Integrand size = 24, antiderivative size = 1164 \[ \int \frac {x^{13}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\frac {\sqrt {d} x^2 \sqrt {c+d x^8}}{8 b (b c-a d) \left (\sqrt {c}+\sqrt {d} x^4\right )}-\frac {x^6 \sqrt {c+d x^8}}{8 (b c-a d) \left (a+b x^8\right )}+\frac {(3 b c-a d) \arctan \left (\frac {\sqrt {b c-a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^8}}\right )}{32 \sqrt [4]{-a} b^{5/4} (b c-a d)^{3/2}}+\frac {(3 b c-a d) \arctan \left (\frac {\sqrt {-b c+a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^8}}\right )}{32 \sqrt [4]{-a} b^{5/4} (-b c+a d)^{3/2}}-\frac {\sqrt [4]{c} \sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 b (b c-a d) \sqrt {c+d x^8}}+\frac {\sqrt [4]{c} \sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{16 b (b c-a d) \sqrt {c+d x^8}}-\frac {\left (\sqrt {c}-\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \sqrt [4]{d} (3 b c-a d) \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{32 b \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt {c+d x^8}}-\frac {\left (\sqrt {c}+\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \sqrt [4]{d} (3 b c-a d) \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{32 b \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt {c+d x^8}}+\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2 (3 b c-a d) \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{64 \sqrt {-a} b^{3/2} \sqrt [4]{c} \sqrt [4]{d} (b c-a d) (b c+a d) \sqrt {c+d x^8}}-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2 (3 b c-a d) \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{64 \sqrt {-a} b^{3/2} \sqrt [4]{c} \sqrt [4]{d} (b c-a d) (b c+a d) \sqrt {c+d x^8}} \]
1/32*(-a*d+3*b*c)*arctan(x^2*(-a*d+b*c)^(1/2)/(-a)^(1/4)/b^(1/4)/(d*x^8+c) ^(1/2))/(-a)^(1/4)/b^(5/4)/(-a*d+b*c)^(3/2)+1/32*(-a*d+3*b*c)*arctan(x^2*( a*d-b*c)^(1/2)/(-a)^(1/4)/b^(1/4)/(d*x^8+c)^(1/2))/(-a)^(1/4)/b^(5/4)/(a*d -b*c)^(3/2)-1/8*x^6*(d*x^8+c)^(1/2)/(-a*d+b*c)/(b*x^8+a)+1/8*x^2*d^(1/2)*( d*x^8+c)^(1/2)/b/(-a*d+b*c)/(c^(1/2)+x^4*d^(1/2))-1/8*c^(1/4)*d^(1/4)*(cos (2*arctan(d^(1/4)*x^2/c^(1/4)))^2)^(1/2)/cos(2*arctan(d^(1/4)*x^2/c^(1/4)) )*EllipticE(sin(2*arctan(d^(1/4)*x^2/c^(1/4))),1/2*2^(1/2))*(c^(1/2)+x^4*d ^(1/2))*((d*x^8+c)/(c^(1/2)+x^4*d^(1/2))^2)^(1/2)/b/(-a*d+b*c)/(d*x^8+c)^( 1/2)+1/16*c^(1/4)*d^(1/4)*(cos(2*arctan(d^(1/4)*x^2/c^(1/4)))^2)^(1/2)/cos (2*arctan(d^(1/4)*x^2/c^(1/4)))*EllipticF(sin(2*arctan(d^(1/4)*x^2/c^(1/4) )),1/2*2^(1/2))*(c^(1/2)+x^4*d^(1/2))*((d*x^8+c)/(c^(1/2)+x^4*d^(1/2))^2)^ (1/2)/b/(-a*d+b*c)/(d*x^8+c)^(1/2)-1/32*d^(1/4)*(-a*d+3*b*c)*(cos(2*arctan (d^(1/4)*x^2/c^(1/4)))^2)^(1/2)/cos(2*arctan(d^(1/4)*x^2/c^(1/4)))*Ellipti cF(sin(2*arctan(d^(1/4)*x^2/c^(1/4))),1/2*2^(1/2))*(c^(1/2)+x^4*d^(1/2))*( b^(1/2)*c^(1/2)-(-a)^(1/2)*d^(1/2))*((d*x^8+c)/(c^(1/2)+x^4*d^(1/2))^2)^(1 /2)/b^(3/2)/c^(1/4)/(-a^2*d^2+b^2*c^2)/(d*x^8+c)^(1/2)-1/64*(-a*d+3*b*c)*( cos(2*arctan(d^(1/4)*x^2/c^(1/4)))^2)^(1/2)/cos(2*arctan(d^(1/4)*x^2/c^(1/ 4)))*EllipticPi(sin(2*arctan(d^(1/4)*x^2/c^(1/4))),1/4*(b^(1/2)*c^(1/2)+(- a)^(1/2)*d^(1/2))^2/(-a)^(1/2)/b^(1/2)/c^(1/2)/d^(1/2),1/2*2^(1/2))*(c^(1/ 2)+x^4*d^(1/2))*(b^(1/2)*c^(1/2)-(-a)^(1/2)*d^(1/2))^2*((d*x^8+c)/(c^(1...
