\(\int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{\sqrt {x}} \, dx\) [153]

Optimal result

Integrand size = 28, antiderivative size = 142 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{\sqrt {x}} \, dx=\frac {4}{7} a^2 c \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}+\frac {2}{7} \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {8 a^{9/2} c^2 \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{7 \sqrt {b} \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:

4/7*a^2*c*x^(1/2)*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)+2/7*x^(1/2)*(b*x+a)^(3/ 
2)*(-b*c*x+a*c)^(3/2)+8/7*a^(9/2)*c^2*(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 = 5.89 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.49 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{\sqrt {x}} \, dx=\frac {2 a^2 c \sqrt {x} \sqrt {c (a-b x)} \sqrt {a+b x} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},\frac {b^2 x^2}{a^2}\right )}{\sqrt {1-\frac {b^2 x^2}{a^2}}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {112, 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.

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

\(\Big \downarrow \) 112

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {6}{7} a^2 c \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{\sqrt {x}}dx+\frac {2}{7} \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}\)

\(\Big \downarrow \) 112

\(\displaystyle \frac {6}{7} a^2 c \left (\frac {2}{3} \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}-\frac {2}{3} \int -\frac {a^2 c}{\sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx\right )+\frac {2}{7} \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {6}{7} a^2 c \left (\frac {2}{3} \int \frac {a^2 c}{\sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx+\frac {2}{3} \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}\right )+\frac {2}{7} \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {6}{7} a^2 c \left (\frac {2}{3} a^2 c \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx+\frac {2}{3} \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}\right )+\frac {2}{7} \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}\)

\(\Big \downarrow \) 127

\(\displaystyle \frac {6}{7} a^2 c \left (\frac {2 a^2 c \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \int \frac {1}{\sqrt {x} \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1}}dx}{3 \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {2}{3} \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}\right )+\frac {2}{7} \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}\)

\(\Big \downarrow \) 126

\(\displaystyle \frac {6}{7} a^2 c \left (\frac {4 a^{5/2} c \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{3 \sqrt {b} \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {2}{3} \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}\right )+\frac {2}{7} \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}\)

Input:

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

Output:

(2*Sqrt[x]*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/7 + (6*a^2*c*((2*Sqrt[x]*S 
qrt[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]*Sq 
rt[a + b*x]*Sqrt[a*c - b*c*x])))/7
 

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 = 0.61 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.87

method	result	size
default	\(\frac {2 \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, c \left (2 \sqrt {2}\, \sqrt {\frac {b x +a}{a}}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{5}+b^{5} x^{5}-4 a^{2} b^{3} x^{3}+3 a^{4} b x \right )}{7 \sqrt {x}\, b \left (-b^{2} x^{2}+a^{2}\right )}\)	\(124\)
risch	\(\frac {2 \left (-b^{2} x^{2}+3 a^{2}\right ) \sqrt {x}\, \sqrt {b x +a}\, \left (-b x +a \right ) c^{2}}{7 \sqrt {-c \left (b x -a \right )}}+\frac {4 a^{5} \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-x \left (b x +a \right ) c \left (b x -a \right )}\, c^{2}}{7 b \sqrt {-b^{2} c \,x^{3}+a^{2} c x}\, \sqrt {x}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\)	\(177\)
elliptic	\(\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \sqrt {c x \left (-b^{2} x^{2}+a^{2}\right )}\, \left (-\frac {2 b^{2} c \,x^{2} \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{7}+\frac {6 a^{2} c \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{7}+\frac {4 a^{5} c^{2} \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{7 b \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{\sqrt {x}\, c \left (-b^{2} x^{2}+a^{2}\right )}\)	\(196\)

Input:

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

Output:

2/7*(b*x+a)^(1/2)*(c*(-b*x+a))^(1/2)/x^(1/2)*c*(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+b^5*x^5-4*a^2*b^3*x^3+3*a^4*b*x)/b/(-b^2*x^2+a^2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.50 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{\sqrt {x}} \, dx=-\frac {2 \, {\left (4 \, \sqrt {-b^{2} c} a^{4} c {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right ) + {\left (b^{4} c x^{2} - 3 \, a^{2} b^{2} c\right )} \sqrt {-b c x + a c} \sqrt {b x + a} \sqrt {x}\right )}}{7 \, b^{2}} \] Input:

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

Output:

-2/7*(4*sqrt(-b^2*c)*a^4*c*weierstrassPInverse(4*a^2/b^2, 0, x) + (b^4*c*x 
^2 - 3*a^2*b^2*c)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*sqrt(x))/b^2
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{\sqrt {x}} \, dx=\frac {2 \sqrt {c}\, c \left (3 \sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{2}-\sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}\, b^{2} x^{2}+2 \left (\int \frac {\sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}}{-b^{2} x^{3}+a^{2} x}d x \right ) a^{4}\right )}{7} \] Input:

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

Output:

(2*sqrt(c)*c*(3*sqrt(x)*sqrt(a + b*x)*sqrt(a - b*x)*a**2 - sqrt(x)*sqrt(a 
+ b*x)*sqrt(a - b*x)*b**2*x**2 + 2*int((sqrt(x)*sqrt(a + b*x)*sqrt(a - b*x 
))/(a**2*x - b**2*x**3),x)*a**4))/7