\(\int \frac {(e x)^{5/2}}{(c+d x)^{5/2} \sqrt {b c^2-b d^2 x^2}} \, dx\) [1027]

Optimal result

Integrand size = 36, antiderivative size = 245 \[ \int \frac {(e x)^{5/2}}{(c+d x)^{5/2} \sqrt {b c^2-b d^2 x^2}} \, dx=\frac {e (e x)^{3/2} \sqrt {b c^2-b d^2 x^2}}{4 b d^2 (c+d x)^{5/2}}+\frac {11 e^2 \sqrt {e x} \sqrt {b c^2-b d^2 x^2}}{16 b d^3 (c+d x)^{3/2}}+\frac {2 e^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {d} \sqrt {e x} \sqrt {c+d x}}{\sqrt {e} \sqrt {b c^2-b d^2 x^2}}\right )}{\sqrt {b} d^{7/2}}-\frac {43 e^{5/2} \arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {d} \sqrt {e x} \sqrt {c+d x}}{\sqrt {e} \sqrt {b c^2-b d^2 x^2}}\right )}{16 \sqrt {2} \sqrt {b} d^{7/2}} \] Output:

1/4*e*(e*x)^(3/2)*(-b*d^2*x^2+b*c^2)^(1/2)/b/d^2/(d*x+c)^(5/2)+11/16*e^2*( 
e*x)^(1/2)*(-b*d^2*x^2+b*c^2)^(1/2)/b/d^3/(d*x+c)^(3/2)+2*e^(5/2)*arctan(1 
/e^(1/2)/(-b*d^2*x^2+b*c^2)^(1/2)*b^(1/2)*d^(1/2)*(e*x)^(1/2)*(d*x+c)^(1/2 
))/b^(1/2)/d^(7/2)-43/32*e^(5/2)*arctan(1/e^(1/2)/(-b*d^2*x^2+b*c^2)^(1/2) 
*2^(1/2)*b^(1/2)*d^(1/2)*(e*x)^(1/2)*(d*x+c)^(1/2))*2^(1/2)/b^(1/2)/d^(7/2 
)
 

Mathematica [A] (verified)

Time = 7.18 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.82 \[ \int \frac {(e x)^{5/2}}{(c+d x)^{5/2} \sqrt {b c^2-b d^2 x^2}} \, dx=-\frac {(e x)^{5/2} \left (-64 (c-d x) (c+d x)^2 \arcsin \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )+\sqrt {c} \sqrt {1-\frac {d x}{c}} \left (2 \sqrt {d} \sqrt {x} \left (-11 c^2-4 c d x+15 d^2 x^2\right )+43 \sqrt {2} \sqrt {c-d x} (c+d x)^2 \arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {x}}{\sqrt {c-d x}}\right )\right )\right )}{32 \sqrt {c} d^{7/2} x^{5/2} (c+d x)^{3/2} \sqrt {1-\frac {d x}{c}} \sqrt {b \left (c^2-d^2 x^2\right )}} \] Input:

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

Output:

-1/32*((e*x)^(5/2)*(-64*(c - d*x)*(c + d*x)^2*ArcSin[(Sqrt[d]*Sqrt[x])/Sqr 
t[c]] + Sqrt[c]*Sqrt[1 - (d*x)/c]*(2*Sqrt[d]*Sqrt[x]*(-11*c^2 - 4*c*d*x + 
15*d^2*x^2) + 43*Sqrt[2]*Sqrt[c - d*x]*(c + d*x)^2*ArcTan[(Sqrt[2]*Sqrt[d] 
*Sqrt[x])/Sqrt[c - d*x]])))/(Sqrt[c]*d^(7/2)*x^(5/2)*(c + d*x)^(3/2)*Sqrt[ 
1 - (d*x)/c]*Sqrt[b*(c^2 - d^2*x^2)])
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {586, 109, 27, 166, 27, 175, 65, 104, 218}

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.

