Integrand size = 24, antiderivative size = 1032 \[ \int \frac {1}{x^3 \left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=-\frac {\sqrt {c+d x^8}}{2 a c x^2}+\frac {\sqrt {d} x^2 \sqrt {c+d x^8}}{2 a c \left (\sqrt {c}+\sqrt {d} x^4\right )}+\frac {b^{3/4} \arctan \left (\frac {\sqrt {b c-a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^8}}\right )}{8 (-a)^{5/4} \sqrt {b c-a d}}-\frac {b^{3/4} \arctan \left (\frac {\sqrt {-b c+a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^8}}\right )}{8 (-a)^{5/4} \sqrt {-b c+a d}}-\frac {\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 )}{2 a c^{3/4} \sqrt {c+d x^8}}+\frac {\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 )}{4 a c^{3/4} \sqrt {c+d x^8}}+\frac {b \left (\sqrt {c}-\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \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 )}{8 a \sqrt [4]{c} (b c+a d) \sqrt {c+d x^8}}+\frac {b \left (\sqrt {c}+\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \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 )}{8 a \sqrt [4]{c} (b c+a d) \sqrt {c+d x^8}}+\frac {\sqrt {b} \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2 \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 )}{16 (-a)^{3/2} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {c+d x^8}}-\frac {\sqrt {b} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2 \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 )}{16 (-a)^{3/2} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {c+d x^8}} \]
1/8*b^(3/4)*arctan(x^2*(-a*d+b*c)^(1/2)/(-a)^(1/4)/b^(1/4)/(d*x^8+c)^(1/2) )/(-a)^(5/4)/(-a*d+b*c)^(1/2)-1/8*b^(3/4)*arctan(x^2*(a*d-b*c)^(1/2)/(-a)^ (1/4)/b^(1/4)/(d*x^8+c)^(1/2))/(-a)^(5/4)/(a*d-b*c)^(1/2)-1/2*(d*x^8+c)^(1 /2)/a/c/x^2+1/2*x^2*d^(1/2)*(d*x^8+c)^(1/2)/a/c/(c^(1/2)+x^4*d^(1/2))-1/2* 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)/a/c^(3/4)/(d *x^8+c)^(1/2)+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)/a/c^(3/4)/(d*x^8+c)^(1/2)-1/16*(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))*b^(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^(1/2)+x^4*d^(1/2))^2)^(1/2)/(-a)^ (3/2)/c^(1/4)/d^(1/4)/(a*d+b*c)/(d*x^8+c)^(1/2)+1/16*(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))*b^(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^(1/2)+x^4*d^(...
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.12 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.14 \[ \int \frac {1}{x^3 \left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\frac {-21 a \left (c+d x^8\right )+7 (-b c+a d) x^8 \sqrt {1+\frac {d x^8}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},-\frac {d x^8}{c},-\frac {b x^8}{a}\right )+3 b d x^{16} \sqrt {1+\frac {d x^8}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},-\frac {d x^8}{c},-\frac {b x^8}{a}\right )}{42 a^2 c x^2 \sqrt {c+d x^8}} \]
(-21*a*(c + d*x^8) + 7*(-(b*c) + a*d)*x^8*Sqrt[1 + (d*x^8)/c]*AppellF1[3/4 , 1/2, 1, 7/4, -((d*x^8)/c), -((b*x^8)/a)] + 3*b*d*x^16*Sqrt[1 + (d*x^8)/c ]*AppellF1[7/4, 1/2, 1, 11/4, -((d*x^8)/c), -((b*x^8)/a)])/(42*a^2*c*x^2*S qrt[c + d*x^8])
Time = 1.49 (sec) , antiderivative size = 1021, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {965, 980, 25, 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.
