Integrand size = 24, antiderivative size = 191 \[ \int \frac {x^{13}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\frac {(b c-2 a d) x^2 \sqrt {c+d x^4}}{4 b^2 d (b c-a d)}+\frac {a x^6 \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}+\frac {a^{3/2} (5 b c-4 a d) \arctan \left (\frac {\sqrt {b c-a d} x^2}{\sqrt {a} \sqrt {c+d x^4}}\right )}{4 b^3 (b c-a d)^{3/2}}-\frac {(b c+4 a d) \text {arctanh}\left (\frac {\sqrt {d} x^2}{\sqrt {c+d x^4}}\right )}{4 b^3 d^{3/2}} \]
1/4*a^(3/2)*(-4*a*d+5*b*c)*arctan(x^2*(-a*d+b*c)^(1/2)/a^(1/2)/(d*x^4+c)^( 1/2))/b^3/(-a*d+b*c)^(3/2)-1/4*(4*a*d+b*c)*arctanh(x^2*d^(1/2)/(d*x^4+c)^( 1/2))/b^3/d^(3/2)+1/4*(-2*a*d+b*c)*x^2*(d*x^4+c)^(1/2)/b^2/d/(-a*d+b*c)+1/ 4*a*x^6*(d*x^4+c)^(1/2)/b/(-a*d+b*c)/(b*x^4+a)
Time = 2.77 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.99 \[ \int \frac {x^{13}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\frac {\frac {b x^2 \sqrt {c+d x^4} \left (-2 a^2 d+b^2 c x^4+a b \left (c-d x^4\right )\right )}{d (b c-a d) \left (a+b x^4\right )}+\frac {a^{3/2} (5 b c-4 a d) \arctan \left (\frac {a \sqrt {d}+b x^2 \left (\sqrt {d} x^2+\sqrt {c+d x^4}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}}-\frac {(b c+4 a d) \log \left (\sqrt {d} x^2+\sqrt {c+d x^4}\right )}{d^{3/2}}}{4 b^3} \]
((b*x^2*Sqrt[c + d*x^4]*(-2*a^2*d + b^2*c*x^4 + a*b*(c - d*x^4)))/(d*(b*c - a*d)*(a + b*x^4)) + (a^(3/2)*(5*b*c - 4*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])])/(b*c - a*d)^(3 /2) - ((b*c + 4*a*d)*Log[Sqrt[d]*x^2 + Sqrt[c + d*x^4]])/d^(3/2))/(4*b^3)
Time = 0.46 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {965, 372, 444, 27, 398, 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.
\(\displaystyle \int \frac {x^{13}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx\) |
\(\Big \downarrow \) 965 |
\(\displaystyle \frac {1}{2} \int \frac {x^{12}}{\left (b x^4+a\right )^2 \sqrt {d x^4+c}}dx^2\) |
\(\Big \downarrow \) 372 |
\(\displaystyle \frac {1}{2} \left (\frac {a x^6 \sqrt {c+d x^4}}{2 b \left (a+b x^4\right ) (b c-a d)}-\frac {\int \frac {x^4 \left (3 a c-2 (b c-2 a d) x^4\right )}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{2 b (b c-a d)}\right )\) |
\(\Big \downarrow \) 444 |
\(\displaystyle \frac {1}{2} \left (\frac {a x^6 \sqrt {c+d x^4}}{2 b \left (a+b x^4\right ) (b c-a d)}-\frac {-\frac {\int -\frac {2 \left ((b c-a d) (b c+4 a d) x^4+a c (b c-2 a d)\right )}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{2 b d}-\frac {x^2 \sqrt {c+d x^4} (b c-2 a d)}{b d}}{2 b (b c-a d)}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (\frac {a x^6 \sqrt {c+d x^4}}{2 b \left (a+b x^4\right ) (b c-a d)}-\frac {\frac {\int \frac {(b c-a d) (b c+4 a d) x^4+a c (b c-2 a d)}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{b d}-\frac {x^2 \sqrt {c+d x^4} (b c-2 a d)}{b d}}{2 b (b c-a d)}\right )\) |
\(\Big \downarrow \) 398 |
