Integrand size = 27, antiderivative size = 460 \[ \int \frac {\left (b+a x^4\right )^{3/4}}{b+2 a x^4+2 x^8} \, dx=\frac {\left (-1-\sqrt [4]{-1}\right ) \arctan \left (\frac {(-1)^{7/8} \sqrt {2+\sqrt {2}} \sqrt [8]{a^2-2 b} x \sqrt [4]{b+a x^4}}{(-1)^{3/4} \sqrt [4]{a^2-2 b} x^2+\sqrt {b+a x^4}}\right )}{8 \sqrt [8]{a^2-2 b}}+\frac {\left (\sqrt {2}+i \sqrt {2 \left (3-2 \sqrt {2}\right )}\right ) \arctan \left (\frac {(-1)^{7/8} \left (-2+\sqrt {2}\right ) \sqrt [8]{a^2-2 b} x \sqrt [4]{b+a x^4}}{(-1)^{3/4} \sqrt {2-\sqrt {2}} \sqrt [4]{a^2-2 b} x^2+\sqrt {2-\sqrt {2}} \sqrt {b+a x^4}}\right )}{16 \sqrt [8]{a^2-2 b}}-\frac {i \left (-i \sqrt {2}+\sqrt {2 \left (3-2 \sqrt {2}\right )}\right ) \text {arctanh}\left (\frac {(-1)^{7/8} \sqrt [4]{a^2-2 b} x^2-\sqrt [8]{-1} \sqrt {b+a x^4}}{\sqrt {2-\sqrt {2}} \sqrt [8]{a^2-2 b} x \sqrt [4]{b+a x^4}}\right )}{16 \sqrt [8]{a^2-2 b}}+\frac {\left (-1-\sqrt [4]{-1}\right ) \text {arctanh}\left (\frac {(-1)^{7/8} \sqrt [4]{a^2-2 b} x^2-\sqrt [8]{-1} \sqrt {b+a x^4}}{\sqrt {2+\sqrt {2}} \sqrt [8]{a^2-2 b} x \sqrt [4]{b+a x^4}}\right )}{8 \sqrt [8]{a^2-2 b}} \]
1/8*(-1-(-1)^(1/4))*arctan((-1)^(7/8)*(2+2^(1/2))^(1/2)*(a^2-2*b)^(1/8)*x* (a*x^4+b)^(1/4)/((-1)^(3/4)*(a^2-2*b)^(1/4)*x^2+(a*x^4+b)^(1/2)))/(a^2-2*b )^(1/8)+1/16*(2^(1/2)+I*(2-2^(1/2)))*arctan((-1)^(7/8)*(-2+2^(1/2))*(a^2-2 *b)^(1/8)*x*(a*x^4+b)^(1/4)/((-1)^(3/4)*(2-2^(1/2))^(1/2)*(a^2-2*b)^(1/4)* x^2+(2-2^(1/2))^(1/2)*(a*x^4+b)^(1/2)))/(a^2-2*b)^(1/8)-1/16*I*(-I*2^(1/2) +2-2^(1/2))*arctanh(((-1)^(7/8)*(a^2-2*b)^(1/4)*x^2-(-1)^(1/8)*(a*x^4+b)^( 1/2))/(2-2^(1/2))^(1/2)/(a^2-2*b)^(1/8)/x/(a*x^4+b)^(1/4))/(a^2-2*b)^(1/8) +1/8*(-1-(-1)^(1/4))*arctanh(((-1)^(7/8)*(a^2-2*b)^(1/4)*x^2-(-1)^(1/8)*(a *x^4+b)^(1/2))/(2+2^(1/2))^(1/2)/(a^2-2*b)^(1/8)/x/(a*x^4+b)^(1/4))/(a^2-2 *b)^(1/8)
\[ \int \frac {\left (b+a x^4\right )^{3/4}}{b+2 a x^4+2 x^8} \, dx=\int \frac {\left (b+a x^4\right )^{3/4}}{b+2 a x^4+2 x^8} \, dx \]
Time = 0.83 (sec) , antiderivative size = 614, normalized size of antiderivative = 1.33, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, 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.
