\(\int \frac {1}{x^{5/2} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx\) [179]

Optimal result

Integrand size = 28, antiderivative size = 147 \[ \int \frac {1}{x^{5/2} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {1}{a^2 c x^{3/2} \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {5 \sqrt {a+b x} \sqrt {a c-b c x}}{3 a^4 c^2 x^{3/2}}+\frac {5 b^{3/2} \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{3 a^{7/2} c \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:

1/a^2/c/x^(3/2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)-5/3*(b*x+a)^(1/2)*(-b*c*x 
+a*c)^(1/2)/a^4/c^2/x^(3/2)+5/3*b^(3/2)*(1-b^2*x^2/a^2)^(1/2)*EllipticF(b^ 
(1/2)*x^(1/2)/a^(1/2),I)/a^(7/2)/c/(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 = 8.54 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.50 \[ \int \frac {1}{x^{5/2} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=-\frac {2 \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {3}{2},\frac {1}{4},\frac {b^2 x^2}{a^2}\right )}{3 a^2 c x^{3/2} \sqrt {c (a-b x)} \sqrt {a+b x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {115, 27, 115, 27, 35, 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.

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

\(\Big \downarrow \) 115

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 115

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 35

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

\(\Big \downarrow \) 127

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

\(\Big \downarrow \) 126

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

Input:

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

Output:

-2/(3*a^2*c*x^(3/2)*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) + (5*b^2*(Sqrt[x]/(a^ 
2*c*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) + (Sqrt[1 - (b*x)/a]*Sqrt[1 + (b*x)/a 
]*EllipticF[ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[a]], -1])/(a^(3/2)*Sqrt[b]*c*Sqr 
t[a + b*x]*Sqrt[a*c - b*c*x])))/(3*a^2)
 

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 35

Int[(u_.)*((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.), x_Symbol] :> 
 Simp[(b/d)^m   Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n} 
, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] &&  !(IntegerQ[n] && SimplerQ[a + 
b*x, c + d*x])
 

rule 115

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 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 
*n, 2*p]
 

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 = 2.89 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.78

method	result	size
default	\(\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (5 \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a b x +10 b^{2} x^{2}-4 a^{2}\right )}{6 c^{2} x^{\frac {3}{2}} a^{4} \left (-b^{2} x^{2}+a^{2}\right )}\)	\(114\)
elliptic	\(\frac {\sqrt {c x \left (-b^{2} x^{2}+a^{2}\right )}\, \left (\frac {b^{2} x}{c \,a^{4} \sqrt {-\left (x^{2}-\frac {a^{2}}{b^{2}}\right ) b^{2} c x}}-\frac {2 \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{3 a^{4} c^{2} x^{2}}+\frac {5 b \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{6 a^{3} c \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{\sqrt {x}\, \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}}\)	\(187\)
risch	\(-\frac {2 \left (-b x +a \right ) \sqrt {b x +a}}{3 a^{4} x^{\frac {3}{2}} \sqrt {-c \left (b x -a \right )}\, c}+\frac {b^{2} \left (\frac {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}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}-\frac {3 a \left (\frac {-b^{2} c \,x^{2}-a b c x}{a^{2} c b \sqrt {\left (x -\frac {a}{b}\right ) \left (-b^{2} c \,x^{2}-a b c x \right )}}-\frac {\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 )}{2 b \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}+\frac {\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 )}{2 a \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{2}+\frac {3 a \left (\frac {-b^{2} c \,x^{2}+a b c x}{a^{2} c b \sqrt {\left (x +\frac {a}{b}\right ) \left (-b^{2} c \,x^{2}+a b c x \right )}}+\frac {\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 )}{2 b \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}+\frac {\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 )}{2 a \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{2}\right ) \sqrt {-x \left (b x +a \right ) c \left (b x -a \right )}}{3 a^{4} \sqrt {x}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}\, c}\)	\(675\)

Input:

int(1/x^(5/2)/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.70 \[ \int \frac {1}{x^{5/2} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=-\frac {{\left (5 \, b^{2} x^{2} - 2 \, a^{2}\right )} \sqrt {-b c x + a c} \sqrt {b x + a} \sqrt {x} + 5 \, {\left (b^{2} x^{4} - a^{2} x^{2}\right )} \sqrt {-b^{2} c} {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right )}{3 \, {\left (a^{4} b^{2} c^{2} x^{4} - a^{6} c^{2} x^{2}\right )}} \] Input:

integrate(1/x^(5/2)/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x, algorithm="fricas" 
)
 

Output:

-1/3*((5*b^2*x^2 - 2*a^2)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*sqrt(x) + 5*(b^ 
2*x^4 - a^2*x^2)*sqrt(-b^2*c)*weierstrassPInverse(4*a^2/b^2, 0, x))/(a^4*b 
^2*c^2*x^4 - a^6*c^2*x^2)
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^{5/2} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(1/x**(5/2)/(b*x+a)**(3/2)/(-b*c*x+a*c)**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{x^{5/2} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\int { \frac {1}{{\left (-b c x + a c\right )}^{\frac {3}{2}} {\left (b x + a\right )}^{\frac {3}{2}} x^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/x^(5/2)/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x, algorithm="maxima" 
)
 

Output:

integrate(1/((-b*c*x + a*c)^(3/2)*(b*x + a)^(3/2)*x^(5/2)), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {1}{x^{5/2} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(1/x^(5/2)/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^{5/2} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\int \frac {1}{x^{5/2}\,{\left (a\,c-b\,c\,x\right )}^{3/2}\,{\left (a+b\,x\right )}^{3/2}} \,d x \] Input:

int(1/(x^(5/2)*(a*c - b*c*x)^(3/2)*(a + b*x)^(3/2)),x)
 

Output:

int(1/(x^(5/2)*(a*c - b*c*x)^(3/2)*(a + b*x)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {1}{x^{5/2} (a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\int \frac {1}{x^{\frac {5}{2}} \left (b x +a \right )^{\frac {3}{2}} \left (-b c x +a c \right )^{\frac {3}{2}}}d x \] Input:

int(1/x^(5/2)/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x)
 

Output:

int(1/x^(5/2)/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x)