\(\int \frac {x^{21/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx\) [191]

Optimal result

Integrand size = 28, antiderivative size = 342 \[ \int \frac {x^{21/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\frac {x^{19/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {19 x^{15/2}}{6 b^4 c^2 \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {1463 a^4 x^{3/2} \sqrt {a+b x} \sqrt {a c-b c x}}{234 b^{10} c^3}-\frac {1045 a^2 x^{7/2} \sqrt {a+b x} \sqrt {a c-b c x}}{234 b^8 c^3}-\frac {95 x^{11/2} \sqrt {a+b x} \sqrt {a c-b c x}}{26 b^6 c^3}+\frac {1463 a^{15/2} \sqrt {1-\frac {b^2 x^2}{a^2}} E\left (\left .\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right |-1\right )}{78 b^{23/2} c^2 \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {1463 a^{15/2} \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{78 b^{23/2} c^2 \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:

1/3*x^(19/2)/b^2/c/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2)-19/6*x^(15/2)/b^4/c^2/ 
(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)-1463/234*a^4*x^(3/2)*(b*x+a)^(1/2)*(-b*c* 
x+a*c)^(1/2)/b^10/c^3-1045/234*a^2*x^(7/2)*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2 
)/b^8/c^3-95/26*x^(11/2)*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/b^6/c^3+1463/78* 
a^(15/2)*(1-b^2*x^2/a^2)^(1/2)*EllipticE(b^(1/2)*x^(1/2)/a^(1/2),I)/b^(23/ 
2)/c^2/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)-1463/78*a^(15/2)*(1-b^2*x^2/a^2)^( 
1/2)*EllipticF(b^(1/2)*x^(1/2)/a^(1/2),I)/b^(23/2)/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 = 10.07 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.44 \[ \int \frac {x^{21/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\frac {2 x^{3/2} \left (-1463 a^8+627 a^6 b^2 x^2+57 a^4 b^4 x^4+19 a^2 b^6 x^6+9 b^8 x^8+1463 a^6 \left (a^2-b^2 x^2\right ) \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{2},\frac {7}{4},\frac {b^2 x^2}{a^2}\right )\right )}{117 b^{10} c^2 \sqrt {c (a-b x)} \sqrt {a+b x} \left (-a^2+b^2 x^2\right )} \] Input:

Integrate[x^(21/2)/((a + b*x)^(5/2)*(a*c - b*c*x)^(5/2)),x]
 

Output:

(2*x^(3/2)*(-1463*a^8 + 627*a^6*b^2*x^2 + 57*a^4*b^4*x^4 + 19*a^2*b^6*x^6 
+ 9*b^8*x^8 + 1463*a^6*(a^2 - b^2*x^2)*Sqrt[1 - (b^2*x^2)/a^2]*Hypergeomet 
ric2F1[3/4, 5/2, 7/4, (b^2*x^2)/a^2]))/(117*b^10*c^2*Sqrt[c*(a - b*x)]*Sqr 
t[a + b*x]*(-a^2 + b^2*x^2))
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.89, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {109, 27, 35, 109, 27, 35, 113, 27, 113, 27, 113, 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.

\(\displaystyle \int \frac {x^{21/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx\)

\(\Big \downarrow \) 109

\(\displaystyle \frac {x^{19/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {\int \frac {19 a c x^{17/2} (a-b x)}{2 (a+b x)^{3/2} (a c-b c x)^{5/2}}dx}{3 a b^2 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^{19/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {19 \int \frac {x^{17/2} (a-b x)}{(a+b x)^{3/2} (a c-b c x)^{5/2}}dx}{6 b^2}\)

\(\Big \downarrow \) 35

\(\displaystyle \frac {x^{19/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {19 \int \frac {x^{17/2}}{(a+b x)^{3/2} (a c-b c x)^{3/2}}dx}{6 b^2 c}\)

\(\Big \downarrow \) 109

\(\displaystyle \frac {x^{19/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {19 \left (\frac {x^{15/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\int \frac {15 a c x^{13/2} (a-b x)}{2 \sqrt {a+b x} (a c-b c x)^{3/2}}dx}{a b^2 c}\right )}{6 b^2 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^{19/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {19 \left (\frac {x^{15/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {15 \int \frac {x^{13/2} (a-b x)}{\sqrt {a+b x} (a c-b c x)^{3/2}}dx}{2 b^2}\right )}{6 b^2 c}\)

\(\Big \downarrow \) 35

\(\displaystyle \frac {x^{19/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {19 \left (\frac {x^{15/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {15 \int \frac {x^{13/2}}{\sqrt {a+b x} \sqrt {a c-b c x}}dx}{2 b^2 c}\right )}{6 b^2 c}\)

