Integrand size = 24, antiderivative size = 208 \[ \int \frac {1}{x^7 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=-\frac {(5 b c-2 a d) \sqrt {c+d x^4}}{12 a^2 c (b c-a d) x^6}+\frac {\left (15 b^2 c^2-8 a b c d-4 a^2 d^2\right ) \sqrt {c+d x^4}}{12 a^3 c^2 (b c-a d) x^2}+\frac {b \sqrt {c+d x^4}}{4 a (b c-a d) x^6 \left (a+b x^4\right )}+\frac {b^2 (5 b c-6 a d) \arctan \left (\frac {\sqrt {b c-a d} x^2}{\sqrt {a} \sqrt {c+d x^4}}\right )}{4 a^{7/2} (b c-a d)^{3/2}} \]
1/4*b^2*(-6*a*d+5*b*c)*arctan(x^2*(-a*d+b*c)^(1/2)/a^(1/2)/(d*x^4+c)^(1/2) )/a^(7/2)/(-a*d+b*c)^(3/2)-1/12*(-2*a*d+5*b*c)*(d*x^4+c)^(1/2)/a^2/c/(-a*d +b*c)/x^6+1/12*(-4*a^2*d^2-8*a*b*c*d+15*b^2*c^2)*(d*x^4+c)^(1/2)/a^3/c^2/( -a*d+b*c)/x^2+1/4*b*(d*x^4+c)^(1/2)/a/(-a*d+b*c)/x^6/(b*x^4+a)
Time = 2.50 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x^7 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=-\frac {\sqrt {c+d x^4} \left (15 b^3 c^2 x^8+2 a b^2 c x^4 \left (5 c-4 d x^4\right )+2 a^3 d \left (c-2 d x^4\right )-2 a^2 b \left (c^2+3 c d x^4+2 d^2 x^8\right )\right )}{12 a^3 c^2 (-b c+a d) x^6 \left (a+b x^4\right )}+\frac {b^2 (5 b c-6 a d) \arctan \left (\frac {a \sqrt {d}+b x^2 \left (\sqrt {d} x^2+\sqrt {c+d x^4}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{4 a^{7/2} (b c-a d)^{3/2}} \]
-1/12*(Sqrt[c + d*x^4]*(15*b^3*c^2*x^8 + 2*a*b^2*c*x^4*(5*c - 4*d*x^4) + 2 *a^3*d*(c - 2*d*x^4) - 2*a^2*b*(c^2 + 3*c*d*x^4 + 2*d^2*x^8)))/(a^3*c^2*(- (b*c) + a*d)*x^6*(a + b*x^4)) + (b^2*(5*b*c - 6*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])])/(4*a^(7/ 2)*(b*c - a*d)^(3/2))
Time = 0.42 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {965, 374, 25, 445, 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.
| \(\displaystyle \int \frac {1}{x^7 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx\) |
| \(\Big \downarrow \) 965 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{x^8 \left (b x^4+a\right )^2 \sqrt {d x^4+c}}dx^2\) |
| \(\Big \downarrow \) 374 |
| \(\displaystyle \frac {1}{2} \left (\frac {b \sqrt {c+d x^4}}{2 a x^6 \left (a+b x^4\right ) (b c-a d)}-\frac {\int -\frac {4 b d x^4+5 b c-2 a d}{x^8 \left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{2 a (b c-a d)}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {4 b d x^4+5 b c-2 a d}{x^8 \left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{2 a (b c-a d)}+\frac {b \sqrt {c+d x^4}}{2 a x^6 \left (a+b x^4\right ) (b c-a d)}\right )\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {1}{2} \left (\frac {-\frac {\int \frac {2 b d (5 b c-2 a d) x^4+15 b^2 c^2-4 a^2 d^2-8 a b c d}{x^4 \left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{3 a c}-\frac {\sqrt {c+d x^4} (5 b c-2 a d)}{3 a c x^6}}{2 a (b c-a d)}+\frac {b \sqrt {c+d x^4}}{2 a x^6 \left (a+b x^4\right ) (b c-a d)}\right )\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {1}{2} \left (\frac {-\frac {-\frac {\int \frac {3 b^2 c^2 (5 b c-6 a d)}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{a c}-\frac {\sqrt {c+d x^4} \left (\frac {15 b^2 c}{a}-\frac {4 a d^2}{c}-8 b d\right )}{x^2}}{3 a c}-\frac {\sqrt {c+d x^4} (5 b c-2 a d)}{3 a c x^6}}{2 a (b c-a d)}+\frac {b \sqrt {c+d x^4}}{2 a x^6 \left (a+b x^4\right ) (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {-\frac {-\frac {3 b^2 c (5 b c-6 a d) \int \frac {1}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{a}-\frac {\sqrt {c+d x^4} \left (\frac {15 b^2 c}{a}-\frac {4 a d^2}{c}-8 b d\right )}{x^2}}{3 a c}-\frac {\sqrt {c+d x^4} (5 b c-2 a d)}{3 a c x^6}}{2 a (b c-a d)}+\frac {b \sqrt {c+d x^4}}{2 a x^6 \left (a+b x^4\right ) (b c-a d)}\right )\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {1}{2} \left (\frac {-\frac {-\frac {3 b^2 c (5 b c-6 a d) \int \frac {1}{a-(a d-b c) x^4}d\frac {x^2}{\sqrt {d x^4+c}}}{a}-\frac {\sqrt {c+d x^4} \left (\frac {15 b^2 c}{a}-\frac {4 a d^2}{c}-8 b d\right )}{x^2}}{3 a c}-\frac {\sqrt {c+d x^4} (5 b c-2 a d)}{3 a c x^6}}{2 a (b c-a d)}+\frac {b \sqrt {c+d x^4}}{2 a x^6 \left (a+b x^4\right ) (b c-a d)}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {1}{2} \left (\frac {-\frac {-\frac {3 b^2 c (5 b c-6 a d) \arctan \left (\frac {x^2 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^4}}\right )}{a^{3/2} \sqrt {b c-a d}}-\frac {\sqrt {c+d x^4} \left (\frac {15 b^2 c}{a}-\frac {4 a d^2}{c}-8 b d\right )}{x^2}}{3 a c}-\frac {\sqrt {c+d x^4} (5 b c-2 a d)}{3 a c x^6}}{2 a (b c-a d)}+\frac {b \sqrt {c+d x^4}}{2 a x^6 \left (a+b x^4\right ) (b c-a d)}\right )\) |
((b*Sqrt[c + d*x^4])/(2*a*(b*c - a*d)*x^6*(a + b*x^4)) + (-1/3*((5*b*c - 2 *a*d)*Sqrt[c + d*x^4])/(a*c*x^6) - (-((((15*b^2*c)/a - 8*b*d - (4*a*d^2)/c )*Sqrt[c + d*x^4])/x^2) - (3*b^2*c*(5*b*c - 6*a*d)*ArcTan[(Sqrt[b*c - a*d] *x^2)/(Sqrt[a]*Sqrt[c + d*x^4])])/(a^(3/2)*Sqrt[b*c - a*d]))/(3*a*c))/(2*a *(b*c - a*d)))/2
3.9.33.3.1 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[(-b)*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*e*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(e*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[b*c*(m + 1) + 2*(b*c - a*d)*(p + 1) + d*b*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -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 = 6.75 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.64
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {\sqrt {d \,x^{4}+c}\, \left (-2 a d \,x^{4}-6 b c \,x^{4}+a c \right )}{3 x^{6}}-\frac {b^{2} c^{2} \left (\frac {b \sqrt {d \,x^{4}+c}\, x^{2}}{b \,x^{4}+a}-\frac {\left (6 a d -5 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{4}+c}\, a}{x^{2} \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )}{2 \left (a d -b c \right )}}{2 a^{3} c^{2}}\) | \(134\) |
| risch | \(-\frac {\sqrt {d \,x^{4}+c}\, \left (-2 a d \,x^{4}-6 b c \,x^{4}+a c \right )}{6 c^{2} a^{3} x^{6}}-\frac {b^{2} \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}}}{8 a^{3} \left (a d -b c \right ) \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}+\frac {b d \sqrt {-a 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 )}{8 a^{3} \left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}-\frac {b^{2} \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}}}{8 a^{3} \left (a d -b c \right ) \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}-\frac {b d \sqrt {-a 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 )}{8 a^{3} \left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}-\frac {5 b^{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 )}{8 a^{3} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}+\frac {5 b^{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 )}{8 a^{3} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}\) | \(913\) |
