\(\int \frac {x^{11}}{(a+b x^8) \sqrt {c+d x^8}} \, dx\) [97]

Optimal result

Integrand size = 24, antiderivative size = 91 \[ \int \frac {x^{11}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=-\frac {\sqrt {a} \arctan \left (\frac {\sqrt {b c-a d} x^4}{\sqrt {a} \sqrt {c+d x^8}}\right )}{4 b \sqrt {b c-a d}}+\frac {\text {arctanh}\left (\frac {\sqrt {d} x^4}{\sqrt {c+d x^8}}\right )}{4 b \sqrt {d}} \] Output:

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

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.19 \[ \int \frac {x^{11}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\frac {-\frac {\sqrt {a} \arctan \left (\frac {a \sqrt {d}+b x^4 \left (\sqrt {d} x^4+\sqrt {c+d x^8}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{\sqrt {b c-a d}}+\frac {\log \left (\sqrt {d} x^4+\sqrt {c+d x^8}\right )}{\sqrt {d}}}{4 b} \] Input:

Integrate[x^11/((a + b*x^8)*Sqrt[c + d*x^8]),x]
 

Output:

(-((Sqrt[a]*ArcTan[(a*Sqrt[d] + b*x^4*(Sqrt[d]*x^4 + Sqrt[c + d*x^8]))/(Sq 
rt[a]*Sqrt[b*c - a*d])])/Sqrt[b*c - a*d]) + Log[Sqrt[d]*x^4 + Sqrt[c + d*x 
^8]]/Sqrt[d])/(4*b)
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {965, 385, 224, 219, 291, 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.

\(\displaystyle \int \frac {x^{11}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx\)

\(\Big \downarrow \) 965

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

\(\Big \downarrow \) 385

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 218

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

Input:

Int[x^11/((a + b*x^8)*Sqrt[c + d*x^8]),x]
 

Output:

(-((Sqrt[a]*ArcTan[(Sqrt[b*c - a*d]*x^4)/(Sqrt[a]*Sqrt[c + d*x^8])])/(b*Sq 
rt[b*c - a*d])) + ArcTanh[(Sqrt[d]*x^4)/Sqrt[c + d*x^8]]/(b*Sqrt[d]))/4
 

Defintions of rubi rules used

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

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

rule 291

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

rule 385

Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2), 
x_Symbol] :> Simp[e^2/b   Int[(e*x)^(m - 2)*(c + d*x^2)^q, x], x] - Simp[a* 
(e^2/b)   Int[(e*x)^(m - 2)*((c + d*x^2)^q/(a + b*x^2)), x], x] /; FreeQ[{a 
, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && LeQ[2, m, 3] && IntBinomial 
Q[a, b, c, d, e, m, 2, -1, q, x]
 

rule 965

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

Maple [A] (verified)

Time = 7.34 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.77

method	result	size
pseudoelliptic	\(-\frac {\frac {a \,\operatorname {arctanh}\left (\frac {a \sqrt {d \,x^{8}+c}}{x^{4} \sqrt {a \left (a d -c b \right )}}\right )}{\sqrt {a \left (a d -c b \right )}}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{8}+c}}{x^{4} \sqrt {d}}\right )}{\sqrt {d}}}{4 b}\)	\(70\)

Input:

int(x^11/(b*x^8+a)/(d*x^8+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 638, normalized size of antiderivative = 7.01 \[ \int \frac {x^{11}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\left [\frac {d \sqrt {-\frac {a}{b c - a d}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{16} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{8} + a^{2} c^{2} - 4 \, {\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{12} - {\left (a b c^{2} - a^{2} c d\right )} x^{4}\right )} \sqrt {d x^{8} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{16} + 2 \, a b x^{8} + a^{2}}\right ) + 2 \, \sqrt {d} \log \left (-2 \, d x^{8} - 2 \, \sqrt {d x^{8} + c} \sqrt {d} x^{4} - c\right )}{16 \, b d}, \frac {d \sqrt {-\frac {a}{b c - a d}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{16} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{8} + a^{2} c^{2} - 4 \, {\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{12} - {\left (a b c^{2} - a^{2} c d\right )} x^{4}\right )} \sqrt {d x^{8} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{16} + 2 \, a b x^{8} + a^{2}}\right ) - 4 \, \sqrt {-d} \arctan \left (\frac {\sqrt {d x^{8} + c} \sqrt {-d}}{d x^{4}}\right )}{16 \, b d}, \frac {d \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{8} - a c\right )} \sqrt {d x^{8} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{12} + a c x^{4}\right )}}\right ) + \sqrt {d} \log \left (-2 \, d x^{8} - 2 \, \sqrt {d x^{8} + c} \sqrt {d} x^{4} - c\right )}{8 \, b d}, \frac {d \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{8} - a c\right )} \sqrt {d x^{8} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{12} + a c x^{4}\right )}}\right ) - 2 \, \sqrt {-d} \arctan \left (\frac {\sqrt {d x^{8} + c} \sqrt {-d}}{d x^{4}}\right )}{8 \, b d}\right ] \] Input:

integrate(x^11/(b*x^8+a)/(d*x^8+c)^(1/2),x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

