\(\int \frac {x^8}{(a+b x^6) \sqrt {c+d x^6}} \, dx\) [63]

Optimal result

Integrand size = 24, antiderivative size = 91 \[ \int \frac {x^8}{\left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=-\frac {\sqrt {a} \arctan \left (\frac {\sqrt {b c-a d} x^3}{\sqrt {a} \sqrt {c+d x^6}}\right )}{3 b \sqrt {b c-a d}}+\frac {\text {arctanh}\left (\frac {\sqrt {d} x^3}{\sqrt {c+d x^6}}\right )}{3 b \sqrt {d}} \] Output:

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

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.19 \[ \int \frac {x^8}{\left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\frac {-\frac {\sqrt {a} \arctan \left (\frac {a \sqrt {d}+b x^3 \left (\sqrt {d} x^3+\sqrt {c+d x^6}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{\sqrt {b c-a d}}+\frac {\log \left (\sqrt {d} x^3+\sqrt {c+d x^6}\right )}{\sqrt {d}}}{3 b} \] Input:

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

Output:

(-((Sqrt[a]*ArcTan[(a*Sqrt[d] + b*x^3*(Sqrt[d]*x^3 + Sqrt[c + d*x^6]))/(Sq 
rt[a]*Sqrt[b*c - a*d])])/Sqrt[b*c - a*d]) + Log[Sqrt[d]*x^3 + Sqrt[c + d*x 
^6]]/Sqrt[d])/(3*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.

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

\(\Big \downarrow \) 965

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

\(\Big \downarrow \) 385

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

(-((Sqrt[a]*ArcTan[(Sqrt[b*c - a*d]*x^3)/(Sqrt[a]*Sqrt[c + d*x^6])])/(b*Sq 
rt[b*c - a*d])) + ArcTanh[(Sqrt[d]*x^3)/Sqrt[c + d*x^6]]/(b*Sqrt[d]))/3
 

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 = 2.20 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.77

method	result	size
pseudoelliptic	\(-\frac {\frac {a \,\operatorname {arctanh}\left (\frac {a \sqrt {d \,x^{6}+c}}{x^{3} \sqrt {a \left (a d -c b \right )}}\right )}{\sqrt {a \left (a d -c b \right )}}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{6}+c}}{x^{3} \sqrt {d}}\right )}{\sqrt {d}}}{3 b}\)	\(70\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 632, normalized size of antiderivative = 6.95 \[ \int \frac {x^8}{\left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\left [\frac {d \sqrt {-\frac {a}{b c - a d}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{12} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{6} + a^{2} c^{2} - 4 \, {\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{9} - {\left (a b c^{2} - a^{2} c d\right )} x^{3}\right )} \sqrt {d x^{6} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{12} + 2 \, a b x^{6} + a^{2}}\right ) + 2 \, \sqrt {d} \log \left (-2 \, d x^{6} - 2 \, \sqrt {d x^{6} + c} \sqrt {d} x^{3} - c\right )}{12 \, b d}, \frac {d \sqrt {-\frac {a}{b c - a d}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{12} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{6} + a^{2} c^{2} - 4 \, {\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{9} - {\left (a b c^{2} - a^{2} c d\right )} x^{3}\right )} \sqrt {d x^{6} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{12} + 2 \, a b x^{6} + a^{2}}\right ) - 4 \, \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x^{3}}{\sqrt {d x^{6} + c}}\right )}{12 \, b d}, \frac {d \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{6} - a c\right )} \sqrt {d x^{6} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{9} + a c x^{3}\right )}}\right ) + \sqrt {d} \log \left (-2 \, d x^{6} - 2 \, \sqrt {d x^{6} + c} \sqrt {d} x^{3} - c\right )}{6 \, b d}, \frac {d \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{6} - a c\right )} \sqrt {d x^{6} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{9} + a c x^{3}\right )}}\right ) - 2 \, \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x^{3}}{\sqrt {d x^{6} + c}}\right )}{6 \, b d}\right ] \] Input:

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

Output:

[1/12*(d*sqrt(-a/(b*c - a*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^2*c^2 - 3*a*b*c*d + 2*a^ 
2*d^2)*x^9 - (a*b*c^2 - a^2*c*d)*x^3)*sqrt(d*x^6 + c)*sqrt(-a/(b*c - a*d)) 
)/(b^2*x^12 + 2*a*b*x^6 + a^2)) + 2*sqrt(d)*log(-2*d*x^6 - 2*sqrt(d*x^6 + 
c)*sqrt(d)*x^3 - c))/(b*d), 1/12*(d*sqrt(-a/(b*c - a*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^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x^9 - (a*b*c^2 - a^2*c*d)*x^3)*sqrt(d*x^ 
6 + c)*sqrt(-a/(b*c - a*d)))/(b^2*x^12 + 2*a*b*x^6 + a^2)) - 4*sqrt(-d)*ar 
ctan(sqrt(-d)*x^3/sqrt(d*x^6 + c)))/(b*d), 1/6*(d*sqrt(a/(b*c - a*d))*arct 
an(-1/2*((b*c - 2*a*d)*x^6 - a*c)*sqrt(d*x^6 + c)*sqrt(a/(b*c - a*d))/(a*d 
*x^9 + a*c*x^3)) + sqrt(d)*log(-2*d*x^6 - 2*sqrt(d*x^6 + c)*sqrt(d)*x^3 - 
c))/(b*d), 1/6*(d*sqrt(a/(b*c - a*d))*arctan(-1/2*((b*c - 2*a*d)*x^6 - a*c 
)*sqrt(d*x^6 + c)*sqrt(a/(b*c - a*d))/(a*d*x^9 + a*c*x^3)) - 2*sqrt(-d)*ar 
ctan(sqrt(-d)*x^3/sqrt(d*x^6 + c)))/(b*d)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (71) = 142\).

Time = 0.13 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.71 \[ \int \frac {x^8}{\left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=-\frac {{\left (a \sqrt {-d} \arctan \left (\frac {a \sqrt {d}}{\sqrt {a b c - a^{2} d}}\right ) - \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {d}}{\sqrt {-d}}\right )\right )} \mathrm {sgn}\left (x\right )}{3 \, \sqrt {a b c - a^{2} d} b \sqrt {-d}} + \frac {a \arctan \left (\frac {a \sqrt {d + \frac {c}{x^{6}}}}{\sqrt {a b c - a^{2} d}}\right )}{3 \, \sqrt {a b c - a^{2} d} b \mathrm {sgn}\left (x\right )} - \frac {\arctan \left (\frac {\sqrt {d + \frac {c}{x^{6}}}}{\sqrt {-d}}\right )}{3 \, b \sqrt {-d} \mathrm {sgn}\left (x\right )} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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