\(\int \frac {1}{x^{3/2} (a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx\) [197]

Optimal result

Integrand size = 28, antiderivative size = 262 \[ \int \frac {1}{x^{3/2} (a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\frac {1}{3 a^2 c \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {7}{6 a^4 c^2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {7 \sqrt {a+b x} \sqrt {a c-b c x}}{2 a^6 c^3 \sqrt {x}}-\frac {7 \sqrt {b} \sqrt {1-\frac {b^2 x^2}{a^2}} E\left (\left .\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right |-1\right )}{2 a^{9/2} c^2 \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {7 \sqrt {b} \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{2 a^{9/2} c^2 \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:

1/3/a^2/c/x^(1/2)/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2)+7/6/a^4/c^2/x^(1/2)/(b* 
x+a)^(1/2)/(-b*c*x+a*c)^(1/2)-7/2*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/a^6/c^3 
/x^(1/2)-7/2*b^(1/2)*(1-b^2*x^2/a^2)^(1/2)*EllipticE(b^(1/2)*x^(1/2)/a^(1/ 
2),I)/a^(9/2)/c^2/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)+7/2*b^(1/2)*(1-b^2*x^2/ 
a^2)^(1/2)*EllipticF(b^(1/2)*x^(1/2)/a^(1/2),I)/a^(9/2)/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 = 71, normalized size of antiderivative = 0.27 \[ \int \frac {1}{x^{3/2} (a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=-\frac {2 \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {5}{2},\frac {3}{4},\frac {b^2 x^2}{a^2}\right )}{a^4 c^2 \sqrt {x} \sqrt {c (a-b x)} \sqrt {a+b x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.12, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {115, 27, 110, 27, 169, 27, 169, 27, 110, 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.

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

\(\Big \downarrow \) 115

\(\displaystyle -\frac {2 \int -\frac {7 b^2 c \sqrt {x}}{2 (a+b x)^{5/2} (a c-b c x)^{5/2}}dx}{a^2 c}-\frac {2}{a^2 c \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7 b^2 \int \frac {\sqrt {x}}{(a+b x)^{5/2} (a c-b c x)^{5/2}}dx}{a^2}-\frac {2}{a^2 c \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}}\)

\(\Big \downarrow \) 110

\(\displaystyle \frac {7 b^2 \left (\frac {\int \frac {c (a+5 b x)}{2 \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{5/2}}dx}{3 a b c}-\frac {\sqrt {x}}{3 a b c (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {2}{a^2 c \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7 b^2 \left (\frac {\int \frac {a+5 b x}{\sqrt {x} (a+b x)^{3/2} (a c-b c x)^{5/2}}dx}{6 a b}-\frac {\sqrt {x}}{3 a b c (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {2}{a^2 c \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {7 b^2 \left (\frac {\frac {\int \frac {3 a b c (a+2 b x)}{\sqrt {x} \sqrt {a+b x} (a c-b c x)^{5/2}}dx}{a^2 b c}-\frac {4 \sqrt {x}}{a c \sqrt {a+b x} (a c-b c x)^{3/2}}}{6 a b}-\frac {\sqrt {x}}{3 a b c (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {2}{a^2 c \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7 b^2 \left (\frac {\frac {3 \int \frac {a+2 b x}{\sqrt {x} \sqrt {a+b x} (a c-b c x)^{5/2}}dx}{a}-\frac {4 \sqrt {x}}{a c \sqrt {a+b x} (a c-b c x)^{3/2}}}{6 a b}-\frac {\sqrt {x}}{3 a b c (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {2}{a^2 c \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {7 b^2 \left (\frac {\frac {3 \left (\frac {\sqrt {x} \sqrt {a+b x}}{a c (a c-b c x)^{3/2}}-\frac {\int -\frac {3 a b c \sqrt {a+b x}}{2 \sqrt {x} (a c-b c x)^{3/2}}dx}{3 a^2 b c^2}\right )}{a}-\frac {4 \sqrt {x}}{a c \sqrt {a+b x} (a c-b c x)^{3/2}}}{6 a b}-\frac {\sqrt {x}}{3 a b c (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {2}{a^2 c \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7 b^2 \left (\frac {\frac {3 \left (\frac {\int \frac {\sqrt {a+b x}}{\sqrt {x} (a c-b c x)^{3/2}}dx}{2 a c}+\frac {\sqrt {x} \sqrt {a+b x}}{a c (a c-b c x)^{3/2}}\right )}{a}-\frac {4 \sqrt {x}}{a c \sqrt {a+b x} (a c-b c x)^{3/2}}}{6 a b}-\frac {\sqrt {x}}{3 a b c (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {2}{a^2 c \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}}\)