| \(\displaystyle \int \frac {1}{x^3 \left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx\) |
| \(\Big \downarrow \) 965 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{x^4 \left (b x^8+a\right ) \sqrt {d x^8+c}}dx^2\) |
| \(\Big \downarrow \) 980 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {x^4 \left (-b d x^8+b c-a d\right )}{\left (b x^8+a\right ) \sqrt {d x^8+c}}dx^2}{a c}-\frac {\sqrt {c+d x^8}}{a c x^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {x^4 \left (-b d x^8+b c-a d\right )}{\left (b x^8+a\right ) \sqrt {d x^8+c}}dx^2}{a c}-\frac {\sqrt {c+d x^8}}{a c x^2}\right )\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \left (\frac {b c x^4}{\left (b x^8+a\right ) \sqrt {d x^8+c}}-\frac {d x^4}{\sqrt {d x^8+c}}\right )dx^2}{a c}-\frac {\sqrt {c+d x^8}}{a c x^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-\frac {\sqrt {b} c^{3/4} \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} \sqrt [4]{d} (b c+a d) \sqrt {d x^8+c}}+\frac {b^{3/4} c \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} \sqrt {b c-a d}}-\frac {b^{3/4} c \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} \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 )}{\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 \sqrt {d x^8+c}}-\frac {b c^{3/4} \left (\sqrt {c}-\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \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 )}{4 (b c+a d) \sqrt {d x^8+c}}-\frac {b c^{3/4} \left (\sqrt {c}+\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \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 )}{4 (b c+a d) \sqrt {d x^8+c}}+\frac {\sqrt {b} c^{3/4} \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2 \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} \sqrt [4]{d} (b c+a d) \sqrt {d x^8+c}}-\frac {\sqrt {d} x^2 \sqrt {d x^8+c}}{\sqrt {d} x^4+\sqrt {c}}}{a c}-\frac {\sqrt {d x^8+c}}{a c x^2}\right )\) |
(-(Sqrt[c + d*x^8]/(a*c*x^2)) - (-((Sqrt[d]*x^2*Sqrt[c + d*x^8])/(Sqrt[c] + Sqrt[d]*x^4)) + (b^(3/4)*c*ArcTan[(Sqrt[b*c - a*d]*x^2)/((-a)^(1/4)*b^(1 /4)*Sqrt[c + d*x^8])])/(4*(-a)^(1/4)*Sqrt[b*c - a*d]) - (b^(3/4)*c*ArcTanh [(Sqrt[b*c - a*d]*x^2)/((-a)^(1/4)*b^(1/4)*Sqrt[c + d*x^8])])/(4*(-a)^(1/4 )*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])/Sqrt[c + d*x^8] - (c^(1/4)*d^(1/4)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticF[2*ArcTan[(d^(1/4)*x^2)/c^(1/ 4)], 1/2])/(2*Sqrt[c + d*x^8]) - (b*c^(3/4)*(Sqrt[c] - (Sqrt[-a]*Sqrt[d])/ Sqrt[b])*d^(1/4)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[ d]*x^4)^2]*EllipticF[2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(4*(b*c + a*d) *Sqrt[c + d*x^8]) - (b*c^(3/4)*(Sqrt[c] + (Sqrt[-a]*Sqrt[d])/Sqrt[b])*d^(1 /4)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*El lipticF[2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(4*(b*c + a*d)*Sqrt[c + d*x ^8]) - (Sqrt[b]*c^(3/4)*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])^2*(Sqrt[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*A rcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(8*Sqrt[-a]*d^(1/4)*(b*c + a*d)*Sqrt[c + d*x^8]) + (Sqrt[b]*c^(3/4)*(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])^2*(Sqrt [c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*Elliptic...
3.10.2.3.1 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*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*e*(m + 1))), x] - Simp[1/(a*c*e^n*(m + 1)) Int[(e*x)^(m + n)*( a + b*x^n)^p*(c + d*x^n)^q*Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) + b*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && 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]
\[\int \frac {1}{x^{3} \left (b \,x^{8}+a \right ) \sqrt {d \,x^{8}+c}}d x\]
Timed out. \[ \int \frac {1}{x^3 \left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{x^3 \left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\int \frac {1}{x^{3} \left (a + b x^{8}\right ) \sqrt {c + d x^{8}}}\, dx \]
\[ \int \frac {1}{x^3 \left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\int { \frac {1}{{\left (b x^{8} + a\right )} \sqrt {d x^{8} + c} x^{3}} \,d x } \]
\[ \int \frac {1}{x^3 \left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\int { \frac {1}{{\left (b x^{8} + a\right )} \sqrt {d x^{8} + c} x^{3}} \,d x } \]
Timed out. \[ \int \frac {1}{x^3 \left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\int \frac {1}{x^3\,\left (b\,x^8+a\right )\,\sqrt {d\,x^8+c}} \,d x \]