\(\int \frac {(e x)^{9/2}}{(c+d x)^{5/2} (c^2-d^2 x^2)^{3/2}} \, dx\) [1079]

Optimal result

Integrand size = 33, antiderivative size = 288 \[ \int \frac {(e x)^{9/2}}{(c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {e (e x)^{7/2}}{d^2 (c+d x)^{5/2} \sqrt {c^2-d^2 x^2}}+\frac {5 e^2 (e x)^{5/2} \sqrt {c^2-d^2 x^2}}{12 d^3 (c+d x)^{7/2}}+\frac {e^3 (e x)^{3/2} \sqrt {c^2-d^2 x^2}}{96 d^4 (c+d x)^{5/2}}-\frac {63 e^4 \sqrt {e x} \sqrt {c^2-d^2 x^2}}{128 d^5 (c+d x)^{3/2}}-\frac {2 e^{9/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x} \sqrt {c+d x}}{\sqrt {e} \sqrt {c^2-d^2 x^2}}\right )}{d^{11/2}}+\frac {319 e^{9/2} \arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {e x} \sqrt {c+d x}}{\sqrt {e} \sqrt {c^2-d^2 x^2}}\right )}{128 \sqrt {2} d^{11/2}} \] Output:

e*(e*x)^(7/2)/d^2/(d*x+c)^(5/2)/(-d^2*x^2+c^2)^(1/2)+5/12*e^2*(e*x)^(5/2)* 
(-d^2*x^2+c^2)^(1/2)/d^3/(d*x+c)^(7/2)+1/96*e^3*(e*x)^(3/2)*(-d^2*x^2+c^2) 
^(1/2)/d^4/(d*x+c)^(5/2)-63/128*e^4*(e*x)^(1/2)*(-d^2*x^2+c^2)^(1/2)/d^5/( 
d*x+c)^(3/2)-2*e^(9/2)*arctan(1/e^(1/2)/(-d^2*x^2+c^2)^(1/2)*d^(1/2)*(e*x) 
^(1/2)*(d*x+c)^(1/2))/d^(11/2)+319/256*e^(9/2)*arctan(1/e^(1/2)/(-d^2*x^2+ 
c^2)^(1/2)*2^(1/2)*d^(1/2)*(e*x)^(1/2)*(d*x+c)^(1/2))*2^(1/2)/d^(11/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 11.33 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.33 \[ \int \frac {(e x)^{9/2}}{(c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {e^4 \sqrt {e x} \left (13104 c^2 d^{5/2} x^{5/2} (-c+d x) (c+d x)^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {5}{2},\frac {7}{2},\frac {d x}{c}\right )-9360 c d^{7/2} x^{7/2} (-c+d x) (c+d x)^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {7}{2},\frac {9}{2},\frac {d x}{c}\right )-35 \left (58689 c^{9/2} (c-d x) (c+d x)^3 \arcsin \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )+3 c^2 \sqrt {1-\frac {d x}{c}} \left (\sqrt {d} \sqrt {x} \left (3405 c^6+1424 c^5 d x-10612 c^4 d^2 x^2-9634 c^3 d^3 x^3+767 c^2 d^4 x^4+338 c d^5 x^5+104 d^6 x^6\right )-11484 \sqrt {2} c^3 \sqrt {c-d x} (c+d x)^3 \arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {x}}{\sqrt {c-d x}}\right )\right )-208 d^{9/2} x^{9/2} (-c+d x) (c+d x)^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {9}{2},\frac {11}{2},\frac {d x}{c}\right )\right )\right )}{967680 c^5 d^{11/2} \sqrt {x} (c+d x)^{5/2} \sqrt {1-\frac {d x}{c}} \sqrt {c^2-d^2 x^2}} \] Input:

Integrate[(e*x)^(9/2)/((c + d*x)^(5/2)*(c^2 - d^2*x^2)^(3/2)),x]
 

Output:

(e^4*Sqrt[e*x]*(13104*c^2*d^(5/2)*x^(5/2)*(-c + d*x)*(c + d*x)^3*Hypergeom 
etric2F1[3/2, 5/2, 7/2, (d*x)/c] - 9360*c*d^(7/2)*x^(7/2)*(-c + d*x)*(c + 
d*x)^3*Hypergeometric2F1[3/2, 7/2, 9/2, (d*x)/c] - 35*(58689*c^(9/2)*(c - 
d*x)*(c + d*x)^3*ArcSin[(Sqrt[d]*Sqrt[x])/Sqrt[c]] + 3*c^2*Sqrt[1 - (d*x)/ 
c]*(Sqrt[d]*Sqrt[x]*(3405*c^6 + 1424*c^5*d*x - 10612*c^4*d^2*x^2 - 9634*c^ 
3*d^3*x^3 + 767*c^2*d^4*x^4 + 338*c*d^5*x^5 + 104*d^6*x^6) - 11484*Sqrt[2] 
*c^3*Sqrt[c - d*x]*(c + d*x)^3*ArcTan[(Sqrt[2]*Sqrt[d]*Sqrt[x])/Sqrt[c - d 
*x]]) - 208*d^(9/2)*x^(9/2)*(-c + d*x)*(c + d*x)^3*Hypergeometric2F1[3/2, 
9/2, 11/2, (d*x)/c])))/(967680*c^5*d^(11/2)*Sqrt[x]*(c + d*x)^(5/2)*Sqrt[1 
 - (d*x)/c]*Sqrt[c^2 - d^2*x^2])
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.394, Rules used = {586, 109, 27, 166, 27, 166, 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.

\(\displaystyle \int \frac {(e x)^{9/2}}{(c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 586

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \int \frac {(e x)^{9/2}}{(c-d x)^{3/2} (c+d x)^4}dx}{\sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 166

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {e (e x)^{7/2}}{d^2 \sqrt {c-d x} (c+d x)^3}-\frac {e^2 \left (\frac {\int \frac {c d e (e x)^{3/2} (25 c+24 d x)}{2 \sqrt {c-d x} (c+d x)^3}dx}{6 c d^2}-\frac {5 (e x)^{5/2} \sqrt {c-d x}}{6 d (c+d x)^3}\right )}{2 d^2}\right )}{\sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {e (e x)^{7/2}}{d^2 \sqrt {c-d x} (c+d x)^3}-\frac {e^2 \left (\frac {e \int \frac {(e x)^{3/2} (25 c+24 d x)}{\sqrt {c-d x} (c+d x)^3}dx}{12 d}-\frac {5 (e x)^{5/2} \sqrt {c-d x}}{6 d (c+d x)^3}\right )}{2 d^2}\right )}{\sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {e (e x)^{7/2}}{d^2 \sqrt {c-d x} (c+d x)^3}-\frac {e^2 \left (\frac {e \left (\frac {\int \frac {3 c d e \sqrt {e x} (c+64 d x)}{2 \sqrt {c-d x} (c+d x)^2}dx}{4 c d^2}-\frac {(e x)^{3/2} \sqrt {c-d x}}{4 d (c+d x)^2}\right )}{12 d}-\frac {5 (e x)^{5/2} \sqrt {c-d x}}{6 d (c+d x)^3}\right )}{2 d^2}\right )}{\sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {e (e x)^{7/2}}{d^2 \sqrt {c-d x} (c+d x)^3}-\frac {e^2 \left (\frac {e \left (\frac {3 e \int \frac {\sqrt {e x} (c+64 d x)}{\sqrt {c-d x} (c+d x)^2}dx}{8 d}-\frac {(e x)^{3/2} \sqrt {c-d x}}{4 d (c+d x)^2}\right )}{12 d}-\frac {5 (e x)^{5/2} \sqrt {c-d x}}{6 d (c+d x)^3}\right )}{2 d^2}\right )}{\sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {e (e x)^{7/2}}{d^2 \sqrt {c-d x} (c+d x)^3}-\frac {e^2 \left (\frac {e \left (\frac {3 e \left (\frac {\int -\frac {c d e (63 c-256 d x)}{2 \sqrt {e x} \sqrt {c-d x} (c+d x)}dx}{2 c d^2}+\frac {63 \sqrt {e x} \sqrt {c-d x}}{2 d (c+d x)}\right )}{8 d}-\frac {(e x)^{3/2} \sqrt {c-d x}}{4 d (c+d x)^2}\right )}{12 d}-\frac {5 (e x)^{5/2} \sqrt {c-d x}}{6 d (c+d x)^3}\right )}{2 d^2}\right )}{\sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {e (e x)^{7/2}}{d^2 \sqrt {c-d x} (c+d x)^3}-\frac {e^2 \left (\frac {e \left (\frac {3 e \left (\frac {63 \sqrt {e x} \sqrt {c-d x}}{2 d (c+d x)}-\frac {e \int \frac {63 c-256 d x}{\sqrt {e x} \sqrt {c-d x} (c+d x)}dx}{4 d}\right )}{8 d}-\frac {(e x)^{3/2} \sqrt {c-d x}}{4 d (c+d x)^2}\right )}{12 d}-\frac {5 (e x)^{5/2} \sqrt {c-d x}}{6 d (c+d x)^3}\right )}{2 d^2}\right )}{\sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 175