\(\Big \downarrow \) 113

\(\displaystyle \frac {x^{19/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {19 \left (\frac {x^{15/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {15 \left (-\frac {2 \int -\frac {11 a^2 c x^{9/2}}{2 \sqrt {a+b x} \sqrt {a c-b c x}}dx}{13 b^2 c}-\frac {2 x^{11/2} \sqrt {a+b x} \sqrt {a c-b c x}}{13 b^2 c}\right )}{2 b^2 c}\right )}{6 b^2 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^{19/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {19 \left (\frac {x^{15/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {15 \left (\frac {11 a^2 \int \frac {x^{9/2}}{\sqrt {a+b x} \sqrt {a c-b c x}}dx}{13 b^2}-\frac {2 x^{11/2} \sqrt {a+b x} \sqrt {a c-b c x}}{13 b^2 c}\right )}{2 b^2 c}\right )}{6 b^2 c}\)

\(\Big \downarrow \) 113

\(\displaystyle \frac {x^{19/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {19 \left (\frac {x^{15/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {15 \left (\frac {11 a^2 \left (-\frac {2 \int -\frac {7 a^2 c x^{5/2}}{2 \sqrt {a+b x} \sqrt {a c-b c x}}dx}{9 b^2 c}-\frac {2 x^{7/2} \sqrt {a+b x} \sqrt {a c-b c x}}{9 b^2 c}\right )}{13 b^2}-\frac {2 x^{11/2} \sqrt {a+b x} \sqrt {a c-b c x}}{13 b^2 c}\right )}{2 b^2 c}\right )}{6 b^2 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^{19/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {19 \left (\frac {x^{15/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {15 \left (\frac {11 a^2 \left (\frac {7 a^2 \int \frac {x^{5/2}}{\sqrt {a+b x} \sqrt {a c-b c x}}dx}{9 b^2}-\frac {2 x^{7/2} \sqrt {a+b x} \sqrt {a c-b c x}}{9 b^2 c}\right )}{13 b^2}-\frac {2 x^{11/2} \sqrt {a+b x} \sqrt {a c-b c x}}{13 b^2 c}\right )}{2 b^2 c}\right )}{6 b^2 c}\)

\(\Big \downarrow \) 113

\(\displaystyle \frac {x^{19/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {19 \left (\frac {x^{15/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {15 \left (\frac {11 a^2 \left (\frac {7 a^2 \left (-\frac {2 \int -\frac {3 a^2 c \sqrt {x}}{2 \sqrt {a+b x} \sqrt {a c-b c x}}dx}{5 b^2 c}-\frac {2 x^{3/2} \sqrt {a+b x} \sqrt {a c-b c x}}{5 b^2 c}\right )}{9 b^2}-\frac {2 x^{7/2} \sqrt {a+b x} \sqrt {a c-b c x}}{9 b^2 c}\right )}{13 b^2}-\frac {2 x^{11/2} \sqrt {a+b x} \sqrt {a c-b c x}}{13 b^2 c}\right )}{2 b^2 c}\right )}{6 b^2 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^{19/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {19 \left (\frac {x^{15/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {15 \left (\frac {11 a^2 \left (\frac {7 a^2 \left (\frac {3 a^2 \int \frac {\sqrt {x}}{\sqrt {a+b x} \sqrt {a c-b c x}}dx}{5 b^2}-\frac {2 x^{3/2} \sqrt {a+b x} \sqrt {a c-b c x}}{5 b^2 c}\right )}{9 b^2}-\frac {2 x^{7/2} \sqrt {a+b x} \sqrt {a c-b c x}}{9 b^2 c}\right )}{13 b^2}-\frac {2 x^{11/2} \sqrt {a+b x} \sqrt {a c-b c x}}{13 b^2 c}\right )}{2 b^2 c}\right )}{6 b^2 c}\)

