Integrand size = 24, antiderivative size = 104 \[ \int \frac {x^{11}}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=-\frac {(b c+a d) \sqrt {c+d x^4}}{2 b^2 d^2}+\frac {\left (c+d x^4\right )^{3/2}}{6 b d^2}-\frac {a^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{2 b^{5/2} \sqrt {b c-a d}} \]
1/6*(d*x^4+c)^(3/2)/b/d^2-1/2*a^2*arctanh(b^(1/2)*(d*x^4+c)^(1/2)/(-a*d+b* c)^(1/2))/b^(5/2)/(-a*d+b*c)^(1/2)-1/2*(a*d+b*c)*(d*x^4+c)^(1/2)/b^2/d^2
Time = 0.29 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.88 \[ \int \frac {x^{11}}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\frac {\sqrt {c+d x^4} \left (-2 b c-3 a d+b d x^4\right )}{6 b^2 d^2}+\frac {a^2 \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {-b c+a d}}\right )}{2 b^{5/2} \sqrt {-b c+a d}} \]
(Sqrt[c + d*x^4]*(-2*b*c - 3*a*d + b*d*x^4))/(6*b^2*d^2) + (a^2*ArcTan[(Sq rt[b]*Sqrt[c + d*x^4])/Sqrt[-(b*c) + a*d]])/(2*b^(5/2)*Sqrt[-(b*c) + a*d])
Time = 0.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {948, 99, 2009}
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.
\(\displaystyle \int \frac {x^{11}}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{4} \int \frac {x^8}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^4\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {1}{4} \int \left (\frac {a^2}{b^2 \left (b x^4+a\right ) \sqrt {d x^4+c}}+\frac {\sqrt {d x^4+c}}{b d}+\frac {-b c-a d}{b^2 d \sqrt {d x^4+c}}\right )dx^4\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \left (-\frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{b^{5/2} \sqrt {b c-a d}}-\frac {2 \sqrt {c+d x^4} (a d+b c)}{b^2 d^2}+\frac {2 \left (c+d x^4\right )^{3/2}}{3 b d^2}\right )\) |
((-2*(b*c + a*d)*Sqrt[c + d*x^4])/(b^2*d^2) + (2*(c + d*x^4)^(3/2))/(3*b*d ^2) - (2*a^2*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])/(b^(5/2)* Sqrt[b*c - a*d]))/4
3.9.5.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Time = 5.06 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.88
method | result | size |
pseudoelliptic | \(\frac {\arctan \left (\frac {b \sqrt {d \,x^{4}+c}}{\sqrt {\left (a d -b c \right ) b}}\right ) a^{2} d^{2}-\left (\left (-\frac {b \,x^{4}}{3}+a \right ) d +\frac {2 b c}{3}\right ) \sqrt {d \,x^{4}+c}\, \sqrt {\left (a d -b c \right ) b}}{2 \sqrt {\left (a d -b c \right ) b}\, b^{2} d^{2}}\) | \(91\) |
risch | \(-\frac {\left (-b d \,x^{4}+3 a d +2 b c \right ) \sqrt {d \,x^{4}+c}}{6 d^{2} b^{2}}-\frac {a^{2} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 b^{3} \sqrt {-\frac {a d -b c}{b}}}-\frac {a^{2} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 b^{3} \sqrt {-\frac {a d -b c}{b}}}\) | \(355\) |
default | \(-\frac {\sqrt {d \,x^{4}+c}\, \left (-d \,x^{4}+2 c \right )}{6 b \,d^{2}}-\frac {a \sqrt {d \,x^{4}+c}}{2 b^{2} d}+\frac {a^{2} \left (-\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 b \sqrt {-\frac {a d -b c}{b}}}-\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 b \sqrt {-\frac {a d -b c}{b}}}\right )}{b^{2}}\) | \(369\) |
