\(\int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^{3/2}} \, dx\) [150]

Optimal result

Integrand size = 28, antiderivative size = 172 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^{3/2}} \, dx=-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{\sqrt {x}}-\frac {4 a^{3/2} \sqrt {b} c \sqrt {1-\frac {b^2 x^2}{a^2}} E\left (\left .\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right |-1\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}+\frac {4 a^{3/2} \sqrt {b} c \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \] Output:

-2*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^(1/2)-4*a^(3/2)*b^(1/2)*c*(1-b^2*x^2 
/a^2)^(1/2)*EllipticE(b^(1/2)*x^(1/2)/a^(1/2),I)/(b*x+a)^(1/2)/(-b*c*x+a*c 
)^(1/2)+4*a^(3/2)*b^(1/2)*c*(1-b^2*x^2/a^2)^(1/2)*EllipticF(b^(1/2)*x^(1/2 
)/a^(1/2),I)/(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 = 1.07 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.38 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^{3/2}} \, dx=-\frac {2 \sqrt {c (a-b x)} \sqrt {a+b x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{4},\frac {3}{4},\frac {b^2 x^2}{a^2}\right )}{\sqrt {x} \sqrt {1-\frac {b^2 x^2}{a^2}}} \] Input:

Integrate[(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/x^(3/2),x]
 

Output:

(-2*Sqrt[c*(a - b*x)]*Sqrt[a + b*x]*Hypergeometric2F1[-1/2, -1/4, 3/4, (b^ 
2*x^2)/a^2])/(Sqrt[x]*Sqrt[1 - (b^2*x^2)/a^2])
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.61, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {108, 25, 27, 124, 27, 123}

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.

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

\(\Big \downarrow \) 108

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 124

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 123

\(\displaystyle -\frac {4 \sqrt {a} b c \sqrt {x} \sqrt {\frac {a-b x}{a}} E\left (\left .\arcsin \left (\frac {\sqrt {a+b x}}{\sqrt {2} \sqrt {a}}\right )\right |2\right )}{\sqrt {-\frac {b x}{a}} \sqrt {a c-b c x}}-\frac {2 \sqrt {a+b x} \sqrt {a c-b c x}}{\sqrt {x}}\)

Input:

Int[(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/x^(3/2),x]
 

Output:

(-2*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/Sqrt[x] - (4*Sqrt[a]*b*c*Sqrt[x]*Sqrt 
[(a - b*x)/a]*EllipticE[ArcSin[Sqrt[a + b*x]/(Sqrt[2]*Sqrt[a])], 2])/(Sqrt 
[-((b*x)/a)]*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 108

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

rule 123

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ 
)]), x_] :> 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] &&  !L 
tQ[-(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 124

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ 
)]), x_] :> Simp[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] && Gt 
Q[b/(b*e - a*f), 0]) &&  !LtQ[-(b*c - a*d)/d, 0]
 

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.95

method	result	size
default	\(\frac {2 \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (2 \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{2}-\sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{2}+b^{2} x^{2}-a^{2}\right )}{\sqrt {x}\, \left (-b^{2} x^{2}+a^{2}\right )}\)	\(164\)
risch	\(-\frac {2 \left (-b x +a \right ) \sqrt {b x +a}\, c}{\sqrt {x}\, \sqrt {-c \left (b x -a \right )}}-\frac {2 b a \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \left (-\frac {2 a \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}\right ) \sqrt {-x \left (b x +a \right ) c \left (b x -a \right )}\, c}{\sqrt {-b^{2} c \,x^{3}+a^{2} c x}\, \sqrt {x}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\)	\(187\)
elliptic	\(\frac {\sqrt {c \left (-b x +a \right )}\, \sqrt {c x \left (-b^{2} x^{2}+a^{2}\right )}\, \left (-\frac {2 \left (-b^{2} c \,x^{2}+a^{2} c \right )}{\sqrt {x \left (-b^{2} c \,x^{2}+a^{2} c \right )}}-\frac {2 b c a \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \left (-\frac {2 a \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{\sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{\sqrt {x}\, \sqrt {b x +a}\, c \left (-b x +a \right )}\)	\(201\)

Input:

int((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^(3/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.35 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^{3/2}} \, dx=-\frac {2 \, {\left (2 \, \sqrt {-b^{2} c} x {\rm weierstrassZeta}\left (\frac {4 \, a^{2}}{b^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right )\right ) + \sqrt {-b c x + a c} \sqrt {b x + a} \sqrt {x}\right )}}{x} \] Input:

integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^(3/2),x, algorithm="fricas")
 

Output:

-2*(2*sqrt(-b^2*c)*x*weierstrassZeta(4*a^2/b^2, 0, weierstrassPInverse(4*a 
^2/b^2, 0, x)) + sqrt(-b*c*x + a*c)*sqrt(b*x + a)*sqrt(x))/x
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^(3/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^{3/2}} \, dx=\int \frac {\sqrt {a\,c-b\,c\,x}\,\sqrt {a+b\,x}}{x^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^{3/2}} \, dx=\frac {2 \sqrt {c}\, \left (\sqrt {b x +a}\, \sqrt {-b x +a}+\sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}}{-b^{2} x^{4}+a^{2} x^{2}}d x \right ) a^{2}\right )}{\sqrt {x}} \] Input:

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

Output:

(2*sqrt(c)*(sqrt(a + b*x)*sqrt(a - b*x) + sqrt(x)*int((sqrt(x)*sqrt(a + b* 
x)*sqrt(a - b*x))/(a**2*x**2 - b**2*x**4),x)*a**2))/sqrt(x)