\(\int x^{3/2} \sqrt {a+b x} \sqrt {a c-b c x} \, dx\) [145]

Optimal result

Integrand size = 28, antiderivative size = 142 \[ \int x^{3/2} \sqrt {a+b x} \sqrt {a c-b c x} \, dx=-\frac {4 a^2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}{21 b^2}+\frac {2}{7} x^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}+\frac {4 a^{9/2} c \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{21 b^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:

-4/21*a^2*x^(1/2)*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/b^2+2/7*x^(5/2)*(b*x+a) 
^(1/2)*(-b*c*x+a*c)^(1/2)+4/21*a^(9/2)*c*(1-b^2*x^2/a^2)^(1/2)*EllipticF(b 
^(1/2)*x^(1/2)/a^(1/2),I)/b^(5/2)/(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 = 1.32 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.75 \[ \int x^{3/2} \sqrt {a+b x} \sqrt {a c-b c x} \, dx=-\frac {2 \sqrt {x} \sqrt {c (a-b x)} \sqrt {a+b x} \left (\left (a^2-b^2 x^2\right ) \sqrt {1-\frac {b^2 x^2}{a^2}}-a^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\frac {b^2 x^2}{a^2}\right )\right )}{7 b^2 \sqrt {1-\frac {b^2 x^2}{a^2}}} \] Input:

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

Output:

(-2*Sqrt[x]*Sqrt[c*(a - b*x)]*Sqrt[a + b*x]*((a^2 - b^2*x^2)*Sqrt[1 - (b^2 
*x^2)/a^2] - a^2*Hypergeometric2F1[-1/2, 1/4, 5/4, (b^2*x^2)/a^2]))/(7*b^2 
*Sqrt[1 - (b^2*x^2)/a^2])
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.41, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {112, 27, 171, 27, 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.

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

\(\Big \downarrow \) 112

\(\displaystyle \frac {2 \int \frac {a c \sqrt {x} (3 a+5 b x) \sqrt {a c-b c x}}{2 \sqrt {a+b x}}dx}{7 b c}-\frac {2 x^{3/2} \sqrt {a+b x} (a c-b c x)^{3/2}}{7 b c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \int \frac {\sqrt {x} (3 a+5 b x) \sqrt {a c-b c x}}{\sqrt {a+b x}}dx}{7 b}-\frac {2 x^{3/2} \sqrt {a+b x} (a c-b c x)^{3/2}}{7 b c}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {a \left (-\frac {2 \int -\frac {5 a b c \sqrt {a+b x} \sqrt {a c-b c x}}{2 \sqrt {x}}dx}{5 b^2 c}-\frac {2 \sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}}{b c}\right )}{7 b}-\frac {2 x^{3/2} \sqrt {a+b x} (a c-b c x)^{3/2}}{7 b c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \left (\frac {a \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{\sqrt {x}}dx}{b}-\frac {2 \sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}}{b c}\right )}{7 b}-\frac {2 x^{3/2} \sqrt {a+b x} (a c-b c x)^{3/2}}{7 b c}\)

\(\Big \downarrow \) 112

\(\displaystyle \frac {a \left (\frac {a \left (\frac {2}{3} \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}-\frac {2}{3} \int -\frac {a^2 c}{\sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx\right )}{b}-\frac {2 \sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}}{b c}\right )}{7 b}-\frac {2 x^{3/2} \sqrt {a+b x} (a c-b c x)^{3/2}}{7 b c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {a \left (\frac {a \left (\frac {2}{3} \int \frac {a^2 c}{\sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx+\frac {2}{3} \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}\right )}{b}-\frac {2 \sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}}{b c}\right )}{7 b}-\frac {2 x^{3/2} \sqrt {a+b x} (a c-b c x)^{3/2}}{7 b c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \left (\frac {a \left (\frac {2}{3} a^2 c \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx+\frac {2}{3} \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}\right )}{b}-\frac {2 \sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}}{b c}\right )}{7 b}-\frac {2 x^{3/2} \sqrt {a+b x} (a c-b c x)^{3/2}}{7 b c}\)

\(\Big \downarrow \) 127

\(\displaystyle \frac {a \left (\frac {a \left (\frac {2 a^2 c \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \int \frac {1}{\sqrt {x} \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1}}dx}{3 \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {2}{3} \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}\right )}{b}-\frac {2 \sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}}{b c}\right )}{7 b}-\frac {2 x^{3/2} \sqrt {a+b x} (a c-b c x)^{3/2}}{7 b c}\)

