\(\int \frac {1}{x (a+b x^8)^2 \sqrt {c+d x^8}} \, dx\) [116]

Optimal result

Integrand size = 24, antiderivative size = 132 \[ \int \frac {1}{x \left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\frac {b \sqrt {c+d x^8}}{8 a (b c-a d) \left (a+b x^8\right )}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x^8}}{\sqrt {c}}\right )}{4 a^2 \sqrt {c}}+\frac {\sqrt {b} (2 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{8 a^2 (b c-a d)^{3/2}} \] Output:

1/8*b*(d*x^8+c)^(1/2)/a/(-a*d+b*c)/(b*x^8+a)-1/4*arctanh((d*x^8+c)^(1/2)/c 
^(1/2))/a^2/c^(1/2)+1/8*b^(1/2)*(-3*a*d+2*b*c)*arctanh(b^(1/2)*(d*x^8+c)^( 
1/2)/(-a*d+b*c)^(1/2))/a^2/(-a*d+b*c)^(3/2)
 

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x \left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\frac {-\frac {a b \sqrt {c+d x^8}}{(-b c+a d) \left (a+b x^8\right )}+\frac {\sqrt {b} (2 b c-3 a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{3/2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c+d x^8}}{\sqrt {c}}\right )}{\sqrt {c}}}{8 a^2} \] Input:

Integrate[1/(x*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]
 

Output:

(-((a*b*Sqrt[c + d*x^8])/((-(b*c) + a*d)*(a + b*x^8))) + (Sqrt[b]*(2*b*c - 
 3*a*d)*ArcTan[(Sqrt[b]*Sqrt[c + d*x^8])/Sqrt[-(b*c) + a*d]])/(-(b*c) + a* 
d)^(3/2) - (2*ArcTanh[Sqrt[c + d*x^8]/Sqrt[c]])/Sqrt[c])/(8*a^2)
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {948, 114, 27, 174, 73, 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.

\(\displaystyle \int \frac {1}{x \left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx\)

\(\Big \downarrow \) 948

\(\displaystyle \frac {1}{8} \int \frac {1}{x^8 \left (b x^8+a\right )^2 \sqrt {d x^8+c}}dx^8\)

\(\Big \downarrow \) 114

\(\displaystyle \frac {1}{8} \left (\frac {\int \frac {b d x^8+2 b c-2 a d}{2 x^8 \left (b x^8+a\right ) \sqrt {d x^8+c}}dx^8}{a (b c-a d)}+\frac {b \sqrt {c+d x^8}}{a \left (a+b x^8\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \left (\frac {\int \frac {b d x^8+2 (b c-a d)}{x^8 \left (b x^8+a\right ) \sqrt {d x^8+c}}dx^8}{2 a (b c-a d)}+\frac {b \sqrt {c+d x^8}}{a \left (a+b x^8\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {1}{8} \left (\frac {\frac {2 (b c-a d) \int \frac {1}{x^8 \sqrt {d x^8+c}}dx^8}{a}-\frac {b (2 b c-3 a d) \int \frac {1}{\left (b x^8+a\right ) \sqrt {d x^8+c}}dx^8}{a}}{2 a (b c-a d)}+\frac {b \sqrt {c+d x^8}}{a \left (a+b x^8\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{8} \left (\frac {\frac {4 (b c-a d) \int \frac {1}{\frac {x^{16}}{d}-\frac {c}{d}}d\sqrt {d x^8+c}}{a d}-\frac {2 b (2 b c-3 a d) \int \frac {1}{\frac {b x^{16}}{d}+a-\frac {b c}{d}}d\sqrt {d x^8+c}}{a d}}{2 a (b c-a d)}+\frac {b \sqrt {c+d x^8}}{a \left (a+b x^8\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{8} \left (\frac {\frac {2 \sqrt {b} (2 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{a \sqrt {b c-a d}}-\frac {4 (b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x^8}}{\sqrt {c}}\right )}{a \sqrt {c}}}{2 a (b c-a d)}+\frac {b \sqrt {c+d x^8}}{a \left (a+b x^8\right ) (b c-a d)}\right )\)

Input:

Int[1/(x*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]
 

Output:

((b*Sqrt[c + d*x^8])/(a*(b*c - a*d)*(a + b*x^8)) + ((-4*(b*c - a*d)*ArcTan 
h[Sqrt[c + d*x^8]/Sqrt[c]])/(a*Sqrt[c]) + (2*Sqrt[b]*(2*b*c - 3*a*d)*ArcTa 
nh[(Sqrt[b]*Sqrt[c + d*x^8])/Sqrt[b*c - a*d]])/(a*Sqrt[b*c - a*d]))/(2*a*( 
b*c - a*d)))/8
 

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 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 114

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

rule 174

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

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 [A] (verified)