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.17 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.14 \[ \int \frac {x^{13}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\frac {x^6 \left (-7 a \left (c+d x^8\right )+7 c \left (a+b x^8\right ) \sqrt {1+\frac {d x^8}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},-\frac {d x^8}{c},-\frac {b x^8}{a}\right )+d x^8 \left (a+b x^8\right ) \sqrt {1+\frac {d x^8}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},-\frac {d x^8}{c},-\frac {b x^8}{a}\right )\right )}{56 a (b c-a d) \left (a+b x^8\right ) \sqrt {c+d x^8}} \]
(x^6*(-7*a*(c + d*x^8) + 7*c*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[3/4, 1/2, 1, 7/4, -((d*x^8)/c), -((b*x^8)/a)] + d*x^8*(a + b*x^8)*Sqrt[1 + (d* x^8)/c]*AppellF1[7/4, 1/2, 1, 11/4, -((d*x^8)/c), -((b*x^8)/a)]))/(56*a*(b *c - a*d)*(a + b*x^8)*Sqrt[c + d*x^8])
Time = 1.55 (sec) , antiderivative size = 1107, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {965, 971, 1054, 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.
| \(\displaystyle \int \frac {x^{13}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx\) |
| \(\Big \downarrow \) 965 |
| \(\displaystyle \frac {1}{2} \int \frac {x^{12}}{\left (b x^8+a\right )^2 \sqrt {d x^8+c}}dx^2\) |
| \(\Big \downarrow \) 971 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {x^4 \left (d x^8+3 c\right )}{\left (b x^8+a\right ) \sqrt {d x^8+c}}dx^2}{4 (b c-a d)}-\frac {x^6 \sqrt {c+d x^8}}{4 \left (a+b x^8\right ) (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \left (\frac {d x^4}{b \sqrt {d x^8+c}}+\frac {(3 b c-a d) x^4}{b \left (b x^8+a\right ) \sqrt {d x^8+c}}\right )dx^2}{4 (b c-a d)}-\frac {x^6 \sqrt {c+d x^8}}{4 \left (a+b x^8\right ) (b c-a d)}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {-\frac {(3 b c-a d) \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right ) \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{8 \sqrt {-a} b^{3/2} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {d x^8+c}}+\frac {(3 b c-a d) \arctan \left (\frac {\sqrt {b c-a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^8+c}}\right )}{4 \sqrt [4]{-a} b^{5/4} \sqrt {b c-a d}}-\frac {(3 b c-a d) \text {arctanh}\left (\frac {\sqrt {b c-a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^8+c}}\right )}{4 \sqrt [4]{-a} b^{5/4} \sqrt {b c-a d}}-\frac {\sqrt [4]{c} \sqrt [4]{d} \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{b \sqrt {d x^8+c}}+\frac {\sqrt [4]{c} \sqrt [4]{d} \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 b \sqrt {d x^8+c}}-\frac {\left (\sqrt {c}-\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \sqrt [4]{d} (3 b c-a d) \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{4 b \sqrt [4]{c} (b c+a d) \sqrt {d x^8+c}}-\frac {\left (\sqrt {c}+\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \sqrt [4]{d} (3 b c-a d) \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{4 b \sqrt [4]{c} (b c+a d) \sqrt {d x^8+c}}+\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2 (3 b c-a d) \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {c} \left (\sqrt {b}-\frac {\sqrt {-a} \sqrt {d}}{\sqrt {c}}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{8 \sqrt {-a} b^{3/2} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {d x^8+c}}+\frac {\sqrt {d} x^2 \sqrt {d x^8+c}}{b \left (\sqrt {d} x^4+\sqrt {c}\right )}}{4 (b c-a d)}-\frac {x^6 \sqrt {d x^8+c}}{4 (b c-a d) \left (b x^8+a\right )}\right )\) |
