\(\int \frac {x^8}{(a+b x^6)^2 \sqrt {c+d x^6}} \, dx\) [80]

Optimal result

Integrand size = 24, antiderivative size = 93 \[ \int \frac {x^8}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=-\frac {x^3 \sqrt {c+d x^6}}{6 (b c-a d) \left (a+b x^6\right )}+\frac {c \arctan \left (\frac {\sqrt {b c-a d} x^3}{\sqrt {a} \sqrt {c+d x^6}}\right )}{6 \sqrt {a} (b c-a d)^{3/2}} \] Output:

-1/6*x^3*(d*x^6+c)^(1/2)/(-a*d+b*c)/(b*x^6+a)+1/6*c*arctan((-a*d+b*c)^(1/2 
)*x^3/a^(1/2)/(d*x^6+c)^(1/2))/a^(1/2)/(-a*d+b*c)^(3/2)
 

Mathematica [A] (verified)

Time = 1.00 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.20 \[ \int \frac {x^8}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\frac {1}{6} \left (-\frac {x^3 \sqrt {c+d x^6}}{(b c-a d) \left (a+b x^6\right )}+\frac {c \arctan \left (\frac {a \sqrt {d}+b x^3 \left (\sqrt {d} x^3+\sqrt {c+d x^6}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{\sqrt {a} (b c-a d)^{3/2}}\right ) \] Input:

Integrate[x^8/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]
 

Output:

(-((x^3*Sqrt[c + d*x^6])/((b*c - a*d)*(a + b*x^6))) + (c*ArcTan[(a*Sqrt[d] 
 + b*x^3*(Sqrt[d]*x^3 + Sqrt[c + d*x^6]))/(Sqrt[a]*Sqrt[b*c - a*d])])/(Sqr 
t[a]*(b*c - a*d)^(3/2)))/6
 

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.

\(\displaystyle \int \frac {x^8}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx\)

\(\Big \downarrow \) 965

\(\displaystyle \frac {1}{3} \int \frac {x^6}{\left (b x^6+a\right )^2 \sqrt {d x^6+c}}dx^3\)

\(\Big \downarrow \) 373

\(\displaystyle \frac {1}{3} \left (\frac {\int \frac {c}{\left (b x^6+a\right ) \sqrt {d x^6+c}}dx^3}{2 (b c-a d)}-\frac {x^3 \sqrt {c+d x^6}}{2 \left (a+b x^6\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (\frac {c \int \frac {1}{\left (b x^6+a\right ) \sqrt {d x^6+c}}dx^3}{2 (b c-a d)}-\frac {x^3 \sqrt {c+d x^6}}{2 \left (a+b x^6\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {1}{3} \left (\frac {c \int \frac {1}{a-(a d-b c) x^6}d\frac {x^3}{\sqrt {d x^6+c}}}{2 (b c-a d)}-\frac {x^3 \sqrt {c+d x^6}}{2 \left (a+b x^6\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {1}{3} \left (\frac {c \arctan \left (\frac {x^3 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^6}}\right )}{2 \sqrt {a} (b c-a d)^{3/2}}-\frac {x^3 \sqrt {c+d x^6}}{2 \left (a+b x^6\right ) (b c-a d)}\right )\)

Input:

Int[x^8/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]
 

Output:

(-1/2*(x^3*Sqrt[c + d*x^6])/((b*c - a*d)*(a + b*x^6)) + (c*ArcTan[(Sqrt[b* 
c - a*d]*x^3)/(Sqrt[a]*Sqrt[c + d*x^6])])/(2*Sqrt[a]*(b*c - a*d)^(3/2)))/3
 

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 = 4.54 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.87

method	result	size
pseudoelliptic	\(-\frac {c \left (-\frac {\sqrt {d \,x^{6}+c}\, x^{3}}{c \left (b \,x^{6}+a \right )}+\frac {\operatorname {arctanh}\left (\frac {a \sqrt {d \,x^{6}+c}}{x^{3} \sqrt {a \left (a d -c b \right )}}\right )}{\sqrt {a \left (a d -c b \right )}}\right )}{6 \left (a d -c b \right )}\)	\(81\)

