Integrand size = 24, antiderivative size = 141 \[ \int \frac {x^{19}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\frac {a x^4 \sqrt {c+d x^8}}{8 b (b c-a d) \left (a+b x^8\right )}-\frac {\sqrt {a} (3 b c-2 a d) \arctan \left (\frac {\sqrt {b c-a d} x^4}{\sqrt {a} \sqrt {c+d x^8}}\right )}{8 b^2 (b c-a d)^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt {d} x^4}{\sqrt {c+d x^8}}\right )}{4 b^2 \sqrt {d}} \]
-1/8*(-2*a*d+3*b*c)*arctan(x^4*(-a*d+b*c)^(1/2)/a^(1/2)/(d*x^8+c)^(1/2))*a ^(1/2)/b^2/(-a*d+b*c)^(3/2)+1/4*arctanh(x^4*d^(1/2)/(d*x^8+c)^(1/2))/b^2/d ^(1/2)+1/8*a*x^4*(d*x^8+c)^(1/2)/b/(-a*d+b*c)/(b*x^8+a)
Time = 4.19 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.09 \[ \int \frac {x^{19}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\frac {\frac {a b x^4 \sqrt {c+d x^8}}{(b c-a d) \left (a+b x^8\right )}+\frac {\sqrt {a} (-3 b c+2 a d) \arctan \left (\frac {a \sqrt {d}+b x^4 \left (\sqrt {d} x^4+\sqrt {c+d x^8}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}}+\frac {2 \log \left (\sqrt {d} x^4+\sqrt {c+d x^8}\right )}{\sqrt {d}}}{8 b^2} \]
((a*b*x^4*Sqrt[c + d*x^8])/((b*c - a*d)*(a + b*x^8)) + (Sqrt[a]*(-3*b*c + 2*a*d)*ArcTan[(a*Sqrt[d] + b*x^4*(Sqrt[d]*x^4 + Sqrt[c + d*x^8]))/(Sqrt[a] *Sqrt[b*c - a*d])])/(b*c - a*d)^(3/2) + (2*Log[Sqrt[d]*x^4 + Sqrt[c + d*x^ 8]])/Sqrt[d])/(8*b^2)
Time = 0.32 (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.
\(\displaystyle \int \frac {x^{19}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx\) |
\(\Big \downarrow \) 965 |
\(\displaystyle \frac {1}{4} \int \frac {x^{16}}{\left (b x^8+a\right )^2 \sqrt {d x^8+c}}dx^4\) |
\(\Big \downarrow \) 372 |
\(\displaystyle \frac {1}{4} \left (\frac {a x^4 \sqrt {c+d x^8}}{2 b \left (a+b x^8\right ) (b c-a d)}-\frac {\int \frac {a c-2 (b c-a d) x^8}{\left (b x^8+a\right ) \sqrt {d x^8+c}}dx^4}{2 b (b c-a d)}\right )\) |
\(\Big \downarrow \) 398 |
\(\displaystyle \frac {1}{4} \left (\frac {a x^4 \sqrt {c+d x^8}}{2 b \left (a+b x^8\right ) (b c-a d)}-\frac {\frac {a (3 b c-2 a d) \int \frac {1}{\left (b x^8+a\right ) \sqrt {d x^8+c}}dx^4}{b}-\frac {2 (b c-a d) \int \frac {1}{\sqrt {d x^8+c}}dx^4}{b}}{2 b (b c-a d)}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {1}{4} \left (\frac {a x^4 \sqrt {c+d x^8}}{2 b \left (a+b x^8\right ) (b c-a d)}-\frac {\frac {a (3 b c-2 a d) \int \frac {1}{\left (b x^8+a\right ) \sqrt {d x^8+c}}dx^4}{b}-\frac {2 (b c-a d) \int \frac {1}{1-d x^8}d\frac {x^4}{\sqrt {d x^8+c}}}{b}}{2 b (b c-a d)}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{4} \left (\frac {a x^4 \sqrt {c+d x^8}}{2 b \left (a+b x^8\right ) (b c-a d)}-\frac {\frac {a (3 b c-2 a d) \int \frac {1}{\left (b x^8+a\right ) \sqrt {d x^8+c}}dx^4}{b}-\frac {2 (b c-a d) \text {arctanh}\left (\frac {\sqrt {d} x^4}{\sqrt {c+d x^8}}\right )}{b \sqrt {d}}}{2 b (b c-a d)}\right )\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \frac {1}{4} \left (\frac {a x^4 \sqrt {c+d x^8}}{2 b \left (a+b x^8\right ) (b c-a d)}-\frac {\frac {a (3 b c-2 a d) \int \frac {1}{a-(a d-b c) x^8}d\frac {x^4}{\sqrt {d x^8+c}}}{b}-\frac {2 (b c-a d) \text {arctanh}\left (\frac {\sqrt {d} x^4}{\sqrt {c+d x^8}}\right )}{b \sqrt {d}}}{2 b (b c-a d)}\right )\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {1}{4} \left (\frac {a x^4 \sqrt {c+d x^8}}{2 b \left (a+b x^8\right ) (b c-a d)}-\frac {\frac {\sqrt {a} (3 b c-2 a d) \arctan \left (\frac {x^4 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^8}}\right )}{b \sqrt {b c-a d}}-\frac {2 (b c-a d) \text {arctanh}\left (\frac {\sqrt {d} x^4}{\sqrt {c+d x^8}}\right )}{b \sqrt {d}}}{2 b (b c-a d)}\right )\) |
