\(\int \frac {\sqrt {x}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx\) [196]

Optimal result

Integrand size = 28, antiderivative size = 225 \[ \int \frac {\sqrt {x}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\frac {x^{3/2}}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {x^{3/2}}{2 a^4 c^2 \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\sqrt {1-\frac {b^2 x^2}{a^2}} E\left (\left .\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right |-1\right )}{2 a^{5/2} b^{3/2} c^2 \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{2 a^{5/2} b^{3/2} c^2 \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:

1/3*x^(3/2)/a^2/c/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2)+1/2*x^(3/2)/a^4/c^2/(b* 
x+a)^(1/2)/(-b*c*x+a*c)^(1/2)-1/2*(1-b^2*x^2/a^2)^(1/2)*EllipticE(b^(1/2)* 
x^(1/2)/a^(1/2),I)/a^(5/2)/b^(3/2)/c^2/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)+1/ 
2*(1-b^2*x^2/a^2)^(1/2)*EllipticF(b^(1/2)*x^(1/2)/a^(1/2),I)/a^(5/2)/b^(3/ 
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 = 8.17 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.32 \[ \int \frac {\sqrt {x}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=\frac {2 x^{3/2} \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{2},\frac {7}{4},\frac {b^2 x^2}{a^2}\right )}{3 a^4 c^2 \sqrt {c (a-b x)} \sqrt {a+b x}} \] Input:

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

Output:

(2*x^(3/2)*Sqrt[1 - (b^2*x^2)/a^2]*Hypergeometric2F1[3/4, 5/2, 7/4, (b^2*x 
^2)/a^2])/(3*a^4*c^2*Sqrt[c*(a - b*x)]*Sqrt[a + b*x])
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {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.

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

\(\Big \downarrow \) 110

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 169

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 169

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 110

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 124

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 123

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

Input:

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

Output:

-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]*Sqr 
t[a])], 2])/(Sqrt[a]*c*Sqrt[-((b*x)/a)]*Sqrt[a*c - b*c*x]))/(2*a*c)))/a)/( 
6*a*b)
 

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 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 = 2.41 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.05

method	result	size
elliptic	\(\frac {\sqrt {c x \left (-b^{2} x^{2}+a^{2}\right )}\, \left (\frac {x \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{3 a^{2} c^{3} b^{4} \left (x^{2}-\frac {a^{2}}{b^{2}}\right )^{2}}+\frac {x^{2}}{2 c^{2} a^{4} \sqrt {-\left (x^{2}-\frac {a^{2}}{b^{2}}\right ) b^{2} c x}}-\frac {\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \left (-\frac {2 a \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{4 a^{3} c^{2} b \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{\sqrt {x}\, \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}}\)	\(236\)
default	\(-\frac {\left (-3 \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{2} b^{2} x^{2}+6 \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{2} b^{2} x^{2}+3 \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{4}-6 \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{4}+6 b^{4} x^{4}-10 a^{2} b^{2} x^{2}\right ) \sqrt {c \left (-b x +a \right )}}{12 c^{3} \sqrt {x}\, \left (-b x +a \right )^{2} a^{4} b^{2} \left (b x +a \right )^{\frac {3}{2}}}\)	\(300\)

Input:

int(x^(1/2)/(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 
^2/c^3*x/b^4*(-b^2*c*x^3+a^2*c*x)^(1/2)/(x^2-a^2/b^2)^2+1/2/c^2*x^2/a^4/(- 
(x^2-a^2/b^2)*b^2*c*x)^(1/2)-1/4/a^3/c^2/b*((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 = 0.09 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {x}}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx=-\frac {{\left (3 \, b^{4} x^{3} - 5 \, a^{2} b^{2} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a} \sqrt {x} + 3 \, {\left (b^{4} x^{4} - 2 \, a^{2} b^{2} x^{2} + a^{4}\right )} \sqrt {-b^{2} c} {\rm weierstrassZeta}\left (\frac {4 \, a^{2}}{b^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right )\right )}{6 \, {\left (a^{4} b^{6} c^{3} x^{4} - 2 \, a^{6} b^{4} c^{3} x^{2} + a^{8} b^{2} c^{3}\right )}} \] Input:

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

Output:

-1/6*((3*b^4*x^3 - 5*a^2*b^2*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*sqrt(x) + 
 3*(b^4*x^4 - 2*a^2*b^2*x^2 + a^4)*sqrt(-b^2*c)*weierstrassZeta(4*a^2/b^2, 
 0, weierstrassPInverse(4*a^2/b^2, 0, x)))/(a^4*b^6*c^3*x^4 - 2*a^6*b^4*c^ 
3*x^2 + a^8*b^2*c^3)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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