[1/16*(d*sqrt(-a/(b*c - a*d))*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^16 
- 2*(3*a*b*c^2 - 4*a^2*c*d)*x^8 + a^2*c^2 - 4*((b^2*c^2 - 3*a*b*c*d + 2*a^ 
2*d^2)*x^12 - (a*b*c^2 - a^2*c*d)*x^4)*sqrt(d*x^8 + c)*sqrt(-a/(b*c - a*d) 
))/(b^2*x^16 + 2*a*b*x^8 + a^2)) + 2*sqrt(d)*log(-2*d*x^8 - 2*sqrt(d*x^8 + 
 c)*sqrt(d)*x^4 - c))/(b*d), 1/16*(d*sqrt(-a/(b*c - a*d))*log(((b^2*c^2 - 
8*a*b*c*d + 8*a^2*d^2)*x^16 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^8 + a^2*c^2 - 4* 
((b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x^12 - (a*b*c^2 - a^2*c*d)*x^4)*sqrt(d* 
x^8 + c)*sqrt(-a/(b*c - a*d)))/(b^2*x^16 + 2*a*b*x^8 + a^2)) - 4*sqrt(-d)* 
arctan(sqrt(d*x^8 + c)*sqrt(-d)/(d*x^4)))/(b*d), 1/8*(d*sqrt(a/(b*c - a*d) 
)*arctan(-1/2*((b*c - 2*a*d)*x^8 - a*c)*sqrt(d*x^8 + c)*sqrt(a/(b*c - a*d) 
)/(a*d*x^12 + a*c*x^4)) + sqrt(d)*log(-2*d*x^8 - 2*sqrt(d*x^8 + c)*sqrt(d) 
*x^4 - c))/(b*d), 1/8*(d*sqrt(a/(b*c - a*d))*arctan(-1/2*((b*c - 2*a*d)*x^ 
8 - a*c)*sqrt(d*x^8 + c)*sqrt(a/(b*c - a*d))/(a*d*x^12 + a*c*x^4)) - 2*sqr 
t(-d)*arctan(sqrt(d*x^8 + c)*sqrt(-d)/(d*x^4)))/(b*d)]
 

Sympy [F]

\[ \int \frac {x^{11}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\int \frac {x^{11}}{\left (a + b x^{8}\right ) \sqrt {c + d x^{8}}}\, dx \] Input:

integrate(x**11/(b*x**8+a)/(d*x**8+c)**(1/2),x)
 

Output:

Integral(x**11/((a + b*x**8)*sqrt(c + d*x**8)), x)
 

Maxima [F]

\[ \int \frac {x^{11}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\int { \frac {x^{11}}{{\left (b x^{8} + a\right )} \sqrt {d x^{8} + c}} \,d x } \] Input:

integrate(x^11/(b*x^8+a)/(d*x^8+c)^(1/2),x, algorithm="maxima")
 

Output:

integrate(x^11/((b*x^8 + a)*sqrt(d*x^8 + c)), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^{11}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^11/(b*x^8+a)/(d*x^8+c)^(1/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{11}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\int \frac {x^{11}}{\left (b\,x^8+a\right )\,\sqrt {d\,x^8+c}} \,d x \] Input:

int(x^11/((a + b*x^8)*(c + d*x^8)^(1/2)),x)
 

Output:

int(x^11/((a + b*x^8)*(c + d*x^8)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {x^{11}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\frac {-\sqrt {d}\, \mathrm {log}\left (\sqrt {d \,x^{8}+c}-\sqrt {d}\, x^{4}\right )+\sqrt {d}\, \mathrm {log}\left (\sqrt {d \,x^{8}+c}+\sqrt {d}\, x^{4}\right )-8 \left (\int \frac {\sqrt {d \,x^{8}+c}\, x^{3}}{b d \,x^{16}+a d \,x^{8}+b c \,x^{8}+a c}d x \right ) a d}{8 b d} \] Input:

int(x^11/(b*x^8+a)/(d*x^8+c)^(1/2),x)
 

Output:

( - sqrt(d)*log(sqrt(c + d*x**8) - sqrt(d)*x**4) + sqrt(d)*log(sqrt(c + d* 
x**8) + sqrt(d)*x**4) - 8*int((sqrt(c + d*x**8)*x**3)/(a*c + a*d*x**8 + b* 
c*x**8 + b*d*x**16),x)*a*d)/(8*b*d)