\(\int \frac {x^{11}}{(a+b x^8)^2 \sqrt {c+d x^8}} \, dx\) [119]

Optimal result

Integrand size = 24, antiderivative size = 93 \[ \int \frac {x^{11}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=-\frac {x^4 \sqrt {c+d x^8}}{8 (b c-a d) \left (a+b x^8\right )}+\frac {c \arctan \left (\frac {\sqrt {b c-a d} x^4}{\sqrt {a} \sqrt {c+d x^8}}\right )}{8 \sqrt {a} (b c-a d)^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.39 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.20 \[ \int \frac {x^{11}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\frac {1}{8} \left (-\frac {x^4 \sqrt {c+d x^8}}{(b c-a d) \left (a+b x^8\right )}+\frac {c \arctan \left (\frac {a \sqrt {d}+b x^4 \left (\sqrt {d} x^4+\sqrt {c+d x^8}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{\sqrt {a} (b c-a d)^{3/2}}\right ) \] Input:

Integrate[x^11/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {965, 373, 27, 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.

\(\displaystyle \int \frac {x^{11}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx\)

\(\Big \downarrow \) 965

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

\(\Big \downarrow \) 373

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 218

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

Input:

Int[x^11/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]
 

Output:

(-1/2*(x^4*Sqrt[c + d*x^8])/((b*c - a*d)*(a + b*x^8)) + (c*ArcTan[(Sqrt[b* 
c - a*d]*x^4)/(Sqrt[a]*Sqrt[c + d*x^8])])/(2*Sqrt[a]*(b*c - a*d)^(3/2)))/4
 

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 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 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 373

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 
1)/(2*(b*c - a*d)*(p + 1))), x] - Simp[e^2/(2*(b*c - a*d)*(p + 1))   Int[(e 
*x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(m - 1) + d*(m + 2*p + 
 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 
 0] && LtQ[p, -1] && GtQ[m, 1] && LeQ[m, 3] && IntBinomialQ[a, b, c, d, e, 
m, 2, p, 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 = 15.38 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.87

method	result	size
pseudoelliptic	\(-\frac {c \left (-\frac {\sqrt {d \,x^{8}+c}\, x^{4}}{c \left (b \,x^{8}+a \right )}+\frac {\operatorname {arctanh}\left (\frac {a \sqrt {d \,x^{8}+c}}{x^{4} \sqrt {a \left (a d -c b \right )}}\right )}{\sqrt {a \left (a d -c b \right )}}\right )}{8 \left (a d -c b \right )}\)	\(81\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (77) = 154\).

Time = 0.20 (sec) , antiderivative size = 426, normalized size of antiderivative = 4.58 \[ \int \frac {x^{11}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\left [-\frac {4 \, \sqrt {d x^{8} + c} {\left (a b c - a^{2} d\right )} x^{4} - {\left (b c x^{8} + a c\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{16} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{8} + a^{2} c^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{12} - a c x^{4}\right )} \sqrt {d x^{8} + c} \sqrt {-a b c + a^{2} d}}{b^{2} x^{16} + 2 \, a b x^{8} + a^{2}}\right )}{32 \, {\left ({\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{8} + a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}}, -\frac {2 \, \sqrt {d x^{8} + c} {\left (a b c - a^{2} d\right )} x^{4} - {\left (b c x^{8} + a c\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{8} - a c\right )} \sqrt {d x^{8} + c} \sqrt {a b c - a^{2} d}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{12} + {\left (a b c^{2} - a^{2} c d\right )} x^{4}\right )}}\right )}{16 \, {\left ({\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{8} + a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}}\right ] \] Input:

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

Output:

[-1/32*(4*sqrt(d*x^8 + c)*(a*b*c - a^2*d)*x^4 - (b*c*x^8 + a*c)*sqrt(-a*b* 
c + a^2*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*c - 2*a*d)*x^12 - a*c*x^4)*sqrt(d*x^8 + c)* 
sqrt(-a*b*c + a^2*d))/(b^2*x^16 + 2*a*b*x^8 + a^2)))/((a*b^3*c^2 - 2*a^2*b 
^2*c*d + a^3*b*d^2)*x^8 + a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2), -1/16*(2*s 
qrt(d*x^8 + c)*(a*b*c - a^2*d)*x^4 - (b*c*x^8 + a*c)*sqrt(a*b*c - a^2*d)*a 
rctan(1/2*((b*c - 2*a*d)*x^8 - a*c)*sqrt(d*x^8 + c)*sqrt(a*b*c - a^2*d)/(( 
a*b*c*d - a^2*d^2)*x^12 + (a*b*c^2 - a^2*c*d)*x^4)))/((a*b^3*c^2 - 2*a^2*b 
^2*c*d + a^3*b*d^2)*x^8 + a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)]
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {x^{11}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int \frac {x^{11}}{\left (a + b x^{8}\right )^{2} \sqrt {c + d x^{8}}}\, dx \] Input:

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

Output:

Integral(x**11/((a + b*x**8)**2*sqrt(c + d*x**8)), x)
 

Maxima [F]

\[ \int \frac {x^{11}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int { \frac {x^{11}}{{\left (b x^{8} + a\right )}^{2} \sqrt {d x^{8} + c}} \,d x } \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (77) = 154\).

