\(\int \frac {x^{-1+3 n} (a+b x^n)^{5/2}}{\sqrt {c+d x^n}} \, dx\) [485]

Optimal result

Integrand size = 30, antiderivative size = 356 \[ \int \frac {x^{-1+3 n} \left (a+b x^n\right )^{5/2}}{\sqrt {c+d x^n}} \, dx=\frac {(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^n} \sqrt {c+d x^n}}{128 b^2 d^5 n}-\frac {(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{192 b^2 d^4 n}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{240 b^2 d^3 n}-\frac {(9 b c+11 a d) \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{40 b^2 d^2 n}+\frac {\left (a+b x^n\right )^{9/2} \sqrt {c+d x^n}}{5 b^2 d n}-\frac {(b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{128 b^{5/2} d^{11/2} n} \] Output:

1/128*(-a*d+b*c)^2*(3*a^2*d^2+14*a*b*c*d+63*b^2*c^2)*(a+b*x^n)^(1/2)*(c+d* 
x^n)^(1/2)/b^2/d^5/n-1/192*(-a*d+b*c)*(3*a^2*d^2+14*a*b*c*d+63*b^2*c^2)*(a 
+b*x^n)^(3/2)*(c+d*x^n)^(1/2)/b^2/d^4/n+1/240*(3*a^2*d^2+14*a*b*c*d+63*b^2 
*c^2)*(a+b*x^n)^(5/2)*(c+d*x^n)^(1/2)/b^2/d^3/n-1/40*(11*a*d+9*b*c)*(a+b*x 
^n)^(7/2)*(c+d*x^n)^(1/2)/b^2/d^2/n+1/5*(a+b*x^n)^(9/2)*(c+d*x^n)^(1/2)/b^ 
2/d/n-1/128*(-a*d+b*c)^3*(3*a^2*d^2+14*a*b*c*d+63*b^2*c^2)*arctanh(d^(1/2) 
*(a+b*x^n)^(1/2)/b^(1/2)/(c+d*x^n)^(1/2))/b^(5/2)/d^(11/2)/n
 

Mathematica [A] (verified)

Time = 2.06 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.77 \[ \int \frac {x^{-1+3 n} \left (a+b x^n\right )^{5/2}}{\sqrt {c+d x^n}} \, dx=\frac {\sqrt {c+d x^n} \left (-\frac {24 (3 b c+a d) \left (a+b x^n\right )^4}{b d}+64 x^n \left (a+b x^n\right )^4+\frac {5 (b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (-\frac {2 d \left (a+b x^n\right )}{-b c+a d}-\frac {4 d^2 \left (a+b x^n\right )^2}{3 (b c-a d)^2}-\frac {16 d^3 \left (a+b x^n\right )^3}{15 (-b c+a d)^3}-\frac {2 \sqrt {d} \sqrt {a+b x^n} \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b c-a d}}\right )}{\sqrt {b c-a d} \sqrt {\frac {b \left (c+d x^n\right )}{b c-a d}}}\right )}{4 b d^5}\right )}{320 b d n \sqrt {a+b x^n}} \] Input:

Integrate[(x^(-1 + 3*n)*(a + b*x^n)^(5/2))/Sqrt[c + d*x^n],x]
 

Output:

(Sqrt[c + d*x^n]*((-24*(3*b*c + a*d)*(a + b*x^n)^4)/(b*d) + 64*x^n*(a + b* 
x^n)^4 + (5*(b*c - a*d)^3*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*((-2*d*(a 
+ b*x^n))/(-(b*c) + a*d) - (4*d^2*(a + b*x^n)^2)/(3*(b*c - a*d)^2) - (16*d 
^3*(a + b*x^n)^3)/(15*(-(b*c) + a*d)^3) - (2*Sqrt[d]*Sqrt[a + b*x^n]*ArcSi 
nh[(Sqrt[d]*Sqrt[a + b*x^n])/Sqrt[b*c - a*d]])/(Sqrt[b*c - a*d]*Sqrt[(b*(c 
 + d*x^n))/(b*c - a*d)])))/(4*b*d^5)))/(320*b*d*n*Sqrt[a + b*x^n])
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.83, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {948, 101, 27, 90, 60, 60, 60, 66, 221}

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.