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {e (e x)^{7/2}}{d^2 \sqrt {c-d x} (c+d x)^3}-\frac {e^2 \left (\frac {e \left (\frac {3 e \left (\frac {63 \sqrt {e x} \sqrt {c-d x}}{2 d (c+d x)}-\frac {e \left (319 c \int \frac {1}{\sqrt {e x} \sqrt {c-d x} (c+d x)}dx-256 \int \frac {1}{\sqrt {e x} \sqrt {c-d x}}dx\right )}{4 d}\right )}{8 d}-\frac {(e x)^{3/2} \sqrt {c-d x}}{4 d (c+d x)^2}\right )}{12 d}-\frac {5 (e x)^{5/2} \sqrt {c-d x}}{6 d (c+d x)^3}\right )}{2 d^2}\right )}{\sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 65

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {e (e x)^{7/2}}{d^2 \sqrt {c-d x} (c+d x)^3}-\frac {e^2 \left (\frac {e \left (\frac {3 e \left (\frac {63 \sqrt {e x} \sqrt {c-d x}}{2 d (c+d x)}-\frac {e \left (319 c \int \frac {1}{\sqrt {e x} \sqrt {c-d x} (c+d x)}dx-512 \int \frac {1}{\frac {d x e}{c-d x}+e}d\frac {\sqrt {e x}}{\sqrt {c-d x}}\right )}{4 d}\right )}{8 d}-\frac {(e x)^{3/2} \sqrt {c-d x}}{4 d (c+d x)^2}\right )}{12 d}-\frac {5 (e x)^{5/2} \sqrt {c-d x}}{6 d (c+d x)^3}\right )}{2 d^2}\right )}{\sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {e (e x)^{7/2}}{d^2 \sqrt {c-d x} (c+d x)^3}-\frac {e^2 \left (\frac {e \left (\frac {3 e \left (\frac {63 \sqrt {e x} \sqrt {c-d x}}{2 d (c+d x)}-\frac {e \left (638 c \int \frac {1}{c e+\frac {2 c d x e}{c-d x}}d\frac {\sqrt {e x}}{\sqrt {c-d x}}-512 \int \frac {1}{\frac {d x e}{c-d x}+e}d\frac {\sqrt {e x}}{\sqrt {c-d x}}\right )}{4 d}\right )}{8 d}-\frac {(e x)^{3/2} \sqrt {c-d x}}{4 d (c+d x)^2}\right )}{12 d}-\frac {5 (e x)^{5/2} \sqrt {c-d x}}{6 d (c+d x)^3}\right )}{2 d^2}\right )}{\sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {e (e x)^{7/2}}{d^2 \sqrt {c-d x} (c+d x)^3}-\frac {e^2 \left (\frac {e \left (\frac {3 e \left (\frac {63 \sqrt {e x} \sqrt {c-d x}}{2 d (c+d x)}-\frac {e \left (\frac {319 \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c-d x}}\right )}{\sqrt {d} \sqrt {e}}-\frac {512 \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c-d x}}\right )}{\sqrt {d} \sqrt {e}}\right )}{4 d}\right )}{8 d}-\frac {(e x)^{3/2} \sqrt {c-d x}}{4 d (c+d x)^2}\right )}{12 d}-\frac {5 (e x)^{5/2} \sqrt {c-d x}}{6 d (c+d x)^3}\right )}{2 d^2}\right )}{\sqrt {c^2-d^2 x^2}}\)