elliptic | \(\frac {x^{4} \sqrt {d \,x^{4}+c}}{6 b d}-\frac {c \sqrt {d \,x^{4}+c}}{3 b \,d^{2}}-\frac {a \sqrt {d \,x^{4}+c}}{2 b^{2} d}-\frac {a^{2} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 b^{3} \sqrt {-\frac {a d -b c}{b}}}-\frac {a^{2} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 b^{3} \sqrt {-\frac {a d -b c}{b}}}\) | \(378\) |
1/2*(arctan(b*(d*x^4+c)^(1/2)/((a*d-b*c)*b)^(1/2))*a^2*d^2-((-1/3*b*x^4+a) *d+2/3*b*c)*(d*x^4+c)^(1/2)*((a*d-b*c)*b)^(1/2))/((a*d-b*c)*b)^(1/2)/b^2/d ^2
Time = 0.31 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.78 \[ \int \frac {x^{11}}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\left [\frac {3 \, \sqrt {b^{2} c - a b d} a^{2} d^{2} \log \left (\frac {b d x^{4} + 2 \, b c - a d - 2 \, \sqrt {d x^{4} + c} \sqrt {b^{2} c - a b d}}{b x^{4} + a}\right ) - 2 \, {\left (2 \, b^{3} c^{2} + a b^{2} c d - 3 \, a^{2} b d^{2} - {\left (b^{3} c d - a b^{2} d^{2}\right )} x^{4}\right )} \sqrt {d x^{4} + c}}{12 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )}}, \frac {3 \, \sqrt {-b^{2} c + a b d} a^{2} d^{2} \arctan \left (\frac {\sqrt {d x^{4} + c} \sqrt {-b^{2} c + a b d}}{b d x^{4} + b c}\right ) - {\left (2 \, b^{3} c^{2} + a b^{2} c d - 3 \, a^{2} b d^{2} - {\left (b^{3} c d - a b^{2} d^{2}\right )} x^{4}\right )} \sqrt {d x^{4} + c}}{6 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )}}\right ] \]
[1/12*(3*sqrt(b^2*c - a*b*d)*a^2*d^2*log((b*d*x^4 + 2*b*c - a*d - 2*sqrt(d *x^4 + c)*sqrt(b^2*c - a*b*d))/(b*x^4 + a)) - 2*(2*b^3*c^2 + a*b^2*c*d - 3 *a^2*b*d^2 - (b^3*c*d - a*b^2*d^2)*x^4)*sqrt(d*x^4 + c))/(b^4*c*d^2 - a*b^ 3*d^3), 1/6*(3*sqrt(-b^2*c + a*b*d)*a^2*d^2*arctan(sqrt(d*x^4 + c)*sqrt(-b ^2*c + a*b*d)/(b*d*x^4 + b*c)) - (2*b^3*c^2 + a*b^2*c*d - 3*a^2*b*d^2 - (b ^3*c*d - a*b^2*d^2)*x^4)*sqrt(d*x^4 + c))/(b^4*c*d^2 - a*b^3*d^3)]
\[ \int \frac {x^{11}}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\int \frac {x^{11}}{\left (a + b x^{4}\right ) \sqrt {c + d x^{4}}}\, dx \]
Exception generated. \[ \int \frac {x^{11}}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Time = 0.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.02 \[ \int \frac {x^{11}}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\frac {a^{2} \arctan \left (\frac {\sqrt {d x^{4} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, \sqrt {-b^{2} c + a b d} b^{2}} + \frac {{\left (d x^{4} + c\right )}^{\frac {3}{2}} b^{2} d^{4} - 3 \, \sqrt {d x^{4} + c} b^{2} c d^{4} - 3 \, \sqrt {d x^{4} + c} a b d^{5}}{6 \, b^{3} d^{6}} \]
1/2*a^2*arctan(sqrt(d*x^4 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b* d)*b^2) + 1/6*((d*x^4 + c)^(3/2)*b^2*d^4 - 3*sqrt(d*x^4 + c)*b^2*c*d^4 - 3 *sqrt(d*x^4 + c)*a*b*d^5)/(b^3*d^6)
Time = 9.27 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.98 \[ \int \frac {x^{11}}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\frac {{\left (d\,x^4+c\right )}^{3/2}}{6\,b\,d^2}-\left (\frac {c}{b\,d^2}+\frac {2\,a\,d^3-2\,b\,c\,d^2}{4\,b^2\,d^4}\right )\,\sqrt {d\,x^4+c}+\frac {a^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^4+c}}{\sqrt {a\,d-b\,c}}\right )}{2\,b^{5/2}\,\sqrt {a\,d-b\,c}} \]