Input:

int(x^8/(b*x^6+a)^2/(d*x^6+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/6*c/(a*d-b*c)*(-(d*x^6+c)^(1/2)*x^3/c/(b*x^6+a)+1/(a*(a*d-b*c))^(1/2)*a 
rctanh(a*(d*x^6+c)^(1/2)/x^3/(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 = 0.16 (sec) , antiderivative size = 426, normalized size of antiderivative = 4.58 \[ \int \frac {x^8}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\left [-\frac {4 \, \sqrt {d x^{6} + c} {\left (a b c - a^{2} d\right )} x^{3} - {\left (b c x^{6} + a c\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{12} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{6} + a^{2} c^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{9} - a c x^{3}\right )} \sqrt {d x^{6} + c} \sqrt {-a b c + a^{2} d}}{b^{2} x^{12} + 2 \, a b x^{6} + a^{2}}\right )}{24 \, {\left ({\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{6} + a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}}, -\frac {2 \, \sqrt {d x^{6} + c} {\left (a b c - a^{2} d\right )} x^{3} - {\left (b c x^{6} + a c\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{6} - a c\right )} \sqrt {d x^{6} + c} \sqrt {a b c - a^{2} d}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{9} + {\left (a b c^{2} - a^{2} c d\right )} x^{3}\right )}}\right )}{12 \, {\left ({\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{6} + a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}}\right ] \] Input:

integrate(x^8/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="fricas")
 

Output:

[-1/24*(4*sqrt(d*x^6 + c)*(a*b*c - a^2*d)*x^3 - (b*c*x^6 + 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^12 - 2*(3*a*b*c^2 - 4* 
a^2*c*d)*x^6 + a^2*c^2 + 4*((b*c - 2*a*d)*x^9 - a*c*x^3)*sqrt(d*x^6 + c)*s 
qrt(-a*b*c + a^2*d))/(b^2*x^12 + 2*a*b*x^6 + a^2)))/((a*b^3*c^2 - 2*a^2*b^ 
2*c*d + a^3*b*d^2)*x^6 + a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2), -1/12*(2*sq 
rt(d*x^6 + c)*(a*b*c - a^2*d)*x^3 - (b*c*x^6 + a*c)*sqrt(a*b*c - a^2*d)*ar 
ctan(1/2*((b*c - 2*a*d)*x^6 - a*c)*sqrt(d*x^6 + c)*sqrt(a*b*c - a^2*d)/((a 
*b*c*d - a^2*d^2)*x^9 + (a*b*c^2 - a^2*c*d)*x^3)))/((a*b^3*c^2 - 2*a^2*b^2 
*c*d + a^3*b*d^2)*x^6 + a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)]
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {x^8}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int \frac {x^{8}}{\left (a + b x^{6}\right )^{2} \sqrt {c + d x^{6}}}\, dx \] Input:

integrate(x**8/(b*x**6+a)**2/(d*x**6+c)**(1/2),x)
 

Output:

Integral(x**8/((a + b*x**6)**2*sqrt(c + d*x**6)), x)
 

Maxima [F]

\[ \int \frac {x^8}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int { \frac {x^{8}}{{\left (b x^{6} + a\right )}^{2} \sqrt {d x^{6} + c}} \,d x } \] Input:

integrate(x^8/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="maxima")
 

Output:

integrate(x^8/((b*x^6 + a)^2*sqrt(d*x^6 + c)), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {x^8}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\text {Timed out} \] Input:

integrate(x^8/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^8}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int \frac {x^8}{{\left (b\,x^6+a\right )}^2\,\sqrt {d\,x^6+c}} \,d x \] Input:

int(x^8/((a + b*x^6)^2*(c + d*x^6)^(1/2)),x)
 

