\(\int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx\) [54]

Optimal result

Integrand size = 28, antiderivative size = 47 \[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\frac {\sqrt {\frac {11}{2}} \sqrt {5-2 x} E\left (\left .\arcsin \left (\frac {\sqrt {1+4 x}}{\sqrt {11}}\right )\right |3\right )}{2 \sqrt {-5+2 x}} \]

[Out]

1/4*EllipticE(1/11*(1+4*x)^(1/2)*11^(1/2),3^(1/2))*22^(1/2)*(5-2*x)^(1/2)/(-5+2*x)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {115, 114} \[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\frac {\sqrt {\frac {11}{2}} \sqrt {5-2 x} E\left (\left .\arcsin \left (\frac {\sqrt {4 x+1}}{\sqrt {11}}\right )\right |3\right )}{2 \sqrt {2 x-5}} \]

[In]

Int[Sqrt[2 - 3*x]/(Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]),x]

[Out]

(Sqrt[11/2]*Sqrt[5 - 2*x]*EllipticE[ArcSin[Sqrt[1 + 4*x]/Sqrt[11]], 3])/(2*Sqrt[-5 + 2*x])

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2/b)*Rt[-(b
*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /;
FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !LtQ[-(b*c - a*d)/d, 0] &&
!(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)/b, 0])

Rule 115

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[Sqrt[e + f*x
]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d*x]*Sqrt[b*((e + f*x)/(b*e - a*f))])), Int[Sqrt[b*(e/(b*e - a*f)
) + b*f*(x/(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]), x], x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !(GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0]) &&  !LtQ[-(b*c - a*d)/d, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {5-2 x} \int \frac {\sqrt {\frac {8}{11}-\frac {12 x}{11}}}{\sqrt {\frac {10}{11}-\frac {4 x}{11}} \sqrt {1+4 x}} \, dx}{\sqrt {2} \sqrt {-5+2 x}} \\ & = \frac {\sqrt {\frac {11}{2}} \sqrt {5-2 x} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {1+4 x}}{\sqrt {11}}\right )\right |3\right )}{2 \sqrt {-5+2 x}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(111\) vs. \(2(47)=94\).

Time = 2.33 (sec) , antiderivative size = 111, normalized size of antiderivative = 2.36 \[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=-\frac {\frac {2 (-5+2 x) (-2+3 x)}{\sqrt {\frac {1}{2}+2 x}}+\sqrt {11} \sqrt {\frac {-5+2 x}{1+4 x}} \sqrt {\frac {-2+3 x}{1+4 x}} (1+4 x) E\left (\left .\arcsin \left (\frac {\sqrt {\frac {11}{3}}}{\sqrt {1+4 x}}\right )\right |3\right )}{2 \sqrt {2-3 x} \sqrt {-10+4 x}} \]

[In]

Integrate[Sqrt[2 - 3*x]/(Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]),x]

[Out]

-1/2*((2*(-5 + 2*x)*(-2 + 3*x))/Sqrt[1/2 + 2*x] + Sqrt[11]*Sqrt[(-5 + 2*x)/(1 + 4*x)]*Sqrt[(-2 + 3*x)/(1 + 4*x
)]*(1 + 4*x)*EllipticE[ArcSin[Sqrt[11/3]/Sqrt[1 + 4*x]], 3])/(Sqrt[2 - 3*x]*Sqrt[-10 + 4*x])

Maple [A] (verified)

Time = 1.58 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.70

method	result	size
default	\(\frac {E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) \sqrt {5-2 x}\, \sqrt {22}}{4 \sqrt {-5+2 x}}\)	\(33\)
elliptic	\(\frac {\sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (\frac {2 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{121 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {3 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \left (-\frac {11 E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{12}+\frac {2 F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{3}\right )}{121 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\)	\(167\)

[In]

int((2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/4*EllipticE(1/11*(11+44*x)^(1/2),3^(1/2))*(5-2*x)^(1/2)*22^(1/2)/(-5+2*x)^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.55 \[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\frac {11}{72} \, \sqrt {-6} {\rm weierstrassPInverse}\left (\frac {847}{108}, \frac {6655}{2916}, x - \frac {35}{36}\right ) - \frac {1}{2} \, \sqrt {-6} {\rm weierstrassZeta}\left (\frac {847}{108}, \frac {6655}{2916}, {\rm weierstrassPInverse}\left (\frac {847}{108}, \frac {6655}{2916}, x - \frac {35}{36}\right )\right ) \]

[In]

integrate((2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x, algorithm="fricas")

[Out]

11/72*sqrt(-6)*weierstrassPInverse(847/108, 6655/2916, x - 35/36) - 1/2*sqrt(-6)*weierstrassZeta(847/108, 6655
/2916, weierstrassPInverse(847/108, 6655/2916, x - 35/36))

Sympy [F]

\[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int \frac {\sqrt {2 - 3 x}}{\sqrt {2 x - 5} \sqrt {4 x + 1}}\, dx \]

[In]

integrate((2-3*x)**(1/2)/(-5+2*x)**(1/2)/(1+4*x)**(1/2),x)

[Out]

Integral(sqrt(2 - 3*x)/(sqrt(2*x - 5)*sqrt(4*x + 1)), x)

Maxima [F]

\[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int { \frac {\sqrt {-3 \, x + 2}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5}} \,d x } \]

[In]

integrate((2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-3*x + 2)/(sqrt(4*x + 1)*sqrt(2*x - 5)), x)

Giac [F]

\[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int { \frac {\sqrt {-3 \, x + 2}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5}} \,d x } \]

[In]

integrate((2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(-3*x + 2)/(sqrt(4*x + 1)*sqrt(2*x - 5)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int \frac {\sqrt {2-3\,x}}{\sqrt {4\,x+1}\,\sqrt {2\,x-5}} \,d x \]

[In]

int((2 - 3*x)^(1/2)/((4*x + 1)^(1/2)*(2*x - 5)^(1/2)),x)

[Out]

int((2 - 3*x)^(1/2)/((4*x + 1)^(1/2)*(2*x - 5)^(1/2)), x)