\(\int \frac {1}{x^{5/2} (a+c x^4)} \, dx\) [140]

Optimal result

Integrand size = 15, antiderivative size = 240 \[ \int \frac {1}{x^{5/2} \left (a+c x^4\right )} \, dx=-\frac {2}{3 a x^{3/2}}-\frac {c^{3/8} \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{11/8}}+\frac {c^{3/8} \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{11/8}}-\frac {c^{3/8} \arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac {c^{3/8} \text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}+\frac {c^{3/8} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{-a}+\sqrt [4]{c} x}\right )}{2 \sqrt {2} (-a)^{11/8}} \] Output:

-2/3/a/x^(3/2)+1/4*c^(3/8)*arctan(-1+2^(1/2)*c^(1/8)*x^(1/2)/(-a)^(1/8))*2 
^(1/2)/(-a)^(11/8)+1/4*c^(3/8)*arctan(1+2^(1/2)*c^(1/8)*x^(1/2)/(-a)^(1/8) 
)*2^(1/2)/(-a)^(11/8)-1/2*c^(3/8)*arctan(c^(1/8)*x^(1/2)/(-a)^(1/8))/(-a)^ 
(11/8)-1/2*c^(3/8)*arctanh(c^(1/8)*x^(1/2)/(-a)^(1/8))/(-a)^(11/8)+1/4*c^( 
3/8)*arctanh(2^(1/2)*(-a)^(1/8)*c^(1/8)*x^(1/2)/((-a)^(1/4)+c^(1/4)*x))*2^ 
(1/2)/(-a)^(11/8)
 

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^{5/2} \left (a+c x^4\right )} \, dx=-\frac {8 a^{3/8}+3 \sqrt {2-\sqrt {2}} c^{3/8} x^{3/2} \arctan \left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )-3 \sqrt {2+\sqrt {2}} c^{3/8} x^{3/2} \arctan \left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )-3 \sqrt {2-\sqrt {2}} c^{3/8} x^{3/2} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )+3 \sqrt {2+\sqrt {2}} c^{3/8} x^{3/2} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [8]{c} \sqrt {-\left (\left (-2+\sqrt {2}\right ) x\right )}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{12 a^{11/8} x^{3/2}} \] Input:

Integrate[1/(x^(5/2)*(a + c*x^4)),x]
 

Output:

-1/12*(8*a^(3/8) + 3*Sqrt[2 - Sqrt[2]]*c^(3/8)*x^(3/2)*ArcTan[(Sqrt[1 - 1/ 
Sqrt[2]]*(a^(1/4) - c^(1/4)*x))/(a^(1/8)*c^(1/8)*Sqrt[x])] - 3*Sqrt[2 + Sq 
rt[2]]*c^(3/8)*x^(3/2)*ArcTan[(Sqrt[1 + 1/Sqrt[2]]*(a^(1/4) - c^(1/4)*x))/ 
(a^(1/8)*c^(1/8)*Sqrt[x])] - 3*Sqrt[2 - Sqrt[2]]*c^(3/8)*x^(3/2)*ArcTanh[( 
Sqrt[2 + Sqrt[2]]*a^(1/8)*c^(1/8)*Sqrt[x])/(a^(1/4) + c^(1/4)*x)] + 3*Sqrt 
[2 + Sqrt[2]]*c^(3/8)*x^(3/2)*ArcTanh[(a^(1/8)*c^(1/8)*Sqrt[-((-2 + Sqrt[2 
])*x)])/(a^(1/4) + c^(1/4)*x)])/(a^(11/8)*x^(3/2))
 

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.44, 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, 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.