\(\displaystyle \int \frac {x^{3 n-1} \left (a+b x^n\right )^{5/2}}{\sqrt {c+d x^n}} \, dx\)

\(\Big \downarrow \) 948

\(\displaystyle \frac {\int \frac {x^{2 n} \left (b x^n+a\right )^{5/2}}{\sqrt {d x^n+c}}dx^n}{n}\)

\(\Big \downarrow \) 101

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {x^n \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{5 b d}-\frac {\int \frac {\left (b x^n+a\right )^{5/2} \left (3 (3 b c+a d) x^n+2 a c\right )}{\sqrt {d x^n+c}}dx^n}{10 b d}}{n}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {\frac {x^n \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{5 b d}-\frac {\frac {3 (a d+3 b c) \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{4 b d}-\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \int \frac {\left (b x^n+a\right )^{5/2}}{\sqrt {d x^n+c}}dx^n}{8 b d}}{10 b d}}{n}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {x^n \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{5 b d}-\frac {\frac {3 (a d+3 b c) \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{4 b d}-\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (\frac {\left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{3 d}-\frac {5 (b c-a d) \int \frac {\left (b x^n+a\right )^{3/2}}{\sqrt {d x^n+c}}dx^n}{6 d}\right )}{8 b d}}{10 b d}}{n}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {x^n \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{5 b d}-\frac {\frac {3 (a d+3 b c) \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{4 b d}-\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (\frac {\left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{3 d}-\frac {5 (b c-a d) \left (\frac {\left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{2 d}-\frac {3 (b c-a d) \int \frac {\sqrt {b x^n+a}}{\sqrt {d x^n+c}}dx^n}{4 d}\right )}{6 d}\right )}{8 b d}}{10 b d}}{n}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {x^n \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{5 b d}-\frac {\frac {3 (a d+3 b c) \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{4 b d}-\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (\frac {\left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{3 d}-\frac {5 (b c-a d) \left (\frac {\left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x^n} \sqrt {c+d x^n}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^n+a} \sqrt {d x^n+c}}dx^n}{2 d}\right )}{4 d}\right )}{6 d}\right )}{8 b d}}{10 b d}}{n}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {\frac {x^n \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{5 b d}-\frac {\frac {3 (a d+3 b c) \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{4 b d}-\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (\frac {\left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{3 d}-\frac {5 (b c-a d) \left (\frac {\left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x^n} \sqrt {c+d x^n}}{d}-\frac {(b c-a d) \int \frac {1}{b-d x^{2 n}}d\frac {\sqrt {b x^n+a}}{\sqrt {d x^n+c}}}{d}\right )}{4 d}\right )}{6 d}\right )}{8 b d}}{10 b d}}{n}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {x^n \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{5 b d}-\frac {\frac {3 (a d+3 b c) \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{4 b d}-\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (\frac {\left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{3 d}-\frac {5 (b c-a d) \left (\frac {\left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x^n} \sqrt {c+d x^n}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{\sqrt {b} d^{3/2}}\right )}{4 d}\right )}{6 d}\right )}{8 b d}}{10 b d}}{n}\)

Input:

Int[(x^(-1 + 3*n)*(a + b*x^n)^(5/2))/Sqrt[c + d*x^n],x]
 

Output:

((x^n*(a + b*x^n)^(7/2)*Sqrt[c + d*x^n])/(5*b*d) - ((3*(3*b*c + a*d)*(a + 
b*x^n)^(7/2)*Sqrt[c + d*x^n])/(4*b*d) - ((63*b^2*c^2 + 14*a*b*c*d + 3*a^2* 
d^2)*(((a + b*x^n)^(5/2)*Sqrt[c + d*x^n])/(3*d) - (5*(b*c - a*d)*(((a + b* 
x^n)^(3/2)*Sqrt[c + d*x^n])/(2*d) - (3*(b*c - a*d)*((Sqrt[a + b*x^n]*Sqrt[ 
c + d*x^n])/d - ((b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^n])/(Sqrt[b]*Sq 
rt[c + d*x^n])])/(Sqrt[b]*d^(3/2))))/(4*d)))/(6*d)))/(8*b*d))/(10*b*d))/n
 

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 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 66

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

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
 

rule 221

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

rule 948

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. 
), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ 
p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ 
[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
 

Maple [F]

\[\int \frac {x^{-1+3 n} \left (a +b \,x^{n}\right )^{\frac {5}{2}}}{\sqrt {c +d \,x^{n}}}d x\]

