Integrand size = 24, antiderivative size = 141 \[ \int \frac {x^{14}}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\frac {a x^3 \sqrt {c+d x^6}}{6 b (b c-a d) \left (a+b x^6\right )}-\frac {\sqrt {a} (3 b c-2 a d) \arctan \left (\frac {\sqrt {b c-a d} x^3}{\sqrt {a} \sqrt {c+d x^6}}\right )}{6 b^2 (b c-a d)^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt {d} x^3}{\sqrt {c+d x^6}}\right )}{3 b^2 \sqrt {d}} \]
-1/6*(-2*a*d+3*b*c)*arctan(x^3*(-a*d+b*c)^(1/2)/a^(1/2)/(d*x^6+c)^(1/2))*a ^(1/2)/b^2/(-a*d+b*c)^(3/2)+1/3*arctanh(x^3*d^(1/2)/(d*x^6+c)^(1/2))/b^2/d ^(1/2)+1/6*a*x^3*(d*x^6+c)^(1/2)/b/(-a*d+b*c)/(b*x^6+a)
Time = 2.99 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.09 \[ \int \frac {x^{14}}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\frac {\frac {a b x^3 \sqrt {c+d x^6}}{(b c-a d) \left (a+b x^6\right )}+\frac {\sqrt {a} (-3 b c+2 a d) \arctan \left (\frac {a \sqrt {d}+b x^3 \left (\sqrt {d} x^3+\sqrt {c+d x^6}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}}+\frac {2 \log \left (\sqrt {d} x^3+\sqrt {c+d x^6}\right )}{\sqrt {d}}}{6 b^2} \]
((a*b*x^3*Sqrt[c + d*x^6])/((b*c - a*d)*(a + b*x^6)) + (Sqrt[a]*(-3*b*c + 2*a*d)*ArcTan[(a*Sqrt[d] + b*x^3*(Sqrt[d]*x^3 + Sqrt[c + d*x^6]))/(Sqrt[a] *Sqrt[b*c - a*d])])/(b*c - a*d)^(3/2) + (2*Log[Sqrt[d]*x^3 + Sqrt[c + d*x^ 6]])/Sqrt[d])/(6*b^2)
Time = 0.31 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {965, 372, 398, 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.
| \(\displaystyle \int \frac {x^{14}}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx\) |
| \(\Big \downarrow \) 965 |
| \(\displaystyle \frac {1}{3} \int \frac {x^{12}}{\left (b x^6+a\right )^2 \sqrt {d x^6+c}}dx^3\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle \frac {1}{3} \left (\frac {a x^3 \sqrt {c+d x^6}}{2 b \left (a+b x^6\right ) (b c-a d)}-\frac {\int \frac {a c-2 (b c-a d) x^6}{\left (b x^6+a\right ) \sqrt {d x^6+c}}dx^3}{2 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {1}{3} \left (\frac {a x^3 \sqrt {c+d x^6}}{2 b \left (a+b x^6\right ) (b c-a d)}-\frac {\frac {a (3 b c-2 a d) \int \frac {1}{\left (b x^6+a\right ) \sqrt {d x^6+c}}dx^3}{b}-\frac {2 (b c-a d) \int \frac {1}{\sqrt {d x^6+c}}dx^3}{b}}{2 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {1}{3} \left (\frac {a x^3 \sqrt {c+d x^6}}{2 b \left (a+b x^6\right ) (b c-a d)}-\frac {\frac {a (3 b c-2 a d) \int \frac {1}{\left (b x^6+a\right ) \sqrt {d x^6+c}}dx^3}{b}-\frac {2 (b c-a d) \int \frac {1}{1-d x^6}d\frac {x^3}{\sqrt {d x^6+c}}}{b}}{2 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{3} \left (\frac {a x^3 \sqrt {c+d x^6}}{2 b \left (a+b x^6\right ) (b c-a d)}-\frac {\frac {a (3 b c-2 a d) \int \frac {1}{\left (b x^6+a\right ) \sqrt {d x^6+c}}dx^3}{b}-\frac {2 (b c-a d) \text {arctanh}\left (\frac {\sqrt {d} x^3}{\sqrt {c+d