\(\displaystyle \int \frac {1}{x^{5/2} \left (a+c x^4\right )} \, dx\)

\(\Big \downarrow \) 847

\(\displaystyle -\frac {c \int \frac {x^{3/2}}{c x^4+a}dx}{a}-\frac {2}{3 a x^{3/2}}\)

\(\Big \downarrow \) 851

\(\displaystyle -\frac {2 c \int \frac {x^2}{c x^4+a}d\sqrt {x}}{a}-\frac {2}{3 a x^{3/2}}\)

\(\Big \downarrow \) 830

\(\displaystyle -\frac {2 c \left (\frac {\int \frac {1}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt {c}}-\frac {\int \frac {1}{\sqrt {-a}-\sqrt {c} x^2}d\sqrt {x}}{2 \sqrt {c}}\right )}{a}-\frac {2}{3 a x^{3/2}}\)

\(\Big \downarrow \) 755

\(\displaystyle -\frac {2 c \left (\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}}{2 \sqrt {c}}-\frac {\int \frac {1}{\sqrt {-a}-\sqrt {c} x^2}d\sqrt {x}}{2 \sqrt {c}}\right )}{a}-\frac {2}{3 a x^{3/2}}\)

\(\Big \downarrow \) 756

\(\displaystyle -\frac {2 c \left (\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\int \frac {1}{\sqrt [4]{c} x+\sqrt [4]{-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{3 a x^{3/2}}\)

\(\Big \downarrow \) 218

\(\displaystyle -\frac {2 c \left (\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{3 a x^{3/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {2 c \left (\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}}{2 \sqrt {c}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{3 a x^{3/2}}\)

\(\Big \downarrow \) 1476

\(\displaystyle -\frac {2 c \left (\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [4]{c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {c}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{3 a x^{3/2}}\)

\(\Big \downarrow \) 1082

\(\displaystyle -\frac {2 c \left (\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {c}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{3 a x^{3/2}}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {2 c \left (\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {c}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{3 a x^{3/2}}\)

\(\Big \downarrow \) 1479

\(\displaystyle -\frac {2 c \left (\frac {\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{c} \left (x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}\right )}{\sqrt [8]{c} \left (x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {c}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{3 a x^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {2 c \left (\frac {\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{c} \left (x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}\right )}{\sqrt [8]{c} \left (x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {c}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{3 a x^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 c \left (\frac {\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}}{x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [8]{-a} \sqrt [4]{c}}}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {c}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{3 a x^{3/2}}\)

\(\Big \downarrow \) 1103

\(\displaystyle -\frac {2 c \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {c}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{3 a x^{3/2}}\)

Input:

Int[1/(x^(5/2)*(a + c*x^4)),x]
 

Output:

-2/(3*a*x^(3/2)) - (2*c*(-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[c] + ((-(ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(Sqr 
t[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) - S 
qrt[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*Sq 
rt[2]*(-a)^(1/8)*c^(1/8)))/(2*(-a)^(1/4)))/(2*Sqrt[c])))/a
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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])
 

rule 218

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]
 

rule 221

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]
 

rule 755

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]]))
 

rule 756

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]
 

rule 830

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]
 

rule 847

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]
 

rule 851

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]
 

rule 1082

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]
 

rule 1103

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]
 

rule 1476

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]
 

rule 1479

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]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.48 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.16

method	result	size
derivativedivides	\(-\frac {2}{3 a \,x^{\frac {3}{2}}}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 a}\)	\(38\)
default	\(-\frac {2}{3 a \,x^{\frac {3}{2}}}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 a}\)	\(38\)
risch	\(-\frac {2}{3 a \,x^{\frac {3}{2}}}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 a}\)	\(38\)

Input:

int(1/x^(5/2)/(c*x^4+a),x,method=_RETURNVERBOSE)
 

Output:

-2/3/a/x^(3/2)-1/4/a*sum(1/_R^3*ln(x^(1/2)-_R),_R=RootOf(_Z^8*c+a))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.49 \[ \int \frac {1}{x^{5/2} \left (a+c x^4\right )} \, dx=-\frac {\left (3 i + 3\right ) \, \sqrt {2} a x^{2} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a^{7} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {5}{8}} + c^{2} \sqrt {x}\right ) - \left (3 i - 3\right ) \, \sqrt {2} a x^{2} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a^{7} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {5}{8}} + c^{2} \sqrt {x}\right ) + \left (3 i - 3\right ) \, \sqrt {2} a x^{2} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a^{7} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {5}{8}} + c^{2} \sqrt {x}\right ) - \left (3 i + 3\right ) \, \sqrt {2} a x^{2} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a^{7} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {5}{8}} + c^{2} \sqrt {x}\right ) - 6 \, a x^{2} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (a^{7} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {5}{8}} + c^{2} \sqrt {x}\right ) - 6 i \, a x^{2} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (i \, a^{7} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {5}{8}} + c^{2} \sqrt {x}\right ) + 6 i \, a x^{2} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (-i \, a^{7} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {5}{8}} + c^{2} \sqrt {x}\right ) + 6 \, a x^{2} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (-a^{7} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {5}{8}} + c^{2} \sqrt {x}\right ) + 16 \, \sqrt {x}}{24 \, a x^{2}} \] Input:

integrate(1/x^(5/2)/(c*x^4+a),x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

-1/24*((3*I + 3)*sqrt(2)*a*x^2*(-c^3/a^11)^(1/8)*log((1/2*I + 1/2)*sqrt(2) 
*a^7*(-c^3/a^11)^(5/8) + c^2*sqrt(x)) - (3*I - 3)*sqrt(2)*a*x^2*(-c^3/a^11 
)^(1/8)*log(-(1/2*I - 1/2)*sqrt(2)*a^7*(-c^3/a^11)^(5/8) + c^2*sqrt(x)) + 
(3*I - 3)*sqrt(2)*a*x^2*(-c^3/a^11)^(1/8)*log((1/2*I - 1/2)*sqrt(2)*a^7*(- 
c^3/a^11)^(5/8) + c^2*sqrt(x)) - (3*I + 3)*sqrt(2)*a*x^2*(-c^3/a^11)^(1/8) 
*log(-(1/2*I + 1/2)*sqrt(2)*a^7*(-c^3/a^11)^(5/8) + c^2*sqrt(x)) - 6*a*x^2 
*(-c^3/a^11)^(1/8)*log(a^7*(-c^3/a^11)^(5/8) + c^2*sqrt(x)) - 6*I*a*x^2*(- 
c^3/a^11)^(1/8)*log(I*a^7*(-c^3/a^11)^(5/8) + c^2*sqrt(x)) + 6*I*a*x^2*(-c 
^3/a^11)^(1/8)*log(-I*a^7*(-c^3/a^11)^(5/8) + c^2*sqrt(x)) + 6*a*x^2*(-c^3 
/a^11)^(1/8)*log(-a^7*(-c^3/a^11)^(5/8) + c^2*sqrt(x)) + 16*sqrt(x))/(a*x^ 
2)
 

Sympy [A] (verification not implemented)

Time = 42.66 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.26 \[ \int \frac {1}{x^{5/2} \left (a+c x^4\right )} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {11}{2}}} & \text {for}\: a = 0 \wedge c = 0 \\- \frac {2}{11 c x^{\frac {11}{2}}} & \text {for}\: a = 0 \\- \frac {2}{3 a x^{\frac {3}{2}}} & \text {for}\: c = 0 \\- \frac {\log {\left (\sqrt {x} - \sqrt [8]{- \frac {a}{c}} \right )}}{4 a \left (- \frac {a}{c}\right )^{\frac {3}{8}}} + \frac {\log {\left (\sqrt {x} + \sqrt [8]{- \frac {a}{c}} \right )}}{4 a \left (- \frac {a}{c}\right )^{\frac {3}{8}}} + \frac {\sqrt {2} \log {\left (- 4 \sqrt {2} \sqrt {x} \sqrt [8]{- \frac {a}{c}} + 4 x + 4 \sqrt [4]{- \frac {a}{c}} \right )}}{8 a \left (- \frac {a}{c}\right )^{\frac {3}{8}}} - \frac {\sqrt {2} \log {\left (4 \sqrt {2} \sqrt {x} \sqrt [8]{- \frac {a}{c}} + 4 x + 4 \sqrt [4]{- \frac {a}{c}} \right )}}{8 a \left (- \frac {a}{c}\right )^{\frac {3}{8}}} + \frac {\operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} \right )}}{2 a \left (- \frac {a}{c}\right )^{\frac {3}{8}}} - \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} - 1 \right )}}{4 a \left (- \frac {a}{c}\right )^{\frac {3}{8}}} - \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} + 1 \right )}}{4 a \left (- \frac {a}{c}\right )^{\frac {3}{8}}} - \frac {2}{3 a x^{\frac {3}{2}}} & \text {otherwise} \end {cases} \] Input:

integrate(1/x**(5/2)/(c*x**4+a),x)
 