\(\Big \downarrow \) 124

\(\displaystyle \frac {x^{19/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {19 \left (\frac {x^{15/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {15 \left (\frac {11 a^2 \left (\frac {7 a^2 \left (\frac {3 a^2 \sqrt {x} \sqrt {\frac {a-b x}{a}} \int \frac {\sqrt {2} \sqrt {-\frac {b x}{a}}}{\sqrt {a+b x} \sqrt {1-\frac {b x}{a}}}dx}{5 \sqrt {2} b^2 \sqrt {-\frac {b x}{a}} \sqrt {a c-b c x}}-\frac {2 x^{3/2} \sqrt {a+b x} \sqrt {a c-b c x}}{5 b^2 c}\right )}{9 b^2}-\frac {2 x^{7/2} \sqrt {a+b x} \sqrt {a c-b c x}}{9 b^2 c}\right )}{13 b^2}-\frac {2 x^{11/2} \sqrt {a+b x} \sqrt {a c-b c x}}{13 b^2 c}\right )}{2 b^2 c}\right )}{6 b^2 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^{19/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {19 \left (\frac {x^{15/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {15 \left (\frac {11 a^2 \left (\frac {7 a^2 \left (\frac {3 a^2 \sqrt {x} \sqrt {\frac {a-b x}{a}} \int \frac {\sqrt {-\frac {b x}{a}}}{\sqrt {a+b x} \sqrt {1-\frac {b x}{a}}}dx}{5 b^2 \sqrt {-\frac {b x}{a}} \sqrt {a c-b c x}}-\frac {2 x^{3/2} \sqrt {a+b x} \sqrt {a c-b c x}}{5 b^2 c}\right )}{9 b^2}-\frac {2 x^{7/2} \sqrt {a+b x} \sqrt {a c-b c x}}{9 b^2 c}\right )}{13 b^2}-\frac {2 x^{11/2} \sqrt {a+b x} \sqrt {a c-b c x}}{13 b^2 c}\right )}{2 b^2 c}\right )}{6 b^2 c}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {x^{19/2}}{3 b^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}-\frac {19 \left (\frac {x^{15/2}}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {15 \left (\frac {11 a^2 \left (\frac {7 a^2 \left (\frac {6 a^{5/2} \sqrt {x} \sqrt {\frac {a-b x}{a}} E\left (\left .\arcsin \left (\frac {\sqrt {a+b x}}{\sqrt {2} \sqrt {a}}\right )\right |2\right )}{5 b^3 \sqrt {-\frac {b x}{a}} \sqrt {a c-b c x}}-\frac {2 x^{3/2} \sqrt {a+b x} \sqrt {a c-b c x}}{5 b^2 c}\right )}{9 b^2}-\frac {2 x^{7/2} \sqrt {a+b x} \sqrt {a c-b c x}}{9 b^2 c}\right )}{13 b^2}-\frac {2 x^{11/2} \sqrt {a+b x} \sqrt {a c-b c x}}{13 b^2 c}\right )}{2 b^2 c}\right )}{6 b^2 c}\)

Input:

Int[x^(21/2)/((a + b*x)^(5/2)*(a*c - b*c*x)^(5/2)),x]
 

Output:

x^(19/2)/(3*b^2*c*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)) - (19*(x^(15/2)/(b^ 
2*c*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) - (15*((-2*x^(11/2)*Sqrt[a + b*x]*Sqr 
t[a*c - b*c*x])/(13*b^2*c) + (11*a^2*((-2*x^(7/2)*Sqrt[a + b*x]*Sqrt[a*c - 
 b*c*x])/(9*b^2*c) + (7*a^2*((-2*x^(3/2)*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/ 
(5*b^2*c) + (6*a^(5/2)*Sqrt[x]*Sqrt[(a - b*x)/a]*EllipticE[ArcSin[Sqrt[a + 
 b*x]/(Sqrt[2]*Sqrt[a])], 2])/(5*b^3*Sqrt[-((b*x)/a)]*Sqrt[a*c - b*c*x]))) 
/(9*b^2)))/(13*b^2)))/(2*b^2*c)))/(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 113

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 
)/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1))   Int[(a + b*x) 
^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m 
 - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m 
 + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & 
& GtQ[m, 1] && NeQ[m + n + p + 1, 0] && 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]
 

Maple [A] (verified)

Time = 5.82 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.96

method	result	size
default	\(-\frac {\left (72 b^{10} x^{10}+152 a^{2} b^{8} x^{8}-8778 \operatorname {EllipticE}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{8} b^{2} x^{2} \sqrt {\frac {b x +a}{a}}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}+4389 \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{8} b^{2} x^{2} \sqrt {\frac {b x +a}{a}}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}+456 a^{4} b^{6} x^{6}+8778 \operatorname {EllipticE}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{10} \sqrt {\frac {b x +a}{a}}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}-4389 \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{10} \sqrt {\frac {b x +a}{a}}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}-3762 a^{6} b^{4} x^{4}+2926 a^{8} b^{2} x^{2}\right ) \sqrt {c \left (-b x +a \right )}}{468 c^{3} \sqrt {x}\, \left (-b x +a \right )^{2} b^{12} \left (b x +a \right )^{\frac {3}{2}}}\)	\(330\)
elliptic	\(\frac {\sqrt {c x \left (-b^{2} x^{2}+a^{2}\right )}\, \left (\frac {a^{8} x \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{3 c^{3} b^{14} \left (x^{2}-\frac {a^{2}}{b^{2}}\right )^{2}}-\frac {9 x^{2} a^{6}}{2 b^{10} c^{2} \sqrt {-\left (x^{2}-\frac {a^{2}}{b^{2}}\right ) b^{2} c x}}-\frac {2 x^{5} \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{13 b^{6} c^{3}}-\frac {74 a^{2} x^{3} \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{117 b^{8} c^{3}}-\frac {244 a^{4} x \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{117 c^{3} b^{10}}+\frac {1463 a^{7} \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \left (-\frac {2 a \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{156 c^{2} b^{11} \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{\sqrt {x}\, \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}}\)	\(330\)
risch	\(\text {Expression too large to display}\)	\(1340\)

