Integrand size = 29, antiderivative size = 452 \[ \int \frac {\left (-b+a x^4\right )^{3/4}}{-b+2 a x^4+x^8} \, dx=\frac {\left (-1-\sqrt [4]{-1}\right ) \arctan \left (\frac {(-1)^{7/8} \sqrt {2+\sqrt {2}} \sqrt [8]{a^2+b} x \sqrt [4]{-b+a x^4}}{(-1)^{3/4} \sqrt [4]{a^2+b} x^2+\sqrt {-b+a x^4}}\right )}{8 \sqrt [8]{a^2+b}}+\frac {\left (\sqrt {2}+i \sqrt {2 \left (3-2 \sqrt {2}\right )}\right ) \arctan \left (\frac {(-1)^{7/8} \left (-2+\sqrt {2}\right ) \sqrt [8]{a^2+b} x \sqrt [4]{-b+a x^4}}{(-1)^{3/4} \sqrt {2-\sqrt {2}} \sqrt [4]{a^2+b} x^2+\sqrt {2-\sqrt {2}} \sqrt {-b+a x^4}}\right )}{16 \sqrt [8]{a^2+b}}-\frac {i \left (-i \sqrt {2}+\sqrt {2 \left (3-2 \sqrt {2}\right )}\right ) \text {arctanh}\left (\frac {(-1)^{7/8} \sqrt [4]{a^2+b} x^2-\sqrt [8]{-1} \sqrt {-b+a x^4}}{\sqrt {2-\sqrt {2}} \sqrt [8]{a^2+b} x \sqrt [4]{-b+a x^4}}\right )}{16 \sqrt [8]{a^2+b}}+\frac {\left (-1-\sqrt [4]{-1}\right ) \text {arctanh}\left (\frac {(-1)^{7/8} \sqrt [4]{a^2+b} x^2-\sqrt [8]{-1} \sqrt {-b+a x^4}}{\sqrt {2+\sqrt {2}} \sqrt [8]{a^2+b} x \sqrt [4]{-b+a x^4}}\right )}{8 \sqrt [8]{a^2+b}} \]
1/8*(-1-(-1)^(1/4))*arctan((-1)^(7/8)*(2+2^(1/2))^(1/2)*(a^2+b)^(1/8)*x*(a *x^4-b)^(1/4)/((-1)^(3/4)*(a^2+b)^(1/4)*x^2+(a*x^4-b)^(1/2)))/(a^2+b)^(1/8 )+1/16*(2^(1/2)+I*(2-2^(1/2)))*arctan((-1)^(7/8)*(-2+2^(1/2))*(a^2+b)^(1/8 )*x*(a*x^4-b)^(1/4)/((-1)^(3/4)*(2-2^(1/2))^(1/2)*(a^2+b)^(1/4)*x^2+(2-2^( 1/2))^(1/2)*(a*x^4-b)^(1/2)))/(a^2+b)^(1/8)-1/16*I*(-I*2^(1/2)+2-2^(1/2))* arctanh(((-1)^(7/8)*(a^2+b)^(1/4)*x^2-(-1)^(1/8)*(a*x^4-b)^(1/2))/(2-2^(1/ 2))^(1/2)/(a^2+b)^(1/8)/x/(a*x^4-b)^(1/4))/(a^2+b)^(1/8)+1/8*(-1-(-1)^(1/4 ))*arctanh(((-1)^(7/8)*(a^2+b)^(1/4)*x^2-(-1)^(1/8)*(a*x^4-b)^(1/2))/(2+2^ (1/2))^(1/2)/(a^2+b)^(1/8)/x/(a*x^4-b)^(1/4))/(a^2+b)^(1/8)
\[ \int \frac {\left (-b+a x^4\right )^{3/4}}{-b+2 a x^4+x^8} \, dx=\int \frac {\left (-b+a x^4\right )^{3/4}}{-b+2 a x^4+x^8} \, dx \]
Time = 0.83 (sec) , antiderivative size = 565, normalized size of antiderivative = 1.25, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {1758, 27, 916, 770, 756, 216, 219, 902, 756, 218, 221}
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.