(-1/4*(x^6*Sqrt[c + d*x^8])/((b*c - a*d)*(a + b*x^8)) + ((Sqrt[d]*x^2*Sqrt [c + d*x^8])/(b*(Sqrt[c] + Sqrt[d]*x^4)) + ((3*b*c - a*d)*ArcTan[(Sqrt[b*c - a*d]*x^2)/((-a)^(1/4)*b^(1/4)*Sqrt[c + d*x^8])])/(4*(-a)^(1/4)*b^(5/4)* Sqrt[b*c - a*d]) - ((3*b*c - a*d)*ArcTanh[(Sqrt[b*c - a*d]*x^2)/((-a)^(1/4 )*b^(1/4)*Sqrt[c + d*x^8])])/(4*(-a)^(1/4)*b^(5/4)*Sqrt[b*c - a*d]) - (c^( 1/4)*d^(1/4)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x ^4)^2]*EllipticE[2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(b*Sqrt[c + d*x^8] ) + (c^(1/4)*d^(1/4)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + S qrt[d]*x^4)^2]*EllipticF[2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(2*b*Sqrt[ c + d*x^8]) - ((Sqrt[c] - (Sqrt[-a]*Sqrt[d])/Sqrt[b])*d^(1/4)*(3*b*c - a*d )*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*Elli pticF[2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(4*b*c^(1/4)*(b*c + a*d)*Sqrt [c + d*x^8]) - ((Sqrt[c] + (Sqrt[-a]*Sqrt[d])/Sqrt[b])*d^(1/4)*(3*b*c - a* d)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*Ell ipticF[2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(4*b*c^(1/4)*(b*c + a*d)*Sqr t[c + d*x^8]) - ((Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])^2*(3*b*c - a*d)*(Sqr t[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticPi [(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])^2/(4*Sqrt[-a]*Sqrt[b]*Sqrt[c]*Sqrt[d ]), 2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(8*Sqrt[-a]*b^(3/2)*c^(1/4)*d^( 1/4)*(b*c + a*d)*Sqrt[c + d*x^8]) + ((Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d...
3.10.21.3.1 Defintions of rubi rules used
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]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[e^(n - 1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)* ((c + d*x^n)^(q + 1)/(n*(b*c - a*d)*(p + 1))), x] - Simp[e^n/(n*(b*c - a*d) *(p + 1)) Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e , q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n _)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0]
\[\int \frac {x^{13}}{\left (b \,x^{8}+a \right )^{2} \sqrt {d \,x^{8}+c}}d x\]
Timed out. \[ \int \frac {x^{13}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {x^{13}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\text {Timed out} \]
\[ \int \frac {x^{13}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int { \frac {x^{13}}{{\left (b x^{8} + a\right )}^{2} \sqrt {d x^{8} + c}} \,d x } \]
\[ \int \frac {x^{13}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int { \frac {x^{13}}{{\left (b x^{8} + a\right )}^{2} \sqrt {d x^{8} + c}} \,d x } \]
Timed out. \[ \int \frac {x^{13}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int \frac {x^{13}}{{\left (b\,x^8+a\right )}^2\,\sqrt {d\,x^8+c}} \,d x \]