3.3.75 \(\int (e \sec (c+d x))^{8/3} \sqrt {a+a \sec (c+d x)} \, dx\) [275]

3.3.75.1 Optimal result

Integrand size = 27, antiderivative size = 716 \[ \int (e \sec (c+d x))^{8/3} \sqrt {a+a \sec (c+d x)} \, dx=\frac {60 a e^2 (e \sec (c+d x))^{2/3} \tan (c+d x)}{91 d \sqrt {a+a \sec (c+d x)}}+\frac {6 a e (e \sec (c+d x))^{5/3} \tan (c+d x)}{13 d \sqrt {a+a \sec (c+d x)}}-\frac {240 a e^3 \tan (c+d x)}{91 d \sqrt {a+a \sec (c+d x)} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )}+\frac {120 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^2 e^{7/3} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}{\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}\right )|-7-4 \sqrt {3}\right ) \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right ) \sqrt {\frac {e^{2/3}+\sqrt [3]{e} \sqrt [3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}} \tan (c+d x)}{91 d (a-a \sec (c+d x)) \sqrt {a+a \sec (c+d x)} \sqrt {\frac {\sqrt [3]{e} \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}}}-\frac {80 \sqrt {2} 3^{3/4} a^2 e^{7/3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}{\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}\right ),-7-4 \sqrt {3}\right ) \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right ) \sqrt {\frac {e^{2/3}+\sqrt [3]{e} \sqrt [3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}} \tan (c+d x)}{91 d (a-a \sec (c+d x)) \sqrt {a+a \sec (c+d x)} \sqrt {\frac {\sqrt [3]{e} \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}}} \]

output

60/91*a*e^2*(e*sec(d*x+c))^(2/3)*tan(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)+6/13* 
a*e*(e*sec(d*x+c))^(5/3)*tan(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)-240/91*a*e^3* 
tan(d*x+c)/d/(-(e*sec(d*x+c))^(1/3)+e^(1/3)*(1+3^(1/2)))/(a+a*sec(d*x+c))^ 
(1/2)-80/91*3^(3/4)*a^2*e^(7/3)*EllipticF((-(e*sec(d*x+c))^(1/3)+e^(1/3)*( 
1-3^(1/2)))/(-(e*sec(d*x+c))^(1/3)+e^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*(e^ 
(1/3)-(e*sec(d*x+c))^(1/3))*2^(1/2)*((e^(2/3)+e^(1/3)*(e*sec(d*x+c))^(1/3) 
+(e*sec(d*x+c))^(2/3))/(-(e*sec(d*x+c))^(1/3)+e^(1/3)*(1+3^(1/2)))^2)^(1/2 
)*tan(d*x+c)/d/(a-a*sec(d*x+c))/(a+a*sec(d*x+c))^(1/2)/(e^(1/3)*(e^(1/3)-( 
e*sec(d*x+c))^(1/3))/(-(e*sec(d*x+c))^(1/3)+e^(1/3)*(1+3^(1/2)))^2)^(1/2)+ 
120/91*3^(1/4)*a^2*e^(7/3)*EllipticE((-(e*sec(d*x+c))^(1/3)+e^(1/3)*(1-3^( 
1/2)))/(-(e*sec(d*x+c))^(1/3)+e^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*(e^(1/3) 
-(e*sec(d*x+c))^(1/3))*(1/2*6^(1/2)-1/2*2^(1/2))*((e^(2/3)+e^(1/3)*(e*sec( 
d*x+c))^(1/3)+(e*sec(d*x+c))^(2/3))/(-(e*sec(d*x+c))^(1/3)+e^(1/3)*(1+3^(1 
/2)))^2)^(1/2)*tan(d*x+c)/d/(a-a*sec(d*x+c))/(a+a*sec(d*x+c))^(1/2)/(e^(1/ 
3)*(e^(1/3)-(e*sec(d*x+c))^(1/3))/(-(e*sec(d*x+c))^(1/3)+e^(1/3)*(1+3^(1/2 
)))^2)^(1/2)
 

3.3.75.2 Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.23 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.10 \[ \int (e \sec (c+d x))^{8/3} \sqrt {a+a \sec (c+d x)} \, dx=\frac {2 \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},\frac {1}{2},\frac {3}{2},1-\sec (c+d x)\right ) (e \sec (c+d x))^{8/3} \sqrt {a (1+\sec (c+d x))} \tan \left (\frac {1}{2} (c+d x)\right )}{d \sec ^{\frac {8}{3}}(c+d x)} \]

input

Integrate[(e*Sec[c + d*x])^(8/3)*Sqrt[a + a*Sec[c + d*x]],x]
 

output

(2*Hypergeometric2F1[-5/3, 1/2, 3/2, 1 - Sec[c + d*x]]*(e*Sec[c + d*x])^(8 
/3)*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(d*Sec[c + d*x]^(8/3))
 

3.3.75.3 Rubi [A] (warning: unable to verify)

Time = 0.65 (sec) , antiderivative size = 708, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {3042, 4293, 60, 60, 73, 832, 759, 2416}

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.