Output:

int(x^8/((a + b*x^6)^2*(c + d*x^6)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {x^8}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\frac {\sqrt {d \,x^{6}+c}\, x^{3}-6 \left (\int \frac {\sqrt {d \,x^{6}+c}\, x^{2}}{2 a \,b^{2} d^{2} x^{18}-b^{3} c d \,x^{18}+4 a^{2} b \,d^{2} x^{12}-b^{3} c^{2} x^{12}+2 a^{3} d^{2} x^{6}+3 a^{2} b c d \,x^{6}-2 a \,b^{2} c^{2} x^{6}+2 a^{3} c d -a^{2} b \,c^{2}}d x \right ) a^{3} c d +3 \left (\int \frac {\sqrt {d \,x^{6}+c}\, x^{2}}{2 a \,b^{2} d^{2} x^{18}-b^{3} c d \,x^{18}+4 a^{2} b \,d^{2} x^{12}-b^{3} c^{2} x^{12}+2 a^{3} d^{2} x^{6}+3 a^{2} b c d \,x^{6}-2 a \,b^{2} c^{2} x^{6}+2 a^{3} c d -a^{2} b \,c^{2}}d x \right ) a^{2} b \,c^{2}-6 \left (\int \frac {\sqrt {d \,x^{6}+c}\, x^{2}}{2 a \,b^{2} d^{2} x^{18}-b^{3} c d \,x^{18}+4 a^{2} b \,d^{2} x^{12}-b^{3} c^{2} x^{12}+2 a^{3} d^{2} x^{6}+3 a^{2} b c d \,x^{6}-2 a \,b^{2} c^{2} x^{6}+2 a^{3} c d -a^{2} b \,c^{2}}d x \right ) a^{2} b c d \,x^{6}+3 \left (\int \frac {\sqrt {d \,x^{6}+c}\, x^{2}}{2 a \,b^{2} d^{2} x^{18}-b^{3} c d \,x^{18}+4 a^{2} b \,d^{2} x^{12}-b^{3} c^{2} x^{12}+2 a^{3} d^{2} x^{6}+3 a^{2} b c d \,x^{6}-2 a \,b^{2} c^{2} x^{6}+2 a^{3} c d -a^{2} b \,c^{2}}d x \right ) a \,b^{2} c^{2} x^{6}}{6 a b d \,x^{6}-3 b^{2} c \,x^{6}+6 a^{2} d -3 a b c} \] Input:

int(x^8/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)
 

Output:

(sqrt(c + d*x**6)*x**3 - 6*int((sqrt(c + d*x**6)*x**2)/(2*a**3*c*d + 2*a** 
3*d**2*x**6 - a**2*b*c**2 + 3*a**2*b*c*d*x**6 + 4*a**2*b*d**2*x**12 - 2*a* 
b**2*c**2*x**6 + 2*a*b**2*d**2*x**18 - b**3*c**2*x**12 - b**3*c*d*x**18),x 
)*a**3*c*d + 3*int((sqrt(c + d*x**6)*x**2)/(2*a**3*c*d + 2*a**3*d**2*x**6 
- a**2*b*c**2 + 3*a**2*b*c*d*x**6 + 4*a**2*b*d**2*x**12 - 2*a*b**2*c**2*x* 
*6 + 2*a*b**2*d**2*x**18 - b**3*c**2*x**12 - b**3*c*d*x**18),x)*a**2*b*c** 
2 - 6*int((sqrt(c + d*x**6)*x**2)/(2*a**3*c*d + 2*a**3*d**2*x**6 - a**2*b* 
c**2 + 3*a**2*b*c*d*x**6 + 4*a**2*b*d**2*x**12 - 2*a*b**2*c**2*x**6 + 2*a* 
b**2*d**2*x**18 - b**3*c**2*x**12 - b**3*c*d*x**18),x)*a**2*b*c*d*x**6 + 3 
*int((sqrt(c + d*x**6)*x**2)/(2*a**3*c*d + 2*a**3*d**2*x**6 - a**2*b*c**2 
+ 3*a**2*b*c*d*x**6 + 4*a**2*b*d**2*x**12 - 2*a*b**2*c**2*x**6 + 2*a*b**2* 
d**2*x**18 - b**3*c**2*x**12 - b**3*c*d*x**18),x)*a*b**2*c**2*x**6)/(3*(2* 
a**2*d - a*b*c + 2*a*b*d*x**6 - b**2*c*x**6))