x^6}}\right )}{b \sqrt {d}}}{2 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {1}{3} \left (\frac {a x^3 \sqrt {c+d x^6}}{2 b \left (a+b x^6\right ) (b c-a d)}-\frac {\frac {a (3 b c-2 a d) \int \frac {1}{a-(a d-b c) x^6}d\frac {x^3}{\sqrt {d x^6+c}}}{b}-\frac {2 (b c-a d) \text {arctanh}\left (\frac {\sqrt {d} x^3}{\sqrt {c+d x^6}}\right )}{b \sqrt {d}}}{2 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {1}{3} \left (\frac {a x^3 \sqrt {c+d x^6}}{2 b \left (a+b x^6\right ) (b c-a d)}-\frac {\frac {\sqrt {a} (3 b c-2 a d) \arctan \left (\frac {x^3 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^6}}\right )}{b \sqrt {b c-a d}}-\frac {2 (b c-a d) \text {arctanh}\left (\frac {\sqrt {d} x^3}{\sqrt {c+d x^6}}\right )}{b \sqrt {d}}}{2 b (b c-a d)}\right )\) |
((a*x^3*Sqrt[c + d*x^6])/(2*b*(b*c - a*d)*(a + b*x^6)) - ((Sqrt[a]*(3*b*c - 2*a*d)*ArcTan[(Sqrt[b*c - a*d]*x^3)/(Sqrt[a]*Sqrt[c + d*x^6])])/(b*Sqrt[ b*c - a*d]) - (2*(b*c - a*d)*ArcTanh[(Sqrt[d]*x^3)/Sqrt[c + d*x^6]])/(b*Sq rt[d]))/(2*b*(b*c - a*d)))/3
3.9.75.3.1 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_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 )^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 )) Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a , b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} , 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 = 10.58 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.83
| method | result | size |
| pseudoelliptic | \(-\frac {-\frac {a \left (-\frac {\sqrt {d \,x^{6}+c}\, b \,x^{3}}{b \,x^{6}+a}-\frac {\left (2 a d -3 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{6}+c}\, a}{x^{3} \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )}{a d -b c}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{6}+c}}{x^{3} \sqrt {d}}\right )}{\sqrt {d}}}{6 b^{2}}\) | \(117\) |
-1/6/b^2*(-a/(a*d-b*c)*(-(d*x^6+c)^(1/2)*b*x^3/(b*x^6+a)-(2*a*d-3*b*c)/((a *d-b*c)*a)^(1/2)*arctanh((d*x^6+c)^(1/2)/x^3*a/((a*d-b*c)*a)^(1/2)))-2/d^( 1/2)*arctanh((d*x^6+c)^(1/2)/x^3/d^(1/2)))
Time = 0.85 (sec) , antiderivative size = 1077, normalized size of antiderivative = 7.64 \[ \int \frac {x^{14}}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\left [\frac {4 \, \sqrt {d x^{6} + c} a b d x^{3} + 4 \, {\left ({\left (b^{2} c - a b d\right )} x^{6} + a b c - a^{2} d\right )} \sqrt {d} \log \left (-2 \, d x^{6} - 2 \, \sqrt {d x^{6} + c} \sqrt {d} x^{3} - c\right ) + {\left ({\left (3 \, b^{2} c d - 2 \, a b d^{2}\right )} x^{6} + 3 \, a b c d - 2 \, a^{2} d^{2}\right )} \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 )}{24 \, {\left ({\left (b^{4} c d - a b^{3} d^{2}\right )} x^{6} + a b^{3} c d - a^{2} b^{2} d^{2}\right )}}, \frac {4 \, \sqrt {d x^{6} + c} a b d x^{3} - 8 \, {\left ({\left (b^{2} c - a b d\right )} x^{6} + a b c - a^{2} d\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x^{3}}{\sqrt {d x^{6} + c}}\right ) + {\left ({\left (3 \, b^{2} c d - 2 \, a b d^{2}\right )} x^{6} + 3 \, a b c d - 2 \, a^{2} d^{2}\right )} \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 )}{24 \, {\left ({\left (b^{4} c d - a b^{3} d^{2}\right )} x^{6} + a b^{3} c d - a^{2} b^{2} d^{2}\right )}}, \frac {2 \, \sqrt {d x^{6} + c} a b d x^{3} + {\left ({\left (3 \, b^{2} c d - 2 \, a b d^{2}\right )} x^{6} + 3 \, a b c d - 2 \, a^{2} d^{2}\right )} \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 \, {\left ({\left (b^{2} c - a b d\right )} x^{6} + a b c - a^{2} d\right )} \sqrt {d} \log \left (-2 \, d x^{6} - 2 \, \sqrt {d x^{6} + c} \sqrt {d} x^{3} - c\right )}{12 \, {\left ({\left (b^{4} c d - a b^{3} d^{2}\right )} x^{6} + a b^{3} c d - a^{2} b^{2} d^{2}\right )}}, \frac {2 \, \sqrt {d x^{6} + c} a b d x^{3} - 4 \, {\left ({\left (b^{2} c - a b d\right )} x^{6} + a b c - a^{2} d\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x^{3}}{\sqrt {d x^{6} + c}}\right ) + {\left ({\left (3 \, b^{2} c d - 2 \, a b d^{2}\right )} x^{6} + 3 \, a b c d - 2 \, a^{2} d^{2}\right )} \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 )}{12 \, {\left ({\left (b^{4} c d - a b^{3} d^{2}\right )} x^{6} + a b^{3} c d - a^{2} b^{2} d^{2}\right )}}\right ] \]
[1/24*(4*sqrt(d*x^6 + c)*a*b*d*x^3 + 4*((b^2*c - a*b*d)*x^6 + a*b*c - a^2* d)*sqrt(d)*log(-2*d*x^6 - 2*sqrt(d*x^6 + c)*sqrt(d)*x^3 - c) + ((3*b^2*c*d - 2*a*b*d^2)*x^6 + 3*a*b*c*d - 2*a^2*d^2)*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)*sq rt(d*x^6 + c)*sqrt(-a/(b*c - a*d)))/(b^2*x^12 + 2*a*b*x^6 + a^2)))/((b^4*c *d - a*b^3*d^2)*x^6 + a*b^3*c*d - a^2*b^2*d^2), 1/24*(4*sqrt(d*x^6 + c)*a* b*d*x^3 - 8*((b^2*c - a*b*d)*x^6 + a*b*c - a^2*d)*sqrt(-d)*arctan(sqrt(-d) *x^3/sqrt(d*x^6 + c)) + ((3*b^2*c*d - 2*a*b*d^2)*x^6 + 3*a*b*c*d - 2*a^2*d ^2)*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)))/((b^4*c*d - a*b^3*d^2)*x^6 + a*b^3*c*d - a^2*b ^2*d^2), 1/12*(2*sqrt(d*x^6 + c)*a*b*d*x^3 + ((3*b^2*c*d - 2*a*b*d^2)*x^6 + 3*a*b*c*d - 2*a^2*d^2)*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*((b^ 2*c - a*b*d)*x^6 + a*b*c - a^2*d)*sqrt(d)*log(-2*d*x^6 - 2*sqrt(d*x^6 + c) *sqrt(d)*x^3 - c))/((b^4*c*d - a*b^3*d^2)*x^6 + a*b^3*c*d - a^2*b^2*d^2), 1/12*(2*sqrt(d*x^6 + c)*a*b*d*x^3 - 4*((b^2*c - a*b*d)*x^6 + a*b*c - a^2*d )*sqrt(-d)*arctan(sqrt(-d)*x^3/sqrt(d*x^6 + c)) + ((3*b^2*c*d - 2*a*b*d...
\[ \int \frac {x^{14}}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int \frac {x^{14}}{\left (a + b x^{6}\right )^{2} \sqrt {c + d x^{6}}}\, dx \]
\[ \int \frac {x^{14}}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int { \frac {x^{14}}{{\left (b x^{6} + a\right )}^{2} \sqrt {d x^{6} + c}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (117) = 234\).
