Integrand size = 24, antiderivative size = 93 \[ \int \frac {x^8}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=-\frac {x^3 \sqrt {c+d x^6}}{6 (b c-a d) \left (a+b x^6\right )}+\frac {c \arctan \left (\frac {\sqrt {b c-a d} x^3}{\sqrt {a} \sqrt {c+d x^6}}\right )}{6 \sqrt {a} (b c-a d)^{3/2}} \]
1/6*c*arctan(x^3*(-a*d+b*c)^(1/2)/a^(1/2)/(d*x^6+c)^(1/2))/(-a*d+b*c)^(3/2 )/a^(1/2)-1/6*x^3*(d*x^6+c)^(1/2)/(-a*d+b*c)/(b*x^6+a)
Time = 1.43 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.20 \[ \int \frac {x^8}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\frac {1}{6} \left (-\frac {x^3 \sqrt {c+d x^6}}{(b c-a d) \left (a+b x^6\right )}+\frac {c \arctan \left (\frac {a \sqrt {d}+b x^3 \left (\sqrt {d} x^3+\sqrt {c+d x^6}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{\sqrt {a} (b c-a d)^{3/2}}\right ) \]
(-((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
Time = 0.23 (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 |
\(\displaystyle \frac {1}{3} \left (\frac {c \arctan \left (\frac {x^3 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^6}}\right )}{2 \sqrt {a} (b c-a d)^{3/2}}-\frac {x^3 \sqrt {c+d x^6}}{2 \left (a+b x^6\right ) (b c-a d)}\right )\) |
(-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
3.9.76.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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]
Time = 9.34 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.87
method | result | size |
pseudoelliptic | \(-\frac {c \left (-\frac {\sqrt {d \,x^{6}+c}\, x^{3}}{c \left (b \,x^{6}+a \right )}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{6}+c}\, a}{x^{3} \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )}{6 \left (a d -b c \right )}\) | \(81\) |
-1/6*c/(a*d-b*c)*(-(d*x^6+c)^(1/2)*x^3/c/(b*x^6+a)+1/((a*d-b*c)*a)^(1/2)*a rctanh((d*x^6+c)^(1/2)/x^3*a/((a*d-b*c)*a)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (77) = 154\).
Time = 0.36 (sec) , antiderivative size = 426, normalized size of antiderivative = 4.58 \[ \int \frac {x^8}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\left [-\frac {4 \, \sqrt {d x^{6} + c} {\left (a b c - a^{2} d\right )} x^{3} - {\left (b c x^{6} + a c\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{12} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{6} + a^{2} c^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{9} - a c x^{3}\right )} \sqrt {d x^{6} + c} \sqrt {-a b c + a^{2} d}}{b^{2} x^{12} + 2 \, a b x^{6} + a^{2}}\right )}{24 \, {\left ({\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{6} + a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}}, -\frac {2 \, \sqrt {d x^{6} + c} {\left (a b c - a^{2} d\right )} x^{3} - {\left (b c x^{6} + a c\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{6} - a c\right )} \sqrt {d x^{6} + c} \sqrt {a b c - a^{2} d}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{9} + {\left (a b c^{2} - a^{2} c d\right )} x^{3}\right )}}\right )}{12 \, {\left ({\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{6} + a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}}\right ] \]
[-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)]
\[ \int \frac {x^8}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int \frac {x^{8}}{\left (a + b x^{6}\right )^{2} \sqrt {c + d x^{6}}}\, dx \]
\[ \int \frac {x^8}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int { \frac {x^{8}}{{\left (b x^{6} + a\right )}^{2} \sqrt {d x^{6} + c}} \,d x } \]
\[ \int \frac {x^8}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int { \frac {x^{8}}{{\left (b x^{6} + a\right )}^{2} \sqrt {d x^{6} + c}} \,d x } \]
Timed out. \[ \int \frac {x^8}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int \frac {x^8}{{\left (b\,x^6+a\right )}^2\,\sqrt {d\,x^6+c}} \,d x \]