Integrand size = 15, antiderivative size = 297 \[ \int \frac {1}{x^{3/2} \left (a+c x^4\right )} \, dx=-\frac {2}{a \sqrt {x}}-\frac {\sqrt [8]{c} \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{c} \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{c} \arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{9/8}}-\frac {\sqrt [8]{c} \text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{9/8}}+\frac {\sqrt [8]{c} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{9/8}}-\frac {\sqrt [8]{c} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{9/8}} \]
1/2*c^(1/8)*arctan(c^(1/8)*x^(1/2)/(-a)^(1/8))/(-a)^(9/8)-1/2*c^(1/8)*arct anh(c^(1/8)*x^(1/2)/(-a)^(1/8))/(-a)^(9/8)+1/4*c^(1/8)*arctan(-1+c^(1/8)*2 ^(1/2)*x^(1/2)/(-a)^(1/8))/(-a)^(9/8)*2^(1/2)+1/4*c^(1/8)*arctan(1+c^(1/8) *2^(1/2)*x^(1/2)/(-a)^(1/8))/(-a)^(9/8)*2^(1/2)+1/8*c^(1/8)*ln((-a)^(1/4)+ c^(1/4)*x-(-a)^(1/8)*c^(1/8)*2^(1/2)*x^(1/2))/(-a)^(9/8)*2^(1/2)-1/8*c^(1/ 8)*ln((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*2^(1/2)*x^(1/2))/(-a)^(9/8)* 2^(1/2)-2/a/x^(1/2)
Time = 0.65 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^{3/2} \left (a+c x^4\right )} \, dx=\frac {-8 \sqrt [8]{a}+\sqrt {2+\sqrt {2}} \sqrt [8]{c} \sqrt {x} \arctan \left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )+\sqrt [8]{c} \sqrt {-\left (\left (-2+\sqrt {2}\right ) x\right )} \arctan \left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )+\sqrt {2+\sqrt {2}} \sqrt [8]{c} \sqrt {x} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )+\sqrt [8]{c} \sqrt {-\left (\left (-2+\sqrt {2}\right ) x\right )} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [8]{c} \sqrt {-\left (\left (-2+\sqrt {2}\right ) x\right )}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{4 a^{9/8} \sqrt {x}} \]
(-8*a^(1/8) + Sqrt[2 + Sqrt[2]]*c^(1/8)*Sqrt[x]*ArcTan[(Sqrt[1 - 1/Sqrt[2] ]*(a^(1/4) - c^(1/4)*x))/(a^(1/8)*c^(1/8)*Sqrt[x])] + c^(1/8)*Sqrt[-((-2 + Sqrt[2])*x)]*ArcTan[(Sqrt[1 + 1/Sqrt[2]]*(a^(1/4) - c^(1/4)*x))/(a^(1/8)* c^(1/8)*Sqrt[x])] + Sqrt[2 + Sqrt[2]]*c^(1/8)*Sqrt[x]*ArcTanh[(Sqrt[2 + Sq rt[2]]*a^(1/8)*c^(1/8)*Sqrt[x])/(a^(1/4) + c^(1/4)*x)] + c^(1/8)*Sqrt[-((- 2 + Sqrt[2])*x)]*ArcTanh[(a^(1/8)*c^(1/8)*Sqrt[-((-2 + Sqrt[2])*x)])/(a^(1 /4) + c^(1/4)*x)])/(4*a^(9/8)*Sqrt[x])
Time = 0.57 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.14, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.933, Rules used = {847, 851, 830, 826, 827, 218, 221, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \frac {1}{x^{3/2} \left (a+c x^4\right )} \, dx\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle -\frac {c \int \frac {x^{5/2}}{c x^4+a}dx}{a}-\frac {2}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle -\frac {2 c \int \frac {x^3}{c x^4+a}d\sqrt {x}}{a}-\frac {2}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 830 |
| \(\displaystyle -\frac {2 c \left (\frac {\int \frac {x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt {c}}-\frac {\int \frac {x}{\sqrt {-a}-\sqrt {c} x^2}d\sqrt {x}}{2 \sqrt {c}}\right )}{a}-\frac {2}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle -\frac {2 c \left (\frac {\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\int \frac {x}{\sqrt {-a}-\sqrt {c} x^2}d\sqrt {x}}{2 \sqrt {c}}\right )}{a}-\frac {2}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle -\frac {2 c \left (\frac {\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\int \frac {1}{\sqrt [4]{c} x+\sqrt [4]{-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {2 c \left (\frac {\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {2 c \left (\frac {\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {2 c \left (\frac {\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [4]{c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {2 c \left (\frac {\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {2 c \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {2 c \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{c} \left (x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}\right )}{\sqrt [8]{c} \left (x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 c \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{c} \left (x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}\right )}{\sqrt [8]{c} \left (x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 c \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}}{x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [8]{-a} \sqrt [4]{c}}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {2 c \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{a \sqrt {x}}\) |
-2/(a*Sqrt[x]) - (2*c*(-1/2*(-1/2*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)]/((- a)^(1/8)*c^(3/8)) + ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(2*(-a)^(1/8)*c^ (3/8)))/Sqrt[c] + ((-(ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(Sq rt[2]*(-a)^(1/8)*c^(1/8))) + ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/ 8)]/(Sqrt[2]*(-a)^(1/8)*c^(1/8)))/(2*c^(1/4)) - (-1/2*Log[(-a)^(1/4) - Sqr t[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x]/(Sqrt[2]*(-a)^(1/8)*c^(1/8)) + Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x]/(2*Sqrt [2]*(-a)^(1/8)*c^(1/8)))/(2*c^(1/4)))/(2*Sqrt[c])))/a
3.8.44.3.1 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, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[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[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Numerator[Rt [-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[x^(m - n/2)/( r + s*x^(n/2)), x], x] - Simp[s/(2*b) Int[x^(m - n/2)/(r - s*x^(n/2)), x] , x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LeQ[n/2, m] && L tQ[m, n] && !GtQ[a/b, 0]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.05 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.13
| method | result | size |
| derivativedivides | \(-\frac {2}{a \sqrt {x}}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}}}{4 a}\) | \(38\) |
| default | \(-\frac {2}{a \sqrt {x}}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}}}{4 a}\) | \(38\) |
| risch | \(-\frac {2}{a \sqrt {x}}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}}}{4 a}\) | \(38\) |
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.99 \[ \int \frac {1}{x^{3/2} \left (a+c x^4\right )} \, dx=-\frac {-\left (i - 1\right ) \, \sqrt {2} a x \left (-\frac {c}{a^{9}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a^{8} \left (-\frac {c}{a^{9}}\right )^{\frac {7}{8}} + c \sqrt {x}\right ) + \left (i + 1\right ) \, \sqrt {2} a x \left (-\frac {c}{a^{9}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a^{8} \left (-\frac {c}{a^{9}}\right )^{\frac {7}{8}} + c \sqrt {x}\right ) - \left (i + 1\right ) \, \sqrt {2} a x \left (-\frac {c}{a^{9}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a^{8} \left (-\frac {c}{a^{9}}\right )^{\frac {7}{8}} + c \sqrt {x}\right ) + \left (i - 1\right ) \, \sqrt {2} a x \left (-\frac {c}{a^{9}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a^{8} \left (-\frac {c}{a^{9}}\right )^{\frac {7}{8}} + c \sqrt {x}\right ) + 2 \, a x \left (-\frac {c}{a^{9}}\right )^{\frac {1}{8}} \log \left (a^{8} \left (-\frac {c}{a^{9}}\right )^{\frac {7}{8}} + c \sqrt {x}\right ) - 2 i \, a x \left (-\frac {c}{a^{9}}\right )^{\frac {1}{8}} \log \left (i \, a^{8} \left (-\frac {c}{a^{9}}\right )^{\frac {7}{8}} + c \sqrt {x}\right ) + 2 i \, a x \left (-\frac {c}{a^{9}}\right )^{\frac {1}{8}} \log \left (-i \, a^{8} \left (-\frac {c}{a^{9}}\right )^{\frac {7}{8}} + c \sqrt {x}\right ) - 2 \, a x \left (-\frac {c}{a^{9}}\right )^{\frac {1}{8}} \log \left (-a^{8} \left (-\frac {c}{a^{9}}\right )^{\frac {7}{8}} + c \sqrt {x}\right ) + 16 \, \sqrt {x}}{8 \, a x} \]
-1/8*(-(I - 1)*sqrt(2)*a*x*(-c/a^9)^(1/8)*log((1/2*I + 1/2)*sqrt(2)*a^8*(- c/a^9)^(7/8) + c*sqrt(x)) + (I + 1)*sqrt(2)*a*x*(-c/a^9)^(1/8)*log(-(1/2*I - 1/2)*sqrt(2)*a^8*(-c/a^9)^(7/8) + c*sqrt(x)) - (I + 1)*sqrt(2)*a*x*(-c/ a^9)^(1/8)*log((1/2*I - 1/2)*sqrt(2)*a^8*(-c/a^9)^(7/8) + c*sqrt(x)) + (I - 1)*sqrt(2)*a*x*(-c/a^9)^(1/8)*log(-(1/2*I + 1/2)*sqrt(2)*a^8*(-c/a^9)^(7 /8) + c*sqrt(x)) + 2*a*x*(-c/a^9)^(1/8)*log(a^8*(-c/a^9)^(7/8) + c*sqrt(x) ) - 2*I*a*x*(-c/a^9)^(1/8)*log(I*a^8*(-c/a^9)^(7/8) + c*sqrt(x)) + 2*I*a*x *(-c/a^9)^(1/8)*log(-I*a^8*(-c/a^9)^(7/8) + c*sqrt(x)) - 2*a*x*(-c/a^9)^(1 /8)*log(-a^8*(-c/a^9)^(7/8) + c*sqrt(x)) + 16*sqrt(x))/(a*x)
Time = 18.69 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^{3/2} \left (a+c x^4\right )} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {9}{2}}} & \text {for}\: a = 0 \wedge c = 0 \\- \frac {2}{9 c x^{\frac {9}{2}}} & \text {for}\: a = 0 \\- \frac {2}{a \sqrt {x}} & \text {for}\: c = 0 \\- \frac {\log {\left (\sqrt {x} - \sqrt [8]{- \frac {a}{c}} \right )}}{4 a \sqrt [8]{- \frac {a}{c}}} + \frac {\log {\left (\sqrt {x} + \sqrt [8]{- \frac {a}{c}} \right )}}{4 a \sqrt [8]{- \frac {a}{c}}} - \frac {\sqrt {2} \log {\left (- 4 \sqrt {2} \sqrt {x} \sqrt [8]{- \frac {a}{c}} + 4 x + 4 \sqrt [4]{- \frac {a}{c}} \right )}}{8 a \sqrt [8]{- \frac {a}{c}}} + \frac {\sqrt {2} \log {\left (4 \sqrt {2} \sqrt {x} \sqrt [8]{- \frac {a}{c}} + 4 x + 4 \sqrt [4]{- \frac {a}{c}} \right )}}{8 a \sqrt [8]{- \frac {a}{c}}} - \frac {\operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} \right )}}{2 a \sqrt [8]{- \frac {a}{c}}} - \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} - 1 \right )}}{4 a \sqrt [8]{- \frac {a}{c}}} - \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} + 1 \right )}}{4 a \sqrt [8]{- \frac {a}{c}}} - \frac {2}{a \sqrt {x}} & \text {otherwise} \end {cases} \]
Piecewise((zoo/x**(9/2), Eq(a, 0) & Eq(c, 0)), (-2/(9*c*x**(9/2)), Eq(a, 0 )), (-2/(a*sqrt(x)), Eq(c, 0)), (-log(sqrt(x) - (-a/c)**(1/8))/(4*a*(-a/c) **(1/8)) + log(sqrt(x) + (-a/c)**(1/8))/(4*a*(-a/c)**(1/8)) - sqrt(2)*log( -4*sqrt(2)*sqrt(x)*(-a/c)**(1/8) + 4*x + 4*(-a/c)**(1/4))/(8*a*(-a/c)**(1/ 8)) + sqrt(2)*log(4*sqrt(2)*sqrt(x)*(-a/c)**(1/8) + 4*x + 4*(-a/c)**(1/4)) /(8*a*(-a/c)**(1/8)) - atan(sqrt(x)/(-a/c)**(1/8))/(2*a*(-a/c)**(1/8)) - s qrt(2)*atan(sqrt(2)*sqrt(x)/(-a/c)**(1/8) - 1)/(4*a*(-a/c)**(1/8)) - sqrt( 2)*atan(sqrt(2)*sqrt(x)/(-a/c)**(1/8) + 1)/(4*a*(-a/c)**(1/8)) - 2/(a*sqrt (x)), True))
\[ \int \frac {1}{x^{3/2} \left (a+c x^4\right )} \, dx=\int { \frac {1}{{\left (c x^{4} + a\right )} x^{\frac {3}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (198) = 396\).