Input:

Int[(e*x)^(9/2)/((c + d*x)^(5/2)*(c^2 - d^2*x^2)^(3/2)),x]
 

Output:

(Sqrt[c - d*x]*Sqrt[c + d*x]*((e*(e*x)^(7/2))/(d^2*Sqrt[c - d*x]*(c + d*x) 
^3) - (e^2*((-5*(e*x)^(5/2)*Sqrt[c - d*x])/(6*d*(c + d*x)^3) + (e*(-1/4*(( 
e*x)^(3/2)*Sqrt[c - d*x])/(d*(c + d*x)^2) + (3*e*((63*Sqrt[e*x]*Sqrt[c - d 
*x])/(2*d*(c + d*x)) - (e*((-512*ArcTan[(Sqrt[d]*Sqrt[e*x])/(Sqrt[e]*Sqrt[ 
c - d*x])])/(Sqrt[d]*Sqrt[e]) + (319*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[d]*Sqrt[ 
e*x])/(Sqrt[e]*Sqrt[c - d*x])])/(Sqrt[d]*Sqrt[e])))/(4*d)))/(8*d)))/(12*d) 
))/(2*d^2)))/Sqrt[c^2 - 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. \(685\) vs. \(2(224)=448\).

Time = 0.27 (sec) , antiderivative size = 686, normalized size of antiderivative = 2.38

method	result	size
default	\(\frac {\sqrt {2}\, \left (-768 \sqrt {2}\, \arctan \left (\frac {\sqrt {d e}\, \left (-2 d x +c \right )}{2 d \sqrt {\left (-d x +c \right ) e x}}\right ) d^{5} e \,x^{4} \sqrt {-\frac {e \,c^{2}}{d}}-1536 \sqrt {2}\, \arctan \left (\frac {\sqrt {d e}\, \left (-2 d x +c \right )}{2 d \sqrt {\left (-d x +c \right ) e x}}\right ) c \,d^{4} e \,x^{3} \sqrt {-\frac {e \,c^{2}}{d}}+957 \ln \left (\frac {3 c d x e +2 \sqrt {2}\, \sqrt {-\frac {e \,c^{2}}{d}}\, \sqrt {\left (-d x +c \right ) e x}\, d -e \,c^{2}}{d x +c}\right ) c \,d^{4} e \,x^{4} \sqrt {d e}+818 x^{3} \sqrt {\left (-d x +c \right ) e x}\, \sqrt {d e}\, \sqrt {2}\, \sqrt {-\frac {e \,c^{2}}{d}}\, d^{4}+1914 \ln \left (\frac {3 c d x e +2 \sqrt {2}\, \sqrt {-\frac {e \,c^{2}}{d}}\, \sqrt {\left (-d x +c \right ) e x}\, d -e \,c^{2}}{d x +c}\right ) c^{2} d^{3} e \,x^{3} \sqrt {d e}+1536 \sqrt {2}\, \arctan \left (\frac {\sqrt {d e}\, \left (-2 d x +c \right )}{2 d \sqrt {\left (-d x +c \right ) e x}}\right ) c^{3} d^{2} e x \sqrt {-\frac {e \,c^{2}}{d}}+698 x^{2} \sqrt {\left (-d x +c \right ) e x}\, \sqrt {d e}\, \sqrt {2}\, \sqrt {-\frac {e \,c^{2}}{d}}\, c \,d^{3}+768 \sqrt {2}\, \arctan \left (\frac {\sqrt {d e}\, \left (-2 d x +c \right )}{2 d \sqrt {\left (-d x +c \right ) e x}}\right ) c^{4} d e \sqrt {-\frac {e \,c^{2}}{d}}-370 \sqrt {\left (-d x +c \right ) e x}\, \sqrt {2}\, \sqrt {-\frac {e \,c^{2}}{d}}\, \sqrt {d e}\, c^{2} d^{2} x -1914 \ln \left (\frac {3 c d x e +2 \sqrt {2}\, \sqrt {-\frac {e \,c^{2}}{d}}\, \sqrt {\left (-d x +c \right ) e x}\, d -e \,c^{2}}{d x +c}\right ) c^{4} d e x \sqrt {d e}-378 \sqrt {2}\, c^{3} d \sqrt {d e}\, \sqrt {-\frac {e \,c^{2}}{d}}\, \sqrt {\left (-d x +c \right ) e x}-957 \ln \left (\frac {3 c d x e +2 \sqrt {2}\, \sqrt {-\frac {e \,c^{2}}{d}}\, \sqrt {\left (-d x +c \right ) e x}\, d -e \,c^{2}}{d x +c}\right ) c^{5} e \sqrt {d e}\right ) \sqrt {-d^{2} x^{2}+c^{2}}\, \sqrt {e x}\, e^{4}}{1536 \sqrt {-\frac {e \,c^{2}}{d}}\, \sqrt {d e}\, \left (-d x +c \right ) \sqrt {\left (-d x +c \right ) e x}\, d^{6} \left (d x +c \right )^{\frac {7}{2}}}\)	\(686\)