Time = 0.30 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.43 \[ \int \frac {x^{14}}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=-\frac {{\left (3 \, a b c \sqrt {-d} \arctan \left (\frac {a \sqrt {d}}{\sqrt {a b c - a^{2} d}}\right ) - 2 \, a^{2} \sqrt {-d} d \arctan \left (\frac {a \sqrt {d}}{\sqrt {a b c - a^{2} d}}\right ) - 2 \, \sqrt {a b c - a^{2} d} b c \arctan \left (\frac {\sqrt {d}}{\sqrt {-d}}\right ) + 2 \, \sqrt {a b c - a^{2} d} a d \arctan \left (\frac {\sqrt {d}}{\sqrt {-d}}\right ) + \sqrt {a b c - a^{2} d} a \sqrt {-d} \sqrt {d}\right )} \mathrm {sgn}\left (x\right )}{6 \, {\left (\sqrt {a b c - a^{2} d} b^{3} c \sqrt {-d} - \sqrt {a b c - a^{2} d} a b^{2} \sqrt {-d} d\right )}} + \frac {a c \sqrt {d + \frac {c}{x^{6}}}}{6 \, {\left (b^{2} c \mathrm {sgn}\left (x\right ) - a b d \mathrm {sgn}\left (x\right )\right )} {\left (b c + a {\left (d + \frac {c}{x^{6}}\right )} - a d\right )}} + \frac {{\left (3 \, a b c - 2 \, a^{2} d\right )} \arctan \left (\frac {a \sqrt {d + \frac {c}{x^{6}}}}{\sqrt {a b c - a^{2} d}}\right )}{6 \, {\left (b^{3} c \mathrm {sgn}\left (x\right ) - a b^{2} d \mathrm {sgn}\left (x\right )\right )} \sqrt {a b c - a^{2} d}} - \frac {\arctan \left (\frac {\sqrt {d + \frac {c}{x^{6}}}}{\sqrt {-d}}\right )}{3 \, b^{2} \sqrt {-d} \mathrm {sgn}\left (x\right )} \]
-1/6*(3*a*b*c*sqrt(-d)*arctan(a*sqrt(d)/sqrt(a*b*c - a^2*d)) - 2*a^2*sqrt( -d)*d*arctan(a*sqrt(d)/sqrt(a*b*c - a^2*d)) - 2*sqrt(a*b*c - a^2*d)*b*c*ar ctan(sqrt(d)/sqrt(-d)) + 2*sqrt(a*b*c - a^2*d)*a*d*arctan(sqrt(d)/sqrt(-d) ) + sqrt(a*b*c - a^2*d)*a*sqrt(-d)*sqrt(d))*sgn(x)/(sqrt(a*b*c - a^2*d)*b^ 3*c*sqrt(-d) - sqrt(a*b*c - a^2*d)*a*b^2*sqrt(-d)*d) + 1/6*a*c*sqrt(d + c/ x^6)/((b^2*c*sgn(x) - a*b*d*sgn(x))*(b*c + a*(d + c/x^6) - a*d)) + 1/6*(3* a*b*c - 2*a^2*d)*arctan(a*sqrt(d + c/x^6)/sqrt(a*b*c - a^2*d))/((b^3*c*sgn (x) - a*b^2*d*sgn(x))*sqrt(a*b*c - a^2*d)) - 1/3*arctan(sqrt(d + c/x^6)/sq rt(-d))/(b^2*sqrt(-d)*sgn(x))
Timed out. \[ \int \frac {x^{14}}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int \frac {x^{14}}{{\left (b\,x^6+a\right )}^2\,\sqrt {d\,x^6+c}} \,d x \]