e x +d}{d}}\, \sqrt {-\frac {\left (-2+3 x \right ) e}{3 d +2 e}}\, \sqrt {-\frac {e x}{d}}\, \left (3 d \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right )+2 \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right ) e -3 \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right ) d -2 \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right ) e \right )}{3 \sqrt {x}\, e \left (3 e \,x^{2}+3 x d -2 e x -2 d \right )}\) | \(212\) |
elliptic | \(\frac {\sqrt {-\left (-2+3 x \right ) x \left (e x +d \right )}\, \left (\frac {2 d^{2} \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {-\frac {2}{3}+x}{-\frac {d}{e}-\frac {2}{3}}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {2}{3}\right )}}\right )}{e \sqrt {-3 e \,x^{3}-3 x^{2} d +2 e \,x^{2}+2 x d}}+\frac {2 d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {-\frac {2}{3}+x}{-\frac {d}{e}-\frac {2}{3}}}\, \sqrt {-\frac {e x}{d}}\, \left (\left (-\frac {d}{e}-\frac {2}{3}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {2}{3}\right )}}\right )+\frac {2 \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {2}{3}\right )}}\right )}{3}\right )}{\sqrt {-3 e \,x^{3}-3 x^{2} d +2 e \,x^{2}+2 x d}}\right )}{\sqrt {e x +d}\, \sqrt {2-3 x}\, \sqrt {x}}\) | \(285\) |
Input:
int((e*x+d)^(1/2)/(2-3*x)^(1/2)/x^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/3*(e*x+d)^(1/2)*(2-3*x)^(1/2)/x^(1/2)*d*((e*x+d)/d)^(1/2)*(-(-2+3*x)*e/ (3*d+2*e))^(1/2)*(-e*x/d)^(1/2)*(3*d*EllipticF(((e*x+d)/d)^(1/2),3^(1/2)*( d/(3*d+2*e))^(1/2))+2*EllipticF(((e*x+d)/d)^(1/2),3^(1/2)*(d/(3*d+2*e))^(1 /2))*e-3*EllipticE(((e*x+d)/d)^(1/2),3^(1/2)*(d/(3*d+2*e))^(1/2))*d-2*Elli pticE(((e*x+d)/d)^(1/2),3^(1/2)*(d/(3*d+2*e))^(1/2))*e)/e/(3*e*x^2+3*d*x-2 *e*x-2*d)
Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (44) = 88\).
Time = 0.09 (sec) , antiderivative size = 211, normalized size of antiderivative = 4.14 \[ \int \frac {\sqrt {d+e x}}{\sqrt {2-3 x} \sqrt {x}} \, dx=-\frac {2 \, {\left (2 \, \sqrt {3} {\left (3 \, d + e\right )} \sqrt {-e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (9 \, d^{2} + 6 \, d e + 4 \, e^{2}\right )}}{27 \, e^{2}}, -\frac {8 \, {\left (27 \, d^{3} + 27 \, d^{2} e - 18 \, d e^{2} - 8 \, e^{3}\right )}}{729 \, e^{3}}, \frac {9 \, e x + 3 \, d - 2 \, e}{9 \, e}\right ) - 9 \, \sqrt {3} \sqrt {-e} e {\rm weierstrassZeta}\left (\frac {4 \, {\left (9 \, d^{2} + 6 \, d e + 4 \, e^{2}\right )}}{27 \, e^{2}}, -\frac {8 \, {\left (27 \, d^{3} + 27 \, d^{2} e - 18 \, d e^{2} - 8 \, e^{3}\right )}}{729 \, e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (9 \, d^{2} + 6 \, d e + 4 \, e^{2}\right )}}{27 \, e^{2}}, -\frac {8 \, {\left (27 \, d^{3} + 27 \, d^{2} e - 18 \, d e^{2} - 8 \, e^{3}\right )}}{729 \, e^{3}}, \frac {9 \, e x + 3 \, d - 2 \, e}{9 \, e}\right )\right )\right )}}{27 \, e} \] Input:
integrate((e*x+d)^(1/2)/(2-3*x)^(1/2)/x^(1/2),x, algorithm="fricas")
Output:
-2/27*(2*sqrt(3)*(3*d + e)*sqrt(-e)*weierstrassPInverse(4/27*(9*d^2 + 6*d* e + 4*e^2)/e^2, -8/729*(27*d^3 + 27*d^2*e - 18*d*e^2 - 8*e^3)/e^3, 1/9*(9* e*x + 3*d - 2*e)/e) - 9*sqrt(3)*sqrt(-e)*e*weierstrassZeta(4/27*(9*d^2 + 6 *d*e + 4*e^2)/e^2, -8/729*(27*d^3 + 27*d^2*e - 18*d*e^2 - 8*e^3)/e^3, weie rstrassPInverse(4/27*(9*d^2 + 6*d*e + 4*e^2)/e^2, -8/729*(27*d^3 + 27*d^2* e - 18*d*e^2 - 8*e^3)/e^3, 1/9*(9*e*x + 3*d - 2*e)/e)))/e
\[ \int \frac {\sqrt {d+e x}}{\sqrt {2-3 x} \sqrt {x}} \, dx=\int \frac {\sqrt {d + e x}}{\sqrt {x} \sqrt {2 - 3 x}}\, dx \] Input:
integrate((e*x+d)**(1/2)/(2-3*x)**(1/2)/x**(1/2),x)
Output:
Integral(sqrt(d + e*x)/(sqrt(x)*sqrt(2 - 3*x)), x)
\[ \int \frac {\sqrt {d+e x}}{\sqrt {2-3 x} \sqrt {x}} \, dx=\int { \frac {\sqrt {e x + d}}{\sqrt {x} \sqrt {-3 \, x + 2}} \,d x } \] Input:
integrate((e*x+d)^(1/2)/(2-3*x)^(1/2)/x^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(e*x + d)/(sqrt(x)*sqrt(-3*x + 2)), x)
Timed out. \[ \int \frac {\sqrt {d+e x}}{\sqrt {2-3 x} \sqrt {x}} \, dx=\text {Timed out} \] Input:
integrate((e*x+d)^(1/2)/(2-3*x)^(1/2)/x^(1/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\sqrt {d+e x}}{\sqrt {2-3 x} \sqrt {x}} \, dx=\int \frac {\sqrt {d+e\,x}}{\sqrt {x}\,\sqrt {2-3\,x}} \,d x \] Input:
int((d + e*x)^(1/2)/(x^(1/2)*(2 - 3*x)^(1/2)),x)
Output:
int((d + e*x)^(1/2)/(x^(1/2)*(2 - 3*x)^(1/2)), x)
\[ \int \frac {\sqrt {d+e x}}{\sqrt {2-3 x} \sqrt {x}} \, dx=-\left (\int \frac {\sqrt {e x +d}\, \sqrt {-3 x +2}}{3 \sqrt {x}\, x -2 \sqrt {x}}d x \right ) \] Input:
int((e*x+d)^(1/2)/(2-3*x)^(1/2)/x^(1/2),x)
Output:
- int((sqrt(d + e*x)*sqrt( - 3*x + 2))/(3*sqrt(x)*x - 2*sqrt(x)),x)