Output:

Piecewise((zoo/x**(11/2), Eq(a, 0) & Eq(c, 0)), (-2/(11*c*x**(11/2)), Eq(a 
, 0)), (-2/(3*a*x**(3/2)), Eq(c, 0)), (-log(sqrt(x) - (-a/c)**(1/8))/(4*a* 
(-a/c)**(3/8)) + log(sqrt(x) + (-a/c)**(1/8))/(4*a*(-a/c)**(3/8)) + sqrt(2 
)*log(-4*sqrt(2)*sqrt(x)*(-a/c)**(1/8) + 4*x + 4*(-a/c)**(1/4))/(8*a*(-a/c 
)**(3/8)) - sqrt(2)*log(4*sqrt(2)*sqrt(x)*(-a/c)**(1/8) + 4*x + 4*(-a/c)** 
(1/4))/(8*a*(-a/c)**(3/8)) + atan(sqrt(x)/(-a/c)**(1/8))/(2*a*(-a/c)**(3/8 
)) - sqrt(2)*atan(sqrt(2)*sqrt(x)/(-a/c)**(1/8) - 1)/(4*a*(-a/c)**(3/8)) - 
 sqrt(2)*atan(sqrt(2)*sqrt(x)/(-a/c)**(1/8) + 1)/(4*a*(-a/c)**(3/8)) - 2/( 
3*a*x**(3/2)), True))
 

Maxima [F]

\[ \int \frac {1}{x^{5/2} \left (a+c x^4\right )} \, dx=\int { \frac {1}{{\left (c x^{4} + a\right )} x^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/x^(5/2)/(c*x^4+a),x, algorithm="maxima")
 

Output:

-c*integrate(x^(3/2)/(a*c*x^4 + a^2), x) - 2/3/(a*x^(3/2))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (159) = 318\).

Time = 0.22 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.92 \[ \int \frac {1}{x^{5/2} \left (a+c x^4\right )} \, dx=\frac {c \left (\frac {a}{c}\right )^{\frac {5}{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 {5}{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 {5}{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 {5}{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 {5}{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 {5}{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 {5}{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 {5}{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}{3 \, a x^{\frac {3}{2}}} \] Input:

integrate(1/x^(5/2)/(c*x^4+a),x, algorithm="giac")
 

Output:

1/2*c*(a/c)^(5/8)*arctan((sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + 2*sqrt(x))/(sqr 
t(sqrt(2) + 2)*(a/c)^(1/8)))/(a^2*sqrt(2*sqrt(2) + 4)) + 1/2*c*(a/c)^(5/8) 
*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)^(5/8)*arctan((sqrt(sq 
rt(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)^(5/8)*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)^(5/8)*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)^(5/8)*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)^(5/8)*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)^(5/8)*log(-sqrt(x)*sqrt( 
-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^2*sqrt(-2*sqrt(2) + 4)) - 
2/3/(a*x^(3/2))
 

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.52 \[ \int \frac {1}{x^{5/2} \left (a+c x^4\right )} \, dx=\frac {{\left (-c\right )}^{3/8}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/8}\,\sqrt {x}}{a^{1/8}}\right )}{2\,a^{11/8}}-\frac {2}{3\,a\,x^{3/2}}-\frac {{\left (-c\right )}^{3/8}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{a^{1/8}}\right )\,1{}\mathrm {i}}{2\,a^{11/8}}+\frac {\sqrt {2}\,{\left (-c\right )}^{3/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-c\right )}^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (-\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )}{a^{11/8}}+\frac {\sqrt {2}\,{\left (-c\right )}^{3/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-c\right )}^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (-\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right )}{a^{11/8}} \] Input:

int(1/(x^(5/2)*(a + c*x^4)),x)
 

