Integrand size = 15, antiderivative size = 297 \[ \int \frac {x^{7/2}}{a+c x^4} \, dx=\frac {2 \sqrt {x}}{c}+\frac {\sqrt [8]{-a} \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} c^{9/8}}-\frac {\sqrt [8]{-a} \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} c^{9/8}}-\frac {\sqrt [8]{-a} \arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{9/8}}-\frac {\sqrt [8]{-a} \text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{9/8}}+\frac {\sqrt [8]{-a} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} c^{9/8}}-\frac {\sqrt [8]{-a} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} c^{9/8}} \]
-1/2*(-a)^(1/8)*arctan(c^(1/8)*x^(1/2)/(-a)^(1/8))/c^(9/8)-1/2*(-a)^(1/8)* arctanh(c^(1/8)*x^(1/2)/(-a)^(1/8))/c^(9/8)-1/4*(-a)^(1/8)*arctan(-1+c^(1/ 8)*2^(1/2)*x^(1/2)/(-a)^(1/8))/c^(9/8)*2^(1/2)-1/4*(-a)^(1/8)*arctan(1+c^( 1/8)*2^(1/2)*x^(1/2)/(-a)^(1/8))/c^(9/8)*2^(1/2)+1/8*(-a)^(1/8)*ln((-a)^(1 /4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*2^(1/2)*x^(1/2))/c^(9/8)*2^(1/2)-1/8*(-a) ^(1/8)*ln((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*2^(1/2)*x^(1/2))/c^(9/8) *2^(1/2)+2*x^(1/2)/c
Time = 0.62 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.90 \[ \int \frac {x^{7/2}}{a+c x^4} \, dx=\frac {8 \sqrt [8]{c} \sqrt {x}+\sqrt {2+\sqrt {2}} \sqrt [8]{a} \arctan \left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )+\sqrt {2-\sqrt {2}} \sqrt [8]{a} \arctan \left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )-\sqrt {2+\sqrt {2}} \sqrt [8]{a} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )-\sqrt {2-\sqrt {2}} \sqrt [8]{a} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [8]{c} \sqrt {-\left (\left (-2+\sqrt {2}\right ) x\right )}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{4 c^{9/8}} \]
(8*c^(1/8)*Sqrt[x] + Sqrt[2 + Sqrt[2]]*a^(1/8)*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]]*a^( 1/8)*ArcTan[(Sqrt[1 + 1/Sqrt[2]]*(a^(1/4) - c^(1/4)*x))/(a^(1/8)*c^(1/8)*S qrt[x])] - Sqrt[2 + Sqrt[2]]*a^(1/8)*ArcTanh[(Sqrt[2 + Sqrt[2]]*a^(1/8)*c^ (1/8)*Sqrt[x])/(a^(1/4) + c^(1/4)*x)] - Sqrt[2 - Sqrt[2]]*a^(1/8)*ArcTanh[ (a^(1/8)*c^(1/8)*Sqrt[-((-2 + Sqrt[2])*x)])/(a^(1/4) + c^(1/4)*x)])/(4*c^( 9/8))
Time = 0.58 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.17, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.933, Rules used = {843, 851, 758, 755, 756, 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.
