Integrand size = 24, antiderivative size = 110 \[ \int \frac {\sqrt {c+d x^4}}{x^7 \left (a+b x^4\right )} \, dx=-\frac {\sqrt {c+d x^4}}{6 a x^6}+\frac {(3 b c-a d) \sqrt {c+d x^4}}{6 a^2 c x^2}+\frac {b \sqrt {b c-a d} \arctan \left (\frac {\sqrt {b c-a d} x^2}{\sqrt {a} \sqrt {c+d x^4}}\right )}{2 a^{5/2}} \]
1/2*b*arctan(x^2*(-a*d+b*c)^(1/2)/a^(1/2)/(d*x^4+c)^(1/2))*(-a*d+b*c)^(1/2 )/a^(5/2)-1/6*(d*x^4+c)^(1/2)/a/x^6+1/6*(-a*d+3*b*c)*(d*x^4+c)^(1/2)/a^2/c /x^2
Time = 0.67 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {c+d x^4}}{x^7 \left (a+b x^4\right )} \, dx=\frac {\sqrt {c+d x^4} \left (3 b c x^4-a \left (c+d x^4\right )\right )}{6 a^2 c x^6}+\frac {b \sqrt {b c-a d} \arctan \left (\frac {a \sqrt {d}+b x^2 \left (\sqrt {d} x^2+\sqrt {c+d x^4}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{2 a^{5/2}} \]
(Sqrt[c + d*x^4]*(3*b*c*x^4 - a*(c + d*x^4)))/(6*a^2*c*x^6) + (b*Sqrt[b*c - a*d]*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^(5/2))
Time = 0.30 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {965, 377, 25, 445, 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.
| \(\displaystyle \int \frac {\sqrt {c+d x^4}}{x^7 \left (a+b x^4\right )} \, dx\) |
| \(\Big \downarrow \) 965 |
| \(\displaystyle \frac {1}{2} \int \frac {\sqrt {d x^4+c}}{x^8 \left (b x^4+a\right )}dx^2\) |
| \(\Big \downarrow \) 377 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {2 b d x^4+3 b c-a d}{x^4 \left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{3 a}-\frac {\sqrt {c+d x^4}}{3 a x^6}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {2 b d x^4+3 b c-a d}{x^4 \left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{3 a}-\frac {\sqrt {c+d x^4}}{3 a x^6}\right )\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-\frac {\int \frac {3 b c (b c-a d)}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{a c}-\frac {\sqrt {c+d x^4} (3 b c-a d)}{a c x^2}}{3 a}-\frac {\sqrt {c+d x^4}}{3 a x^6}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-\frac {3 b (b c-a d) \int \frac {1}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{a}-\frac {\sqrt {c+d x^4} (3 b c-a d)}{a c x^2}}{3 a}-\frac {\sqrt {c+d x^4}}{3 a x^6}\right )\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-\frac {3 b (b c-a d) \int \frac {1}{a-(a d-b c) x^4}d\frac {x^2}{\sqrt {d x^4+c}}}{a}-\frac {\sqrt {c+d x^4} (3 b c-a d)}{a c x^2}}{3 a}-\frac {\sqrt {c+d x^4}}{3 a x^6}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-\frac {3 b \sqrt {b c-a d} \arctan \left (\frac {x^2 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^4}}\right )}{a^{3/2}}-\frac {\sqrt {c+d x^4} (3 b c-a d)}{a c x^2}}{3 a}-\frac {\sqrt {c+d x^4}}{3 a x^6}\right )\) |
(-1/3*Sqrt[c + d*x^4]/(a*x^6) - (-(((3*b*c - a*d)*Sqrt[c + d*x^4])/(a*c*x^ 2)) - (3*b*Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x^2)/(Sqrt[a]*Sqrt[c + d*x^4])])/a^(3/2))/(3*a))/2
3.8.94.3.1 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/(a*e*( m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[b*c*(m + 1) + 2*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) + 2*b*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b *c - a*d, 0] && LtQ[0, q, 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m , 2, p, q, x]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
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 = 5.68 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.84
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {\sqrt {d \,x^{4}+c}\, \left (a d \,x^{4}-3 b c \,x^{4}+a c \right )}{3 x^{6}}-\frac {b c \left (a d -b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{4}+c}\, a}{x^{2} \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}}{2 a^{2} c}\) | \(92\) |
| risch | \(-\frac {\sqrt {d \,x^{4}+c}\, \left (a d \,x^{4}-3 b c \,x^{4}+a c \right )}{6 c \,a^{2} x^{6}}-\frac {\left (a d -b c \right ) b \left (\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}-\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}\right )}{a^{2}}\) | \(374\) |
| default | \(-\frac {\left (d \,x^{4}+c \right )^{\frac {3}{2}}}{6 a \,x^{6} c}-\frac {b \left (-\frac {\left (d \,x^{4}+c \right )^{\frac {3}{2}}}{2 c \,x^{2}}+\frac {d \,x^{2} \sqrt {d \,x^{4}+c}}{2 c}+\frac {\sqrt {d}\, \ln \left (x^{2} \sqrt {d}+\sqrt {d \,x^{4}+c}\right )}{2}\right )}{a^{2}}+\frac {b^{2} \left (-\frac {\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}}-\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {-\frac {d \sqrt {-a b}}{b}+d \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{\sqrt {d}}+\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}}\right )}{b}+\frac {\left (a d -b c \right ) \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 )}{b \sqrt {-\frac {a d -b c}{b}}}}{4 \sqrt {-a b}}+\frac {\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}}+\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {\frac {d \sqrt {-a b}}{b}+d \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{\sqrt {d}}+\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}}\right )}{b}+\frac {\left (a d -b c \right ) \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 )}{b \sqrt {-\frac {a d -b c}{b}}}}{4 \sqrt {-a b}}\right )}{a^{2}}\) | \(782\) |