Input:

int((e*x)^(9/2)/(d*x+c)^(5/2)/(-d^2*x^2+c^2)^(3/2),x,method=_RETURNVERBOSE 
)
 

Output:

1/1536*2^(1/2)*(-768*2^(1/2)*arctan(1/2*(d*e)^(1/2)*(-2*d*x+c)/d/((-d*x+c) 
*e*x)^(1/2))*d^5*e*x^4*(-1/d*e*c^2)^(1/2)-1536*2^(1/2)*arctan(1/2*(d*e)^(1 
/2)*(-2*d*x+c)/d/((-d*x+c)*e*x)^(1/2))*c*d^4*e*x^3*(-1/d*e*c^2)^(1/2)+957* 
ln((3*c*d*x*e+2*2^(1/2)*(-1/d*e*c^2)^(1/2)*((-d*x+c)*e*x)^(1/2)*d-e*c^2)/( 
d*x+c))*c*d^4*e*x^4*(d*e)^(1/2)+818*x^3*((-d*x+c)*e*x)^(1/2)*(d*e)^(1/2)*2 
^(1/2)*(-1/d*e*c^2)^(1/2)*d^4+1914*ln((3*c*d*x*e+2*2^(1/2)*(-1/d*e*c^2)^(1 
/2)*((-d*x+c)*e*x)^(1/2)*d-e*c^2)/(d*x+c))*c^2*d^3*e*x^3*(d*e)^(1/2)+1536* 
2^(1/2)*arctan(1/2*(d*e)^(1/2)*(-2*d*x+c)/d/((-d*x+c)*e*x)^(1/2))*c^3*d^2* 
e*x*(-1/d*e*c^2)^(1/2)+698*x^2*((-d*x+c)*e*x)^(1/2)*(d*e)^(1/2)*2^(1/2)*(- 
1/d*e*c^2)^(1/2)*c*d^3+768*2^(1/2)*arctan(1/2*(d*e)^(1/2)*(-2*d*x+c)/d/((- 
d*x+c)*e*x)^(1/2))*c^4*d*e*(-1/d*e*c^2)^(1/2)-370*((-d*x+c)*e*x)^(1/2)*2^( 
1/2)*(-1/d*e*c^2)^(1/2)*(d*e)^(1/2)*c^2*d^2*x-1914*ln((3*c*d*x*e+2*2^(1/2) 
*(-1/d*e*c^2)^(1/2)*((-d*x+c)*e*x)^(1/2)*d-e*c^2)/(d*x+c))*c^4*d*e*x*(d*e) 
^(1/2)-378*2^(1/2)*c^3*d*(d*e)^(1/2)*(-1/d*e*c^2)^(1/2)*((-d*x+c)*e*x)^(1/ 
2)-957*ln((3*c*d*x*e+2*2^(1/2)*(-1/d*e*c^2)^(1/2)*((-d*x+c)*e*x)^(1/2)*d-e 
*c^2)/(d*x+c))*c^5*e*(d*e)^(1/2))*(-d^2*x^2+c^2)^(1/2)*(e*x)^(1/2)*e^4/(-1 
/d*e*c^2)^(1/2)/(d*e)^(1/2)/(-d*x+c)/((-d*x+c)*e*x)^(1/2)/d^6/(d*x+c)^(7/2 
)
 