| \(\displaystyle \int \frac {x^{7/2}}{a+c x^4} \, dx\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {2 \sqrt {x}}{c}-\frac {a \int \frac {1}{\sqrt {x} \left (c x^4+a\right )}dx}{c}\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle \frac {2 \sqrt {x}}{c}-\frac {2 a \int \frac {1}{c x^4+a}d\sqrt {x}}{c}\) |
| \(\Big \downarrow \) 758 |
| \(\displaystyle \frac {2 \sqrt {x}}{c}-\frac {2 a \left (-\frac {\int \frac {1}{\sqrt {-a}-\sqrt {c} x^2}d\sqrt {x}}{2 \sqrt {-a}}-\frac {\int \frac {1}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt {-a}}\right )}{c}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {2 \sqrt {x}}{c}-\frac {2 a \left (-\frac {\int \frac {1}{\sqrt {-a}-\sqrt {c} x^2}d\sqrt {x}}{2 \sqrt {-a}}-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{c}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {2 \sqrt {x}}{c}-\frac {2 a \left (-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\int \frac {1}{\sqrt [4]{c} x+\sqrt [4]{-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{c}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2 \sqrt {x}}{c}-\frac {2 a \left (-\frac {\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {-a}}-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 \sqrt {x}}{c}-\frac {2 a \left (-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {-a}}\right )}{c}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 \sqrt {x}}{c}-\frac {2 a \left (-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [4]{c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {-a}}\right )}{c}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 \sqrt {x}}{c}-\frac {2 a \left (-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {-a}}\right )}{c}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 \sqrt {x}}{c}-\frac {2 a \left (-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {-a}}\right )}{c}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 \sqrt {x}}{c}-\frac {2 a \left (-\frac {\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{c} \left (x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}\right )}{\sqrt [8]{c} \left (x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {-a}}\right )}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \sqrt {x}}{c}-\frac {2 a \left (-\frac {\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{c} \left (x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}\right )}{\sqrt [8]{c} \left (x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {-a}}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {x}}{c}-\frac {2 a \left (-\frac {\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}}{x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [8]{-a} \sqrt [4]{c}}}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {-a}}\right )}{c}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 \sqrt {x}}{c}-\frac {2 a \left (-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {-a}}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{c}\) |
(2*Sqrt[x])/c - (2*a*(-1/2*(ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(2*(-a)^( 3/8)*c^(1/8)) + ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(2*(-a)^(3/8)*c^(1/8 )))/Sqrt[-a] - ((-(ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(Sqrt[ 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*(-a)^(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*(-a)^(1/4)))/(2*Sqrt[-a])))/c
3.8.39.3.1 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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) 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[((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_))^(-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^(n/2)), x], x] + Simp[r/(2*a) Int[1/(r + s*x^(n/2)), x], x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 1] && !GtQ[a/b, 0]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && 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 = 4.13 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.13
| method | result | size |
| derivativedivides | \(\frac {2 \sqrt {x}}{c}-\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{7}}\right )}{4 c^{2}}\) | \(39\) |
| default | \(\frac {2 \sqrt {x}}{c}-\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{7}}\right )}{4 c^{2}}\) | \(39\) |