Input:

int(x^(-1+3*n)*(a+b*x^n)^(5/2)/(c+d*x^n)^(1/2),x)
 

Output:

int(x^(-1+3*n)*(a+b*x^n)^(5/2)/(c+d*x^n)^(1/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 771, normalized size of antiderivative = 2.17 \[ \int \frac {x^{-1+3 n} \left (a+b x^n\right )^{5/2}}{\sqrt {c+d x^n}} \, dx =\text {Too large to display} \] Input:

integrate(x^(-1+3*n)*(a+b*x^n)^(5/2)/(c+d*x^n)^(1/2),x, algorithm="fricas" 
)
 

Output:

[-1/7680*(15*(63*b^5*c^5 - 175*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 - 30*a^3* 
b^2*c^2*d^3 - 5*a^4*b*c*d^4 - 3*a^5*d^5)*sqrt(b*d)*log(8*b^2*d^2*x^(2*n) + 
 b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*sqrt(b*d)*b*d*x^n + (b*c + a*d)*sqrt 
(b*d))*sqrt(b*x^n + a)*sqrt(d*x^n + c) + 8*(b^2*c*d + a*b*d^2)*x^n) - 4*(3 
84*b^5*d^5*x^(4*n) + 945*b^5*c^4*d - 2310*a*b^4*c^3*d^2 + 1564*a^2*b^3*c^2 
*d^3 - 90*a^3*b^2*c*d^4 - 45*a^4*b*d^5 - 144*(3*b^5*c*d^4 - 7*a*b^4*d^5)*x 
^(3*n) + 8*(63*b^5*c^2*d^3 - 148*a*b^4*c*d^4 + 93*a^2*b^3*d^5)*x^(2*n) - 2 
*(315*b^5*c^3*d^2 - 749*a*b^4*c^2*d^3 + 481*a^2*b^3*c*d^4 - 15*a^3*b^2*d^5 
)*x^n)*sqrt(b*x^n + a)*sqrt(d*x^n + c))/(b^3*d^6*n), 1/3840*(15*(63*b^5*c^ 
5 - 175*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 - 30*a^3*b^2*c^2*d^3 - 5*a^4*b*c 
*d^4 - 3*a^5*d^5)*sqrt(-b*d)*arctan(1/2*(2*sqrt(-b*d)*b*d*x^n + (b*c + a*d 
)*sqrt(-b*d))*sqrt(b*x^n + a)*sqrt(d*x^n + c)/(b^2*d^2*x^(2*n) + a*b*c*d + 
 (b^2*c*d + a*b*d^2)*x^n)) + 2*(384*b^5*d^5*x^(4*n) + 945*b^5*c^4*d - 2310 
*a*b^4*c^3*d^2 + 1564*a^2*b^3*c^2*d^3 - 90*a^3*b^2*c*d^4 - 45*a^4*b*d^5 - 
144*(3*b^5*c*d^4 - 7*a*b^4*d^5)*x^(3*n) + 8*(63*b^5*c^2*d^3 - 148*a*b^4*c* 
d^4 + 93*a^2*b^3*d^5)*x^(2*n) - 2*(315*b^5*c^3*d^2 - 749*a*b^4*c^2*d^3 + 4 
81*a^2*b^3*c*d^4 - 15*a^3*b^2*d^5)*x^n)*sqrt(b*x^n + a)*sqrt(d*x^n + c))/( 
b^3*d^6*n)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{-1+3 n} \left (a+b x^n\right )^{5/2}}{\sqrt {c+d x^n}} \, dx=\text {Timed out} \] Input:

integrate(x**(-1+3*n)*(a+b*x**n)**(5/2)/(c+d*x**n)**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^{-1+3 n} \left (a+b x^n\right )^{5/2}}{\sqrt {c+d x^n}} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{\frac {5}{2}} x^{3 \, n - 1}}{\sqrt {d x^{n} + c}} \,d x } \] Input:

integrate(x^(-1+3*n)*(a+b*x^n)^(5/2)/(c+d*x^n)^(1/2),x, algorithm="maxima" 
)
 

Output:

integrate((b*x^n + a)^(5/2)*x^(3*n - 1)/sqrt(d*x^n + c), x)
 