| \(\displaystyle \int \frac {\left (a x^4-b\right )^{3/4}}{2 a x^4-b+x^8} \, dx\) |
| \(\Big \downarrow \) 1758 |
| \(\displaystyle \frac {\int \frac {\left (a x^4-b\right )^{3/4}}{2 \left (x^4+a-\sqrt {a^2+b}\right )}dx}{\sqrt {a^2+b}}-\frac {\int \frac {\left (a x^4-b\right )^{3/4}}{2 \left (x^4+a+\sqrt {a^2+b}\right )}dx}{\sqrt {a^2+b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (a x^4-b\right )^{3/4}}{x^4+a-\sqrt {a^2+b}}dx}{2 \sqrt {a^2+b}}-\frac {\int \frac {\left (a x^4-b\right )^{3/4}}{x^4+a+\sqrt {a^2+b}}dx}{2 \sqrt {a^2+b}}\) |
| \(\Big \downarrow \) 916 |
| \(\displaystyle \frac {a \int \frac {1}{\sqrt [4]{a x^4-b}}dx-\left (-a \sqrt {a^2+b}+a^2+b\right ) \int \frac {1}{\left (x^4+a-\sqrt {a^2+b}\right ) \sqrt [4]{a x^4-b}}dx}{2 \sqrt {a^2+b}}-\frac {a \int \frac {1}{\sqrt [4]{a x^4-b}}dx-\left (a \left (\sqrt {a^2+b}+a\right )+b\right ) \int \frac {1}{\left (x^4+a+\sqrt {a^2+b}\right ) \sqrt [4]{a x^4-b}}dx}{2 \sqrt {a^2+b}}\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle \frac {a \int \frac {1}{1-\frac {a x^4}{a x^4-b}}d\frac {x}{\sqrt [4]{a x^4-b}}-\left (-a \sqrt {a^2+b}+a^2+b\right ) \int \frac {1}{\left (x^4+a-\sqrt {a^2+b}\right ) \sqrt [4]{a x^4-b}}dx}{2 \sqrt {a^2+b}}-\frac {a \int \frac {1}{1-\frac {a x^4}{a x^4-b}}d\frac {x}{\sqrt [4]{a x^4-b}}-\left (a \left (\sqrt {a^2+b}+a\right )+b\right ) \int \frac {1}{\left (x^4+a+\sqrt {a^2+b}\right ) \sqrt [4]{a x^4-b}}dx}{2 \sqrt {a^2+b}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {a \left (\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a} x^2}{\sqrt {a x^4-b}}}d\frac {x}{\sqrt [4]{a x^4-b}}+\frac {1}{2} \int \frac {1}{\frac {\sqrt {a} x^2}{\sqrt {a x^4-b}}+1}d\frac {x}{\sqrt [4]{a x^4-b}}\right )-\left (-a \sqrt {a^2+b}+a^2+b\right ) \int \frac {1}{\left (x^4+a-\sqrt {a^2+b}\right ) \sqrt [4]{a x^4-b}}dx}{2 \sqrt {a^2+b}}-\frac {a \left (\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a} x^2}{\sqrt {a x^4-b}}}d\frac {x}{\sqrt [4]{a x^4-b}}+\frac {1}{2} \int \frac {1}{\frac {\sqrt {a} x^2}{\sqrt {a x^4-b}}+1}d\frac {x}{\sqrt [4]{a x^4-b}}\right )-\left (a \left (\sqrt {a^2+b}+a\right )+b\right ) \int \frac {1}{\left (x^4+a+\sqrt {a^2+b}\right ) \sqrt [4]{a x^4-b}}dx}{2 \sqrt {a^2+b}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {a \left (\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a} x^2}{\sqrt {a x^4-b}}}d\frac {x}{\sqrt [4]{a x^4-b}}+\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}}\right )-\left (-a \sqrt {a^2+b}+a^2+b\right ) \int \frac {1}{\left (x^4+a-\sqrt {a^2+b}\right ) \sqrt [4]{a x^4-b}}dx}{2 \sqrt {a^2+b}}-\frac {a \left (\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a} x^2}{\sqrt {a x^4-b}}}d\frac {x}{\sqrt [4]{a x^4-b}}+\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}}\right )-\left (a \left (\sqrt {a^2+b}+a\right )+b\right ) \int \frac {1}{\left (x^4+a+\sqrt {a^2+b}\right ) \sqrt [4]{a x^4-b}}dx}{2 \sqrt {a^2+b}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}}\right )-\left (-a \sqrt {a^2+b}+a^2+b\right ) \int \frac {1}{\left (x^4+a-\sqrt {a^2+b}\right ) \sqrt [4]{a x^4-b}}dx}{2 \sqrt {a^2+b}}-\frac {a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}}\right )-\left (a \left (\sqrt {a^2+b}+a\right )+b\right ) \int \frac {1}{\left (x^4+a+\sqrt {a^2+b}\right ) \sqrt [4]{a x^4-b}}dx}{2 \sqrt {a^2+b}}\) |
| \(\Big \downarrow \) 902 |