\(\displaystyle \int \sqrt {a \sec (c+d x)+a} (e \sec (c+d x))^{8/3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sqrt {a \csc \left (c+d x+\frac {\pi }{2}\right )+a} \left (e \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{8/3}dx\)

\(\Big \downarrow \) 4293

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

\(\Big \downarrow \) 60

\(\displaystyle -\frac {a^2 e \tan (c+d x) \left (\frac {10}{13} e \int \frac {(e \sec (c+d x))^{2/3}}{\sqrt {a-a \sec (c+d x)}}d\sec (c+d x)-\frac {6 \sqrt {a-a \sec (c+d x)} (e \sec (c+d x))^{5/3}}{13 a}\right )}{d \sqrt {a-a \sec (c+d x)} \sqrt {a \sec (c+d x)+a}}\)

\(\Big \downarrow \) 60

\(\displaystyle -\frac {a^2 e \tan (c+d x) \left (\frac {10}{13} e \left (\frac {4}{7} e \int \frac {1}{\sqrt [3]{e \sec (c+d x)} \sqrt {a-a \sec (c+d x)}}d\sec (c+d x)-\frac {6 \sqrt {a-a \sec (c+d x)} (e \sec (c+d x))^{2/3}}{7 a}\right )-\frac {6 \sqrt {a-a \sec (c+d x)} (e \sec (c+d x))^{5/3}}{13 a}\right )}{d \sqrt {a-a \sec (c+d x)} \sqrt {a \sec (c+d x)+a}}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {a^2 e \tan (c+d x) \left (\frac {10}{13} e \left (\frac {12}{7} \int \frac {\sqrt [3]{e \sec (c+d x)}}{\sqrt {a-a \sec (c+d x)}}d\sqrt [3]{e \sec (c+d x)}-\frac {6 \sqrt {a-a \sec (c+d x)} (e \sec (c+d x))^{2/3}}{7 a}\right )-\frac {6 \sqrt {a-a \sec (c+d x)} (e \sec (c+d x))^{5/3}}{13 a}\right )}{d \sqrt {a-a \sec (c+d x)} \sqrt {a \sec (c+d x)+a}}\)

\(\Big \downarrow \) 832

\(\displaystyle -\frac {a^2 e \tan (c+d x) \left (\frac {10}{13} e \left (\frac {12}{7} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{e} \int \frac {1}{\sqrt {a-a \sec (c+d x)}}d\sqrt [3]{e \sec (c+d x)}-\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}{\sqrt {a-a \sec (c+d x)}}d\sqrt [3]{e \sec (c+d x)}\right )-\frac {6 \sqrt {a-a \sec (c+d x)} (e \sec (c+d x))^{2/3}}{7 a}\right )-\frac {6 \sqrt {a-a \sec (c+d x)} (e \sec (c+d x))^{5/3}}{13 a}\right )}{d \sqrt {a-a \sec (c+d x)} \sqrt {a \sec (c+d x)+a}}\)

\(\Big \downarrow \) 759

\(\displaystyle -\frac {a^2 e \tan (c+d x) \left (\frac {10}{13} e \left (\frac {12}{7} \left (-\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}{\sqrt {a-a \sec (c+d x)}}d\sqrt [3]{e \sec (c+d x)}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{e} \sqrt {\frac {\sqrt [3]{e} \sqrt [3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}} \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}{\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {a-a \sec (c+d x)} \sqrt {\frac {\sqrt [3]{e} \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}}}\right )-\frac {6 \sqrt {a-a \sec (c+d x)} (e \sec (c+d x))^{2/3}}{7 a}\right )-\frac {6 \sqrt {a-a \sec (c+d x)} (e \sec (c+d x))^{5/3}}{13 a}\right )}{d \sqrt {a-a \sec (c+d x)} \sqrt {a \sec (c+d x)+a}}\)

\(\Big \downarrow \) 2416

\(\displaystyle -\frac {a^2 e \tan (c+d x) \left (\frac {10}{13} e \left (\frac {12}{7} \left (-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{e} \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right ) \sqrt {\frac {\sqrt [3]{e} \sqrt [3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}{\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {a-a \sec (c+d x)} \sqrt {\frac {\sqrt [3]{e} \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}}}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{e} \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right ) \sqrt {\frac {\sqrt [3]{e} \sqrt [3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}{\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}\right )|-7-4 \sqrt {3}\right )}{\sqrt {a-a \sec (c+d x)} \sqrt {\frac {\sqrt [3]{e} \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}}}+\frac {2 e \sqrt {a-a \sec (c+d x)}}{a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )}\right )-\frac {6 \sqrt {a-a \sec (c+d x)} (e \sec (c+d x))^{2/3}}{7 a}\right )-\frac {6 \sqrt {a-a \sec (c+d x)} (e \sec (c+d x))^{5/3}}{13 a}\right )}{d \sqrt {a-a \sec (c+d x)} \sqrt {a \sec (c+d x)+a}}\)