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

\(\Big \downarrow \) 586

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

\(\Big \downarrow \) 109

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {e (e x)^{3/2} \sqrt {b c-b d x}}{4 b d^2 (c+d x)^2}-\frac {\int \frac {b c e^2 \sqrt {e x} (3 c-8 d x)}{2 (c+d x)^2 \sqrt {b c-b d x}}dx}{4 b c d^2}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {e (e x)^{3/2} \sqrt {b c-b d x}}{4 b d^2 (c+d x)^2}-\frac {e^2 \int \frac {\sqrt {e x} (3 c-8 d x)}{(c+d x)^2 \sqrt {b c-b d x}}dx}{8 d^2}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {e (e x)^{3/2} \sqrt {b c-b d x}}{4 b d^2 (c+d x)^2}-\frac {e^2 \left (\frac {\int \frac {b c d e (11 c-32 d x)}{2 \sqrt {e x} (c+d x) \sqrt {b c-b d x}}dx}{2 b c d^2}-\frac {11 \sqrt {e x} \sqrt {b c-b d x}}{2 b d (c+d x)}\right )}{8 d^2}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {e (e x)^{3/2} \sqrt {b c-b d x}}{4 b d^2 (c+d x)^2}-\frac {e^2 \left (\frac {e \int \frac {11 c-32 d x}{\sqrt {e x} (c+d x) \sqrt {b c-b d x}}dx}{4 d}-\frac {11 \sqrt {e x} \sqrt {b c-b d x}}{2 b d (c+d x)}\right )}{8 d^2}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 175

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {e (e x)^{3/2} \sqrt {b c-b d x}}{4 b d^2 (c+d x)^2}-\frac {e^2 \left (\frac {e \left (43 c \int \frac {1}{\sqrt {e x} (c+d x) \sqrt {b c-b d x}}dx-32 \int \frac {1}{\sqrt {e x} \sqrt {b c-b d x}}dx\right )}{4 d}-\frac {11 \sqrt {e x} \sqrt {b c-b d x}}{2 b d (c+d x)}\right )}{8 d^2}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 65

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {e (e x)^{3/2} \sqrt {b c-b d x}}{4 b d^2 (c+d x)^2}-\frac {e^2 \left (\frac {e \left (43 c \int \frac {1}{\sqrt {e x} (c+d x) \sqrt {b c-b d x}}dx-64 \int \frac {1}{\frac {b d x e}{b c-b d x}+e}d\frac {\sqrt {e x}}{\sqrt {b c-b d x}}\right )}{4 d}-\frac {11 \sqrt {e x} \sqrt {b c-b d x}}{2 b d (c+d x)}\right )}{8 d^2}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {e (e x)^{3/2} \sqrt {b c-b d x}}{4 b d^2 (c+d x)^2}-\frac {e^2 \left (\frac {e \left (86 c \int \frac {1}{c e+\frac {2 b c d x e}{b c-b d x}}d\frac {\sqrt {e x}}{\sqrt {b c-b d x}}-64 \int \frac {1}{\frac {b d x e}{b c-b d x}+e}d\frac {\sqrt {e x}}{\sqrt {b c-b d x}}\right )}{4 d}-\frac {11 \sqrt {e x} \sqrt {b c-b d x}}{2 b d (c+d x)}\right )}{8 d^2}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {e (e x)^{3/2} \sqrt {b c-b d x}}{4 b d^2 (c+d x)^2}-\frac {e^2 \left (\frac {e \left (\frac {43 \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {b c-b d x}}\right )}{\sqrt {b} \sqrt {d} \sqrt {e}}-\frac {64 \arctan \left (\frac {\sqrt {b} \sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {b c-b d x}}\right )}{\sqrt {b} \sqrt {d} \sqrt {e}}\right )}{4 d}-\frac {11 \sqrt {e x} \sqrt {b c-b d x}}{2 b d (c+d x)}\right )}{8 d^2}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

Input:

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

Output:

(Sqrt[c + d*x]*Sqrt[b*c - b*d*x]*((e*(e*x)^(3/2)*Sqrt[b*c - b*d*x])/(4*b*d 
^2*(c + d*x)^2) - (e^2*((-11*Sqrt[e*x]*Sqrt[b*c - b*d*x])/(2*b*d*(c + d*x) 
) + (e*((-64*ArcTan[(Sqrt[b]*Sqrt[d]*Sqrt[e*x])/(Sqrt[e]*Sqrt[b*c - b*d*x] 
)])/(Sqrt[b]*Sqrt[d]*Sqrt[e]) + (43*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[b]*Sqrt[d 
]*Sqrt[e*x])/(Sqrt[e]*Sqrt[b*c - b*d*x])])/(Sqrt[b]*Sqrt[d]*Sqrt[e])))/(4* 
d)))/(8*d^2)))/Sqrt[b*c^2 - b*d^2*x^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 65

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2   Sub 
st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d 
}, x] &&  !GtQ[c, 0]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && 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 166

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

rule 175

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

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 586

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_) 
, x_Symbol] :> Simp[(a + b*x^2)^FracPart[p]/((c + d*x)^FracPart[p]*(a/c + ( 
b*x)/d)^FracPart[p])   Int[(e*x)^m*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(518\) vs. \(2(187)=374\).

Time = 0.27 (sec) , antiderivative size = 519, normalized size of antiderivative = 2.12

method	result	size
default	\(-\frac {e^{2} \sqrt {e x}\, \sqrt {b \left (-d^{2} x^{2}+c^{2}\right )}\, \left (32 \sqrt {2}\, \arctan \left (\frac {\sqrt {b d e}\, \left (-2 d x +c \right )}{2 d \sqrt {\left (-d x +c \right ) b e x}}\right ) b \,d^{3} e \,x^{2} \sqrt {-\frac {b \,c^{2} e}{d}}+64 \sqrt {2}\, \arctan \left (\frac {\sqrt {b d e}\, \left (-2 d x +c \right )}{2 d \sqrt {\left (-d x +c \right ) b e x}}\right ) b c \,d^{2} e x \sqrt {-\frac {b \,c^{2} e}{d}}-43 \ln \left (\frac {3 b c d e x -b \,c^{2} e +2 \sqrt {2}\, \sqrt {-\frac {b \,c^{2} e}{d}}\, \sqrt {\left (-d x +c \right ) b e x}\, d}{d x +c}\right ) b c \,d^{2} e \,x^{2} \sqrt {b d e}+32 \arctan \left (\frac {\sqrt {b d e}\, \left (-2 d x +c \right )}{2 d \sqrt {\left (-d x +c \right ) b e x}}\right ) \sqrt {2}\, \sqrt {-\frac {b \,c^{2} e}{d}}\, b \,c^{2} d e -86 \ln \left (\frac {3 b c d e x -b \,c^{2} e +2 \sqrt {2}\, \sqrt {-\frac {b \,c^{2} e}{d}}\, \sqrt {\left (-d x +c \right ) b e x}\, d}{d x +c}\right ) b \,c^{2} d e x \sqrt {b d e}-30 \sqrt {2}\, \sqrt {-\frac {b \,c^{2} e}{d}}\, \sqrt {b d e}\, \sqrt {\left (-d x +c \right ) b e x}\, d^{2} x -43 \ln \left (\frac {3 b c d e x -b \,c^{2} e +2 \sqrt {2}\, \sqrt {-\frac {b \,c^{2} e}{d}}\, \sqrt {\left (-d x +c \right ) b e x}\, d}{d x +c}\right ) \sqrt {b d e}\, b \,c^{3} e -22 \sqrt {\left (-d x +c \right ) b e x}\, \sqrt {2}\, \sqrt {-\frac {b \,c^{2} e}{d}}\, \sqrt {b d e}\, c d \right ) \sqrt {2}}{64 b \left (d x +c \right )^{\frac {5}{2}} d^{4} \sqrt {\left (-d x +c \right ) b e x}\, \sqrt {b d e}\, \sqrt {-\frac {b \,c^{2} e}{d}}}\)	\(519\)

Input:

int((e*x)^(5/2)/(d*x+c)^(5/2)/(-b*d^2*x^2+b*c^2)^(1/2),x,method=_RETURNVER 
BOSE)
 

Output:

-1/64*e^2*(e*x)^(1/2)*(b*(-d^2*x^2+c^2))^(1/2)/b*(32*2^(1/2)*arctan(1/2*(b 
*d*e)^(1/2)*(-2*d*x+c)/d/((-d*x+c)*b*e*x)^(1/2))*b*d^3*e*x^2*(-1/d*b*c^2*e 
)^(1/2)+64*2^(1/2)*arctan(1/2*(b*d*e)^(1/2)*(-2*d*x+c)/d/((-d*x+c)*b*e*x)^ 
(1/2))*b*c*d^2*e*x*(-1/d*b*c^2*e)^(1/2)-43*ln((3*b*c*d*e*x-b*c^2*e+2*2^(1/ 
2)*(-1/d*b*c^2*e)^(1/2)*((-d*x+c)*b*e*x)^(1/2)*d)/(d*x+c))*b*c*d^2*e*x^2*( 
b*d*e)^(1/2)+32*arctan(1/2*(b*d*e)^(1/2)*(-2*d*x+c)/d/((-d*x+c)*b*e*x)^(1/ 
2))*2^(1/2)*(-1/d*b*c^2*e)^(1/2)*b*c^2*d*e-86*ln((3*b*c*d*e*x-b*c^2*e+2*2^ 
(1/2)*(-1/d*b*c^2*e)^(1/2)*((-d*x+c)*b*e*x)^(1/2)*d)/(d*x+c))*b*c^2*d*e*x* 
(b*d*e)^(1/2)-30*2^(1/2)*(-1/d*b*c^2*e)^(1/2)*(b*d*e)^(1/2)*((-d*x+c)*b*e* 
x)^(1/2)*d^2*x-43*ln((3*b*c*d*e*x-b*c^2*e+2*2^(1/2)*(-1/d*b*c^2*e)^(1/2)*( 
(-d*x+c)*b*e*x)^(1/2)*d)/(d*x+c))*(b*d*e)^(1/2)*b*c^3*e-22*((-d*x+c)*b*e*x 
)^(1/2)*2^(1/2)*(-1/d*b*c^2*e)^(1/2)*(b*d*e)^(1/2)*c*d)*2^(1/2)/(d*x+c)^(5 
/2)/d^4/((-d*x+c)*b*e*x)^(1/2)/(b*d*e)^(1/2)/(-1/d*b*c^2*e)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 750, normalized size of antiderivative = 3.06 \[ \int \frac {(e x)^{5/2}}{(c+d x)^{5/2} \sqrt {b c^2-b d^2 x^2}} \, dx=\left [\frac {43 \, \sqrt {\frac {1}{2}} {\left (b d^{3} e^{2} x^{3} + 3 \, b c d^{2} e^{2} x^{2} + 3 \, b c^{2} d e^{2} x + b c^{3} e^{2}\right )} \sqrt {-\frac {e}{b d}} \log \left (-\frac {17 \, d^{3} e x^{3} + 3 \, c d^{2} e x^{2} - 13 \, c^{2} d e x + c^{3} e - 8 \, \sqrt {\frac {1}{2}} \sqrt {-b d^{2} x^{2} + b c^{2}} {\left (3 \, d^{2} x - c d\right )} \sqrt {d x + c} \sqrt {e x} \sqrt {-\frac {e}{b d}}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) + 4 \, \sqrt {-b d^{2} x^{2} + b c^{2}} {\left (15 \, d e^{2} x + 11 \, c e^{2}\right )} \sqrt {d x + c} \sqrt {e x} + 32 \, {\left (b d^{3} e^{2} x^{3} + 3 \, b c d^{2} e^{2} x^{2} + 3 \, b c^{2} d e^{2} x + b c^{3} e^{2}\right )} \sqrt {-\frac {e}{b d}} \log \left (-\frac {8 \, d^{3} e x^{3} - 7 \, c^{2} d e x + c^{3} e + 4 \, \sqrt {-b d^{2} x^{2} + b c^{2}} {\left (2 \, d^{2} x - c d\right )} \sqrt {d x + c} \sqrt {e x} \sqrt {-\frac {e}{b d}}}{d x + c}\right )}{64 \, {\left (b d^{6} x^{3} + 3 \, b c d^{5} x^{2} + 3 \, b c^{2} d^{4} x + b c^{3} d^{3}\right )}}, \frac {43 \, \sqrt {\frac {1}{2}} {\left (b d^{3} e^{2} x^{3} + 3 \, b c d^{2} e^{2} x^{2} + 3 \, b c^{2} d e^{2} x + b c^{3} e^{2}\right )} \sqrt {\frac {e}{b d}} \arctan \left (\frac {4 \, \sqrt {\frac {1}{2}} \sqrt {-b d^{2} x^{2} + b c^{2}} \sqrt {d x + c} \sqrt {e x} d \sqrt {\frac {e}{b d}}}{3 \, d^{2} e x^{2} + 2 \, c d e x - c^{2} e}\right ) + 2 \, \sqrt {-b d^{2} x^{2} + b c^{2}} {\left (15 \, d e^{2} x + 11 \, c e^{2}\right )} \sqrt {d x + c} \sqrt {e x} - 32 \, {\left (b d^{3} e^{2} x^{3} + 3 \, b c d^{2} e^{2} x^{2} + 3 \, b c^{2} d e^{2} x + b c^{3} e^{2}\right )} \sqrt {\frac {e}{b d}} \arctan \left (\frac {2 \, \sqrt {-b d^{2} x^{2} + b c^{2}} \sqrt {d x + c} \sqrt {e x} d \sqrt {\frac {e}{b d}}}{2 \, d^{2} e x^{2} + c d e x - c^{2} e}\right )}{32 \, {\left (b d^{6} x^{3} + 3 \, b c d^{5} x^{2} + 3 \, b c^{2} d^{4} x + b c^{3} d^{3}\right )}}\right ] \] Input:

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

Output:

[1/64*(43*sqrt(1/2)*(b*d^3*e^2*x^3 + 3*b*c*d^2*e^2*x^2 + 3*b*c^2*d*e^2*x + 
 b*c^3*e^2)*sqrt(-e/(b*d))*log(-(17*d^3*e*x^3 + 3*c*d^2*e*x^2 - 13*c^2*d*e 
*x + c^3*e - 8*sqrt(1/2)*sqrt(-b*d^2*x^2 + b*c^2)*(3*d^2*x - c*d)*sqrt(d*x 
 + c)*sqrt(e*x)*sqrt(-e/(b*d)))/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) 
 + 4*sqrt(-b*d^2*x^2 + b*c^2)*(15*d*e^2*x + 11*c*e^2)*sqrt(d*x + c)*sqrt(e 
*x) + 32*(b*d^3*e^2*x^3 + 3*b*c*d^2*e^2*x^2 + 3*b*c^2*d*e^2*x + b*c^3*e^2) 
*sqrt(-e/(b*d))*log(-(8*d^3*e*x^3 - 7*c^2*d*e*x + c^3*e + 4*sqrt(-b*d^2*x^ 
2 + b*c^2)*(2*d^2*x - c*d)*sqrt(d*x + c)*sqrt(e*x)*sqrt(-e/(b*d)))/(d*x + 
c)))/(b*d^6*x^3 + 3*b*c*d^5*x^2 + 3*b*c^2*d^4*x + b*c^3*d^3), 1/32*(43*sqr 
t(1/2)*(b*d^3*e^2*x^3 + 3*b*c*d^2*e^2*x^2 + 3*b*c^2*d*e^2*x + b*c^3*e^2)*s 
qrt(e/(b*d))*arctan(4*sqrt(1/2)*sqrt(-b*d^2*x^2 + b*c^2)*sqrt(d*x + c)*sqr 
t(e*x)*d*sqrt(e/(b*d))/(3*d^2*e*x^2 + 2*c*d*e*x - c^2*e)) + 2*sqrt(-b*d^2* 
x^2 + b*c^2)*(15*d*e^2*x + 11*c*e^2)*sqrt(d*x + c)*sqrt(e*x) - 32*(b*d^3*e 
^2*x^3 + 3*b*c*d^2*e^2*x^2 + 3*b*c^2*d*e^2*x + b*c^3*e^2)*sqrt(e/(b*d))*ar 
ctan(2*sqrt(-b*d^2*x^2 + b*c^2)*sqrt(d*x + c)*sqrt(e*x)*d*sqrt(e/(b*d))/(2 
*d^2*e*x^2 + c*d*e*x - c^2*e)))/(b*d^6*x^3 + 3*b*c*d^5*x^2 + 3*b*c^2*d^4*x 
 + b*c^3*d^3)]
 