Input:

int(x^(21/2)/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/468*(72*b^10*x^10+152*a^2*b^8*x^8-8778*EllipticE(((b*x+a)/a)^(1/2),1/2* 
2^(1/2))*2^(1/2)*a^8*b^2*x^2*((b*x+a)/a)^(1/2)*((-b*x+a)/a)^(1/2)*(-b*x/a) 
^(1/2)+4389*EllipticF(((b*x+a)/a)^(1/2),1/2*2^(1/2))*2^(1/2)*a^8*b^2*x^2*( 
(b*x+a)/a)^(1/2)*((-b*x+a)/a)^(1/2)*(-b*x/a)^(1/2)+456*a^4*b^6*x^6+8778*El 
lipticE(((b*x+a)/a)^(1/2),1/2*2^(1/2))*2^(1/2)*a^10*((b*x+a)/a)^(1/2)*((-b 
*x+a)/a)^(1/2)*(-b*x/a)^(1/2)-4389*EllipticF(((b*x+a)/a)^(1/2),1/2*2^(1/2) 
)*2^(1/2)*a^10*((b*x+a)/a)^(1/2)*((-b*x+a)/a)^(1/2)*(-b*x/a)^(1/2)-3762*a^ 
6*b^4*x^4+2926*a^8*b^2*x^2)*(c*(-b*x+a))^(1/2)/c^3/x^(1/2)/(-b*x+a)^2/b^12 
/(b*x+a)^(3/2)
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.49 \[ \int \frac {x^{21/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=-\frac {{\left (36 \, b^{10} x^{9} + 76 \, a^{2} b^{8} x^{7} + 228 \, a^{4} b^{6} x^{5} - 1881 \, a^{6} b^{4} x^{3} + 1463 \, a^{8} b^{2} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a} \sqrt {x} - 4389 \, {\left (a^{6} b^{4} x^{4} - 2 \, a^{8} b^{2} x^{2} + a^{10}\right )} \sqrt {-b^{2} c} {\rm weierstrassZeta}\left (\frac {4 \, a^{2}}{b^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right )\right )}{234 \, {\left (b^{16} c^{3} x^{4} - 2 \, a^{2} b^{14} c^{3} x^{2} + a^{4} b^{12} c^{3}\right )}} \] Input:

integrate(x^(21/2)/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x, algorithm="fricas")
 

Output:

-1/234*((36*b^10*x^9 + 76*a^2*b^8*x^7 + 228*a^4*b^6*x^5 - 1881*a^6*b^4*x^3 
 + 1463*a^8*b^2*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*sqrt(x) - 4389*(a^6*b^ 
4*x^4 - 2*a^8*b^2*x^2 + a^10)*sqrt(-b^2*c)*weierstrassZeta(4*a^2/b^2, 0, w 
eierstrassPInverse(4*a^2/b^2, 0, x)))/(b^16*c^3*x^4 - 2*a^2*b^14*c^3*x^2 + 
 a^4*b^12*c^3)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{21/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(x**(21/2)/(b*x+a)**(5/2)/(-b*c*x+a*c)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^{21/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\int { \frac {x^{\frac {21}{2}}}{{\left (-b c x + a c\right )}^{\frac {5}{2}} {\left (b x + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^(21/2)/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x, algorithm="maxima")
 

Output:

integrate(x^(21/2)/((-b*c*x + a*c)^(5/2)*(b*x + a)^(5/2)), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {x^{21/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(x^(21/2)/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{21/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\int \frac {x^{21/2}}{{\left (a\,c-b\,c\,x\right )}^{5/2}\,{\left (a+b\,x\right )}^{5/2}} \,d x \] Input:

int(x^(21/2)/((a*c - b*c*x)^(5/2)*(a + b*x)^(5/2)),x)
 

Output:

int(x^(21/2)/((a*c - b*c*x)^(5/2)*(a + b*x)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {x^{21/2}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\int \frac {x^{\frac {21}{2}}}{\left (b x +a \right )^{\frac {5}{2}} \left (-b c x +a c \right )^{\frac {5}{2}}}d x \] Input:

int(x^(21/2)/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x)
 

Output:

int(x^(21/2)/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2),x)