\(\int \frac {1}{x^{5/2} (a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx\) [189]

Optimal result

Integrand size = 28, antiderivative size = 187 \[ \int \frac {1}{x^{5/2} (a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\frac {1}{3 a^2 c x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {3}{2 a^4 c^2 x^{3/2} \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {5 \sqrt {a+b x} \sqrt {a c-b c x}}{2 a^6 c^3 x^{3/2}}+\frac {5 b^{3/2} \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{2 a^{11/2} c^2 \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:

1/3/a^2/c/x^(3/2)/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2)+3/2/a^4/c^2/x^(3/2)/(b* 
x+a)^(1/2)/(-b*c*x+a*c)^(1/2)-5/2*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/a^6/c^3 
/x^(3/2)+5/2*b^(3/2)*(1-b^2*x^2/a^2)^(1/2)*EllipticF(b^(1/2)*x^(1/2)/a^(1/ 
2),I)/a^(11/2)/c^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 = 10.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.39 \[ \int \frac {1}{x^{5/2} (a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=-\frac {2 \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {5}{2},\frac {1}{4},\frac {b^2 x^2}{a^2}\right )}{3 a^4 c^2 x^{3/2} \sqrt {c (a-b x)} \sqrt {a+b x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {115, 27, 115, 27, 35, 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.

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

\(\Big \downarrow \) 115

\(\displaystyle -\frac {2 \int -\frac {9 b^2 c}{2 \sqrt {x} (a+b x)^{5/2} (a c-b c x)^{5/2}}dx}{3 a^2 c}-\frac {2}{3 a^2 c x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 115

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 b^2 \left (\frac {5 \int \frac {a-b x}{\sqrt {x} (a+b x)^{3/2} (a c-b c x)^{5/2}}dx}{6 a^2}+\frac {\sqrt {x}}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {2}{3 a^2 c x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\)

\(\Big \downarrow \) 35

\(\displaystyle \frac {3 b^2 \left (\frac {5 \int \frac {1}{\sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}}dx}{6 a^2 c}+\frac {\sqrt {x}}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {2}{3 a^2 c x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\)

\(\Big \downarrow \) 115

\(\displaystyle \frac {3 b^2 \left (\frac {5 \left (\frac {\int \frac {b c (a-b x)}{2 \sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}}dx}{a^2 b c}+\frac {\sqrt {x}}{a^2 c \sqrt {a+b x} \sqrt {a c-b c x}}\right )}{6 a^2 c}+\frac {\sqrt {x}}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {2}{3 a^2 c x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 b^2 \left (\frac {5 \left (\frac {\int \frac {a-b x}{\sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}}dx}{2 a^2}+\frac {\sqrt {x}}{a^2 c \sqrt {a+b x} \sqrt {a c-b c x}}\right )}{6 a^2 c}+\frac {\sqrt {x}}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {2}{3 a^2 c x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\)

\(\Big \downarrow \) 35

\(\displaystyle \frac {3 b^2 \left (\frac {5 \left (\frac {\int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}dx}{2 a^2 c}+\frac {\sqrt {x}}{a^2 c \sqrt {a+b x} \sqrt {a c-b c x}}\right )}{6 a^2 c}+\frac {\sqrt {x}}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {2}{3 a^2 c x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\)

\(\Big \downarrow \) 127

\(\displaystyle \frac {3 b^2 \left (\frac {5 \left (\frac {\sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \int \frac {1}{\sqrt {x} \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1}}dx}{2 a^2 c \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\sqrt {x}}{a^2 c \sqrt {a+b x} \sqrt {a c-b c x}}\right )}{6 a^2 c}+\frac {\sqrt {x}}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {2}{3 a^2 c x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}}\)

\(\Big \downarrow \) 126

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

Input:

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

Output:

-2/(3*a^2*c*x^(3/2)*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)) + (3*b^2*(Sqrt[x] 
/(3*a^2*c*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)) + (5*(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]*Elliptic 
F[ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[a]], -1])/(a^(3/2)*Sqrt[b]*c*Sqrt[a + b*x] 
*Sqrt[a*c - b*c*x])))/(6*a^2*c)))/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 = 5.18 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.97

method	result	size
default	\(\frac {\left (-15 \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a \,b^{3} x^{3}+15 \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{3} b x -30 b^{4} x^{4}+42 a^{2} b^{2} x^{2}-8 a^{4}\right ) \sqrt {c \left (-b x +a \right )}}{12 c^{3} \left (-b x +a \right )^{2} a^{6} \left (b x +a \right )^{\frac {3}{2}} x^{\frac {3}{2}}}\)	\(182\)
elliptic	\(\frac {\sqrt {c x \left (-b^{2} x^{2}+a^{2}\right )}\, \left (\frac {\sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{3 a^{4} c^{3} b^{2} \left (x^{2}-\frac {a^{2}}{b^{2}}\right )^{2}}+\frac {11 b^{2} x}{6 c^{2} a^{6} \sqrt {-\left (x^{2}-\frac {a^{2}}{b^{2}}\right ) b^{2} c x}}-\frac {2 \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{3 a^{6} c^{3} x^{2}}+\frac {5 b \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{4 a^{5} c^{2} \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{\sqrt {x}\, \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}}\)	\(231\)
risch	\(\text {Expression too large to display}\)	\(1266\)

Input:

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

Output:

1/12*(-15*((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*b^3*x^3+15*((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^3*b*x-30*b^4*x^4+42*a^2*b^2*x^2-8*a^4)*(c*(-b*x+a))^(1/2)/c^3/(-b*x+a 
)^2/a^6/(b*x+a)^(3/2)/x^(3/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.73 \[ \int \frac {1}{x^{5/2} (a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=-\frac {{\left (15 \, b^{4} x^{4} - 21 \, a^{2} b^{2} x^{2} + 4 \, a^{4}\right )} \sqrt {-b c x + a c} \sqrt {b x + a} \sqrt {x} + 15 \, {\left (b^{4} x^{6} - 2 \, a^{2} b^{2} x^{4} + a^{4} x^{2}\right )} \sqrt {-b^{2} c} {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right )}{6 \, {\left (a^{6} b^{4} c^{3} x^{6} - 2 \, a^{8} b^{2} c^{3} x^{4} + a^{10} c^{3} x^{2}\right )}} \] Input:

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

Output:

-1/6*((15*b^4*x^4 - 21*a^2*b^2*x^2 + 4*a^4)*sqrt(-b*c*x + a*c)*sqrt(b*x + 
a)*sqrt(x) + 15*(b^4*x^6 - 2*a^2*b^2*x^4 + a^4*x^2)*sqrt(-b^2*c)*weierstra 
ssPInverse(4*a^2/b^2, 0, x))/(a^6*b^4*c^3*x^6 - 2*a^8*b^2*c^3*x^4 + a^10*c 
^3*x^2)
                                                                                    
                                                                                    
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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