| \(\displaystyle \int \frac {\left (a x^4+b\right )^{3/4}}{2 a x^4+b+2 x^8} \, dx\) |
| \(\Big \downarrow \) 1758 |
| \(\displaystyle \frac {2 \int \frac {\left (a x^4+b\right )^{3/4}}{2 \left (2 x^4+a-\sqrt {a^2-2 b}\right )}dx}{\sqrt {a^2-2 b}}-\frac {2 \int \frac {\left (a x^4+b\right )^{3/4}}{2 \left (2 x^4+a+\sqrt {a^2-2 b}\right )}dx}{\sqrt {a^2-2 b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (a x^4+b\right )^{3/4}}{2 x^4+a-\sqrt {a^2-2 b}}dx}{\sqrt {a^2-2 b}}-\frac {\int \frac {\left (a x^4+b\right )^{3/4}}{2 x^4+a+\sqrt {a^2-2 b}}dx}{\sqrt {a^2-2 b}}\) |
| \(\Big \downarrow \) 916 |
| \(\displaystyle \frac {\frac {1}{2} a \int \frac {1}{\sqrt [4]{a x^4+b}}dx-\frac {1}{2} \left (-a \sqrt {a^2-2 b}+a^2-2 b\right ) \int \frac {1}{\left (2 x^4+a-\sqrt {a^2-2 b}\right ) \sqrt [4]{a x^4+b}}dx}{\sqrt {a^2-2 b}}-\frac {\frac {1}{2} a \int \frac {1}{\sqrt [4]{a x^4+b}}dx-\frac {1}{2} \left (a \left (\sqrt {a^2-2 b}+a\right )-2 b\right ) \int \frac {1}{\left (2 x^4+a+\sqrt {a^2-2 b}\right ) \sqrt [4]{a x^4+b}}dx}{\sqrt {a^2-2 b}}\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle \frac {\frac {1}{2} a \int \frac {1}{1-\frac {a x^4}{a x^4+b}}d\frac {x}{\sqrt [4]{a x^4+b}}-\frac {1}{2} \left (-a \sqrt {a^2-2 b}+a^2-2 b\right ) \int \frac {1}{\left (2 x^4+a-\sqrt {a^2-2 b}\right ) \sqrt [4]{a x^4+b}}dx}{\sqrt {a^2-2 b}}-\frac {\frac {1}{2} a \int \frac {1}{1-\frac {a x^4}{a x^4+b}}d\frac {x}{\sqrt [4]{a x^4+b}}-\frac {1}{2} \left (a \left (\sqrt {a^2-2 b}+a\right )-2 b\right ) \int \frac {1}{\left (2 x^4+a+\sqrt {a^2-2 b}\right ) \sqrt [4]{a x^4+b}}dx}{\sqrt {a^2-2 b}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {\frac {1}{2} a \left (\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a} x^2}{\sqrt {a x^4+b}}}d\frac {x}{\sqrt [4]{a x^4+b}}+\frac {1}{2} \int \frac {1}{\frac {\sqrt {a} x^2}{\sqrt {a x^4+b}}+1}d\frac {x}{\sqrt [4]{a x^4+b}}\right )-\frac {1}{2} \left (-a \sqrt {a^2-2 b}+a^2-2 b\right ) \int \frac {1}{\left (2 x^4+a-\sqrt {a^2-2 b}\right ) \sqrt [4]{a x^4+b}}dx}{\sqrt {a^2-2 b}}-\frac {\frac {1}{2} a \left (\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a} x^2}{\sqrt {a x^4+b}}}d\frac {x}{\sqrt [4]{a x^4+b}}+\frac {1}{2} \int \frac {1}{\frac {\sqrt {a} x^2}{\sqrt {a x^4+b}}+1}d\frac {x}{\sqrt [4]{a x^4+b}}\right )-\frac {1}{2} \left (a \left (\sqrt {a^2-2 b}+a\right )-2 b\right ) \int \frac {1}{\left (2 x^4+a+\sqrt {a^2-2 b}\right ) \sqrt [4]{a x^4+b}}dx}{\sqrt {a^2-2 b}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {1}{2} a \left (\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a} x^2}{\sqrt {a x^4+b}}}d\frac {x}{\sqrt [4]{a x^4+b}}+\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}\right )-\frac {1}{2} \left (-a \sqrt {a^2-2 b}+a^2-2 b\right ) \int \frac {1}{\left (2 x^4+a-\sqrt {a^2-2 b}\right ) \sqrt [4]{a x^4+b}}dx}{\sqrt {a^2-2 b}}-\frac {\frac {1}{2} a \left (\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a} x^2}{\sqrt {a x^4+b}}}d\frac {x}{\sqrt [4]{a x^4+b}}+\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}\right )-\frac {1}{2} \left (a \left (\sqrt {a^2-2 b}+a\right )-2 b\right ) \int \frac {1}{\left (2 x^4+a+\sqrt {a^2-2 b}\right ) \sqrt [4]{a x^4+b}}dx}{\sqrt {a^2-2 b}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}\right )-\frac {1}{2} \left (-a \sqrt {a^2-2 b}+a^2-2 b\right ) \int \frac {1}{\left (2 x^4+a-\sqrt {a^2-2 b}\right ) \sqrt [4]{a x^4+b}}dx}{\sqrt {a^2-2 b}}-\frac {\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}\right )-\frac {1}{2} \left (a \left (\sqrt {a^2-2 b}+a\right )-2 b\right ) \int \frac {1}{\left (2 x^4+a+\sqrt {a^2-2 b}\right ) \sqrt [4]{a x^4+b}}dx}{\sqrt {a^2-2 b}}\) |