| risch | \(\frac {2 \sqrt {x}}{c}-\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{7}}\right )}{4 c^{2}}\) | \(39\) |
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.84 \[ \int \frac {x^{7/2}}{a+c x^4} \, dx=-\frac {\left (i + 1\right ) \, \sqrt {2} c \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} c \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} + \sqrt {x}\right ) - \left (i - 1\right ) \, \sqrt {2} c \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} c \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} + \sqrt {x}\right ) + \left (i - 1\right ) \, \sqrt {2} c \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} c \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} + \sqrt {x}\right ) - \left (i + 1\right ) \, \sqrt {2} c \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} c \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} + \sqrt {x}\right ) + 2 \, c \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} \log \left (c \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} + \sqrt {x}\right ) + 2 i \, c \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} \log \left (i \, c \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} + \sqrt {x}\right ) - 2 i \, c \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} \log \left (-i \, c \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} + \sqrt {x}\right ) - 2 \, c \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} \log \left (-c \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} + \sqrt {x}\right ) - 16 \, \sqrt {x}}{8 \, c} \]
-1/8*((I + 1)*sqrt(2)*c*(-a/c^9)^(1/8)*log((1/2*I + 1/2)*sqrt(2)*c*(-a/c^9 )^(1/8) + sqrt(x)) - (I - 1)*sqrt(2)*c*(-a/c^9)^(1/8)*log(-(1/2*I - 1/2)*s qrt(2)*c*(-a/c^9)^(1/8) + sqrt(x)) + (I - 1)*sqrt(2)*c*(-a/c^9)^(1/8)*log( (1/2*I - 1/2)*sqrt(2)*c*(-a/c^9)^(1/8) + sqrt(x)) - (I + 1)*sqrt(2)*c*(-a/ c^9)^(1/8)*log(-(1/2*I + 1/2)*sqrt(2)*c*(-a/c^9)^(1/8) + sqrt(x)) + 2*c*(- a/c^9)^(1/8)*log(c*(-a/c^9)^(1/8) + sqrt(x)) + 2*I*c*(-a/c^9)^(1/8)*log(I* c*(-a/c^9)^(1/8) + sqrt(x)) - 2*I*c*(-a/c^9)^(1/8)*log(-I*c*(-a/c^9)^(1/8) + sqrt(x)) - 2*c*(-a/c^9)^(1/8)*log(-c*(-a/c^9)^(1/8) + sqrt(x)) - 16*sqr t(x))/c
Time = 26.65 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.00 \[ \int \frac {x^{7/2}}{a+c x^4} \, dx=\begin {cases} \tilde {\infty } \sqrt {x} & \text {for}\: a = 0 \wedge c = 0 \\\frac {2 x^{\frac {9}{2}}}{9 a} & \text {for}\: c = 0 \\\frac {2 \sqrt {x}}{c} & \text {for}\: a = 0 \\\frac {2 \sqrt {x}}{c} + \frac {\sqrt [8]{- \frac {a}{c}} \log {\left (\sqrt {x} - \sqrt [8]{- \frac {a}{c}} \right )}}{4 c} - \frac {\sqrt [8]{- \frac {a}{c}} \log {\left (\sqrt {x} + \sqrt [8]{- \frac {a}{c}} \right )}}{4 c} + \frac {\sqrt {2} \sqrt [8]{- \frac {a}{c}} \log {\left (- 4 \sqrt {2} \sqrt {x} \sqrt [8]{- \frac {a}{c}} + 4 x + 4 \sqrt [4]{- \frac {a}{c}} \right )}}{8 c} - \frac {\sqrt {2} \sqrt [8]{- \frac {a}{c}} \log {\left (4 \sqrt {2} \sqrt {x} \sqrt [8]{- \frac {a}{c}} + 4 x + 4 \sqrt [4]{- \frac {a}{c}} \right )}}{8 c} - \frac {\sqrt [8]{- \frac {a}{c}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} \right )}}{2 c} - \frac {\sqrt {2} \sqrt [8]{- \frac {a}{c}} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} - 1 \right )}}{4 c} - \frac {\sqrt {2} \sqrt [8]{- \frac {a}{c}} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} + 1 \right )}}{4 c} & \text {otherwise} \end {cases} \]
Piecewise((zoo*sqrt(x), Eq(a, 0) & Eq(c, 0)), (2*x**(9/2)/(9*a), Eq(c, 0)) , (2*sqrt(x)/c, Eq(a, 0)), (2*sqrt(x)/c + (-a/c)**(1/8)*log(sqrt(x) - (-a/ c)**(1/8))/(4*c) - (-a/c)**(1/8)*log(sqrt(x) + (-a/c)**(1/8))/(4*c) + sqrt (2)*(-a/c)**(1/8)*log(-4*sqrt(2)*sqrt(x)*(-a/c)**(1/8) + 4*x + 4*(-a/c)**( 1/4))/(8*c) - sqrt(2)*(-a/c)**(1/8)*log(4*sqrt(2)*sqrt(x)*(-a/c)**(1/8) + 4*x + 4*(-a/c)**(1/4))/(8*c) - (-a/c)**(1/8)*atan(sqrt(x)/(-a/c)**(1/8))/( 2*c) - sqrt(2)*(-a/c)**(1/8)*atan(sqrt(2)*sqrt(x)/(-a/c)**(1/8) - 1)/(4*c) - sqrt(2)*(-a/c)**(1/8)*atan(sqrt(2)*sqrt(x)/(-a/c)**(1/8) + 1)/(4*c), Tr ue))
\[ \int \frac {x^{7/2}}{a+c x^4} \, dx=\int { \frac {x^{\frac {7}{2}}}{c x^{4} + a} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 453 vs. \(2 (198) = 396\).