Fricas [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 902, normalized size of antiderivative = 3.13 \[ \int \frac {(e x)^{9/2}}{(c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

integrate((e*x)^(9/2)/(d*x+c)^(5/2)/(-d^2*x^2+c^2)^(3/2),x, algorithm="fri 
cas")
 

Output:

[1/1536*(957*sqrt(1/2)*(d^5*e^4*x^5 + 3*c*d^4*e^4*x^4 + 2*c^2*d^3*e^4*x^3 
- 2*c^3*d^2*e^4*x^2 - 3*c^4*d*e^4*x - c^5*e^4)*sqrt(-e/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(-d^2*x^2 + c^ 
2)*(3*d^2*x - c*d)*sqrt(d*x + c)*sqrt(e*x)*sqrt(-e/d))/(d^3*x^3 + 3*c*d^2* 
x^2 + 3*c^2*d*x + c^3)) - 4*(409*d^3*e^4*x^3 + 349*c*d^2*e^4*x^2 - 185*c^2 
*d*e^4*x - 189*c^3*e^4)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c)*sqrt(e*x) + 768 
*(d^5*e^4*x^5 + 3*c*d^4*e^4*x^4 + 2*c^2*d^3*e^4*x^3 - 2*c^3*d^2*e^4*x^2 - 
3*c^4*d*e^4*x - c^5*e^4)*sqrt(-e/d)*log(-(8*d^3*e*x^3 - 7*c^2*d*e*x + c^3* 
e - 4*sqrt(-d^2*x^2 + c^2)*(2*d^2*x - c*d)*sqrt(d*x + c)*sqrt(e*x)*sqrt(-e 
/d))/(d*x + c)))/(d^10*x^5 + 3*c*d^9*x^4 + 2*c^2*d^8*x^3 - 2*c^3*d^7*x^2 - 
 3*c^4*d^6*x - c^5*d^5), -1/768*(957*sqrt(1/2)*(d^5*e^4*x^5 + 3*c*d^4*e^4* 
x^4 + 2*c^2*d^3*e^4*x^3 - 2*c^3*d^2*e^4*x^2 - 3*c^4*d*e^4*x - c^5*e^4)*sqr 
t(e/d)*arctan(4*sqrt(1/2)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c)*sqrt(e*x)*d*s 
qrt(e/d)/(3*d^2*e*x^2 + 2*c*d*e*x - c^2*e)) + 2*(409*d^3*e^4*x^3 + 349*c*d 
^2*e^4*x^2 - 185*c^2*d*e^4*x - 189*c^3*e^4)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x 
+ c)*sqrt(e*x) - 768*(d^5*e^4*x^5 + 3*c*d^4*e^4*x^4 + 2*c^2*d^3*e^4*x^3 - 
2*c^3*d^2*e^4*x^2 - 3*c^4*d*e^4*x - c^5*e^4)*sqrt(e/d)*arctan(2*sqrt(-d^2* 
x^2 + c^2)*sqrt(d*x + c)*sqrt(e*x)*d*sqrt(e/d)/(2*d^2*e*x^2 + c*d*e*x - c^ 
2*e)))/(d^10*x^5 + 3*c*d^9*x^4 + 2*c^2*d^8*x^3 - 2*c^3*d^7*x^2 - 3*c^4*d^6 
*x - c^5*d^5)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(e x)^{9/2}}{(c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((e*x)**(9/2)/(d*x+c)**(5/2)/(-d**2*x**2+c**2)**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(e x)^{9/2}}{(c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {9}{2}}}{{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((e*x)^(9/2)/(d*x+c)^(5/2)/(-d^2*x^2+c^2)^(3/2),x, algorithm="max 
ima")
 