| \(\Big \downarrow \) 902 |
| \(\displaystyle \frac {\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}\right )-\frac {1}{2} \left (-a \sqrt {a^2-2 b}+a^2-2 b\right ) \int \frac {1}{-\frac {\left (a \left (a-\sqrt {a^2-2 b}\right )-2 b\right ) x^4}{a x^4+b}+a-\sqrt {a^2-2 b}}d\frac {x}{\sqrt [4]{a x^4+b}}}{\sqrt {a^2-2 b}}-\frac {\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}\right )-\frac {1}{2} \left (a \left (\sqrt {a^2-2 b}+a\right )-2 b\right ) \int \frac {1}{-\frac {\left (a \left (a+\sqrt {a^2-2 b}\right )-2 b\right ) x^4}{a x^4+b}+a+\sqrt {a^2-2 b}}d\frac {x}{\sqrt [4]{a x^4+b}}}{\sqrt {a^2-2 b}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}\right )-\frac {1}{2} \left (-a \sqrt {a^2-2 b}+a^2-2 b\right ) \left (\frac {\int \frac {1}{\sqrt {a-\sqrt {a^2-2 b}}-\frac {\sqrt {a^2-\sqrt {a^2-2 b} a-2 b} x^2}{\sqrt {a x^4+b}}}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 \sqrt {a-\sqrt {a^2-2 b}}}+\frac {\int \frac {1}{\frac {\sqrt {a^2-\sqrt {a^2-2 b} a-2 b} x^2}{\sqrt {a x^4+b}}+\sqrt {a-\sqrt {a^2-2 b}}}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 \sqrt {a-\sqrt {a^2-2 b}}}\right )}{\sqrt {a^2-2 b}}-\frac {\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}\right )-\frac {1}{2} \left (a \left (\sqrt {a^2-2 b}+a\right )-2 b\right ) \left (\frac {\int \frac {1}{\sqrt {a+\sqrt {a^2-2 b}}-\frac {\sqrt {a^2+\sqrt {a^2-2 b} a-2 b} x^2}{\sqrt {a x^4+b}}}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 \sqrt {\sqrt {a^2-2 b}+a}}+\frac {\int \frac {1}{\frac {\sqrt {a^2+\sqrt {a^2-2 b} a-2 b} x^2}{\sqrt {a x^4+b}}+\sqrt {a+\sqrt {a^2-2 b}}}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 \sqrt {\sqrt {a^2-2 b}+a}}\right )}{\sqrt {a^2-2 b}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}\right )-\frac {1}{2} \left (-a \sqrt {a^2-2 b}+a^2-2 b\right ) \left (\frac {\int \frac {1}{\sqrt {a-\sqrt {a^2-2 b}}-\frac {\sqrt {a^2-\sqrt {a^2-2 b} a-2 b} x^2}{\sqrt {a x^4+b}}}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 \sqrt {a-\sqrt {a^2-2 b}}}+\frac {\arctan \left (\frac {x \sqrt [4]{-a \sqrt {a^2-2 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2-2 b}} \sqrt [4]{a x^4+b}}\right )}{2 \left (a-\sqrt {a^2-2 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2-2 b}+a^2-2 b}}\right )}{\sqrt {a^2-2 b}}-\frac {\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}\right )-\frac {1}{2} \left (a \left (\sqrt {a^2-2 b}+a\right )-2 b\right ) \left (\frac {\int \frac {1}{\sqrt {a+\sqrt {a^2-2 b}}-\frac {\sqrt {a^2+\sqrt {a^2-2 b} a-2 b} x^2}{\sqrt {a x^4+b}}}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 \sqrt {\sqrt {a^2-2 b}+a}}+\frac {\arctan \left (\frac {x \sqrt [4]{a \sqrt {a^2-2 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2-2 b}+a} \sqrt [4]{a x^4+b}}\right )}{2 \left (\sqrt {a^2-2 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2-2 b}+a^2-2 b}}\right )}{\sqrt {a^2-2 