| elliptic | \(-\frac {\left (d \,x^{4}+c \right )^{\frac {3}{2}}}{6 a \,x^{6} c}-\frac {b \left (-\frac {\left (d \,x^{4}+c \right )^{\frac {3}{2}}}{c \,x^{2}}+\frac {2 d \left (\frac {x^{2} \sqrt {d \,x^{4}+c}}{2}+\frac {c \ln \left (x^{2} \sqrt {d}+\sqrt {d \,x^{4}+c}\right )}{2 \sqrt {d}}\right )}{c}\right )}{2 a^{2}}-\frac {b^{2} \left (\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}}-\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {-\frac {d \sqrt {-a b}}{b}+d \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{\sqrt {d}}+\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}}\right )}{b}+\frac {\left (a d -b c \right ) \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 )}{b \sqrt {-\frac {a d -b c}{b}}}\right )}{4 a^{2} \sqrt {-a b}}+\frac {b^{2} \left (\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}}+\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {\frac {d \sqrt {-a b}}{b}+d \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{\sqrt {d}}+\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}}\right )}{b}+\frac {\left (a d -b c \right ) \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 )}{b \sqrt {-\frac {a d -b c}{b}}}\right )}{4 a^{2} \sqrt {-a b}}\) | \(790\) |
1/2/a^2*(-1/3*(d*x^4+c)^(1/2)*(a*d*x^4-3*b*c*x^4+a*c)/x^6-b*c*(a*d-b*c)/(( a*d-b*c)*a)^(1/2)*arctanh((d*x^4+c)^(1/2)/x^2*a/((a*d-b*c)*a)^(1/2)))/c
Time = 0.28 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.99 \[ \int \frac {\sqrt {c+d x^4}}{x^7 \left (a+b x^4\right )} \, dx=\left [\frac {3 \, b c x^{6} \sqrt {-\frac {b c - a d}{a}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} + 4 \, {\left ({\left (a b c - 2 \, a^{2} d\right )} x^{6} - a^{2} c x^{2}\right )} \sqrt {d x^{4} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right ) + 4 \, {\left ({\left (3 \, b c - a d\right )} x^{4} - a c\right )} \sqrt {d x^{4} + c}}{24 \, a^{2} c x^{6}}, \frac {3 \, b c x^{6} \sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{4} - a c\right )} \sqrt {d x^{4} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{6} + {\left (b c^{2} - a c d\right )} x^{2}\right )}}\right ) + 2 \, {\left ({\left (3 \, b c - a d\right )} x^{4} - a c\right )} \sqrt {d x^{4} + c}}{12 \, a^{2} c x^{6}}\right ] \]
[1/24*(3*b*c*x^6*sqrt(-(b*c - a*d)/a)*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*((a*b*c - 2*a^2*d)*x^ 6 - a^2*c*x^2)*sqrt(d*x^4 + c)*sqrt(-(b*c - a*d)/a))/(b^2*x^8 + 2*a*b*x^4 + a^2)) + 4*((3*b*c - a*d)*x^4 - a*c)*sqrt(d*x^4 + c))/(a^2*c*x^6), 1/12*( 3*b*c*x^6*sqrt((b*c - a*d)/a)*arctan(1/2*((b*c - 2*a*d)*x^4 - a*c)*sqrt(d* x^4 + c)*sqrt((b*c - a*d)/a)/((b*c*d - a*d^2)*x^6 + (b*c^2 - a*c*d)*x^2)) + 2*((3*b*c - a*d)*x^4 - a*c)*sqrt(d*x^4 + c))/(a^2*c*x^6)]
\[ \int \frac {\sqrt {c+d x^4}}{x^7 \left (a+b x^4\right )} \, dx=\int \frac {\sqrt {c + d x^{4}}}{x^{7} \left (a + b x^{4}\right )}\, dx \]
\[ \int \frac {\sqrt {c+d x^4}}{x^7 \left (a+b x^4\right )} \, dx=\int { \frac {\sqrt {d x^{4} + c}}{{\left (b x^{4} + a\right )} x^{7}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (90) = 180\).
Time = 1.11 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.05 \[ \int \frac {\sqrt {c+d x^4}}{x^7 \left (a+b x^4\right )} \, dx=-\frac {{\left (b^{2} c \sqrt {d} - a b d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}} a^{2}} - \frac {3 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{4} b c \sqrt {d} - 3 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{4} a d^{\frac {3}{2}} - 6 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b c^{2} \sqrt {d} + 3 \, b c^{3} \sqrt {d} - a c^{2} d^{\frac {3}{2}}}{3 \, {\left ({\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} - c\right )}^{3} a^{2}} \]
-1/2*(b^2*c*sqrt(d) - a*b*d^(3/2))*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^2) - 1/3*(3*(sqrt(d)*x^2 - sqrt(d*x^4 + c))^4*b*c*sqrt(d) - 3*(sqrt(d)*x ^2 - sqrt(d*x^4 + c))^4*a*d^(3/2) - 6*(sqrt(d)*x^2 - sqrt(d*x^4 + c))^2*b* c^2*sqrt(d) + 3*b*c^3*sqrt(d) - a*c^2*d^(3/2))/(((sqrt(d)*x^2 - sqrt(d*x^4 + c))^2 - c)^3*a^2)
Timed out. \[ \int \frac {\sqrt {c+d x^4}}{x^7 \left (a+b x^4\right )} \, dx=\int \frac {\sqrt {d\,x^4+c}}{x^7\,\left (b\,x^4+a\right )} \,d x \]