\(\int \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x} \, dx\) [149]

Optimal result

Integrand size = 28, antiderivative size = 178 \[ \int \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x} \, dx=\frac {2}{5} x^{3/2} \sqrt {a+b x} \sqrt {a c-b c x}+\frac {4 a^{7/2} c \sqrt {1-\frac {b^2 x^2}{a^2}} E\left (\left .\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right |-1\right )}{5 b^{3/2} \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {4 a^{7/2} c \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{5 b^{3/2} \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:

2/5*x^(3/2)*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)+4/5*a^(7/2)*c*(1-b^2*x^2/a^2) 
^(1/2)*EllipticE(b^(1/2)*x^(1/2)/a^(1/2),I)/b^(3/2)/(b*x+a)^(1/2)/(-b*c*x+ 
a*c)^(1/2)-4/5*a^(7/2)*c*(1-b^2*x^2/a^2)^(1/2)*EllipticF(b^(1/2)*x^(1/2)/a 
^(1/2),I)/b^(3/2)/(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 = 0.92 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.38 \[ \int \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x} \, dx=\frac {2 x^{3/2} \sqrt {c (a-b x)} \sqrt {a+b x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {b^2 x^2}{a^2}\right )}{3 \sqrt {1-\frac {b^2 x^2}{a^2}}} \] Input:

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

Output:

(2*x^(3/2)*Sqrt[c*(a - b*x)]*Sqrt[a + b*x]*Hypergeometric2F1[-1/2, 3/4, 7/ 
4, (b^2*x^2)/a^2])/(3*Sqrt[1 - (b^2*x^2)/a^2])
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.84, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {112, 27, 171, 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.

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

\(\Big \downarrow \) 112

\(\displaystyle \frac {2 \int \frac {a c (a+3 b x) \sqrt {a c-b c x}}{2 \sqrt {x} \sqrt {a+b x}}dx}{5 b c}-\frac {2 \sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}}{5 b c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \int \frac {(a+3 b x) \sqrt {a c-b c x}}{\sqrt {x} \sqrt {a+b x}}dx}{5 b}-\frac {2 \sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}}{5 b c}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {a \left (\frac {2 \int \frac {3 a b^2 c \sqrt {x}}{\sqrt {a+b x} \sqrt {a c-b c x}}dx}{3 b}+2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}\right )}{5 b}-\frac {2 \sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}}{5 b c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \left (2 a b c \int \frac {\sqrt {x}}{\sqrt {a+b x} \sqrt {a c-b c x}}dx+2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}\right )}{5 b}-\frac {2 \sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}}{5 b c}\)

\(\Big \downarrow \) 124

\(\displaystyle \frac {a \left (\frac {\sqrt {2} a b c \sqrt {x} \sqrt {\frac {a-b x}{a}} \int \frac {\sqrt {2} \sqrt {-\frac {b x}{a}}}{\sqrt {a+b x} \sqrt {1-\frac {b x}{a}}}dx}{\sqrt {-\frac {b x}{a}} \sqrt {a c-b c x}}+2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}\right )}{5 b}-\frac {2 \sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}}{5 b c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \left (\frac {2 a b c \sqrt {x} \sqrt {\frac {a-b x}{a}} \int \frac {\sqrt {-\frac {b x}{a}}}{\sqrt {a+b x} \sqrt {1-\frac {b x}{a}}}dx}{\sqrt {-\frac {b x}{a}} \sqrt {a c-b c x}}+2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}\right )}{5 b}-\frac {2 \sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}}{5 b c}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {a \left (\frac {4 a^{3/2} c \sqrt {x} \sqrt {\frac {a-b x}{a}} E\left (\left .\arcsin \left (\frac {\sqrt {a+b x}}{\sqrt {2} \sqrt {a}}\right )\right |2\right )}{\sqrt {-\frac {b x}{a}} \sqrt {a c-b c x}}+2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}\right )}{5 b}-\frac {2 \sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}}{5 b c}\)

Input:

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

Output:

(-2*Sqrt[x]*Sqrt[a + b*x]*(a*c - b*c*x)^(3/2))/(5*b*c) + (a*(2*Sqrt[x]*Sqr 
t[a + b*x]*Sqrt[a*c - b*c*x] + (4*a^(3/2)*c*Sqrt[x]*Sqrt[(a - b*x)/a]*Elli 
pticE[ArcSin[Sqrt[a + b*x]/(Sqrt[2]*Sqrt[a])], 2])/(Sqrt[-((b*x)/a)]*Sqrt[ 
a*c - b*c*x])))/(5*b)
 

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 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 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]
 

rule 171

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

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.97

method	result	size
default	\(-\frac {2 \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (2 \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{4}-\sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{4}+b^{4} x^{4}-a^{2} b^{2} x^{2}\right )}{5 \sqrt {x}\, b^{2} \left (-b^{2} x^{2}+a^{2}\right )}\)	\(173\)
elliptic	\(\frac {\sqrt {c \left (-b x +a \right )}\, \sqrt {c x \left (-b^{2} x^{2}+a^{2}\right )}\, \left (\frac {2 x \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{5}+\frac {2 a^{3} c \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \left (-\frac {2 a \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{5 b \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{\sqrt {x}\, \sqrt {b x +a}\, c \left (-b x +a \right )}\)	\(190\)
risch	\(\frac {2 \left (-b x +a \right ) x^{\frac {3}{2}} \sqrt {b x +a}\, c}{5 \sqrt {-c \left (b x -a \right )}}+\frac {2 a^{3} \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \left (-\frac {2 a \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}\right ) \sqrt {-x \left (b x +a \right ) c \left (b x -a \right )}\, c}{5 b \sqrt {-b^{2} c \,x^{3}+a^{2} c x}\, \sqrt {x}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\)	\(191\)

Input:

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

Output:

-2/5/x^(1/2)*(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^4-((b*x+a)/a)^(1/2)*2^(1/2)*((-b*x+a)/a)^(1/2)*(-b*x/a)^(1/2)*Elliptic 
F(((b*x+a)/a)^(1/2),1/2*2^(1/2))*a^4+b^4*x^4-a^2*b^2*x^2)/b^2/(-b^2*x^2+a^ 
2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.37 \[ \int \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x} \, dx=\frac {2 \, {\left (\sqrt {-b c x + a c} \sqrt {b x + a} b^{2} x^{\frac {3}{2}} + 2 \, \sqrt {-b^{2} c} a^{2} {\rm weierstrassZeta}\left (\frac {4 \, a^{2}}{b^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right )\right )\right )}}{5 \, b^{2}} \] Input:

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

Output:

2/5*(sqrt(-b*c*x + a*c)*sqrt(b*x + a)*b^2*x^(3/2) + 2*sqrt(-b^2*c)*a^2*wei 
erstrassZeta(4*a^2/b^2, 0, weierstrassPInverse(4*a^2/b^2, 0, x)))/b^2
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x} \, dx=\frac {2 \sqrt {c}\, \left (\sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}\, x +\left (\int \frac {\sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}}{-b^{2} x^{2}+a^{2}}d x \right ) a^{2}\right )}{5} \] Input:

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

Output:

(2*sqrt(c)*(sqrt(x)*sqrt(a + b*x)*sqrt(a - b*x)*x + int((sqrt(x)*sqrt(a + 
b*x)*sqrt(a - b*x))/(a**2 - b**2*x**2),x)*a**2))/5