Time = 0.37 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.53 \[ \int \frac {x^{7/2}}{a+c x^4} \, dx=-\frac {\left (\frac {a}{c}\right )^{\frac {1}{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 \, c \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{c}\right )^{\frac {1}{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 \, c \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{c}\right )^{\frac {1}{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 \, c \sqrt {2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{c}\right )^{\frac {1}{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 \, c \sqrt {2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{c}\right )^{\frac {1}{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 \, c \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {1}{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 \, c \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{c}\right )^{\frac {1}{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 \, c \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {1}{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 \, c \sqrt {2 \, \sqrt {2} + 4}} + \frac {2 \, \sqrt {x}}{c} \]
-1/2*(a/c)^(1/8)*arctan((sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + 2*sqrt(x))/(sqrt (sqrt(2) + 2)*(a/c)^(1/8)))/(c*sqrt(-2*sqrt(2) + 4)) - 1/2*(a/c)^(1/8)*arc tan(-(sqrt(-sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(sqrt(2) + 2)*(a/c) ^(1/8)))/(c*sqrt(-2*sqrt(2) + 4)) - 1/2*(a/c)^(1/8)*arctan((sqrt(sqrt(2) + 2)*(a/c)^(1/8) + 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/(c*sqrt(2*s qrt(2) + 4)) - 1/2*(a/c)^(1/8)*arctan(-(sqrt(sqrt(2) + 2)*(a/c)^(1/8) - 2* sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/(c*sqrt(2*sqrt(2) + 4)) - 1/4*( a/c)^(1/8)*log(sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(c *sqrt(-2*sqrt(2) + 4)) + 1/4*(a/c)^(1/8)*log(-sqrt(x)*sqrt(sqrt(2) + 2)*(a /c)^(1/8) + x + (a/c)^(1/4))/(c*sqrt(-2*sqrt(2) + 4)) - 1/4*(a/c)^(1/8)*lo g(sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(c*sqrt(2*sqrt (2) + 4)) + 1/4*(a/c)^(1/8)*log(-sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(c*sqrt(2*sqrt(2) + 4)) + 2*sqrt(x)/c
Time = 5.70 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.42 \[ \int \frac {x^{7/2}}{a+c x^4} \, dx=\frac {2\,\sqrt {x}}{c}-\frac {{\left (-a\right )}^{1/8}\,\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{2\,c^{9/8}}+\frac {{\left (-a\right )}^{1/8}\,\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,1{}\mathrm {i}}{2\,c^{9/8}}+\frac {\sqrt {2}\,{\left (-a\right )}^{1/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )}{c^{9/8}}+\frac {\sqrt {2}\,{\left (-a\right )}^{1/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right )}{c^{9/8}} \]
(2*x^(1/2))/c - ((-a)^(1/8)*atan((c^(1/8)*x^(1/2))/(-a)^(1/8)))/(2*c^(9/8) ) + ((-a)^(1/8)*atan((c^(1/8)*x^(1/2)*1i)/(-a)^(1/8))*1i)/(2*c^(9/8)) - (2 ^(1/2)*(-a)^(1/8)*atan((2^(1/2)*c^(1/8)*x^(1/2)*(1/2 - 1i/2))/(-a)^(1/8))* (1/4 + 1i/4))/c^(9/8) - (2^(1/2)*(-a)^(1/8)*atan((2^(1/2)*c^(1/8)*x^(1/2)* (1/2 + 1i/2))/(-a)^(1/8))*(1/4 - 1i/4))/c^(9/8)