Integrand size = 24, antiderivative size = 91 \[ \int \frac {x^{11}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=-\frac {\sqrt {a} \arctan \left (\frac {\sqrt {b c-a d} x^4}{\sqrt {a} \sqrt {c+d x^8}}\right )}{4 b \sqrt {b c-a d}}+\frac {\text {arctanh}\left (\frac {\sqrt {d} x^4}{\sqrt {c+d x^8}}\right )}{4 b \sqrt {d}} \]
1/4*arctanh(x^4*d^(1/2)/(d*x^8+c)^(1/2))/b/d^(1/2)-1/4*arctan(x^4*(-a*d+b* c)^(1/2)/a^(1/2)/(d*x^8+c)^(1/2))*a^(1/2)/b/(-a*d+b*c)^(1/2)
Time = 1.09 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.19 \[ \int \frac {x^{11}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\frac {-\frac {\sqrt {a} \arctan \left (\frac {a \sqrt {d}+b x^4 \left (\sqrt {d} x^4+\sqrt {c+d x^8}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{\sqrt {b c-a d}}+\frac {\log \left (\sqrt {d} x^4+\sqrt {c+d x^8}\right )}{\sqrt {d}}}{4 b} \]
(-((Sqrt[a]*ArcTan[(a*Sqrt[d] + b*x^4*(Sqrt[d]*x^4 + Sqrt[c + d*x^8]))/(Sq rt[a]*Sqrt[b*c - a*d])])/Sqrt[b*c - a*d]) + Log[Sqrt[d]*x^4 + Sqrt[c + d*x ^8]]/Sqrt[d])/(4*b)
Time = 0.24 (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.
\(\displaystyle \int \frac {x^{11}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx\) |
\(\Big \downarrow \) 965 |
\(\displaystyle \frac {1}{4} \int \frac {x^8}{\left (b x^8+a\right ) \sqrt {d x^8+c}}dx^4\) |
\(\Big \downarrow \) 385 |
\(\displaystyle \frac {1}{4} \left (\frac {\int \frac {1}{\sqrt {d x^8+c}}dx^4}{b}-\frac {a \int \frac {1}{\left (b x^8+a\right ) \sqrt {d x^8+c}}dx^4}{b}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {1}{4} \left (\frac {\int \frac {1}{1-d x^8}d\frac {x^4}{\sqrt {d x^8+c}}}{b}-\frac {a \int \frac {1}{\left (b x^8+a\right ) \sqrt {d x^8+c}}dx^4}{b}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{4} \left (\frac {\text {arctanh}\left (\frac {\sqrt {d} x^4}{\sqrt {c+d x^8}}\right )}{b \sqrt {d}}-\frac {a \int \frac {1}{\left (b x^8+a\right ) \sqrt {d x^8+c}}dx^4}{b}\right )\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \frac {1}{4} \left (\frac {\text {arctanh}\left (\frac {\sqrt {d} x^4}{\sqrt {c+d x^8}}\right )}{b \sqrt {d}}-\frac {a \int \frac {1}{a-(a d-b c) x^8}d\frac {x^4}{\sqrt {d x^8+c}}}{b}\right )\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {1}{4} \left (\frac {\text {arctanh}\left (\frac {\sqrt {d} x^4}{\sqrt {c+d x^8}}\right )}{b \sqrt {d}}-\frac {\sqrt {a} \arctan \left (\frac {x^4 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^8}}\right )}{b \sqrt {b c-a d}}\right )\) |
(-((Sqrt[a]*ArcTan[(Sqrt[b*c - a*d]*x^4)/(Sqrt[a]*Sqrt[c + d*x^8])])/(b*Sq rt[b*c - a*d])) + ArcTanh[(Sqrt[d]*x^4)/Sqrt[c + d*x^8]]/(b*Sqrt[d]))/4
3.9.93.3.1 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 = 12.91 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.93
method | result | size |
pseudoelliptic | \(\frac {-a \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{8}+c}\, a}{x^{4} \sqrt {\left (a d -b c \right ) a}}\right ) \sqrt {d}+\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{8}+c}}{x^{4} \sqrt {d}}\right ) \sqrt {\left (a d -b c \right ) a}}{4 b \sqrt {\left (a d -b c \right ) a}\, \sqrt {d}}\) | \(85\) |
1/4*(-a*arctanh((d*x^8+c)^(1/2)/x^4*a/((a*d-b*c)*a)^(1/2))*d^(1/2)+arctanh ((d*x^8+c)^(1/2)/x^4/d^(1/2))*((a*d-b*c)*a)^(1/2))/b/((a*d-b*c)*a)^(1/2)/d ^(1/2)
Time = 0.40 (sec) , antiderivative size = 632, normalized size of antiderivative = 6.95 \[ \int \frac {x^{11}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\left [\frac {d \sqrt {-\frac {a}{b c - a d}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{16} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{8} + a^{2} c^{2} - 4 \, {\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{12} - {\left (a b c^{2} - a^{2} c d\right )} x^{4}\right )} \sqrt {d x^{8} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{16} + 2 \, a b x^{8} + a^{2}}\right ) + 2 \, \sqrt {d} \log \left (-2 \, d x^{8} - 2 \, \sqrt {d x^{8} + c} \sqrt {d} x^{4} - c\right )}{16 \, b d}, \frac {d \sqrt {-\frac {a}{b c - a d}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{16} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{8} + a^{2} c^{2} - 4 \, {\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{12} - {\left (a b c^{2} - a^{2} c d\right )} x^{4}\right )} \sqrt {d x^{8} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{16} + 2 \, a b x^{8} + a^{2}}\right ) - 4 \, \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x^{4}}{\sqrt {d x^{8} + c}}\right )}{16 \, b d}, \frac {d \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{8} - a c\right )} \sqrt {d x^{8} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{12} + a c x^{4}\right )}}\right ) + \sqrt {d} \log \left (-2 \, d x^{8} - 2 \, \sqrt {d x^{8} + c} \sqrt {d} x^{4} - c\right )}{8 \, b d}, \frac {d \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{8} - a c\right )} \sqrt {d x^{8} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{12} + a c x^{4}\right )}}\right ) - 2 \, \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x^{4}}{\sqrt {d x^{8} + c}}\right )}{8 \, b d}\right ] \]
[1/16*(d*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)) + 2*sqrt(d)*log(-2*d*x^8 - 2*sqrt(d*x^8 + c)*sqrt(d)*x^4 - c))/(b*d), 1/16*(d*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)) - 4*sqrt(-d)* arctan(sqrt(-d)*x^4/sqrt(d*x^8 + c)))/(b*d), 1/8*(d*sqrt(a/(b*c - a*d))*ar ctan(-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)) + sqrt(d)*log(-2*d*x^8 - 2*sqrt(d*x^8 + c)*sqrt(d)*x^4 - c))/(b*d), 1/8*(d*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*sqrt(-d )*arctan(sqrt(-d)*x^4/sqrt(d*x^8 + c)))/(b*d)]
\[ \int \frac {x^{11}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\int \frac {x^{11}}{\left (a + b x^{8}\right ) \sqrt {c + d x^{8}}}\, dx \]
\[ \int \frac {x^{11}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\int { \frac {x^{11}}{{\left (b x^{8} + a\right )} \sqrt {d x^{8} + c}} \,d x } \]
Exception generated. \[ \int \frac {x^{11}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {x^{11}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\int \frac {x^{11}}{\left (b\,x^8+a\right )\,\sqrt {d\,x^8+c}} \,d x \]