((a*x^4*Sqrt[c + d*x^8])/(2*b*(b*c - a*d)*(a + b*x^8)) - ((Sqrt[a]*(3*b*c - 2*a*d)*ArcTan[(Sqrt[b*c - a*d]*x^4)/(Sqrt[a]*Sqrt[c + d*x^8])])/(b*Sqrt[ b*c - a*d]) - (2*(b*c - a*d)*ArcTanh[(Sqrt[d]*x^4)/Sqrt[c + d*x^8]])/(b*Sq rt[d]))/(2*b*(b*c - a*d)))/4
3.10.13.3.1 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 = 24.90 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.83
method | result | size |
pseudoelliptic | \(-\frac {-\frac {a \left (-\frac {b \sqrt {d \,x^{8}+c}\, x^{4}}{b \,x^{8}+a}-\frac {\left (2 a d -3 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{8}+c}\, a}{x^{4} \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )}{a d -b c}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{8}+c}}{x^{4} \sqrt {d}}\right )}{\sqrt {d}}}{8 b^{2}}\) | \(117\) |
-1/8/b^2*(-a/(a*d-b*c)*(-b*(d*x^8+c)^(1/2)*x^4/(b*x^8+a)-(2*a*d-3*b*c)/((a *d-b*c)*a)^(1/2)*arctanh((d*x^8+c)^(1/2)/x^4*a/((a*d-b*c)*a)^(1/2)))-2/d^( 1/2)*arctanh((d*x^8+c)^(1/2)/x^4/d^(1/2)))
Time = 1.19 (sec) , antiderivative size = 1077, normalized size of antiderivative = 7.64 \[ \int \frac {x^{19}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\left [\frac {4 \, \sqrt {d x^{8} + c} a b d x^{4} + 4 \, {\left ({\left (b^{2} c - a b d\right )} x^{8} + a b c - a^{2} d\right )} \sqrt {d} \log \left (-2 \, d x^{8} - 2 \, \sqrt {d x^{8} + c} \sqrt {d} x^{4} - c\right ) + {\left ({\left (3 \, b^{2} c d - 2 \, a b d^{2}\right )} x^{8} + 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^{16} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{8} + a^{2} c^{2} - 4 \, {\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{12} - {\left (a b c^{2} - a^{2} c d\right )} x^{4}\right )} \sqrt {d x^{8} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{16} + 2 \, a b x^{8} + a^{2}}\right )}{32 \, {\left ({\left (b^{4} c d - a b^{3} d^{2}\right )} x^{8} + a b^{3} c d - a^{2} b^{2} d^{2}\right )}}, \frac {4 \, \sqrt {d x^{8} + c} a b d x^{4} - 8 \, {\left ({\left (b^{2} c - a b d\right )} x^{8} + a b c - a^{2} d\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x^{4}}{\sqrt {d x^{8} + c}}\right ) + {\left ({\left (3 \, b^{2} c d - 2 \, a b d^{2}\right )} x^{8} + 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^{16} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{8} + a^{2} c^{2} - 4 \, {\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{12} - {\left (a b c^{2} - a^{2} c d\right )} x^{4}\right )} \sqrt {d x^{8} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{16} + 2 \, a b x^{8} + a^{2}}\right )}{32 \, {\left ({\left (b^{4} c d - a b^{3} d^{2}\right )} x^{8} + a b^{3} c d - a^{2} b^{2} d^{2}\right )}}, \frac {2 \, \sqrt {d x^{8} + c} a b d x^{4} + {\left ({\left (3 \, b^{2} c d - 2 \, a b d^{2}\right )} x^{8} + 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^{8} - a c\right )} \sqrt {d x^{8} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{12} + a c x^{4}\right )}}\right ) + 2 \, {\left ({\left (b^{2} c - a b d\right )} x^{8} + a b c - a^{2} d\right )} \sqrt {d} \log \left (-2 \, d x^{8} - 2 \, \sqrt {d x^{8} + c} \sqrt {d} x^{4} - c\right )}{16 \, {\left ({\left (b^{4} c d - a b^{3} d^{2}\right )} x^{8} + a b^{3} c d - a^{2} b^{2} d^{2}\right )}}, \frac {2 \, \sqrt {d x^{8} + c} a b d x^{4} - 4 \, {\left ({\left (b^{2} c - a b d\right )} x^{8} + a b c - a^{2} d\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x^{4}}{\sqrt {d x^{8} + c}}\right ) + {\left ({\left (3 \, b^{2} c d - 2 \, a b d^{2}\right )} x^{8} + 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^{8} - a c\right )} \sqrt {d x^{8} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{12} + a c x^{4}\right )}}\right )}{16 \, {\left ({\left (b^{4} c d - a b^{3} d^{2}\right )} x^{8} + a b^{3} c d - a^{2} b^{2} d^{2}\right )}}\right ] \]