\(\Big \downarrow \) 110

\(\displaystyle \frac {7 b^2 \left (\frac {\frac {3 \left (\frac {\frac {2 \sqrt {x} \sqrt {a+b x}}{a c \sqrt {a c-b c x}}-\frac {2 \int \frac {b \sqrt {x}}{2 \sqrt {a+b x} \sqrt {a c-b c x}}dx}{a c}}{2 a c}+\frac {\sqrt {x} \sqrt {a+b x}}{a c (a c-b c x)^{3/2}}\right )}{a}-\frac {4 \sqrt {x}}{a c \sqrt {a+b x} (a c-b c x)^{3/2}}}{6 a b}-\frac {\sqrt {x}}{3 a b c (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {2}{a^2 c \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7 b^2 \left (\frac {\frac {3 \left (\frac {\frac {2 \sqrt {x} \sqrt {a+b x}}{a c \sqrt {a c-b c x}}-\frac {b \int \frac {\sqrt {x}}{\sqrt {a+b x} \sqrt {a c-b c x}}dx}{a c}}{2 a c}+\frac {\sqrt {x} \sqrt {a+b x}}{a c (a c-b c x)^{3/2}}\right )}{a}-\frac {4 \sqrt {x}}{a c \sqrt {a+b x} (a c-b c x)^{3/2}}}{6 a b}-\frac {\sqrt {x}}{3 a b c (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {2}{a^2 c \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}}\)

\(\Big \downarrow \) 124

\(\displaystyle \frac {7 b^2 \left (\frac {\frac {3 \left (\frac {\frac {2 \sqrt {x} \sqrt {a+b x}}{a c \sqrt {a c-b c x}}-\frac {b \sqrt {x} \sqrt {\frac {a-b x}{a}} \int \frac {\sqrt {2} \sqrt {-\frac {b x}{a}}}{\sqrt {a+b x} \sqrt {1-\frac {b x}{a}}}dx}{\sqrt {2} a c \sqrt {-\frac {b x}{a}} \sqrt {a c-b c x}}}{2 a c}+\frac {\sqrt {x} \sqrt {a+b x}}{a c (a c-b c x)^{3/2}}\right )}{a}-\frac {4 \sqrt {x}}{a c \sqrt {a+b x} (a c-b c x)^{3/2}}}{6 a b}-\frac {\sqrt {x}}{3 a b c (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {2}{a^2 c \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7 b^2 \left (\frac {\frac {3 \left (\frac {\frac {2 \sqrt {x} \sqrt {a+b x}}{a c \sqrt {a c-b c x}}-\frac {b \sqrt {x} \sqrt {\frac {a-b x}{a}} \int \frac {\sqrt {-\frac {b x}{a}}}{\sqrt {a+b x} \sqrt {1-\frac {b x}{a}}}dx}{a c \sqrt {-\frac {b x}{a}} \sqrt {a c-b c x}}}{2 a c}+\frac {\sqrt {x} \sqrt {a+b x}}{a c (a c-b c x)^{3/2}}\right )}{a}-\frac {4 \sqrt {x}}{a c \sqrt {a+b x} (a c-b c x)^{3/2}}}{6 a b}-\frac {\sqrt {x}}{3 a b c (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {2}{a^2 c \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {7 b^2 \left (\frac {\frac {3 \left (\frac {\frac {2 \sqrt {x} \sqrt {a+b x}}{a c \sqrt {a c-b c x}}-\frac {2 \sqrt {x} \sqrt {\frac {a-b x}{a}} E\left (\left .\arcsin \left (\frac {\sqrt {a+b x}}{\sqrt {2} \sqrt {a}}\right )\right |2\right )}{\sqrt {a} c \sqrt {-\frac {b x}{a}} \sqrt {a c-b c x}}}{2 a c}+\frac {\sqrt {x} \sqrt {a+b x}}{a c (a c-b c x)^{3/2}}\right )}{a}-\frac {4 \sqrt {x}}{a c \sqrt {a+b x} (a c-b c x)^{3/2}}}{6 a b}-\frac {\sqrt {x}}{3 a b c (a+b x)^{3/2} (a c-b c x)^{3/2}}\right )}{a^2}-\frac {2}{a^2 c \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2}}\)

Input:

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

Output:

-2/(a^2*c*Sqrt[x]*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)) + (7*b^2*(-1/3*Sqrt 
[x]/(a*b*c*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)) + ((-4*Sqrt[x])/(a*c*Sqrt[ 
a + b*x]*(a*c - b*c*x)^(3/2)) + (3*((Sqrt[x]*Sqrt[a + b*x])/(a*c*(a*c - b* 
c*x)^(3/2)) + ((2*Sqrt[x]*Sqrt[a + b*x])/(a*c*Sqrt[a*c - b*c*x]) - (2*Sqrt 
[x]*Sqrt[(a - b*x)/a]*EllipticE[ArcSin[Sqrt[a + b*x]/(Sqrt[2]*Sqrt[a])], 2 
])/(Sqrt[a]*c*Sqrt[-((b*x)/a)]*Sqrt[a*c - b*c*x]))/(2*a*c)))/a)/(6*a*b)))/ 
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 110

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

rule 169

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(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] + S 
imp[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*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

Maple [A] (verified)

Time = 4.76 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.06

method	result	size
elliptic	\(\frac {\sqrt {c x \left (-b^{2} x^{2}+a^{2}\right )}\, \left (\frac {x \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{3 a^{4} c^{3} b^{2} \left (x^{2}-\frac {a^{2}}{b^{2}}\right )^{2}}+\frac {3 b^{2} x^{2}}{2 c^{2} a^{6} \sqrt {-\left (x^{2}-\frac {a^{2}}{b^{2}}\right ) b^{2} c x}}-\frac {2 \left (-b^{2} c \,x^{2}+a^{2} c \right )}{a^{6} c^{3} \sqrt {x \left (-b^{2} c \,x^{2}+a^{2} c \right )}}-\frac {7 b \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \left (-\frac {2 a \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{4 a^{5} c^{2} \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{\sqrt {x}\, \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}}\)	\(279\)
default	\(\frac {\left (-42 \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{2} b^{2} x^{2}+21 \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{2} b^{2} x^{2}+42 \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{4}-21 \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{4}-42 b^{4} x^{4}+70 a^{2} b^{2} x^{2}-24 a^{4}\right ) \sqrt {c \left (-b x +a \right )}}{12 c^{3} \left (b x +a \right )^{\frac {3}{2}} a^{6} \left (-b x +a \right )^{2} \sqrt {x}}\)	\(302\)
risch	\(\text {Expression too large to display}\)	\(1314\)

Input:

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

Output:

(c*x*(-b^2*x^2+a^2))^(1/2)/x^(1/2)/(b*x+a)^(1/2)/(c*(-b*x+a))^(1/2)*(1/3/a 
^4/c^3/b^2*x*(-b^2*c*x^3+a^2*c*x)^(1/2)/(x^2-a^2/b^2)^2+3/2*b^2/c^2*x^2/a^ 
6/(-(x^2-a^2/b^2)*b^2*c*x)^(1/2)-2*(-b^2*c*x^2+a^2*c)/a^6/c^3/(x*(-b^2*c*x 
^2+a^2*c))^(1/2)-7/4*b/a^5/c^2*((x+a/b)/a*b)^(1/2)*(-2*(x-a/b)/a*b)^(1/2)* 
(-b*x/a)^(1/2)/(-b^2*c*x^3+a^2*c*x)^(1/2)*(-2*a/b*EllipticE(((x+a/b)/a*b)^ 
(1/2),1/2*2^(1/2))+a/b*EllipticF(((x+a/b)/a*b)^(1/2),1/2*2^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.55 \[ \int \frac {1}{x^{3/2} (a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=-\frac {{\left (21 \, b^{4} x^{4} - 35 \, a^{2} b^{2} x^{2} + 12 \, a^{4}\right )} \sqrt {-b c x + a c} \sqrt {b x + a} \sqrt {x} + 21 \, {\left (b^{4} x^{5} - 2 \, a^{2} b^{2} x^{3} + a^{4} x\right )} \sqrt {-b^{2} c} {\rm weierstrassZeta}\left (\frac {4 \, a^{2}}{b^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right )\right )}{6 \, {\left (a^{6} b^{4} c^{3} x^{5} - 2 \, a^{8} b^{2} c^{3} x^{3} + a^{10} c^{3} x\right )}} \] Input:

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

Output:

-1/6*((21*b^4*x^4 - 35*a^2*b^2*x^2 + 12*a^4)*sqrt(-b*c*x + a*c)*sqrt(b*x + 
 a)*sqrt(x) + 21*(b^4*x^5 - 2*a^2*b^2*x^3 + a^4*x)*sqrt(-b^2*c)*weierstras 
sZeta(4*a^2/b^2, 0, weierstrassPInverse(4*a^2/b^2, 0, x)))/(a^6*b^4*c^3*x^ 
5 - 2*a^8*b^2*c^3*x^3 + a^10*c^3*x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate(1/x^(3/2)/(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^(3/2)), x)
 

Giac [F(-1)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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