Giac [F]

\[ \int \frac {x^{-1+3 n} \left (a+b x^n\right )^{5/2}}{\sqrt {c+d x^n}} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{\frac {5}{2}} x^{3 \, n - 1}}{\sqrt {d x^{n} + c}} \,d x } \] Input:

integrate(x^(-1+3*n)*(a+b*x^n)^(5/2)/(c+d*x^n)^(1/2),x, algorithm="giac")
 

Output:

integrate((b*x^n + a)^(5/2)*x^(3*n - 1)/sqrt(d*x^n + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{-1+3 n} \left (a+b x^n\right )^{5/2}}{\sqrt {c+d x^n}} \, dx=\int \frac {x^{3\,n-1}\,{\left (a+b\,x^n\right )}^{5/2}}{\sqrt {c+d\,x^n}} \,d x \] Input:

int((x^(3*n - 1)*(a + b*x^n)^(5/2))/(c + d*x^n)^(1/2),x)
 

Output:

int((x^(3*n - 1)*(a + b*x^n)^(5/2))/(c + d*x^n)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^{-1+3 n} \left (a+b x^n\right )^{5/2}}{\sqrt {c+d x^n}} \, dx =\text {Too large to display} \] Input:

int(x^(-1+3*n)*(a+b*x^n)^(5/2)/(c+d*x^n)^(1/2),x)
 

Output:

(384*x**(4*n)*sqrt(x**n*d + c)*sqrt(x**n*b + a)*a*b**3*d**4 + 384*x**(4*n) 
*sqrt(x**n*d + c)*sqrt(x**n*b + a)*b**4*c*d**3 + 1008*x**(3*n)*sqrt(x**n*d 
 + c)*sqrt(x**n*b + a)*a**2*b**2*d**4 + 576*x**(3*n)*sqrt(x**n*d + c)*sqrt 
(x**n*b + a)*a*b**3*c*d**3 - 432*x**(3*n)*sqrt(x**n*d + c)*sqrt(x**n*b + a 
)*b**4*c**2*d**2 + 744*x**(2*n)*sqrt(x**n*d + c)*sqrt(x**n*b + a)*a**3*b*d 
**4 - 440*x**(2*n)*sqrt(x**n*d + c)*sqrt(x**n*b + a)*a**2*b**2*c*d**3 - 68 
0*x**(2*n)*sqrt(x**n*d + c)*sqrt(x**n*b + a)*a*b**3*c**2*d**2 + 504*x**(2* 
n)*sqrt(x**n*d + c)*sqrt(x**n*b + a)*b**4*c**3*d + 30*x**n*sqrt(x**n*d + c 
)*sqrt(x**n*b + a)*a**4*d**4 - 932*x**n*sqrt(x**n*d + c)*sqrt(x**n*b + a)* 
a**3*b*c*d**3 + 536*x**n*sqrt(x**n*d + c)*sqrt(x**n*b + a)*a**2*b**2*c**2* 
d**2 + 868*x**n*sqrt(x**n*d + c)*sqrt(x**n*b + a)*a*b**3*c**3*d - 630*x**n 
*sqrt(x**n*d + c)*sqrt(x**n*b + a)*b**4*c**4 - 60*sqrt(x**n*d + c)*sqrt(x* 
*n*b + a)*a**4*c*d**3 + 1924*sqrt(x**n*d + c)*sqrt(x**n*b + a)*a**3*b*c**2 
*d**2 - 2996*sqrt(x**n*d + c)*sqrt(x**n*b + a)*a**2*b**2*c**3*d + 1260*sqr 
t(x**n*d + c)*sqrt(x**n*b + a)*a*b**3*c**4 - 45*int((x**(2*n)*sqrt(x**n*d 
+ c)*sqrt(x**n*b + a))/(x**(2*n)*a*b*d**2*x + x**(2*n)*b**2*c*d*x + x**n*a 
**2*d**2*x + 2*x**n*a*b*c*d*x + x**n*b**2*c**2*x + a**2*c*d*x + a*b*c**2*x 
),x)*a**6*d**6*n - 120*int((x**(2*n)*sqrt(x**n*d + c)*sqrt(x**n*b + a))/(x 
**(2*n)*a*b*d**2*x + x**(2*n)*b**2*c*d*x + x**n*a**2*d**2*x + 2*x**n*a*b*c 
*d*x + x**n*b**2*c**2*x + a**2*c*d*x + a*b*c**2*x),x)*a**5*b*c*d**5*n -...