Output:

integrate((e*x)^(9/2)/((-d^2*x^2 + c^2)^(3/2)*(d*x + c)^(5/2)), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 877 vs. \(2 (224) = 448\).

Time = 1.07 (sec) , antiderivative size = 877, normalized size of antiderivative = 3.05 \[ \int \frac {(e x)^{9/2}}{(c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

integrate((e*x)^(9/2)/(d*x+c)^(5/2)/(-d^2*x^2+c^2)^(3/2),x, algorithm="gia 
c")
 

Output:

1/1536*(1536*abs(d)*abs(e)*log((sqrt((d*x + c)*d*e - c*d*e)*sqrt(-d*e) - s 
qrt(c*d^2*e^2 - ((d*x + c)*d*e - c*d*e)*d*e))^2)/(sqrt(-d*e)*d) + 192*sqrt 
((d*x + c)*d*e - c*d*e)*abs(d)*abs(e)/(sqrt(c*d^2*e^2 - ((d*x + c)*d*e - c 
*d*e)*d*e)*d) - 957*sqrt(2)*sqrt(-d*e)*c*abs(d)*abs(e)*log(abs(-4*sqrt(2)* 
d^2*e^2*abs(c) - 6*c*d^2*e^2 + 2*(sqrt((d*x + c)*d*e - c*d*e)*sqrt(-d*e) - 
 sqrt(c*d^2*e^2 - ((d*x + c)*d*e - c*d*e)*d*e))^2)/abs(4*sqrt(2)*d^2*e^2*a 
bs(c) - 6*c*d^2*e^2 + 2*(sqrt((d*x + c)*d*e - c*d*e)*sqrt(-d*e) - sqrt(c*d 
^2*e^2 - ((d*x + c)*d*e - c*d*e)*d*e))^2))/(d^2*e*abs(c)) - 8*(361*sqrt(-d 
*e)*c^6*d^10*e^11*abs(d)*abs(e) - 5103*sqrt(-d*e)*(sqrt((d*x + c)*d*e - c* 
d*e)*sqrt(-d*e) - sqrt(c*d^2*e^2 - ((d*x + c)*d*e - c*d*e)*d*e))^2*c^5*d^8 
*e^9*abs(d)*abs(e) + 25098*sqrt(-d*e)*(sqrt((d*x + c)*d*e - c*d*e)*sqrt(-d 
*e) - sqrt(c*d^2*e^2 - ((d*x + c)*d*e - c*d*e)*d*e))^4*c^4*d^6*e^7*abs(d)* 
abs(e) - 45486*sqrt(-d*e)*(sqrt((d*x + c)*d*e - c*d*e)*sqrt(-d*e) - sqrt(c 
*d^2*e^2 - ((d*x + c)*d*e - c*d*e)*d*e))^6*c^3*d^4*e^5*abs(d)*abs(e) + 149 
73*sqrt(-d*e)*(sqrt((d*x + c)*d*e - c*d*e)*sqrt(-d*e) - sqrt(c*d^2*e^2 - ( 
(d*x + c)*d*e - c*d*e)*d*e))^8*c^2*d^2*e^3*abs(d)*abs(e) - 1395*sqrt(-d*e) 
*(sqrt((d*x + c)*d*e - c*d*e)*sqrt(-d*e) - sqrt(c*d^2*e^2 - ((d*x + c)*d*e 
 - c*d*e)*d*e))^10*c*e*abs(d)*abs(e))/(c^2*d^4*e^4 - 6*(sqrt((d*x + c)*d*e 
 - c*d*e)*sqrt(-d*e) - sqrt(c*d^2*e^2 - ((d*x + c)*d*e - c*d*e)*d*e))^2*c* 
d^2*e^2 + (sqrt((d*x + c)*d*e - c*d*e)*sqrt(-d*e) - sqrt(c*d^2*e^2 - ((...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{9/2}}{(c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\int \frac {{\left (e\,x\right )}^{9/2}}{{\left (c^2-d^2\,x^2\right )}^{3/2}\,{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:

int((e*x)^(9/2)/((c^2 - d^2*x^2)^(3/2)*(c + d*x)^(5/2)),x)
 

Output:

int((e*x)^(9/2)/((c^2 - d^2*x^2)^(3/2)*(c + d*x)^(5/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 1071, normalized size of antiderivative = 3.72 \[ \int \frac {(e x)^{9/2}}{(c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

int((e*x)^(9/2)/(d*x+c)^(5/2)/(-d^2*x^2+c^2)^(3/2),x)
 

Output:

(sqrt(e)*e**4*(16269*sqrt(d)*sqrt(c - d*x)*sqrt(2)*log((sqrt(c - d*x) - sq 
rt(c)*sqrt(2) - sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))*c**3*i + 48807*sqrt( 
d)*sqrt(c - d*x)*sqrt(2)*log((sqrt(c - d*x) - sqrt(c)*sqrt(2) - sqrt(c) + 
sqrt(x)*sqrt(d)*i)/sqrt(c))*c**2*d*i*x + 48807*sqrt(d)*sqrt(c - d*x)*sqrt( 
2)*log((sqrt(c - d*x) - sqrt(c)*sqrt(2) - sqrt(c) + sqrt(x)*sqrt(d)*i)/sqr 
t(c))*c*d**2*i*x**2 + 16269*sqrt(d)*sqrt(c - d*x)*sqrt(2)*log((sqrt(c - d* 
x) - sqrt(c)*sqrt(2) - sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))*d**3*i*x**3 - 
 16269*sqrt(d)*sqrt(c - d*x)*sqrt(2)*log((sqrt(c - d*x) - sqrt(c)*sqrt(2) 
+ sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))*c**3*i - 48807*sqrt(d)*sqrt(c - d* 
x)*sqrt(2)*log((sqrt(c - d*x) - sqrt(c)*sqrt(2) + sqrt(c) + sqrt(x)*sqrt(d 
)*i)/sqrt(c))*c**2*d*i*x - 48807*sqrt(d)*sqrt(c - d*x)*sqrt(2)*log((sqrt(c 
 - d*x) - sqrt(c)*sqrt(2) + sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))*c*d**2*i 
*x**2 - 16269*sqrt(d)*sqrt(c - d*x)*sqrt(2)*log((sqrt(c - d*x) - sqrt(c)*s 
qrt(2) + sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))*d**3*i*x**3 - 16269*sqrt(d) 
*sqrt(c - d*x)*sqrt(2)*log((sqrt(c - d*x) + sqrt(c)*sqrt(2) - sqrt(c) + sq 
rt(x)*sqrt(d)*i)/sqrt(c))*c**3*i - 48807*sqrt(d)*sqrt(c - d*x)*sqrt(2)*log 
((sqrt(c - d*x) + sqrt(c)*sqrt(2) - sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))* 
c**2*d*i*x - 48807*sqrt(d)*sqrt(c - d*x)*sqrt(2)*log((sqrt(c - d*x) + sqrt 
(c)*sqrt(2) - sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))*c*d**2*i*x**2 - 16269* 
sqrt(d)*sqrt(c - d*x)*sqrt(2)*log((sqrt(c - d*x) + sqrt(c)*sqrt(2) - sq...