\(\displaystyle \frac {1}{2} \left (\frac {a x^6 \sqrt {c+d x^4}}{2 b \left (a+b x^4\right ) (b c-a d)}-\frac {\frac {\frac {(b c-a d) (4 a d+b c) \int \frac {1}{\sqrt {d x^4+c}}dx^2}{b}-\frac {a^2 d (5 b c-4 a d) \int \frac {1}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{b}}{b d}-\frac {x^2 \sqrt {c+d x^4} (b c-2 a d)}{b d}}{2 b (b c-a d)}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {1}{2} \left (\frac {a x^6 \sqrt {c+d x^4}}{2 b \left (a+b x^4\right ) (b c-a d)}-\frac {\frac {\frac {(b c-a d) (4 a d+b c) \int \frac {1}{1-d x^4}d\frac {x^2}{\sqrt {d x^4+c}}}{b}-\frac {a^2 d (5 b c-4 a d) \int \frac {1}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{b}}{b d}-\frac {x^2 \sqrt {c+d x^4} (b c-2 a d)}{b d}}{2 b (b c-a d)}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (\frac {a x^6 \sqrt {c+d x^4}}{2 b \left (a+b x^4\right ) (b c-a d)}-\frac {\frac {\frac {(b c-a d) (4 a d+b c) \text {arctanh}\left (\frac {\sqrt {d} x^2}{\sqrt {c+d x^4}}\right )}{b \sqrt {d}}-\frac {a^2 d (5 b c-4 a d) \int \frac {1}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{b}}{b d}-\frac {x^2 \sqrt {c+d x^4} (b c-2 a d)}{b d}}{2 b (b c-a d)}\right )\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \frac {1}{2} \left (\frac {a x^6 \sqrt {c+d x^4}}{2 b \left (a+b x^4\right ) (b c-a d)}-\frac {\frac {\frac {(b c-a d) (4 a d+b c) \text {arctanh}\left (\frac {\sqrt {d} x^2}{\sqrt {c+d x^4}}\right )}{b \sqrt {d}}-\frac {a^2 d (5 b c-4 a d) \int \frac {1}{a-(a d-b c) x^4}d\frac {x^2}{\sqrt {d x^4+c}}}{b}}{b d}-\frac {x^2 \sqrt {c+d x^4} (b c-2 a d)}{b d}}{2 b (b c-a d)}\right )\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {1}{2} \left (\frac {a x^6 \sqrt {c+d x^4}}{2 b \left (a+b x^4\right ) (b c-a d)}-\frac {\frac {\frac {(b c-a d) (4 a d+b c) \text {arctanh}\left (\frac {\sqrt {d} x^2}{\sqrt {c+d x^4}}\right )}{b \sqrt {d}}-\frac {a^{3/2} d (5 b c-4 a d) \arctan \left (\frac {x^2 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^4}}\right )}{b \sqrt {b c-a d}}}{b d}-\frac {x^2 \sqrt {c+d x^4} (b c-2 a d)}{b d}}{2 b (b c-a d)}\right )\) |
((a*x^6*Sqrt[c + d*x^4])/(2*b*(b*c - a*d)*(a + b*x^4)) - (-(((b*c - 2*a*d) *x^2*Sqrt[c + d*x^4])/(b*d)) + (-((a^(3/2)*d*(5*b*c - 4*a*d)*ArcTan[(Sqrt[ b*c - a*d]*x^2)/(Sqrt[a]*Sqrt[c + d*x^4])])/(b*Sqrt[b*c - a*d])) + ((b*c - a*d)*(b*c + 4*a*d)*ArcTanh[(Sqrt[d]*x^2)/Sqrt[c + d*x^4]])/(b*Sqrt[d]))/( b*d))/(2*b*(b*c - a*d)))/2
3.9.28.3.1 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[((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_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 )^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 )) Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a , b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} , x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[f*g*(g*x)^(m - 1)*(a + b*x^2)^ (p + 1)*((c + d*x^2)^(q + 1)/(b*d*(m + 2*(p + q + 1) + 1))), x] - Simp[g^2/ (b*d*(m + 2*(p + q + 1) + 1)) Int[(g*x)^(m - 2)*(a + b*x^2)^p*(c + d*x^2) ^q*Simp[a*f*c*(m - 1) + (a*f*d*(m + 2*q + 1) + b*(f*c*(m + 2*p + 1) - e*d*( m + 2*(p + q + 1) + 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && GtQ[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 = 6.40 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.86
method | result | size |