\(\Big \downarrow \) 126

\(\displaystyle \frac {a \left (\frac {a \left (\frac {4 a^{5/2} c \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{3 \sqrt {b} \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {2}{3} \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}\right )}{b}-\frac {2 \sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}}{b c}\right )}{7 b}-\frac {2 x^{3/2} \sqrt {a+b x} (a c-b c x)^{3/2}}{7 b c}\)

Input:

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

Output:

(-2*x^(3/2)*Sqrt[a + b*x]*(a*c - b*c*x)^(3/2))/(7*b*c) + (a*((-2*Sqrt[x]*S 
qrt[a + b*x]*(a*c - b*c*x)^(3/2))/(b*c) + (a*((2*Sqrt[x]*Sqrt[a + b*x]*Sqr 
t[a*c - b*c*x])/3 + (4*a^(5/2)*c*Sqrt[1 - (b*x)/a]*Sqrt[1 + (b*x)/a]*Ellip 
ticF[ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[a]], -1])/(3*Sqrt[b]*Sqrt[a + b*x]*Sqrt 
[a*c - b*c*x])))/b))/(7*b)
 

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])
 

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 = 0.47 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.87

method	result	size
default	\(\frac {2 \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (\sqrt {2}\, \sqrt {\frac {b x +a}{a}}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{5}-3 b^{5} x^{5}+5 a^{2} b^{3} x^{3}-2 a^{4} b x \right )}{21 \sqrt {x}\, \left (-b^{2} x^{2}+a^{2}\right ) b^{3}}\)	\(123\)
risch	\(-\frac {2 \left (-3 b^{2} x^{2}+2 a^{2}\right ) \sqrt {x}\, \sqrt {b x +a}\, \left (-b x +a \right ) c}{21 b^{2} \sqrt {-c \left (b x -a \right )}}+\frac {2 a^{5} \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-x \left (b x +a \right ) c \left (b x -a \right )}\, c}{21 b^{3} \sqrt {-b^{2} c \,x^{3}+a^{2} c x}\, \sqrt {x}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\)	\(176\)
elliptic	\(\frac {\sqrt {c \left (-b x +a \right )}\, \sqrt {c x \left (-b^{2} x^{2}+a^{2}\right )}\, \left (\frac {2 x^{2} \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{7}-\frac {4 a^{2} \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{21 b^{2}}+\frac {2 a^{5} c \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{21 b^{3} \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{\sqrt {x}\, \sqrt {b x +a}\, c \left (-b x +a \right )}\)	\(186\)

Input:

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

Output:

2/21/x^(1/2)*(b*x+a)^(1/2)*(c*(-b*x+a))^(1/2)*(2^(1/2)*((b*x+a)/a)^(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^5-3*b^5*x^5+5*a^2*b^3*x^3-2*a^4*b*x)/(-b^2*x^2+a^2)/b^3
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.49 \[ \int x^{3/2} \sqrt {a+b x} \sqrt {a c-b c x} \, dx=-\frac {2 \, {\left (2 \, \sqrt {-b^{2} c} a^{4} {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right ) - {\left (3 \, b^{4} x^{2} - 2 \, a^{2} b^{2}\right )} \sqrt {-b c x + a c} \sqrt {b x + a} \sqrt {x}\right )}}{21 \, b^{4}} \] Input:

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

Output:

-2/21*(2*sqrt(-b^2*c)*a^4*weierstrassPInverse(4*a^2/b^2, 0, x) - (3*b^4*x^ 
2 - 2*a^2*b^2)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*sqrt(x))/b^4
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(x^(3/2)*(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)*x^(3/2), x)
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^{3/2} \sqrt {a+b x} \sqrt {a c-b c x} \, dx=\frac {2 \sqrt {c}\, \left (-2 \sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{2}+3 \sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}\, b^{2} x^{2}+\left (\int \frac {\sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}}{-b^{2} x^{3}+a^{2} x}d x \right ) a^{4}\right )}{21 b^{2}} \] Input:

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

Output:

(2*sqrt(c)*( - 2*sqrt(x)*sqrt(a + b*x)*sqrt(a - b*x)*a**2 + 3*sqrt(x)*sqrt 
(a + b*x)*sqrt(a - b*x)*b**2*x**2 + int((sqrt(x)*sqrt(a + b*x)*sqrt(a - b* 
x))/(a**2*x - b**2*x**3),x)*a**4))/(21*b**2)