\(\int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx\) [55]

Optimal result

Integrand size = 35, antiderivative size = 103 \[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx=-\frac {\sqrt {\frac {6}{11}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{5 \sqrt {-5+2 x}}-\frac {3 \sqrt {5-2 x} \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )}{5 \sqrt {11} \sqrt {-5+2 x}} \]

[Out]

-1/55*EllipticF(1/11*33^(1/2)*(1+4*x)^(1/2),1/3*3^(1/2))*66^(1/2)*(5-2*x)^(1/2)/(-5+2*x)^(1/2)-3/55*EllipticPi
(2/11*(2-3*x)^(1/2)*11^(1/2),55/124,1/2*I*2^(1/2))*(5-2*x)^(1/2)*11^(1/2)/(-5+2*x)^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {181, 122, 120, 174, 552, 551} \[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx=-\frac {\sqrt {\frac {6}{11}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{5 \sqrt {2 x-5}}-\frac {3 \sqrt {5-2 x} \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )}{5 \sqrt {11} \sqrt {2 x-5}} \]

[In]

Int[Sqrt[2 - 3*x]/(Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)),x]

[Out]

-1/5*(Sqrt[6/11]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/Sqrt[-5 + 2*x] - (3*Sqrt[5 -
2*x]*EllipticPi[55/124, ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(5*Sqrt[11]*Sqrt[-5 + 2*x])

Rule 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[2*(Rt[-b/d,
2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)
/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] && Po
sQ[-b/d] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(d*e - c*f)/d, 0] && GtQ[-d/b, 0]) &&  !(SimplerQ[c + d*x, a
+ b*x] && GtQ[((-b)*e + a*f)/f, 0] && GtQ[-f/b, 0]) &&  !(SimplerQ[e + f*x, a + b*x] && GtQ[((-d)*e + c*f)/f,
0] && GtQ[((-b)*e + a*f)/f, 0] && (PosQ[-f/d] || PosQ[-f/b]))

Rule 122

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[b*((c
+ d*x)/(b*c - a*d))]/Sqrt[c + d*x], Int[1/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]*Sqrt[e
+ f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !GtQ[(b*c - a*d)/b, 0] && SimplerQ[a + b*x, c + d*x] && Si
mplerQ[a + b*x, e + f*x]

Rule 174

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d
*g - c*h)/d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c
*f)/d, 0]

Rule 181

Int[Sqrt[(c_.) + (d_.)*(x_)]/(((a_.) + (b_.)*(x_))*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbo
l] :> Dist[d/b, Int[1/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] + Dist[(b*c - a*d)/b, Int[1/((a + b*
x)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-f/e, -d/c])

Rule 552

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d/c)*x^2]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rubi steps \begin{align*} \text {integral}& = -\left (\frac {3}{5} \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx\right )+\frac {31}{5} \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx \\ & = -\left (\frac {62}{5} \text {Subst}\left (\int \frac {1}{\left (31-5 x^2\right ) \sqrt {\frac {11}{3}-\frac {4 x^2}{3}} \sqrt {-\frac {11}{3}-\frac {2 x^2}{3}}} \, dx,x,\sqrt {2-3 x}\right )\right )-\frac {\left (3 \sqrt {\frac {2}{11}} \sqrt {5-2 x}\right ) \int \frac {1}{\sqrt {2-3 x} \sqrt {\frac {10}{11}-\frac {4 x}{11}} \sqrt {1+4 x}} \, dx}{5 \sqrt {-5+2 x}} \\ & = -\frac {\sqrt {\frac {6}{11}} \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )}{5 \sqrt {-5+2 x}}-\frac {\left (62 \sqrt {\frac {3}{11}} \sqrt {5-2 x}\right ) \text {Subst}\left (\int \frac {1}{\left (31-5 x^2\right ) \sqrt {\frac {11}{3}-\frac {4 x^2}{3}} \sqrt {1+\frac {2 x^2}{11}}} \, dx,x,\sqrt {2-3 x}\right )}{5 \sqrt {-5+2 x}} \\ & = -\frac {\sqrt {\frac {6}{11}} \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )}{5 \sqrt {-5+2 x}}-\frac {3 \sqrt {5-2 x} \Pi \left (\frac {55}{124};\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{5 \sqrt {11} \sqrt {-5+2 x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.38 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx=\frac {3 \sqrt {5-2 x} \left (\operatorname {EllipticF}\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )-\operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )\right )}{5 \sqrt {-55+22 x}} \]

[In]

Integrate[Sqrt[2 - 3*x]/(Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)),x]

[Out]

(3*Sqrt[5 - 2*x]*(EllipticF[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2] - EllipticPi[55/124, ArcSin[(2*Sqrt[2 -
3*x])/Sqrt[11]], -1/2]))/(5*Sqrt[-55 + 22*x])

Maple [A] (verified)

Time = 5.37 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.50

method	result	size
default	\(-\frac {\left (69 F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-124 \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right )\right ) \sqrt {5-2 x}\, \sqrt {22}}{1265 \sqrt {-5+2 x}}\)	\(52\)
elliptic	\(\frac {\sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (-\frac {3 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{605 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {124 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right )}{13915 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\)	\(150\)

[In]

int((2-3*x)^(1/2)/(7+5*x)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/1265*(69*EllipticF(1/11*(11+44*x)^(1/2),3^(1/2))-124*EllipticPi(1/11*(11+44*x)^(1/2),-55/23,3^(1/2)))*(5-2*
x)^(1/2)*22^(1/2)/(-5+2*x)^(1/2)

Fricas [F]

\[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx=\int { \frac {\sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5}} \,d x } \]

[In]

integrate((2-3*x)^(1/2)/(7+5*x)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(40*x^3 - 34*x^2 - 151*x - 35), x)

Sympy [F]

\[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx=\int \frac {\sqrt {2 - 3 x}}{\sqrt {2 x - 5} \sqrt {4 x + 1} \cdot \left (5 x + 7\right )}\, dx \]

[In]

integrate((2-3*x)**(1/2)/(7+5*x)/(-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)*(5*x + 7)), x)

Maxima [F]

\[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx=\int { \frac {\sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5}} \,d x } \]

[In]

integrate((2-3*x)^(1/2)/(7+5*x)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-3*x + 2)/((5*x + 7)*sqrt(4*x + 1)*sqrt(2*x - 5)), x)

Giac [F]

\[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx=\int { \frac {\sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5}} \,d x } \]

[In]

integrate((2-3*x)^(1/2)/(7+5*x)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(-3*x + 2)/((5*x + 7)*sqrt(4*x + 1)*sqrt(2*x - 5)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx=\int \frac {\sqrt {2-3\,x}}{\sqrt {4\,x+1}\,\sqrt {2\,x-5}\,\left (5\,x+7\right )} \,d x \]

[In]

int((2 - 3*x)^(1/2)/((4*x + 1)^(1/2)*(2*x - 5)^(1/2)*(5*x + 7)),x)

[Out]

int((2 - 3*x)^(1/2)/((4*x + 1)^(1/2)*(2*x - 5)^(1/2)*(5*x + 7)), x)