pseudoelliptic | \(-\frac {\frac {a^{2} \left (-\frac {b \sqrt {d \,x^{4}+c}\, x^{2}}{b \,x^{4}+a}-\frac {\left (4 a d -5 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{4}+c}\, a}{x^{2} \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )}{a d -b c}-\frac {b \sqrt {d \,x^{4}+c}\, x^{2} d^{\frac {3}{2}}-4 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{4}+c}}{x^{2} \sqrt {d}}\right ) a \,d^{2}-\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{4}+c}}{x^{2} \sqrt {d}}\right ) b c d}{d^{\frac {5}{2}}}}{4 b^{3}}\) | \(164\) |
risch | \(\frac {x^{2} \sqrt {d \,x^{4}+c}}{4 b^{2} d}-\frac {a \ln \left (x^{2} \sqrt {d}+\sqrt {d \,x^{4}+c}\right )}{b^{3} \sqrt {d}}-\frac {\ln \left (x^{2} \sqrt {d}+\sqrt {d \,x^{4}+c}\right ) c}{4 b^{2} d^{\frac {3}{2}}}+\frac {a^{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 b^{3} \left (a d -b c \right ) \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}-\frac {d \,a^{2} \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 b^{4} \left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}+\frac {a^{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 b^{3} \left (a d -b c \right ) \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}+\frac {d \,a^{2} \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 b^{4} \left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}-\frac {5 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 )}{8 b^{3} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}+\frac {5 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 )}{8 b^{3} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}\) | \(953\) |
elliptic | \(\frac {x^{2} \sqrt {d \,x^{4}+c}}{4 b^{2} d}-\frac {a \ln \left (x^{2} \sqrt {d}+\sqrt {d \,x^{4}+c}\right )}{b^{3} \sqrt {d}}-\frac {\ln \left (x^{2} \sqrt {d}+\sqrt {d \,x^{4}+c}\right ) c}{4 b^{2} d^{\frac {3}{2}}}+\frac {a^{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 b^{3} \left (a d -b c \right ) \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}-\frac {d \,a^{2} \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 b^{4} \left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}+\frac {a^{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 b^{3} \left (a d -b c \right ) \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}+\frac {d \,a^{2} \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 b^{4} \left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}-\frac {5 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 )}{8 b^{3} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}+\frac {5 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 )}{8 b^{3} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}\) | \(953\) |
default | \(\text {Expression too large to display}\) | \(1278\) |
-1/4/b^3*(a^2/(a*d-b*c)*(-b*(d*x^4+c)^(1/2)*x^2/(b*x^4+a)-(4*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)))-(b*( d*x^4+c)^(1/2)*x^2*d^(3/2)-4*arctanh((d*x^4+c)^(1/2)/x^2/d^(1/2))*a*d^2-ar ctanh((d*x^4+c)^(1/2)/x^2/d^(1/2))*b*c*d)/d^(5/2))
Time = 1.45 (sec) , antiderivative size = 1386, normalized size of antiderivative = 7.26 \[ \int \frac {x^{13}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\text {Too large to display} \]
