Integrand size = 24, antiderivative size = 91 \[ \int \frac {x^8}{\left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=-\frac {\sqrt {a} \arctan \left (\frac {\sqrt {b c-a d} x^3}{\sqrt {a} \sqrt {c+d x^6}}\right )}{3 b \sqrt {b c-a d}}+\frac {\text {arctanh}\left (\frac {\sqrt {d} x^3}{\sqrt {c+d x^6}}\right )}{3 b \sqrt {d}} \]
1/3*arctanh(x^3*d^(1/2)/(d*x^6+c)^(1/2))/b/d^(1/2)-1/3*arctan(x^3*(-a*d+b* c)^(1/2)/a^(1/2)/(d*x^6+c)^(1/2))*a^(1/2)/b/(-a*d+b*c)^(1/2)
Time = 0.70 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.19 \[ \int \frac {x^8}{\left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\frac {-\frac {\sqrt {a} \arctan \left (\frac {a \sqrt {d}+b x^3 \left (\sqrt {d} x^3+\sqrt {c+d x^6}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{\sqrt {b c-a d}}+\frac {\log \left (\sqrt {d} x^3+\sqrt {c+d x^6}\right )}{\sqrt {d}}}{3 b} \]
(-((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)
Time = 0.23 (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 |
\(\displaystyle \frac {1}{3} \left (\frac {\text {arctanh}\left (\frac {\sqrt {d} x^3}{\sqrt {c+d x^6}}\right )}{b \sqrt {d}}-\frac {\sqrt {a} \arctan \left (\frac {x^3 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^6}}\right )}{b \sqrt {b c-a d}}\right )\) |
(-((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
3.9.59.3.1 Defintions of rubi rules used
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[((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])
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]
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_)*((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]
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 = 7.22 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.93
method | result | size |
pseudoelliptic | \(\frac {-a \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{6}+c}\, a}{x^{3} \sqrt {\left (a d -b c \right ) a}}\right ) \sqrt {d}+\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{6}+c}}{x^{3} \sqrt {d}}\right ) \sqrt {\left (a d -b c \right ) a}}{3 b \sqrt {\left (a d -b c \right ) a}\, \sqrt {d}}\) | \(85\) |
1/3*(-a*arctanh((d*x^6+c)^(1/2)/x^3*a/((a*d-b*c)*a)^(1/2))*d^(1/2)+arctanh ((d*x^6+c)^(1/2)/x^3/d^(1/2))*((a*d-b*c)*a)^(1/2))/b/((a*d-b*c)*a)^(1/2)/d ^(1/2)
Time = 0.61 (sec) , antiderivative size = 632, normalized size of antiderivative = 6.95 \[ \int \frac {x^8}{\left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\left [\frac {d \sqrt {-\frac {a}{b c - a d}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{12} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{6} + a^{2} c^{2} - 4 \, {\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{9} - {\left (a b c^{2} - a^{2} c d\right )} x^{3}\right )} \sqrt {d x^{6} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{12} + 2 \, a b x^{6} + a^{2}}\right ) + 2 \, \sqrt {d} \log \left (-2 \, d x^{6} - 2 \, \sqrt {d x^{6} + c} \sqrt {d} x^{3} - c\right )}{12 \, b d}, \frac {d \sqrt {-\frac {a}{b c - a d}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{12} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{6} + a^{2} c^{2} - 4 \, {\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{9} - {\left (a b c^{2} - a^{2} c d\right )} x^{3}\right )} \sqrt {d x^{6} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{12} + 2 \, a b x^{6} + a^{2}}\right ) - 4 \, \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x^{3}}{\sqrt {d x^{6} + c}}\right )}{12 \, b d}, \frac {d \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{6} - a c\right )} \sqrt {d x^{6} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{9} + a c x^{3}\right )}}\right ) + \sqrt {d} \log \left (-2 \, d x^{6} - 2 \, \sqrt {d x^{6} + c} \sqrt {d} x^{3} - c\right )}{6 \, b d}, \frac {d \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{6} - a c\right )} \sqrt {d x^{6} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{9} + a c x^{3}\right )}}\right ) - 2 \, \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x^{3}}{\sqrt {d x^{6} + c}}\right )}{6 \, b d}\right ] \]
[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)]
\[ \int \frac {x^8}{\left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\int \frac {x^{8}}{\left (a + b x^{6}\right ) \sqrt {c + d x^{6}}}\, dx \]
\[ \int \frac {x^8}{\left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\int { \frac {x^{8}}{{\left (b x^{6} + a\right )} \sqrt {d x^{6} + c}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (71) = 142\).
Time = 0.30 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.71 \[ \int \frac {x^8}{\left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=-\frac {{\left (a \sqrt {-d} \arctan \left (\frac {a \sqrt {d}}{\sqrt {a b c - a^{2} d}}\right ) - \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {d}}{\sqrt {-d}}\right )\right )} \mathrm {sgn}\left (x\right )}{3 \, \sqrt {a b c - a^{2} d} b \sqrt {-d}} + \frac {a \arctan \left (\frac {a \sqrt {d + \frac {c}{x^{6}}}}{\sqrt {a b c - a^{2} d}}\right )}{3 \, \sqrt {a b c - a^{2} d} b \mathrm {sgn}\left (x\right )} - \frac {\arctan \left (\frac {\sqrt {d + \frac {c}{x^{6}}}}{\sqrt {-d}}\right )}{3 \, b \sqrt {-d} \mathrm {sgn}\left (x\right )} \]
-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))
Timed out. \[ \int \frac {x^8}{\left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\int \frac {x^8}{\left (b\,x^6+a\right )\,\sqrt {d\,x^6+c}} \,d x \]