| elliptic | \(-\frac {\sqrt {d \,x^{4}+c}}{6 a^{2} c \,x^{6}}+\frac {d \sqrt {d \,x^{4}+c}}{3 a^{2} c^{2} x^{2}}+\frac {b \sqrt {d \,x^{4}+c}}{a^{3} c \,x^{2}}-\frac {b^{2} \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}}}{8 a^{3} \left (a d -b c \right ) \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}-\frac {b d \sqrt {-a 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 )}{8 a^{3} \left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}-\frac {b^{2} \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}}}{8 a^{3} \left (a d -b c \right ) \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}+\frac {b d \sqrt {-a 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 )}{8 a^{3} \left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}+\frac {5 b^{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 )}{8 a^{3} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}-\frac {5 b^{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 )}{8 a^{3} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}\) | \(936\) |
| default | \(\text {Expression too large to display}\) | \(1252\) |
1/2/a^3*(-1/3*(d*x^4+c)^(1/2)*(-2*a*d*x^4-6*b*c*x^4+a*c)/x^6-1/2*b^2*c^2/( a*d-b*c)*(b*(d*x^4+c)^(1/2)*x^2/(b*x^4+a)-(6*a*d-5*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^2
Time = 0.74 (sec) , antiderivative size = 760, normalized size of antiderivative = 3.65 \[ \int \frac {1}{x^7 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\left [-\frac {3 \, {\left ({\left (5 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d\right )} x^{10} + {\left (5 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d\right )} x^{6}\right )} \sqrt {-a b c + a^{2} d} \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 \, {\left ({\left (15 \, a b^{4} c^{3} - 23 \, a^{2} b^{3} c^{2} d + 4 \, a^{3} b^{2} c d^{2} + 4 \, a^{4} b d^{3}\right )} x^{8} - 2 \, a^{3} b^{2} c^{3} + 4 \, a^{4} b c^{2} d - 2 \, a^{5} c d^{2} + 2 \, {\left (5 \, a^{2} b^{3} c^{3} - 8 \, a^{3} b^{2} c^{2} d + a^{4} b c d^{2} + 2 \, a^{5} d^{3}\right )} x^{4}\right )} \sqrt {d x^{4} + c}}{48 \, {\left ({\left (a^{4} b^{3} c^{4} - 2 \, a^{5} b^{2} c^{3} d + a^{6} b c^{2} d^{2}\right )} x^{10} + {\left (a^{5} b^{2} c^{4} - 2 \, a^{6} b c^{3} d + a^{7} c^{2} d^{2}\right )} x^{6}\right )}}, \frac {3 \, {\left ({\left (5 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d\right )} x^{10} + {\left (5 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d\right )} x^{6}\right )} \sqrt {a b c - a^{2} d} \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 \, {\left ({\left (15 \, a b^{4} c^{3} - 23 \, a^{2} b^{3} c^{2} d + 4 \, a^{3} b^{2} c d^{2} + 4 \, a^{4} b d^{3}\right )} x^{8} - 2 \, a^{3} b^{2} c^{3} + 4 \, a^{4} b c^{2} d - 2 \, a^{5} c d^{2} + 2 \, {\left (5 \, a^{2} b^{3} c^{3} - 8 \, a^{3} b^{2} c^{2} d + a^{4} b c d^{2} + 2 \, a^{5} d^{3}\right )} x^{4}\right )} \sqrt {d x^{4} + c}}{24 \, {\left ({\left (a^{4} b^{3} c^{4} - 2 \, a^{5} b^{2} c^{3} d + a^{6} b c^{2} d^{2}\right )} x^{10} + {\left (a^{5} b^{2} c^{4} - 2 \, a^{6} b c^{3} d + a^{7} c^{2} d^{2}\right )} x^{6}\right )}}\right ] \]
