\(\int \frac {x^{5/2}}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx\) [173]

Optimal result

Integrand size = 28, antiderivative size = 182 \[ \int \frac {x^{5/2}}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {2 x^{3/2} \sqrt {a+b x} \sqrt {a c-b c x}}{5 b^2 c}+\frac {6 a^{7/2} \sqrt {1-\frac {b^2 x^2}{a^2}} E\left (\left .\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right |-1\right )}{5 b^{7/2} \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {6 a^{7/2} \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{5 b^{7/2} \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:

-2/5*x^(3/2)*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/b^2/c+6/5*a^(7/2)*(1-b^2*x^2 
/a^2)^(1/2)*EllipticE(b^(1/2)*x^(1/2)/a^(1/2),I)/b^(7/2)/(b*x+a)^(1/2)/(-b 
*c*x+a*c)^(1/2)-6/5*a^(7/2)*(1-b^2*x^2/a^2)^(1/2)*EllipticF(b^(1/2)*x^(1/2 
)/a^(1/2),I)/b^(7/2)/(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 = 1.90 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.48 \[ \int \frac {x^{5/2}}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\frac {2 x^{3/2} \left (-a^2+b^2 x^2+a^2 \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {b^2 x^2}{a^2}\right )\right )}{5 b^2 \sqrt {c (a-b x)} \sqrt {a+b x}} \] Input:

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

Output:

(2*x^(3/2)*(-a^2 + b^2*x^2 + a^2*Sqrt[1 - (b^2*x^2)/a^2]*Hypergeometric2F1 
[1/2, 3/4, 7/4, (b^2*x^2)/a^2]))/(5*b^2*Sqrt[c*(a - b*x)]*Sqrt[a + b*x])
 

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.64, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {113, 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.

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

\(\Big \downarrow \) 113

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 124

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 123

\(\displaystyle \frac {6 a^{5/2} \sqrt {x} \sqrt {\frac {a-b x}{a}} E\left (\left .\arcsin \left (\frac {\sqrt {a+b x}}{\sqrt {2} \sqrt {a}}\right )\right |2\right )}{5 b^3 \sqrt {-\frac {b x}{a}} \sqrt {a c-b c x}}-\frac {2 x^{3/2} \sqrt {a+b x} \sqrt {a c-b c x}}{5 b^2 c}\)

Input:

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

Output:

(-2*x^(3/2)*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/(5*b^2*c) + (6*a^(5/2)*Sqrt[x 
]*Sqrt[(a - b*x)/a]*EllipticE[ArcSin[Sqrt[a + b*x]/(Sqrt[2]*Sqrt[a])], 2]) 
/(5*b^3*Sqrt[-((b*x)/a)]*Sqrt[a*c - b*c*x])
 

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 113

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

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 = 0.83 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.97

method	result	size
default	\(-\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (6 \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{4}-3 \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{4}-2 b^{4} x^{4}+2 a^{2} b^{2} x^{2}\right )}{5 \sqrt {x}\, c \left (-b^{2} x^{2}+a^{2}\right ) b^{4}}\)	\(177\)
elliptic	\(\frac {\sqrt {c x \left (-b^{2} x^{2}+a^{2}\right )}\, \left (-\frac {2 x \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{5 b^{2} c}+\frac {3 a^{3} \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \left (-\frac {2 a \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{5 b^{3} \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{\sqrt {x}\, \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}}\)	\(184\)
risch	\(-\frac {2 \left (-b x +a \right ) x^{\frac {3}{2}} \sqrt {b x +a}}{5 b^{2} \sqrt {-c \left (b x -a \right )}}+\frac {3 a^{3} \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \left (-\frac {2 a \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}\right ) \sqrt {-x \left (b x +a \right ) c \left (b x -a \right )}}{5 b^{3} \sqrt {-b^{2} c \,x^{3}+a^{2} c x}\, \sqrt {x}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\)	\(192\)

Input:

int(x^(5/2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/5/x^(1/2)*(b*x+a)^(1/2)*(c*(-b*x+a))^(1/2)/c*(6*((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^4-3*((b*x+a)/a)^(1/2)*2^(1/2)*((-b*x+a)/a)^(1/2)*(-b*x/a)^(1/2)*Elli 
pticF(((b*x+a)/a)^(1/2),1/2*2^(1/2))*a^4-2*b^4*x^4+2*a^2*b^2*x^2)/(-b^2*x^ 
2+a^2)/b^4
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.37 \[ \int \frac {x^{5/2}}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {2 \, {\left (\sqrt {-b c x + a c} \sqrt {b x + a} b^{2} x^{\frac {3}{2}} - 3 \, \sqrt {-b^{2} c} a^{2} {\rm weierstrassZeta}\left (\frac {4 \, a^{2}}{b^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right )\right )\right )}}{5 \, b^{4} c} \] Input:

integrate(x^(5/2)/(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) - 3*sqrt(-b^2*c)*a^2*we 
ierstrassZeta(4*a^2/b^2, 0, weierstrassPInverse(4*a^2/b^2, 0, x)))/(b^4*c)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{5/2}}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\text {Timed out} \] Input:

integrate(x**(5/2)/(b*x+a)**(1/2)/(-b*c*x+a*c)**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(x^(5/2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F(-1)]

Timed out. \[ \int \frac {x^{5/2}}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\text {Timed out} \] Input:

integrate(x^(5/2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{5/2}}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\int \frac {x^{5/2}}{\sqrt {a\,c-b\,c\,x}\,\sqrt {a+b\,x}} \,d x \] Input:

int(x^(5/2)/((a*c - b*c*x)^(1/2)*(a + b*x)^(1/2)),x)
 

Output:

int(x^(5/2)/((a*c - b*c*x)^(1/2)*(a + b*x)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {x^{5/2}}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\frac {\sqrt {c}\, \left (-2 \sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}\, x +3 \left (\int \frac {\sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}}{-b^{2} x^{2}+a^{2}}d x \right ) a^{2}\right )}{5 b^{2} c} \] Input:

int(x^(5/2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x)
                                                                                    
                                                                                    
 

Output:

(sqrt(c)*( - 2*sqrt(x)*sqrt(a + b*x)*sqrt(a - b*x)*x + 3*int((sqrt(x)*sqrt 
(a + b*x)*sqrt(a - b*x))/(a**2 - b**2*x**2),x)*a**2))/(5*b**2*c)