input

Int[(e*Sec[c + d*x])^(8/3)*Sqrt[a + a*Sec[c + d*x]],x]
 

output

-((a^2*e*((-6*(e*Sec[c + d*x])^(5/3)*Sqrt[a - a*Sec[c + d*x]])/(13*a) + (1 
0*e*((-6*(e*Sec[c + d*x])^(2/3)*Sqrt[a - a*Sec[c + d*x]])/(7*a) + (12*((2* 
e*Sqrt[a - a*Sec[c + d*x]])/(a*((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^( 
1/3))) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*e^(1/3)*EllipticE[ArcSin[((1 - Sqrt[3] 
)*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d* 
x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2 
/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[ 
3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2])/(Sqrt[a - a*Sec[c + d*x]]*Sqrt[( 
e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Se 
c[c + d*x])^(1/3))^2]) - (2*(1 - Sqrt[3])*Sqrt[2 + Sqrt[3]]*e^(1/3)*Ellipt 
icF[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3]) 
*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + 
 d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + 
d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2])/(3^(1/4) 
*Sqrt[a - a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)) 
)/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2])))/7))/13)*Tan[c + d 
*x])/(d*Sqrt[a - a*Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]))
 

3.3.75.3.1 Defintions of rubi rules used

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 759

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* 
((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s 
+ r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & 
& PosQ[a]
 

rule 832

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] 
], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 - Sqrt[3]))*(s/r)   Int[1/Sqrt[a + b*x 
^3], x], x] + Simp[1/r   Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x 
]] /; FreeQ[{a, b}, x] && PosQ[a]
 

rule 2416

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 - Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 - Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x] - S 
imp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 + Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt 
[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) 
*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq 
Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4293

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) 
 + (a_)], x_Symbol] :> Simp[a^2*d*(Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]] 
*Sqrt[a - b*Csc[e + f*x]]))   Subst[Int[(d*x)^(n - 1)/Sqrt[a - b*x], x], x, 
 Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0]
 

3.3.75.4 Maple [F]

\[\int \left (e \sec \left (d x +c \right )\right )^{\frac {8}{3}} \sqrt {a +a \sec \left (d x +c \right )}d x\]

input

int((e*sec(d*x+c))^(8/3)*(a+a*sec(d*x+c))^(1/2),x)
 

output

int((e*sec(d*x+c))^(8/3)*(a+a*sec(d*x+c))^(1/2),x)
 

3.3.75.5 Fricas [F]

\[ \int (e \sec (c+d x))^{8/3} \sqrt {a+a \sec (c+d x)} \, dx=\int { \sqrt {a \sec \left (d x + c\right ) + a} \left (e \sec \left (d x + c\right )\right )^{\frac {8}{3}} \,d x } \]

input

integrate((e*sec(d*x+c))^(8/3)*(a+a*sec(d*x+c))^(1/2),x, algorithm="fricas 
")
 

output

integral(sqrt(a*sec(d*x + c) + a)*(e*sec(d*x + c))^(2/3)*e^2*sec(d*x + c)^ 
2, x)
 

3.3.75.6 Sympy [F(-1)]

Timed out. \[ \int (e \sec (c+d x))^{8/3} \sqrt {a+a \sec (c+d x)} \, dx=\text {Timed out} \]

input

integrate((e*sec(d*x+c))**(8/3)*(a+a*sec(d*x+c))**(1/2),x)
 

output

Timed out
 

3.3.75.7 Maxima [F]

\[ \int (e \sec (c+d x))^{8/3} \sqrt {a+a \sec (c+d x)} \, dx=\int { \sqrt {a \sec \left (d x + c\right ) + a} \left (e \sec \left (d x + c\right )\right )^{\frac {8}{3}} \,d x } \]

input

integrate((e*sec(d*x+c))^(8/3)*(a+a*sec(d*x+c))^(1/2),x, algorithm="maxima 
")
 

output

integrate(sqrt(a*sec(d*x + c) + a)*(e*sec(d*x + c))^(8/3), x)
 

3.3.75.8 Giac [F]

\[ \int (e \sec (c+d x))^{8/3} \sqrt {a+a \sec (c+d x)} \, dx=\int { \sqrt {a \sec \left (d x + c\right ) + a} \left (e \sec \left (d x + c\right )\right )^{\frac {8}{3}} \,d x } \]

input

integrate((e*sec(d*x+c))^(8/3)*(a+a*sec(d*x+c))^(1/2),x, algorithm="giac")
 

output

sage0*x
 

3.3.75.9 Mupad [F(-1)]

Timed out. \[ \int (e \sec (c+d x))^{8/3} \sqrt {a+a \sec (c+d x)} \, dx=\int \sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}\,{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{8/3} \,d x \]

input

int((a + a/cos(c + d*x))^(1/2)*(e/cos(c + d*x))^(8/3),x)
 

output

int((a + a/cos(c + d*x))^(1/2)*(e/cos(c + d*x))^(8/3), x)