| \(\displaystyle \frac {a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}}\right )-\left (-a \sqrt {a^2+b}+a^2+b\right ) \int \frac {1}{-\frac {\left (b+a \left (a-\sqrt {a^2+b}\right )\right ) x^4}{a x^4-b}+a-\sqrt {a^2+b}}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 \sqrt {a^2+b}}-\frac {a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}}\right )-\left (a \left (\sqrt {a^2+b}+a\right )+b\right ) \int \frac {1}{-\frac {\left (b+a \left (a+\sqrt {a^2+b}\right )\right ) x^4}{a x^4-b}+a+\sqrt {a^2+b}}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 \sqrt {a^2+b}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}}\right )-\left (-a \sqrt {a^2+b}+a^2+b\right ) \left (\frac {\int \frac {1}{\sqrt {a-\sqrt {a^2+b}}-\frac {\sqrt {a^2-\sqrt {a^2+b} a+b} x^2}{\sqrt {a x^4-b}}}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 \sqrt {a-\sqrt {a^2+b}}}+\frac {\int \frac {1}{\frac {\sqrt {a^2-\sqrt {a^2+b} a+b} x^2}{\sqrt {a x^4-b}}+\sqrt {a-\sqrt {a^2+b}}}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 \sqrt {a-\sqrt {a^2+b}}}\right )}{2 \sqrt {a^2+b}}-\frac {a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}}\right )-\left (a \left (\sqrt {a^2+b}+a\right )+b\right ) \left (\frac {\int \frac {1}{\sqrt {a+\sqrt {a^2+b}}-\frac {\sqrt {a^2+\sqrt {a^2+b} a+b} x^2}{\sqrt {a x^4-b}}}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 \sqrt {\sqrt {a^2+b}+a}}+\frac {\int \frac {1}{\frac {\sqrt {a^2+\sqrt {a^2+b} a+b} x^2}{\sqrt {a x^4-b}}+\sqrt {a+\sqrt {a^2+b}}}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 \sqrt {\sqrt {a^2+b}+a}}\right )}{2 \sqrt {a^2+b}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}}\right )-\left (-a \sqrt {a^2+b}+a^2+b\right ) \left (\frac {\int \frac {1}{\sqrt {a-\sqrt {a^2+b}}-\frac {\sqrt {a^2-\sqrt {a^2+b} a+b} x^2}{\sqrt {a x^4-b}}}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 \sqrt {a-\sqrt {a^2+b}}}+\frac {\arctan \left (\frac {x \sqrt [4]{-a \sqrt {a^2+b}+a^2+b}}{\sqrt [4]{a-\sqrt {a^2+b}} \sqrt [4]{a x^4-b}}\right )}{2 \left (a-\sqrt {a^2+b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+b}+a^2+b}}\right )}{2 \sqrt {a^2+b}}-\frac {a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}}\right )-\left (a \left (\sqrt {a^2+b}+a\right )+b\right ) \left (\frac {\int \frac {1}{\sqrt {a+\sqrt {a^2+b}}-\frac {\sqrt {a^2+\sqrt {a^2+b} a+b} x^2}{\sqrt {a x^4-b}}}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 \sqrt {\sqrt {a^2+b}+a}}+\frac {\arctan \left (\frac {x \sqrt [4]{a \sqrt {a^2+b}+a^2+b}}{\sqrt [4]{\sqrt {a^2+b}+a} \sqrt [4]{a x^4-b}}\right )}{2 \left (\sqrt {a^2+b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+b}+a^2+b}}\right )}{2 \sqrt {a^2+b}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}}\right )-\left (-a \sqrt {a^2+b}+a^2+b\right ) \left (\frac {\arctan \left (\frac {x \sqrt [4]{-a \sqrt {a^2+b}+a^2+b}}{\sqrt [4]{a-\sqrt {a^2+b}} \sqrt [4]{a x^4-b}}\right )}{2 \left (a-\sqrt {a^2+b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+b}+a^2+b}}+\frac {\text {arctanh}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+b}+a^2+b}}{\sqrt [4]{a-\sqrt {a^2+b}} \sqrt [4]{a x^4-b}}\right )}{2 \left (a-\sqrt {a^2+b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+b}+a^2+b}}\right )}{2 \sqrt {a^2+b}}-\frac {a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}}\right )-\left (a \left (\sqrt {a^2+b}+a\right )+b\right ) \left (\frac {\arctan \left (\frac {x \sqrt [4]{a \sqrt {a^2+b}+a^2+b}}{\sqrt [4]{\sqrt {a^2+b}+a} \sqrt [4]{a x^4-b}}\right )}{2 \left (\sqrt {a^2+b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+b}+a^2+b}}+\frac {\text {arctanh}\left (\frac {x \sqrt [4]{a \sqrt {a^2+b}+a^2+b}}{\sqrt [4]{\sqrt {a^2+b}+a} \sqrt [4]{a x^4-b}}\right )}{2 \left (\sqrt {a^2+b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+b}+a^2+b}}\right )}{2 \sqrt {a^2+b}}\) |
