\(\int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{\sqrt {x}} \, dx\) [146]

Optimal result

Integrand size = 28, antiderivative size = 105 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{\sqrt {x}} \, dx=\frac {2}{3} \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}+\frac {4 a^{5/2} c \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{3 \sqrt {b} \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:

2/3*x^(1/2)*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)+4/3*a^(5/2)*c*(1-b^2*x^2/a^2) 
^(1/2)*EllipticF(b^(1/2)*x^(1/2)/a^(1/2),I)/b^(1/2)/(b*x+a)^(1/2)/(-b*c*x+ 
a*c)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.99 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{\sqrt {x}} \, dx=\frac {2 \sqrt {x} \sqrt {c (a-b x)} \sqrt {a+b x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\frac {b^2 x^2}{a^2}\right )}{\sqrt {1-\frac {b^2 x^2}{a^2}}} \] Input:

Integrate[(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/Sqrt[x],x]
 

Output:

(2*Sqrt[x]*Sqrt[c*(a - b*x)]*Sqrt[a + b*x]*Hypergeometric2F1[-1/2, 1/4, 5/ 
4, (b^2*x^2)/a^2])/Sqrt[1 - (b^2*x^2)/a^2]
 

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {112, 25, 27, 127, 126}

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.

\(\displaystyle \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{\sqrt {x}} \, dx\)

\(\Big \downarrow \) 112

\(\displaystyle \frac {2}{3} \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}-\frac {2}{3} \int -\frac {a^2 c}{\sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2}{3} \int \frac {a^2 c}{\sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx+\frac {2}{3} \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{3} a^2 c \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx+\frac {2}{3} \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}\)

\(\Big \downarrow \) 127

\(\displaystyle \frac {2 a^2 c \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \int \frac {1}{\sqrt {x} \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1}}dx}{3 \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {2}{3} \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}\)

\(\Big \downarrow \) 126

\(\displaystyle \frac {4 a^{5/2} c \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{3 \sqrt {b} \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {2}{3} \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}\)

Input:

Int[(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/Sqrt[x],x]
 

Output:

(2*Sqrt[x]*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/3 + (4*a^(5/2)*c*Sqrt[1 - (b*x 
)/a]*Sqrt[1 + (b*x)/a]*EllipticF[ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[a]], -1])/( 
3*Sqrt[b]*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 112

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + 
p + 1))), x] - Simp[1/(f*(m + n + p + 1))   Int[(a + b*x)^(m - 1)*(c + d*x) 
^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a 
*f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && 
GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (IntegersQ[2*m, 2*n, 2*p 
] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))
 

rule 126

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] && (PosQ[-b/d] || NegQ[-b/f])
 

rule 127

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x 
_] :> Simp[Sqrt[1 + d*(x/c)]*(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x 
]))   Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x], x] /; Free 
Q[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])
 

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.06

method	result	size
default	\(\frac {2 \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (\sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{3}-b^{3} x^{3}+a^{2} b x \right )}{3 \sqrt {x}\, b \left (-b^{2} x^{2}+a^{2}\right )}\)	\(111\)
elliptic	\(\frac {\sqrt {c \left (-b x +a \right )}\, \sqrt {c x \left (-b^{2} x^{2}+a^{2}\right )}\, \left (\frac {2 \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{3}+\frac {2 a^{3} c \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{3 b \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{\sqrt {x}\, \sqrt {b x +a}\, c \left (-b x +a \right )}\)	\(157\)
risch	\(\frac {2 \sqrt {x}\, \sqrt {b x +a}\, \left (-b x +a \right ) c}{3 \sqrt {-c \left (b x -a \right )}}+\frac {2 a^{3} \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-x \left (b x +a \right ) c \left (b x -a \right )}\, c}{3 b \sqrt {-b^{2} c \,x^{3}+a^{2} c x}\, \sqrt {x}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\)	\(159\)

Input:

int((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^(1/2),x,method=_RETURNVERBOSE)
 

Output:

2/3*(b*x+a)^(1/2)*(c*(-b*x+a))^(1/2)/x^(1/2)*(((b*x+a)/a)^(1/2)*2^(1/2)*(( 
-b*x+a)/a)^(1/2)*(-b*x/a)^(1/2)*EllipticF(((b*x+a)/a)^(1/2),1/2*2^(1/2))*a 
^3-b^3*x^3+a^2*b*x)/b/(-b^2*x^2+a^2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.52 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{\sqrt {x}} \, dx=\frac {2 \, {\left (\sqrt {-b c x + a c} \sqrt {b x + a} b^{2} \sqrt {x} - 2 \, \sqrt {-b^{2} c} a^{2} {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right )\right )}}{3 \, b^{2}} \] Input:

integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^(1/2),x, algorithm="fricas")
 

Output:

2/3*(sqrt(-b*c*x + a*c)*sqrt(b*x + a)*b^2*sqrt(x) - 2*sqrt(-b^2*c)*a^2*wei 
erstrassPInverse(4*a^2/b^2, 0, x))/b^2
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{\sqrt {x}} \, dx=\int \frac {\sqrt {- c \left (- a + b x\right )} \sqrt {a + b x}}{\sqrt {x}}\, dx \] Input:

integrate((b*x+a)**(1/2)*(-b*c*x+a*c)**(1/2)/x**(1/2),x)
 

Output:

Integral(sqrt(-c*(-a + b*x))*sqrt(a + b*x)/sqrt(x), x)
 

Maxima [F]

\[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{\sqrt {x}} \, dx=\int { \frac {\sqrt {-b c x + a c} \sqrt {b x + a}}{\sqrt {x}} \,d x } \] Input:

integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(-b*c*x + a*c)*sqrt(b*x + a)/sqrt(x), x)
 

Giac [F]

\[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{\sqrt {x}} \, dx=\int { \frac {\sqrt {-b c x + a c} \sqrt {b x + a}}{\sqrt {x}} \,d x } \] Input:

integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(-b*c*x + a*c)*sqrt(b*x + a)/sqrt(x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{\sqrt {x}} \, dx=\int \frac {\sqrt {a\,c-b\,c\,x}\,\sqrt {a+b\,x}}{\sqrt {x}} \,d x \] Input:

int(((a*c - b*c*x)^(1/2)*(a + b*x)^(1/2))/x^(1/2),x)
 

Output:

int(((a*c - b*c*x)^(1/2)*(a + b*x)^(1/2))/x^(1/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{\sqrt {x}} \, dx=\frac {2 \sqrt {c}\, \left (\sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}+\left (\int \frac {\sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}}{-b^{2} x^{3}+a^{2} x}d x \right ) a^{2}\right )}{3} \] Input:

int((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^(1/2),x)
 

Output:

(2*sqrt(c)*(sqrt(x)*sqrt(a + b*x)*sqrt(a - b*x) + int((sqrt(x)*sqrt(a + b* 
x)*sqrt(a - b*x))/(a**2*x - b**2*x**3),x)*a**2))/3