[1/32*(4*sqrt(d*x^8 + c)*a*b*d*x^4 + 4*((b^2*c - a*b*d)*x^8 + a*b*c - a^2* d)*sqrt(d)*log(-2*d*x^8 - 2*sqrt(d*x^8 + c)*sqrt(d)*x^4 - c) + ((3*b^2*c*d - 2*a*b*d^2)*x^8 + 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^16 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^8 + a^2*c^ 2 - 4*((b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x^12 - (a*b*c^2 - a^2*c*d)*x^4)*s qrt(d*x^8 + c)*sqrt(-a/(b*c - a*d)))/(b^2*x^16 + 2*a*b*x^8 + a^2)))/((b^4* c*d - a*b^3*d^2)*x^8 + a*b^3*c*d - a^2*b^2*d^2), 1/32*(4*sqrt(d*x^8 + c)*a *b*d*x^4 - 8*((b^2*c - a*b*d)*x^8 + a*b*c - a^2*d)*sqrt(-d)*arctan(sqrt(-d )*x^4/sqrt(d*x^8 + c)) + ((3*b^2*c*d - 2*a*b*d^2)*x^8 + 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^16 - 2* (3*a*b*c^2 - 4*a^2*c*d)*x^8 + a^2*c^2 - 4*((b^2*c^2 - 3*a*b*c*d + 2*a^2*d^ 2)*x^12 - (a*b*c^2 - a^2*c*d)*x^4)*sqrt(d*x^8 + c)*sqrt(-a/(b*c - a*d)))/( b^2*x^16 + 2*a*b*x^8 + a^2)))/((b^4*c*d - a*b^3*d^2)*x^8 + a*b^3*c*d - a^2 *b^2*d^2), 1/16*(2*sqrt(d*x^8 + c)*a*b*d*x^4 + ((3*b^2*c*d - 2*a*b*d^2)*x^ 8 + 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^8 - a*c)*sqrt(d*x^8 + c)*sqrt(a/(b*c - a*d))/(a*d*x^12 + a*c*x^4)) + 2*( (b^2*c - a*b*d)*x^8 + a*b*c - a^2*d)*sqrt(d)*log(-2*d*x^8 - 2*sqrt(d*x^8 + c)*sqrt(d)*x^4 - c))/((b^4*c*d - a*b^3*d^2)*x^8 + a*b^3*c*d - a^2*b^2*d^2 ), 1/16*(2*sqrt(d*x^8 + c)*a*b*d*x^4 - 4*((b^2*c - a*b*d)*x^8 + a*b*c - a^ 2*d)*sqrt(-d)*arctan(sqrt(-d)*x^4/sqrt(d*x^8 + c)) + ((3*b^2*c*d - 2*a*...
Timed out. \[ \int \frac {x^{19}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\text {Timed out} \]
\[ \int \frac {x^{19}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int { \frac {x^{19}}{{\left (b x^{8} + a\right )}^{2} \sqrt {d x^{8} + c}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (117) = 234\).
Time = 0.37 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.11 \[ \int \frac {x^{19}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=-\frac {{\left (3 \, a b c \sqrt {d} - 2 \, a^{2} d^{\frac {3}{2}}\right )} \arctan \left (-\frac {{\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{8 \, {\left (b^{3} c - a b^{2} d\right )} \sqrt {a b c d - a^{2} d^{2}}} - \frac {{\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} a b c \sqrt {d} - 2 \, {\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} a^{2} d^{\frac {3}{2}} - a b c^{2} \sqrt {d}}{4 \, {\left ({\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} a d + b c^{2}\right )} {\left (b^{3} c - a b^{2} d\right )}} - \frac {\log \left ({\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2}\right )}{8 \, b^{2} \sqrt {d}} \]
-1/8*(3*a*b*c*sqrt(d) - 2*a^2*d^(3/2))*arctan(-1/2*((sqrt(d)*x^4 - sqrt(d* x^8 + c))^2*b - b*c + 2*a*d)/sqrt(a*b*c*d - a^2*d^2))/((b^3*c - a*b^2*d)*s qrt(a*b*c*d - a^2*d^2)) - 1/4*((sqrt(d)*x^4 - sqrt(d*x^8 + c))^2*a*b*c*sqr t(d) - 2*(sqrt(d)*x^4 - sqrt(d*x^8 + c))^2*a^2*d^(3/2) - a*b*c^2*sqrt(d))/ (((sqrt(d)*x^4 - sqrt(d*x^8 + c))^4*b - 2*(sqrt(d)*x^4 - sqrt(d*x^8 + c))^ 2*b*c + 4*(sqrt(d)*x^4 - sqrt(d*x^8 + c))^2*a*d + b*c^2)*(b^3*c - a*b^2*d) ) - 1/8*log((sqrt(d)*x^4 - sqrt(d*x^8 + c))^2)/(b^2*sqrt(d))
Timed out. \[ \int \frac {x^{19}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int \frac {x^{19}}{{\left (b\,x^8+a\right )}^2\,\sqrt {d\,x^8+c}} \,d x \]