Sympy [F]

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

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

Output:

Integral((e*x)**(5/2)/(sqrt(-b*(-c + d*x)*(c + d*x))*(c + d*x)**(5/2)), x)
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 481 vs. \(2 (187) = 374\).

Time = 0.36 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.96 \[ \int \frac {(e x)^{5/2}}{(c+d x)^{5/2} \sqrt {b c^2-b d^2 x^2}} \, dx=\frac {1}{64} \, e^{2} {\left (\frac {43 \, \sqrt {2} \sqrt {-b d e} c e \log \left (\frac {{\left | -4 \, \sqrt {2} e^{2} {\left | b \right |} {\left | c \right |} - 6 \, b c e^{2} + 2 \, {\left (\sqrt {-b d e} \sqrt {e x} - \sqrt {-b d e^{2} x + b c e^{2}}\right )}^{2} \right |}}{{\left | 4 \, \sqrt {2} e^{2} {\left | b \right |} {\left | c \right |} - 6 \, b c e^{2} + 2 \, {\left (\sqrt {-b d e} \sqrt {e x} - \sqrt {-b d e^{2} x + b c e^{2}}\right )}^{2} \right |}}\right )}{d^{4} {\left | b \right |} {\left | c \right |} {\left | e \right |}} + \frac {64 \, \sqrt {-b d e} e \log \left ({\left (\sqrt {-b d e} \sqrt {e x} - \sqrt {-b d e^{2} x + b c e^{2}}\right )}^{2}\right )}{b d^{4} {\left | e \right |}} + \frac {8 \, {\left (15 \, \sqrt {-b d e} b^{3} c^{4} e^{9} - 127 \, \sqrt {-b d e} {\left (\sqrt {-b d e} \sqrt {e x} - \sqrt {-b d e^{2} x + b c e^{2}}\right )}^{2} b^{2} c^{3} e^{7} + 285 \, \sqrt {-b d e} {\left (\sqrt {-b d e} \sqrt {e x} - \sqrt {-b d e^{2} x + b c e^{2}}\right )}^{4} b c^{2} e^{5} - 53 \, \sqrt {-b d e} {\left (\sqrt {-b d e} \sqrt {e x} - \sqrt {-b d e^{2} x + b c e^{2}}\right )}^{6} c e^{3}\right )}}{{\left (b^{2} c^{2} e^{4} - 6 \, {\left (\sqrt {-b d e} \sqrt {e x} - \sqrt {-b d e^{2} x + b c e^{2}}\right )}^{2} b c e^{2} + {\left (\sqrt {-b d e} \sqrt {e x} - \sqrt {-b d e^{2} x + b c e^{2}}\right )}^{4}\right )}^{2} d^{4} {\left | e \right |}}\right )} \] Input:

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

Output:

1/64*e^2*(43*sqrt(2)*sqrt(-b*d*e)*c*e*log(abs(-4*sqrt(2)*e^2*abs(b)*abs(c) 
 - 6*b*c*e^2 + 2*(sqrt(-b*d*e)*sqrt(e*x) - sqrt(-b*d*e^2*x + b*c*e^2))^2)/ 
abs(4*sqrt(2)*e^2*abs(b)*abs(c) - 6*b*c*e^2 + 2*(sqrt(-b*d*e)*sqrt(e*x) - 
sqrt(-b*d*e^2*x + b*c*e^2))^2))/(d^4*abs(b)*abs(c)*abs(e)) + 64*sqrt(-b*d* 
e)*e*log((sqrt(-b*d*e)*sqrt(e*x) - sqrt(-b*d*e^2*x + b*c*e^2))^2)/(b*d^4*a 
bs(e)) + 8*(15*sqrt(-b*d*e)*b^3*c^4*e^9 - 127*sqrt(-b*d*e)*(sqrt(-b*d*e)*s 
qrt(e*x) - sqrt(-b*d*e^2*x + b*c*e^2))^2*b^2*c^3*e^7 + 285*sqrt(-b*d*e)*(s 
qrt(-b*d*e)*sqrt(e*x) - sqrt(-b*d*e^2*x + b*c*e^2))^4*b*c^2*e^5 - 53*sqrt( 
-b*d*e)*(sqrt(-b*d*e)*sqrt(e*x) - sqrt(-b*d*e^2*x + b*c*e^2))^6*c*e^3)/((b 
^2*c^2*e^4 - 6*(sqrt(-b*d*e)*sqrt(e*x) - sqrt(-b*d*e^2*x + b*c*e^2))^2*b*c 
*e^2 + (sqrt(-b*d*e)*sqrt(e*x) - sqrt(-b*d*e^2*x + b*c*e^2))^4)^2*d^4*abs( 
e)))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 650, normalized size of antiderivative = 2.65 \[ \int \frac {(e x)^{5/2}}{(c+d x)^{5/2} \sqrt {b c^2-b d^2 x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(sqrt(e)*sqrt(b)*e**2*(132*sqrt(x)*sqrt(c - d*x)*c*d + 180*sqrt(x)*sqrt(c 
- d*x)*d**2*x - 129*sqrt(d)*sqrt(2)*log((sqrt(c - d*x) - sqrt(c)*sqrt(2) - 
 sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))*c**2*i - 258*sqrt(d)*sqrt(2)*log((s 
qrt(c - d*x) - sqrt(c)*sqrt(2) - sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))*c*d 
*i*x - 129*sqrt(d)*sqrt(2)*log((sqrt(c - d*x) - sqrt(c)*sqrt(2) - sqrt(c) 
+ sqrt(x)*sqrt(d)*i)/sqrt(c))*d**2*i*x**2 + 129*sqrt(d)*sqrt(2)*log((sqrt( 
c - d*x) - sqrt(c)*sqrt(2) + sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))*c**2*i 
+ 258*sqrt(d)*sqrt(2)*log((sqrt(c - d*x) - sqrt(c)*sqrt(2) + sqrt(c) + sqr 
t(x)*sqrt(d)*i)/sqrt(c))*c*d*i*x + 129*sqrt(d)*sqrt(2)*log((sqrt(c - d*x) 
- sqrt(c)*sqrt(2) + sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))*d**2*i*x**2 + 12 
9*sqrt(d)*sqrt(2)*log((sqrt(c - d*x) + sqrt(c)*sqrt(2) - sqrt(c) + sqrt(x) 
*sqrt(d)*i)/sqrt(c))*c**2*i + 258*sqrt(d)*sqrt(2)*log((sqrt(c - d*x) + sqr 
t(c)*sqrt(2) - sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))*c*d*i*x + 129*sqrt(d) 
*sqrt(2)*log((sqrt(c - d*x) + sqrt(c)*sqrt(2) - sqrt(c) + sqrt(x)*sqrt(d)* 
i)/sqrt(c))*d**2*i*x**2 - 129*sqrt(d)*sqrt(2)*log((sqrt(c - d*x) + sqrt(c) 
*sqrt(2) + sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))*c**2*i - 258*sqrt(d)*sqrt 
(2)*log((sqrt(c - d*x) + sqrt(c)*sqrt(2) + sqrt(c) + sqrt(x)*sqrt(d)*i)/sq 
rt(c))*c*d*i*x - 129*sqrt(d)*sqrt(2)*log((sqrt(c - d*x) + sqrt(c)*sqrt(2) 
+ sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))*d**2*i*x**2 - 384*sqrt(d)*log((sqr 
t(c - d*x) + sqrt(x)*sqrt(d)*i)/sqrt(c))*c**2*i - 768*sqrt(d)*log((sqrt...