[1/16*(2*(a*b^2*c^2 + 3*a^2*b*c*d - 4*a^3*d^2 + (b^3*c^2 + 3*a*b^2*c*d - 4 *a^2*b*d^2)*x^4)*sqrt(d)*log(-2*d*x^4 + 2*sqrt(d*x^4 + c)*sqrt(d)*x^2 - c) + (5*a^2*b*c*d^2 - 4*a^3*d^3 + (5*a*b^2*c*d^2 - 4*a^2*b*d^3)*x^4)*sqrt(-a /(b*c - a*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^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x^6 - (a*b *c^2 - a^2*c*d)*x^2)*sqrt(d*x^4 + c)*sqrt(-a/(b*c - a*d)))/(b^2*x^8 + 2*a* b*x^4 + a^2)) + 4*((b^3*c*d - a*b^2*d^2)*x^6 + (a*b^2*c*d - 2*a^2*b*d^2)*x ^2)*sqrt(d*x^4 + c))/(a*b^4*c*d^2 - a^2*b^3*d^3 + (b^5*c*d^2 - a*b^4*d^3)* x^4), 1/16*(4*(a*b^2*c^2 + 3*a^2*b*c*d - 4*a^3*d^2 + (b^3*c^2 + 3*a*b^2*c* d - 4*a^2*b*d^2)*x^4)*sqrt(-d)*arctan(sqrt(-d)*x^2/sqrt(d*x^4 + c)) + (5*a ^2*b*c*d^2 - 4*a^3*d^3 + (5*a*b^2*c*d^2 - 4*a^2*b*d^3)*x^4)*sqrt(-a/(b*c - a*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^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x^6 - (a*b*c^2 - a^2*c*d)*x^2)*sqrt(d*x^4 + c)*sqrt(-a/(b*c - a*d)))/(b^2*x^8 + 2*a*b*x^4 + a^2)) + 4*((b^3*c*d - a*b^2*d^2)*x^6 + (a*b^2*c*d - 2*a^2*b*d^2)*x^2)*sqr t(d*x^4 + c))/(a*b^4*c*d^2 - a^2*b^3*d^3 + (b^5*c*d^2 - a*b^4*d^3)*x^4), - 1/8*((5*a^2*b*c*d^2 - 4*a^3*d^3 + (5*a*b^2*c*d^2 - 4*a^2*b*d^3)*x^4)*sqrt( a/(b*c - a*d))*arctan(-1/2*((b*c - 2*a*d)*x^4 - a*c)*sqrt(d*x^4 + c)*sqrt( a/(b*c - a*d))/(a*d*x^6 + a*c*x^2)) - (a*b^2*c^2 + 3*a^2*b*c*d - 4*a^3*d^2 + (b^3*c^2 + 3*a*b^2*c*d - 4*a^2*b*d^2)*x^4)*sqrt(d)*log(-2*d*x^4 + 2*...
\[ \int \frac {x^{13}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\int \frac {x^{13}}{\left (a + b x^{4}\right )^{2} \sqrt {c + d x^{4}}}\, dx \]
\[ \int \frac {x^{13}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\int { \frac {x^{13}}{{\left (b x^{4} + a\right )}^{2} \sqrt {d x^{4} + c}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (163) = 326\).
Time = 0.39 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.74 \[ \int \frac {x^{13}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=-\frac {{\left (5 \, a^{2} b c \sqrt {d} - 4 \, a^{3} d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{4 \, {\left (b^{4} c - a b^{3} d\right )} \sqrt {a b c d - a^{2} d^{2}}} + \frac {\sqrt {d x^{4} + c} x^{2}}{4 \, b^{2} d} + \frac {{\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} a^{2} b c d - 2 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} a^{3} d^{2} - a^{2} b c^{2} d}{2 \, {\left (b^{4} c \sqrt {d} - a b^{3} d^{\frac {3}{2}}\right )} {\left ({\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} a d + b c^{2}\right )}} + \frac {{\left (b c + 4 \, a d\right )} \log \left ({\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2}\right )}{8 \, b^{3} d^{\frac {3}{2}}} \]
-1/4*(5*a^2*b*c*sqrt(d) - 4*a^3*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))/((b^4*c - a*b^3*d)* sqrt(a*b*c*d - a^2*d^2)) + 1/4*sqrt(d*x^4 + c)*x^2/(b^2*d) + 1/2*((sqrt(d) *x^2 - sqrt(d*x^4 + c))^2*a^2*b*c*d - 2*(sqrt(d)*x^2 - sqrt(d*x^4 + c))^2* a^3*d^2 - a^2*b*c^2*d)/((b^4*c*sqrt(d) - a*b^3*d^(3/2))*((sqrt(d)*x^2 - sq rt(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)) + 1/8*(b*c + 4*a*d)*log((sqrt(d)*x^ 2 - sqrt(d*x^4 + c))^2)/(b^3*d^(3/2))
Timed out. \[ \int \frac {x^{13}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\int \frac {x^{13}}{{\left (b\,x^4+a\right )}^2\,\sqrt {d\,x^4+c}} \,d x \]