[-1/48*(3*((5*b^4*c^3 - 6*a*b^3*c^2*d)*x^10 + (5*a*b^3*c^3 - 6*a^2*b^2*c^2 *d)*x^6)*sqrt(-a*b*c + a^2*d)*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*(( 15*a*b^4*c^3 - 23*a^2*b^3*c^2*d + 4*a^3*b^2*c*d^2 + 4*a^4*b*d^3)*x^8 - 2*a ^3*b^2*c^3 + 4*a^4*b*c^2*d - 2*a^5*c*d^2 + 2*(5*a^2*b^3*c^3 - 8*a^3*b^2*c^ 2*d + a^4*b*c*d^2 + 2*a^5*d^3)*x^4)*sqrt(d*x^4 + c))/((a^4*b^3*c^4 - 2*a^5 *b^2*c^3*d + a^6*b*c^2*d^2)*x^10 + (a^5*b^2*c^4 - 2*a^6*b*c^3*d + a^7*c^2* d^2)*x^6), 1/24*(3*((5*b^4*c^3 - 6*a*b^3*c^2*d)*x^10 + (5*a*b^3*c^3 - 6*a^ 2*b^2*c^2*d)*x^6)*sqrt(a*b*c - a^2*d)*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*((15*a*b^4*c^3 - 23*a^2*b^3*c^2*d + 4*a^3*b^2*c*d^2 + 4*a^4*b*d^3)*x^8 - 2*a^3*b^2*c^3 + 4*a^4*b*c^2*d - 2*a^5*c*d^2 + 2*(5*a^2* b^3*c^3 - 8*a^3*b^2*c^2*d + a^4*b*c*d^2 + 2*a^5*d^3)*x^4)*sqrt(d*x^4 + c)) /((a^4*b^3*c^4 - 2*a^5*b^2*c^3*d + a^6*b*c^2*d^2)*x^10 + (a^5*b^2*c^4 - 2* a^6*b*c^3*d + a^7*c^2*d^2)*x^6)]
\[ \int \frac {1}{x^7 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\int \frac {1}{x^{7} \left (a + b x^{4}\right )^{2} \sqrt {c + d x^{4}}}\, dx \]
\[ \int \frac {1}{x^7 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{2} \sqrt {d x^{4} + c} x^{7}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 395 vs. \(2 (184) = 368\).
Time = 0.94 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.90 \[ \int \frac {1}{x^7 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\frac {1}{12} \, d^{\frac {7}{2}} {\left (\frac {3 \, {\left (5 \, b^{3} c - 6 \, a b^{2} d\right )} \arctan \left (-\frac {{\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{{\left (a^{3} b c d^{3} - a^{4} d^{4}\right )} \sqrt {a b c d - a^{2} d^{2}}} - \frac {6 \, {\left ({\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b^{3} c - 2 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} a b^{2} d - b^{3} c^{2}\right )}}{{\left (a^{3} b c d^{3} - a^{4} d^{4}\right )} {\left ({\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} a d + b c^{2}\right )}} - \frac {8 \, {\left (3 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{4} b - 6 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b c - 3 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} a d + 3 \, b c^{2} + a c d\right )}}{{\left ({\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} - c\right )}^{3} a^{3} d^{3}}\right )} \]
1/12*d^(7/2)*(3*(5*b^3*c - 6*a*b^2*d)*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))/((a^3*b*c*d^3 - a^4*d ^4)*sqrt(a*b*c*d - a^2*d^2)) - 6*((sqrt(d)*x^2 - sqrt(d*x^4 + c))^2*b^3*c - 2*(sqrt(d)*x^2 - sqrt(d*x^4 + c))^2*a*b^2*d - b^3*c^2)/((a^3*b*c*d^3 - a ^4*d^4)*((sqrt(d)*x^2 - sqrt(d*x^4 + c))^4*b - 2*(sqrt(d)*x^2 - sqrt(d*x^4 + c))^2*b*c + 4*(sqrt(d)*x^2 - sqrt(d*x^4 + c))^2*a*d + b*c^2)) - 8*(3*(s qrt(d)*x^2 - sqrt(d*x^4 + c))^4*b - 6*(sqrt(d)*x^2 - sqrt(d*x^4 + c))^2*b* c - 3*(sqrt(d)*x^2 - sqrt(d*x^4 + c))^2*a*d + 3*b*c^2 + a*c*d)/(((sqrt(d)* x^2 - sqrt(d*x^4 + c))^2 - c)^3*a^3*d^3))
Timed out. \[ \int \frac {1}{x^7 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\int \frac {1}{x^7\,{\left (b\,x^4+a\right )}^2\,\sqrt {d\,x^4+c}} \,d x \]