Time = 0.36 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.62 \[ \int \frac {x^{11}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\frac {c \sqrt {d} \arctan \left (-\frac {{\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{8 \, \sqrt {a b c d - a^{2} d^{2}} {\left (b c - a d\right )}} + \frac {{\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} b c \sqrt {d} - 2 \, {\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} a d^{\frac {3}{2}} - b c^{2} \sqrt {d}}{4 \, {\left ({\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} a d + b c^{2}\right )} {\left (b^{2} c - a b d\right )}} \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{11}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int \frac {x^{11}}{{\left (b\,x^8+a\right )}^2\,\sqrt {d\,x^8+c}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {x^{11}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\frac {\sqrt {d \,x^{8}+c}\, x^{4}-8 \left (\int \frac {\sqrt {d \,x^{8}+c}\, x^{3}}{2 a \,b^{2} d^{2} x^{24}-b^{3} c d \,x^{24}+4 a^{2} b \,d^{2} x^{16}-b^{3} c^{2} x^{16}+2 a^{3} d^{2} x^{8}+3 a^{2} b c d \,x^{8}-2 a \,b^{2} c^{2} x^{8}+2 a^{3} c d -a^{2} b \,c^{2}}d x \right ) a^{3} c d +4 \left (\int \frac {\sqrt {d \,x^{8}+c}\, x^{3}}{2 a \,b^{2} d^{2} x^{24}-b^{3} c d \,x^{24}+4 a^{2} b \,d^{2} x^{16}-b^{3} c^{2} x^{16}+2 a^{3} d^{2} x^{8}+3 a^{2} b c d \,x^{8}-2 a \,b^{2} c^{2} x^{8}+2 a^{3} c d -a^{2} b \,c^{2}}d x \right ) a^{2} b \,c^{2}-8 \left (\int \frac {\sqrt {d \,x^{8}+c}\, x^{3}}{2 a \,b^{2} d^{2} x^{24}-b^{3} c d \,x^{24}+4 a^{2} b \,d^{2} x^{16}-b^{3} c^{2} x^{16}+2 a^{3} d^{2} x^{8}+3 a^{2} b c d \,x^{8}-2 a \,b^{2} c^{2} x^{8}+2 a^{3} c d -a^{2} b \,c^{2}}d x \right ) a^{2} b c d \,x^{8}+4 \left (\int \frac {\sqrt {d \,x^{8}+c}\, x^{3}}{2 a \,b^{2} d^{2} x^{24}-b^{3} c d \,x^{24}+4 a^{2} b \,d^{2} x^{16}-b^{3} c^{2} x^{16}+2 a^{3} d^{2} x^{8}+3 a^{2} b c d \,x^{8}-2 a \,b^{2} c^{2} x^{8}+2 a^{3} c d -a^{2} b \,c^{2}}d x \right ) a \,b^{2} c^{2} x^{8}}{8 a b d \,x^{8}-4 b^{2} c \,x^{8}+8 a^{2} d -4 a b c} \] Input:

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

Output:

(sqrt(c + d*x**8)*x**4 - 8*int((sqrt(c + d*x**8)*x**3)/(2*a**3*c*d + 2*a** 
3*d**2*x**8 - a**2*b*c**2 + 3*a**2*b*c*d*x**8 + 4*a**2*b*d**2*x**16 - 2*a* 
b**2*c**2*x**8 + 2*a*b**2*d**2*x**24 - b**3*c**2*x**16 - b**3*c*d*x**24),x 
)*a**3*c*d + 4*int((sqrt(c + d*x**8)*x**3)/(2*a**3*c*d + 2*a**3*d**2*x**8 
- a**2*b*c**2 + 3*a**2*b*c*d*x**8 + 4*a**2*b*d**2*x**16 - 2*a*b**2*c**2*x* 
*8 + 2*a*b**2*d**2*x**24 - b**3*c**2*x**16 - b**3*c*d*x**24),x)*a**2*b*c** 
2 - 8*int((sqrt(c + d*x**8)*x**3)/(2*a**3*c*d + 2*a**3*d**2*x**8 - a**2*b* 
c**2 + 3*a**2*b*c*d*x**8 + 4*a**2*b*d**2*x**16 - 2*a*b**2*c**2*x**8 + 2*a* 
b**2*d**2*x**24 - b**3*c**2*x**16 - b**3*c*d*x**24),x)*a**2*b*c*d*x**8 + 4 
*int((sqrt(c + d*x**8)*x**3)/(2*a**3*c*d + 2*a**3*d**2*x**8 - a**2*b*c**2 
+ 3*a**2*b*c*d*x**8 + 4*a**2*b*d**2*x**16 - 2*a*b**2*c**2*x**8 + 2*a*b**2* 
d**2*x**24 - b**3*c**2*x**16 - b**3*c*d*x**24),x)*a*b**2*c**2*x**8)/(4*(2* 
a**2*d - a*b*c + 2*a*b*d*x**8 - b**2*c*x**8))