\(\int \frac {x^{3/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx\) [187]

Optimal result

Integrand size = 28, antiderivative size = 153 \[ \int \frac {x^{3/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\frac {\sqrt {x}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {\sqrt {x}}{6 a^2 b^2 c^2 \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{6 a^{3/2} b^{5/2} c^2 \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:

1/3*x^(1/2)/b^2/c/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2)-1/6*x^(1/2)/a^2/b^2/c^2 
/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)-1/6*(1-b^2*x^2/a^2)^(1/2)*EllipticF(b^(1 
/2)*x^(1/2)/a^(1/2),I)/a^(3/2)/b^(5/2)/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 = 8.78 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.66 \[ \int \frac {x^{3/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\frac {\sqrt {x} \left (a^2+b^2 x^2+\left (-a^2+b^2 x^2\right ) \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {b^2 x^2}{a^2}\right )\right )}{6 a^2 b^2 c (c (a-b x))^{3/2} (a+b x)^{3/2}} \] Input:

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

Output:

(Sqrt[x]*(a^2 + b^2*x^2 + (-a^2 + b^2*x^2)*Sqrt[1 - (b^2*x^2)/a^2]*Hyperge 
ometric2F1[1/4, 1/2, 5/4, (b^2*x^2)/a^2]))/(6*a^2*b^2*c*(c*(a - b*x))^(3/2 
)*(a + b*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {109, 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.

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 35

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

\(\Big \downarrow \) 115

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 35

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

\(\Big \downarrow \) 127

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

\(\Big \downarrow \) 126

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

Input:

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

Output:

Sqrt[x]/(3*b^2*c*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)) - (Sqrt[x]/(a^2*c*Sq 
rt[a + b*x]*Sqrt[a*c - b*c*x]) + (Sqrt[1 - (b*x)/a]*Sqrt[1 + (b*x)/a]*Elli 
pticF[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*b^2*c)
 

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 109

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

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 = 2.02 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.13

method	result	size
default	\(-\frac {\left (-\operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a \,b^{2} x^{2} \sqrt {\frac {b x +a}{a}}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}+\sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{3}-2 b^{3} x^{3}-2 a^{2} b x \right ) \sqrt {c \left (-b x +a \right )}}{12 c^{3} \sqrt {x}\, \left (-b x +a \right )^{2} a^{2} b^{3} \left (b x +a \right )^{\frac {3}{2}}}\)	\(173\)
elliptic	\(\frac {\sqrt {c x \left (-b^{2} x^{2}+a^{2}\right )}\, \left (\frac {\sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{3 c^{3} b^{6} \left (x^{2}-\frac {a^{2}}{b^{2}}\right )^{2}}-\frac {x}{6 b^{2} c^{2} a^{2} \sqrt {-\left (x^{2}-\frac {a^{2}}{b^{2}}\right ) b^{2} c x}}-\frac {\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{12 b^{3} a \,c^{2} \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{\sqrt {x}\, \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}}\)	\(201\)

Input:

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

Output:

-1/12*(-EllipticF(((b*x+a)/a)^(1/2),1/2*2^(1/2))*2^(1/2)*a*b^2*x^2*((b*x+a 
)/a)^(1/2)*((-b*x+a)/a)^(1/2)*(-b*x/a)^(1/2)+((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^ 
3-2*b^3*x^3-2*a^2*b*x)/c^3*(c*(-b*x+a))^(1/2)/x^(1/2)/(-b*x+a)^2/a^2/b^3/( 
b*x+a)^(3/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.80 \[ \int \frac {x^{3/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\frac {{\left (b^{4} x^{2} + a^{2} b^{2}\right )} \sqrt {-b c x + a c} \sqrt {b x + a} \sqrt {x} + {\left (b^{4} x^{4} - 2 \, a^{2} b^{2} x^{2} + a^{4}\right )} \sqrt {-b^{2} c} {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right )}{6 \, {\left (a^{2} b^{8} c^{3} x^{4} - 2 \, a^{4} b^{6} c^{3} x^{2} + a^{6} b^{4} c^{3}\right )}} \] Input:

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

Output:

1/6*((b^4*x^2 + a^2*b^2)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*sqrt(x) + (b^4*x 
^4 - 2*a^2*b^2*x^2 + a^4)*sqrt(-b^2*c)*weierstrassPInverse(4*a^2/b^2, 0, x 
))/(a^2*b^8*c^3*x^4 - 2*a^4*b^6*c^3*x^2 + a^6*b^4*c^3)
 

Sympy [A] (verification not implemented)

Time = 135.80 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.76 \[ \int \frac {x^{3/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & - \frac {1}{4}, \frac {7}{4}, \frac {9}{4} \\\frac {1}{2}, 1, \frac {5}{4}, \frac {7}{4}, \frac {9}{4} & 0 \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{2}}} \right )}}{3 \pi ^{\frac {3}{2}} a^{\frac {5}{2}} b^{\frac {5}{2}} c^{\frac {5}{2}}} - \frac {i {G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {5}{4}, - \frac {3}{4}, - \frac {1}{4}, 0, \frac {1}{2} & 1 \\0, \frac {1}{2}, 0 & - \frac {5}{4}, - \frac {3}{4}, \frac {5}{4} \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{3 \pi ^{\frac {3}{2}} a^{\frac {5}{2}} b^{\frac {5}{2}} c^{\frac {5}{2}}} \] Input:

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

Output:

I*meijerg(((1/2, 1, 1), (-1/4, 7/4, 9/4)), ((1/2, 1, 5/4, 7/4, 9/4), (0,)) 
, a**2/(b**2*x**2))/(3*pi**(3/2)*a**(5/2)*b**(5/2)*c**(5/2)) - I*meijerg(( 
(-5/4, -3/4, -1/4, 0, 1/2), (1,)), ((0, 1/2, 0), (-5/4, -3/4, 5/4)), a**2* 
exp_polar(-2*I*pi)/(b**2*x**2))/(3*pi**(3/2)*a**(5/2)*b**(5/2)*c**(5/2))
 

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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