3.1.96 \(\int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx\) [96]

3.1.96.1 Optimal result

Integrand size = 37, antiderivative size = 101 \[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx=\frac {62 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {1+4 x}{2-3 x}} \operatorname {EllipticPi}\left (-\frac {69}{55},\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{5 \sqrt {429} \sqrt {-5+2 x} \sqrt {1+4 x}} \]

output

62/2145*(2-3*x)*EllipticPi(1/23*253^(1/2)*(7+5*x)^(1/2)/(2-3*x)^(1/2),-69/ 
55,1/39*I*897^(1/2))*((5-2*x)/(2-3*x))^(1/2)*((-1-4*x)/(2-3*x))^(1/2)*429^ 
(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2)
 

3.1.96.2 Mathematica [A] (warning: unable to verify)

Time = 5.03 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.68 \[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx=\frac {\sqrt {\frac {1+4 x}{7+5 x}} (7+5 x)^{3/2} \left (-62 \sqrt {\frac {5-2 x}{7+5 x}} \sqrt {\frac {-2+3 x}{7+5 x}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {155-62 x}{77+55 x}}\right ),\frac {23}{62}\right )+117 \sqrt {\frac {-10+19 x-6 x^2}{(7+5 x)^2}} \operatorname {EllipticPi}\left (-\frac {55}{62},\arcsin \left (\sqrt {\frac {155-62 x}{77+55 x}}\right ),\frac {23}{62}\right )\right )}{5 \sqrt {682} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \]

input

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

output

(Sqrt[(1 + 4*x)/(7 + 5*x)]*(7 + 5*x)^(3/2)*(-62*Sqrt[(5 - 2*x)/(7 + 5*x)]* 
Sqrt[(-2 + 3*x)/(7 + 5*x)]*EllipticF[ArcSin[Sqrt[(155 - 62*x)/(77 + 55*x)] 
], 23/62] + 117*Sqrt[(-10 + 19*x - 6*x^2)/(7 + 5*x)^2]*EllipticPi[-55/62, 
ArcSin[Sqrt[(155 - 62*x)/(77 + 55*x)]], 23/62]))/(5*Sqrt[682]*Sqrt[2 - 3*x 
]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])
 

3.1.96.3 Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {183, 27, 412}

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.

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

\(\Big \downarrow \) 183

\(\displaystyle \frac {62 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \int \frac {\sqrt {897}}{\sqrt {23-\frac {11 (5 x+7)}{2-3 x}} \left (\frac {3 (5 x+7)}{2-3 x}+5\right ) \sqrt {\frac {11 (5 x+7)}{2-3 x}+39}}d\frac {\sqrt {5 x+7}}{\sqrt {2-3 x}}}{\sqrt {897} \sqrt {2 x-5} \sqrt {4 x+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {62 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \int \frac {1}{\sqrt {23-\frac {11 (5 x+7)}{2-3 x}} \left (\frac {3 (5 x+7)}{2-3 x}+5\right ) \sqrt {\frac {11 (5 x+7)}{2-3 x}+39}}d\frac {\sqrt {5 x+7}}{\sqrt {2-3 x}}}{\sqrt {2 x-5} \sqrt {4 x+1}}\)

\(\Big \downarrow \) 412

\(\displaystyle \frac {62 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \operatorname {EllipticPi}\left (-\frac {69}{55},\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {5 x+7}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{5 \sqrt {429} \sqrt {2 x-5} \sqrt {4 x+1}}\)

input

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

output

(62*(2 - 3*x)*Sqrt[(5 - 2*x)/(2 - 3*x)]*Sqrt[-((1 + 4*x)/(2 - 3*x))]*Ellip 
ticPi[-69/55, ArcSin[(Sqrt[11/23]*Sqrt[7 + 5*x])/Sqrt[2 - 3*x]], -23/39])/ 
(5*Sqrt[429]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])
 

3.1.96.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 183

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

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[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] && S 
implerSqrtQ[-f/e, -d/c])
 

3.1.96.4 Maple [A] (verified)

Time = 1.61 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.33

method	result	size
default	\(-\frac {62 \Pi \left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, -\frac {69}{55}, \frac {i \sqrt {897}}{39}\right ) \sqrt {\frac {1+4 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {3}\, \sqrt {13}\, \left (-2+3 x \right ) \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {1+4 x}\, \sqrt {-5+2 x}\, \sqrt {7+5 x}\, \sqrt {2-3 x}}{49335 \left (40 x^{3}-34 x^{2}-151 x -35\right )}\)	\(134\)
elliptic	\(\frac {\sqrt {-\left (7+5 x \right ) \left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (\frac {4 \sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}\, \left (-\frac {2}{3}+x \right )^{2} \sqrt {806}\, \sqrt {\frac {x -\frac {5}{2}}{-\frac {2}{3}+x}}\, \sqrt {2139}\, \sqrt {\frac {x +\frac {1}{4}}{-\frac {2}{3}+x}}\, F\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{305877 \sqrt {-30 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}}-\frac {2 \sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}\, \left (-\frac {2}{3}+x \right )^{2} \sqrt {806}\, \sqrt {\frac {x -\frac {5}{2}}{-\frac {2}{3}+x}}\, \sqrt {2139}\, \sqrt {\frac {x +\frac {1}{4}}{-\frac {2}{3}+x}}\, \left (\frac {2 F\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{3}-\frac {31 \Pi \left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, -\frac {69}{55}, \frac {i \sqrt {897}}{39}\right )}{15}\right )}{101959 \sqrt {-30 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {7+5 x}}\)	\(250\)

input

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

output

-62/49335*EllipticPi(1/23*(-253*(7+5*x)/(-2+3*x))^(1/2),-69/55,1/39*I*897^ 
(1/2))*((1+4*x)/(-2+3*x))^(1/2)*23^(1/2)*((-5+2*x)/(-2+3*x))^(1/2)*3^(1/2) 
*13^(1/2)*(-2+3*x)*(-253*(7+5*x)/(-2+3*x))^(1/2)*(1+4*x)^(1/2)*(-5+2*x)^(1 
/2)*(7+5*x)^(1/2)*(2-3*x)^(1/2)/(40*x^3-34*x^2-151*x-35)
 

3.1.96.5 Fricas [F]

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

input

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

output

integral(sqrt(5*x + 7)*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(40*x^3 
- 34*x^2 - 151*x - 35), x)
 

3.1.96.6 Sympy [F]

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

input

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

output

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

3.1.96.7 Maxima [F]

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

input

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

output

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

3.1.96.8 Giac [F]

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

input

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

output

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

3.1.96.9 Mupad [F(-1)]

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

input

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

output

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