(a*(ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(2*a^(1/4)) + ArcTanh[(a^(1/4)* x)/(-b + a*x^4)^(1/4)]/(2*a^(1/4))) - (a^2 + b - a*Sqrt[a^2 + b])*(ArcTan[ ((a^2 + b - a*Sqrt[a^2 + b])^(1/4)*x)/((a - Sqrt[a^2 + b])^(1/4)*(-b + a*x ^4)^(1/4))]/(2*(a - Sqrt[a^2 + b])^(3/4)*(a^2 + b - a*Sqrt[a^2 + b])^(1/4) ) + ArcTanh[((a^2 + b - a*Sqrt[a^2 + b])^(1/4)*x)/((a - Sqrt[a^2 + b])^(1/ 4)*(-b + a*x^4)^(1/4))]/(2*(a - Sqrt[a^2 + b])^(3/4)*(a^2 + b - a*Sqrt[a^2 + b])^(1/4))))/(2*Sqrt[a^2 + b]) - (a*(ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1 /4)]/(2*a^(1/4)) + ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(2*a^(1/4))) - (b + a*(a + Sqrt[a^2 + b]))*(ArcTan[((a^2 + b + a*Sqrt[a^2 + b])^(1/4)*x)/ ((a + Sqrt[a^2 + b])^(1/4)*(-b + a*x^4)^(1/4))]/(2*(a + Sqrt[a^2 + b])^(3/ 4)*(a^2 + b + a*Sqrt[a^2 + b])^(1/4)) + ArcTanh[((a^2 + b + a*Sqrt[a^2 + b ])^(1/4)*x)/((a + Sqrt[a^2 + b])^(1/4)*(-b + a*x^4)^(1/4))]/(2*(a + Sqrt[a ^2 + b])^(3/4)*(a^2 + b + a*Sqrt[a^2 + b])^(1/4))))/(2*Sqrt[a^2 + b])
3.31.40.3.1 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[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n) Subst[In t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 /n]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Su bst[Int[1/(c - (b*c - a*d)*x^n), x], x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b , c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[b/d Int[(a + b*x^n)^(p - 1), x], x] - Simp[(b*c - a*d)/d Int[(a + b* x^n)^(p - 1)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p - 1) + 1, 0] && IntegerQ[n]
Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_ )), x_Symbol] :> With[{r = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/r) Int[(d + e*x ^n)^q/(b - r + 2*c*x^n), x], x] - Simp[2*(c/r) Int[(d + e*x^n)^q/(b + r + 2*c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && Ne Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[q]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.21 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.10
| method | result | size |
| pseudoelliptic | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-a^{2}-b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right )}{8}\) | \(44\) |
Timed out. \[ \int \frac {\left (-b+a x^4\right )^{3/4}}{-b+2 a x^4+x^8} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (-b+a x^4\right )^{3/4}}{-b+2 a x^4+x^8} \, dx=\text {Timed out} \]
\[ \int \frac {\left (-b+a x^4\right )^{3/4}}{-b+2 a x^4+x^8} \, dx=\int { \frac {{\left (a x^{4} - b\right )}^{\frac {3}{4}}}{x^{8} + 2 \, a x^{4} - b} \,d x } \]
\[ \int \frac {\left (-b+a x^4\right )^{3/4}}{-b+2 a x^4+x^8} \, dx=\int { \frac {{\left (a x^{4} - b\right )}^{\frac {3}{4}}}{x^{8} + 2 \, a x^{4} - b} \,d x } \]
Timed out. \[ \int \frac {\left (-b+a x^4\right )^{3/4}}{-b+2 a x^4+x^8} \, dx=\int \frac {{\left (a\,x^4-b\right )}^{3/4}}{x^8+2\,a\,x^4-b} \,d x \]