Time = 0.45 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.11

method	result	size
pseudoelliptic	\(-\frac {-2 \left (b \,x^{8}+a \right ) \sqrt {c}\, b \left (c b -\frac {3 a d}{2}\right ) \arctan \left (\frac {\sqrt {d \,x^{8}+c}\, b}{\sqrt {\left (a d -c b \right ) b}}\right )+\left (2 \left (a d -c b \right ) \left (b \,x^{8}+a \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{8}+c}}{\sqrt {c}}\right )+\sqrt {d \,x^{8}+c}\, \sqrt {c}\, a b \right ) \sqrt {\left (a d -c b \right ) b}}{8 \sqrt {c}\, \sqrt {\left (a d -c b \right ) b}\, a^{2} \left (a d -c b \right ) \left (b \,x^{8}+a \right )}\)	\(146\)

Input:

int(1/x/(b*x^8+a)^2/(d*x^8+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/8/c^(1/2)*(-2*(b*x^8+a)*c^(1/2)*b*(c*b-3/2*a*d)*arctan((d*x^8+c)^(1/2)* 
b/((a*d-b*c)*b)^(1/2))+(2*(a*d-b*c)*(b*x^8+a)*arctanh((d*x^8+c)^(1/2)/c^(1 
/2))+(d*x^8+c)^(1/2)*c^(1/2)*a*b)*((a*d-b*c)*b)^(1/2))/((a*d-b*c)*b)^(1/2) 
/a^2/(a*d-b*c)/(b*x^8+a)
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 816, normalized size of antiderivative = 6.18 \[ \int \frac {1}{x \left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/16*(2*sqrt(d*x^8 + c)*a*b*c + ((2*b^2*c^2 - 3*a*b*c*d)*x^8 + 2*a*b*c^2 
- 3*a^2*c*d)*sqrt(b/(b*c - a*d))*log((b*d*x^8 + 2*b*c - a*d + 2*sqrt(d*x^8 
 + c)*(b*c - a*d)*sqrt(b/(b*c - a*d)))/(b*x^8 + a)) + 2*((b^2*c - a*b*d)*x 
^8 + a*b*c - a^2*d)*sqrt(c)*log((d*x^8 - 2*sqrt(d*x^8 + c)*sqrt(c) + 2*c)/ 
x^8))/((a^2*b^2*c^2 - a^3*b*c*d)*x^8 + a^3*b*c^2 - a^4*c*d), 1/8*(sqrt(d*x 
^8 + c)*a*b*c - ((2*b^2*c^2 - 3*a*b*c*d)*x^8 + 2*a*b*c^2 - 3*a^2*c*d)*sqrt 
(-b/(b*c - a*d))*arctan(sqrt(d*x^8 + c)*sqrt(-b/(b*c - a*d))) + ((b^2*c - 
a*b*d)*x^8 + a*b*c - a^2*d)*sqrt(c)*log((d*x^8 - 2*sqrt(d*x^8 + c)*sqrt(c) 
 + 2*c)/x^8))/((a^2*b^2*c^2 - a^3*b*c*d)*x^8 + a^3*b*c^2 - a^4*c*d), 1/16* 
(2*sqrt(d*x^8 + c)*a*b*c + 4*((b^2*c - a*b*d)*x^8 + a*b*c - a^2*d)*sqrt(-c 
)*arctan(sqrt(-c)/sqrt(d*x^8 + c)) + ((2*b^2*c^2 - 3*a*b*c*d)*x^8 + 2*a*b* 
c^2 - 3*a^2*c*d)*sqrt(b/(b*c - a*d))*log((b*d*x^8 + 2*b*c - a*d + 2*sqrt(d 
*x^8 + c)*(b*c - a*d)*sqrt(b/(b*c - a*d)))/(b*x^8 + a)))/((a^2*b^2*c^2 - a 
^3*b*c*d)*x^8 + a^3*b*c^2 - a^4*c*d), 1/8*(sqrt(d*x^8 + c)*a*b*c - ((2*b^2 
*c^2 - 3*a*b*c*d)*x^8 + 2*a*b*c^2 - 3*a^2*c*d)*sqrt(-b/(b*c - a*d))*arctan 
(sqrt(d*x^8 + c)*sqrt(-b/(b*c - a*d))) + 2*((b^2*c - a*b*d)*x^8 + a*b*c - 
a^2*d)*sqrt(-c)*arctan(sqrt(-c)/sqrt(d*x^8 + c)))/((a^2*b^2*c^2 - a^3*b*c* 
d)*x^8 + a^3*b*c^2 - a^4*c*d)]
 

Sympy [F]

\[ \int \frac {1}{x \left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int \frac {1}{x \left (a + b x^{8}\right )^{2} \sqrt {c + d x^{8}}}\, dx \] Input:

integrate(1/x/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)
 

Output:

Integral(1/(x*(a + b*x**8)**2*sqrt(c + d*x**8)), x)
 

Maxima [F]

\[ \int \frac {1}{x \left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int { \frac {1}{{\left (b x^{8} + a\right )}^{2} \sqrt {d x^{8} + c} x} \,d x } \] Input:

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

Output:

integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x), x)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x \left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\frac {\sqrt {d x^{8} + c} b d}{8 \, {\left (a b c - a^{2} d\right )} {\left ({\left (d x^{8} + c\right )} b - b c + a d\right )}} - \frac {{\left (2 \, b^{2} c - 3 \, a b d\right )} \arctan \left (\frac {\sqrt {d x^{8} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{8 \, {\left (a^{2} b c - a^{3} d\right )} \sqrt {-b^{2} c + a b d}} + \frac {\arctan \left (\frac {\sqrt {d x^{8} + c}}{\sqrt {-c}}\right )}{4 \, a^{2} \sqrt {-c}} \] Input:

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

Output:

1/8*sqrt(d*x^8 + c)*b*d/((a*b*c - a^2*d)*((d*x^8 + c)*b - b*c + a*d)) - 1/ 
8*(2*b^2*c - 3*a*b*d)*arctan(sqrt(d*x^8 + c)*b/sqrt(-b^2*c + a*b*d))/((a^2 
*b*c - a^3*d)*sqrt(-b^2*c + a*b*d)) + 1/4*arctan(sqrt(d*x^8 + c)/sqrt(-c)) 
/(a^2*sqrt(-c))
 

Mupad [B] (verification not implemented)

Time = 5.08 (sec) , antiderivative size = 3017, normalized size of antiderivative = 22.86 \[ \int \frac {1}{x \left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\text {Too large to display} \] Input:

int(1/(x*(a + b*x^8)^2*(c + d*x^8)^(1/2)),x)
 

Output:

(atan((((((c + d*x^8)^(1/2)*(13*a^2*b^3*d^4 + 8*b^5*c^2*d^2 - 20*a*b^4*c*d 
^3))/(32*(a^4*d^2 + a^2*b^2*c^2 - 2*a^3*b*c*d)) - (((a^6*b^2*d^5 - (3*a^5* 
b^3*c*d^4)/2 + (a^4*b^4*c^2*d^3)/2)/(a^5*d^2 + a^3*b^2*c^2 - 2*a^4*b*c*d) 
- ((c + d*x^8)^(1/2)*(3*a*d - 2*b*c)*(-b*(a*d - b*c)^3)^(1/2)*(256*a^7*b^2 
*d^5 - 1024*a^6*b^3*c*d^4 - 512*a^4*b^5*c^3*d^2 + 1280*a^5*b^4*c^2*d^3))/( 
512*(a^4*d^2 + a^2*b^2*c^2 - 2*a^3*b*c*d)*(a^5*d^3 - a^2*b^3*c^3 + 3*a^3*b 
^2*c^2*d - 3*a^4*b*c*d^2)))*(3*a*d - 2*b*c)*(-b*(a*d - b*c)^3)^(1/2))/(16* 
(a^5*d^3 - a^2*b^3*c^3 + 3*a^3*b^2*c^2*d - 3*a^4*b*c*d^2)))*(3*a*d - 2*b*c 
)*(-b*(a*d - b*c)^3)^(1/2)*1i)/(16*(a^5*d^3 - a^2*b^3*c^3 + 3*a^3*b^2*c^2* 
d - 3*a^4*b*c*d^2)) + ((((c + d*x^8)^(1/2)*(13*a^2*b^3*d^4 + 8*b^5*c^2*d^2 
 - 20*a*b^4*c*d^3))/(32*(a^4*d^2 + a^2*b^2*c^2 - 2*a^3*b*c*d)) + (((a^6*b^ 
2*d^5 - (3*a^5*b^3*c*d^4)/2 + (a^4*b^4*c^2*d^3)/2)/(a^5*d^2 + a^3*b^2*c^2 
- 2*a^4*b*c*d) + ((c + d*x^8)^(1/2)*(3*a*d - 2*b*c)*(-b*(a*d - b*c)^3)^(1/ 
2)*(256*a^7*b^2*d^5 - 1024*a^6*b^3*c*d^4 - 512*a^4*b^5*c^3*d^2 + 1280*a^5* 
b^4*c^2*d^3))/(512*(a^4*d^2 + a^2*b^2*c^2 - 2*a^3*b*c*d)*(a^5*d^3 - a^2*b^ 
3*c^3 + 3*a^3*b^2*c^2*d - 3*a^4*b*c*d^2)))*(3*a*d - 2*b*c)*(-b*(a*d - b*c) 
^3)^(1/2))/(16*(a^5*d^3 - a^2*b^3*c^3 + 3*a^3*b^2*c^2*d - 3*a^4*b*c*d^2))) 
*(3*a*d - 2*b*c)*(-b*(a*d - b*c)^3)^(1/2)*1i)/(16*(a^5*d^3 - a^2*b^3*c^3 + 
 3*a^3*b^2*c^2*d - 3*a^4*b*c*d^2)))/(((3*a*b^3*d^4)/128 - (b^4*c*d^3)/64)/ 
(a^5*d^2 + a^3*b^2*c^2 - 2*a^4*b*c*d) - ((((c + d*x^8)^(1/2)*(13*a^2*b^...
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {1}{x \left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int \frac {1}{x \left (b \,x^{8}+a \right )^{2} \sqrt {d \,x^{8}+c}}d x \] Input:

int(1/x/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)
 

Output:

int(1/x/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)