Time = 0.43 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.55 \[ \int \frac {1}{x^{3/2} \left (a+c x^4\right )} \, dx=-\frac {c \left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (\frac {\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + 2 \, \sqrt {x}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{2 \, a^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {c \left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (-\frac {\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} - 2 \, \sqrt {x}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{2 \, a^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {c \left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (\frac {\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + 2 \, \sqrt {x}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{2 \, a^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {c \left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (-\frac {\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} - 2 \, \sqrt {x}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{2 \, a^{2} \sqrt {2 \, \sqrt {2} + 4}} + \frac {c \left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (\sqrt {x} \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{4 \, a^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {c \left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (-\sqrt {x} \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{4 \, a^{2} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {c \left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (\sqrt {x} \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{4 \, a^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {c \left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (-\sqrt {x} \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{4 \, a^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {2}{a \sqrt {x}} \]
-1/2*c*(a/c)^(7/8)*arctan((sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + 2*sqrt(x))/(sq rt(sqrt(2) + 2)*(a/c)^(1/8)))/(a^2*sqrt(-2*sqrt(2) + 4)) - 1/2*c*(a/c)^(7/ 8)*arctan(-(sqrt(-sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(sqrt(2) + 2) *(a/c)^(1/8)))/(a^2*sqrt(-2*sqrt(2) + 4)) - 1/2*c*(a/c)^(7/8)*arctan((sqrt (sqrt(2) + 2)*(a/c)^(1/8) + 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/( a^2*sqrt(2*sqrt(2) + 4)) - 1/2*c*(a/c)^(7/8)*arctan(-(sqrt(sqrt(2) + 2)*(a /c)^(1/8) - 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/(a^2*sqrt(2*sqrt( 2) + 4)) + 1/4*c*(a/c)^(7/8)*log(sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^2*sqrt(-2*sqrt(2) + 4)) - 1/4*c*(a/c)^(7/8)*log(-sqrt(x )*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^2*sqrt(-2*sqrt(2) + 4)) + 1/4*c*(a/c)^(7/8)*log(sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + ( a/c)^(1/4))/(a^2*sqrt(2*sqrt(2) + 4)) - 1/4*c*(a/c)^(7/8)*log(-sqrt(x)*sqr t(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^2*sqrt(2*sqrt(2) + 4)) - 2/(a*sqrt(x))
Time = 5.55 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.42 \[ \int \frac {1}{x^{3/2} \left (a+c x^4\right )} \, dx=-\frac {2}{a\,\sqrt {x}}-\frac {{\left (-c\right )}^{1/8}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/8}\,\sqrt {x}}{a^{1/8}}\right )}{2\,a^{9/8}}-\frac {{\left (-c\right )}^{1/8}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{a^{1/8}}\right )\,1{}\mathrm {i}}{2\,a^{9/8}}+\frac {\sqrt {2}\,{\left (-c\right )}^{1/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-c\right )}^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (-\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right )}{a^{9/8}}+\frac {\sqrt {2}\,{\left (-c\right )}^{1/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-c\right )}^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (-\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )}{a^{9/8}} \]
- 2/(a*x^(1/2)) - ((-c)^(1/8)*atan(((-c)^(1/8)*x^(1/2))/a^(1/8)))/(2*a^(9/ 8)) - ((-c)^(1/8)*atan(((-c)^(1/8)*x^(1/2)*1i)/a^(1/8))*1i)/(2*a^(9/8)) - (2^(1/2)*(-c)^(1/8)*atan((2^(1/2)*(-c)^(1/8)*x^(1/2)*(1/2 - 1i/2))/a^(1/8) )*(1/4 - 1i/4))/a^(9/8) - (2^(1/2)*(-c)^(1/8)*atan((2^(1/2)*(-c)^(1/8)*x^( 1/2)*(1/2 + 1i/2))/a^(1/8))*(1/4 + 1i/4))/a^(9/8)