b}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}\right )-\frac {1}{2} \left (-a \sqrt {a^2-2 b}+a^2-2 b\right ) \left (\frac {\arctan \left (\frac {x \sqrt [4]{-a \sqrt {a^2-2 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2-2 b}} \sqrt [4]{a x^4+b}}\right )}{2 \left (a-\sqrt {a^2-2 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2-2 b}+a^2-2 b}}+\frac {\text {arctanh}\left (\frac {x \sqrt [4]{-a \sqrt {a^2-2 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2-2 b}} \sqrt [4]{a x^4+b}}\right )}{2 \left (a-\sqrt {a^2-2 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2-2 b}+a^2-2 b}}\right )}{\sqrt {a^2-2 b}}-\frac {\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}\right )-\frac {1}{2} \left (a \left (\sqrt {a^2-2 b}+a\right )-2 b\right ) \left (\frac {\arctan \left (\frac {x \sqrt [4]{a \sqrt {a^2-2 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2-2 b}+a} \sqrt [4]{a x^4+b}}\right )}{2 \left (\sqrt {a^2-2 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2-2 b}+a^2-2 b}}+\frac {\text {arctanh}\left (\frac {x \sqrt [4]{a \sqrt {a^2-2 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2-2 b}+a} \sqrt [4]{a x^4+b}}\right )}{2 \left (\sqrt {a^2-2 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2-2 b}+a^2-2 b}}\right )}{\sqrt {a^2-2 b}}\) |
((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))))/2 - ((a^2 - a*Sqrt[a^2 - 2*b] - 2*b)*( ArcTan[((a^2 - a*Sqrt[a^2 - 2*b] - 2*b)^(1/4)*x)/((a - Sqrt[a^2 - 2*b])^(1 /4)*(b + a*x^4)^(1/4))]/(2*(a - Sqrt[a^2 - 2*b])^(3/4)*(a^2 - a*Sqrt[a^2 - 2*b] - 2*b)^(1/4)) + ArcTanh[((a^2 - a*Sqrt[a^2 - 2*b] - 2*b)^(1/4)*x)/(( a - Sqrt[a^2 - 2*b])^(1/4)*(b + a*x^4)^(1/4))]/(2*(a - Sqrt[a^2 - 2*b])^(3 /4)*(a^2 - a*Sqrt[a^2 - 2*b] - 2*b)^(1/4))))/2)/Sqrt[a^2 - 2*b] - ((a*(Arc Tan[(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))))/2 - ((a*(a + Sqrt[a^2 - 2*b]) - 2*b)*(ArcTan[( (a^2 + a*Sqrt[a^2 - 2*b] - 2*b)^(1/4)*x)/((a + Sqrt[a^2 - 2*b])^(1/4)*(b + a*x^4)^(1/4))]/(2*(a + Sqrt[a^2 - 2*b])^(3/4)*(a^2 + a*Sqrt[a^2 - 2*b] - 2*b)^(1/4)) + ArcTanh[((a^2 + a*Sqrt[a^2 - 2*b] - 2*b)^(1/4)*x)/((a + Sqrt [a^2 - 2*b])^(1/4)*(b + a*x^4)^(1/4))]/(2*(a + Sqrt[a^2 - 2*b])^(3/4)*(a^2 + a*Sqrt[a^2 - 2*b] - 2*b)^(1/4))))/2)/Sqrt[a^2 - 2*b]
3.31.49.3.1 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 = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.09
| method | result | size |
| pseudoelliptic | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-a^{2}+2 b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right )}{8}\) | \(42\) |
Timed out. \[ \int \frac {\left (b+a x^4\right )^{3/4}}{b+2 a x^4+2 x^8} \, dx=\text {Timed out} \]
\[ \int \frac {\left (b+a x^4\right )^{3/4}}{b+2 a x^4+2 x^8} \, dx=\int \frac {\left (a x^{4} + b\right )^{\frac {3}{4}}}{2 a x^{4} + b + 2 x^{8}}\, dx \]
\[ \int \frac {\left (b+a x^4\right )^{3/4}}{b+2 a x^4+2 x^8} \, dx=\int { \frac {{\left (a x^{4} + b\right )}^{\frac {3}{4}}}{2 \, x^{8} + 2 \, a x^{4} + b} \,d x } \]
\[ \int \frac {\left (b+a x^4\right )^{3/4}}{b+2 a x^4+2 x^8} \, dx=\int { \frac {{\left (a x^{4} + b\right )}^{\frac {3}{4}}}{2 \, x^{8} + 2 \, a x^{4} + b} \,d x } \]
Timed out. \[ \int \frac {\left (b+a x^4\right )^{3/4}}{b+2 a x^4+2 x^8} \, dx=\int \frac {{\left (a\,x^4+b\right )}^{3/4}}{2\,x^8+2\,a\,x^4+b} \,d x \]