Integrand size = 24, antiderivative size = 80 \[ \int \frac {1}{x^3 \left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=-\frac {\sqrt {c+d x^4}}{2 a c x^2}-\frac {b \arctan \left (\frac {\sqrt {b c-a d} x^2}{\sqrt {a} \sqrt {c+d x^4}}\right )}{2 a^{3/2} \sqrt {b c-a d}} \]
-1/2*b*arctan(x^2*(-a*d+b*c)^(1/2)/a^(1/2)/(d*x^4+c)^(1/2))/a^(3/2)/(-a*d+ b*c)^(1/2)-1/2*(d*x^4+c)^(1/2)/a/c/x^2
Time = 0.57 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.25 \[ \int \frac {1}{x^3 \left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=-\frac {\sqrt {c+d x^4}}{2 a c x^2}-\frac {b \arctan \left (\frac {a \sqrt {d}+b x^2 \left (\sqrt {d} x^2+\sqrt {c+d x^4}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{2 a^{3/2} \sqrt {b c-a d}} \]
-1/2*Sqrt[c + d*x^4]/(a*c*x^2) - (b*ArcTan[(a*Sqrt[d] + b*x^2*(Sqrt[d]*x^2 + Sqrt[c + d*x^4]))/(Sqrt[a]*Sqrt[b*c - a*d])])/(2*a^(3/2)*Sqrt[b*c - a*d ])
Time = 0.23 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {965, 382, 25, 27, 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.
\(\displaystyle \int \frac {1}{x^3 \left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx\) |
\(\Big \downarrow \) 965 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^4 \left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2\) |
\(\Big \downarrow \) 382 |
\(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {b c}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{a c}-\frac {\sqrt {c+d x^4}}{a c x^2}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {b c}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{a c}-\frac {\sqrt {c+d x^4}}{a c x^2}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (-\frac {b \int \frac {1}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{a}-\frac {\sqrt {c+d x^4}}{a c x^2}\right )\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \frac {1}{2} \left (-\frac {b \int \frac {1}{a-(a d-b c) x^4}d\frac {x^2}{\sqrt {d x^4+c}}}{a}-\frac {\sqrt {c+d x^4}}{a c x^2}\right )\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {1}{2} \left (-\frac {b \arctan \left (\frac {x^2 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^4}}\right )}{a^{3/2} \sqrt {b c-a d}}-\frac {\sqrt {c+d x^4}}{a c x^2}\right )\) |
(-(Sqrt[c + d*x^4]/(a*c*x^2)) - (b*ArcTan[(Sqrt[b*c - a*d]*x^2)/(Sqrt[a]*S qrt[c + d*x^4])])/(a^(3/2)*Sqrt[b*c - a*d]))/2
3.9.13.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[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[(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/ (a*c*e*(m + 1))), x] - Simp[1/(a*c*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b* x^2)^p*(c + d*x^2)^q*Simp[(b*c + a*d)*(m + 3) + 2*(b*c*p + a*d*q) + b*d*(m + 2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[ b*c - a*d, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, 2, p, 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 = 5.35 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00
method | result | size |
pseudoelliptic | \(-\frac {\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{4}+c}\, a}{x^{2} \sqrt {\left (a d -b c \right ) a}}\right ) b c \,x^{2}+\sqrt {d \,x^{4}+c}\, \sqrt {\left (a d -b c \right ) a}}{2 a \,x^{2} \sqrt {\left (a d -b c \right ) a}\, c}\) | \(80\) |
default | \(-\frac {\sqrt {d \,x^{4}+c}}{2 a c \,x^{2}}-\frac {b \left (\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}-\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}\right )}{a}\) | \(349\) |
risch | \(-\frac {\sqrt {d \,x^{4}+c}}{2 a c \,x^{2}}-\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 a \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}+\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 a \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}\) | \(350\) |
elliptic | \(-\frac {\sqrt {d \,x^{4}+c}}{2 a c \,x^{2}}-\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 a \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}+\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 a \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}\) | \(350\) |
-1/2*(arctanh((d*x^4+c)^(1/2)/x^2*a/((a*d-b*c)*a)^(1/2))*b*c*x^2+(d*x^4+c) ^(1/2)*((a*d-b*c)*a)^(1/2))/a/x^2/((a*d-b*c)*a)^(1/2)/c
Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (64) = 128\).
Time = 0.30 (sec) , antiderivative size = 332, normalized size of antiderivative = 4.15 \[ \int \frac {1}{x^3 \left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\left [-\frac {\sqrt {-a b c + a^{2} d} b c x^{2} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{6} - a c x^{2}\right )} \sqrt {d x^{4} + c} \sqrt {-a b c + a^{2} d}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right ) + 4 \, \sqrt {d x^{4} + c} {\left (a b c - a^{2} d\right )}}{8 \, {\left (a^{2} b c^{2} - a^{3} c d\right )} x^{2}}, -\frac {\sqrt {a b c - a^{2} d} b c x^{2} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{4} - a c\right )} \sqrt {d x^{4} + c} \sqrt {a b c - a^{2} d}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{6} + {\left (a b c^{2} - a^{2} c d\right )} x^{2}\right )}}\right ) + 2 \, \sqrt {d x^{4} + c} {\left (a b c - a^{2} d\right )}}{4 \, {\left (a^{2} b c^{2} - a^{3} c d\right )} x^{2}}\right ] \]
[-1/8*(sqrt(-a*b*c + a^2*d)*b*c*x^2*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2) *x^8 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^4 + a^2*c^2 + 4*((b*c - 2*a*d)*x^6 - a* c*x^2)*sqrt(d*x^4 + c)*sqrt(-a*b*c + a^2*d))/(b^2*x^8 + 2*a*b*x^4 + a^2)) + 4*sqrt(d*x^4 + c)*(a*b*c - a^2*d))/((a^2*b*c^2 - a^3*c*d)*x^2), -1/4*(sq rt(a*b*c - a^2*d)*b*c*x^2*arctan(1/2*((b*c - 2*a*d)*x^4 - a*c)*sqrt(d*x^4 + c)*sqrt(a*b*c - a^2*d)/((a*b*c*d - a^2*d^2)*x^6 + (a*b*c^2 - a^2*c*d)*x^ 2)) + 2*sqrt(d*x^4 + c)*(a*b*c - a^2*d))/((a^2*b*c^2 - a^3*c*d)*x^2)]
\[ \int \frac {1}{x^3 \left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\int \frac {1}{x^{3} \left (a + b x^{4}\right ) \sqrt {c + d x^{4}}}\, dx \]
\[ \int \frac {1}{x^3 \left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )} \sqrt {d x^{4} + c} x^{3}} \,d x } \]
Time = 0.31 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.45 \[ \int \frac {1}{x^3 \left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\frac {1}{2} \, d^{\frac {3}{2}} {\left (\frac {b \arctan \left (\frac {{\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{\sqrt {a b c d - a^{2} d^{2}} a d} + \frac {2}{{\left ({\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} - c\right )} a d}\right )} \]
1/2*d^(3/2)*(b*arctan(1/2*((sqrt(d)*x^2 - sqrt(d*x^4 + c))^2*b - b*c + 2*a *d)/sqrt(a*b*c*d - a^2*d^2))/(sqrt(a*b*c*d - a^2*d^2)*a*d) + 2/(((sqrt(d)* x^2 - sqrt(d*x^4 + c))^2 - c)*a*d))
Timed out. \[ \int \frac {1}{x^3 \left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\int \frac {1}{x^3\,\left (b\,x^4+a\right )\,\sqrt {d\,x^4+c}} \,d x \]