Output:

((-c)^(3/8)*atan(((-c)^(1/8)*x^(1/2))/a^(1/8)))/(2*a^(11/8)) - 2/(3*a*x^(3 
/2)) - ((-c)^(3/8)*atan(((-c)^(1/8)*x^(1/2)*1i)/a^(1/8))*1i)/(2*a^(11/8)) 
- (2^(1/2)*(-c)^(3/8)*atan((2^(1/2)*(-c)^(1/8)*x^(1/2)*(1/2 - 1i/2))/a^(1/ 
8))*(1/4 + 1i/4))/a^(11/8) - (2^(1/2)*(-c)^(3/8)*atan((2^(1/2)*(-c)^(1/8)* 
x^(1/2)*(1/2 + 1i/2))/a^(1/8))*(1/4 - 1i/4))/a^(11/8)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 804, normalized size of antiderivative = 3.35 \[ \int \frac {1}{x^{5/2} \left (a+c x^4\right )} \, dx =\text {Too large to display} \] Input:

int(1/x^(5/2)/(c*x^4+a),x)
 

Output:

( - 6*sqrt(x)*c**(3/8)*a**(5/8)*sqrt(sqrt(2) + 2)*sqrt(2)*atan((c**(1/8)*a 
**(1/8)*sqrt( - sqrt(2) + 2) - 2*sqrt(x)*c**(1/4))/(c**(1/8)*a**(1/8)*sqrt 
(sqrt(2) + 2)))*x + 6*sqrt(x)*c**(3/8)*a**(5/8)*sqrt(sqrt(2) + 2)*atan((c* 
*(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) - 2*sqrt(x)*c**(1/4))/(c**(1/8)*a**(1 
/8)*sqrt(sqrt(2) + 2)))*x + 6*sqrt(x)*c**(3/8)*a**(5/8)*sqrt(sqrt(2) + 2)* 
sqrt(2)*atan((c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) + 2*sqrt(x)*c**(1/4)) 
/(c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2)))*x - 6*sqrt(x)*c**(3/8)*a**(5/8)*sq 
rt(sqrt(2) + 2)*atan((c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) + 2*sqrt(x)*c 
**(1/4))/(c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2)))*x + 6*sqrt(x)*c**(3/8)*a** 
(5/8)*sqrt( - sqrt(2) + 2)*sqrt(2)*atan((c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 
2) - 2*sqrt(x)*c**(1/4))/(c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2)))*x + 6*s 
qrt(x)*c**(3/8)*a**(5/8)*sqrt( - sqrt(2) + 2)*atan((c**(1/8)*a**(1/8)*sqrt 
(sqrt(2) + 2) - 2*sqrt(x)*c**(1/4))/(c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2 
)))*x - 6*sqrt(x)*c**(3/8)*a**(5/8)*sqrt( - sqrt(2) + 2)*sqrt(2)*atan((c** 
(1/8)*a**(1/8)*sqrt(sqrt(2) + 2) + 2*sqrt(x)*c**(1/4))/(c**(1/8)*a**(1/8)* 
sqrt( - sqrt(2) + 2)))*x - 6*sqrt(x)*c**(3/8)*a**(5/8)*sqrt( - sqrt(2) + 2 
)*atan((c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2) + 2*sqrt(x)*c**(1/4))/(c**(1/8 
)*a**(1/8)*sqrt( - sqrt(2) + 2)))*x + 3*sqrt(x)*c**(3/8)*a**(5/8)*sqrt( - 
sqrt(2) + 2)*sqrt(2)*log( - sqrt(x)*c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) 
 + a**(1/4) + c**(1/4)*x)*x - 3*sqrt(x)*c**(3/8)*a**(5/8)*sqrt( - sqrt(...