\(\int \frac {1}{x^{3/2} (a+c x^4)^2} \, dx\) [149]

Optimal result

Integrand size = 15, antiderivative size = 261 \[ \int \frac {1}{x^{3/2} \left (a+c x^4\right )^2} \, dx=-\frac {9}{4 a^2 \sqrt {x}}+\frac {1}{4 a \sqrt {x} \left (a+c x^4\right )}+\frac {9 \sqrt [8]{c} \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{17/8}}-\frac {9 \sqrt [8]{c} \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{17/8}}-\frac {9 \sqrt [8]{c} \arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{17/8}}+\frac {9 \sqrt [8]{c} \text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{17/8}}+\frac {9 \sqrt [8]{c} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{-a}+\sqrt [4]{c} x}\right )}{16 \sqrt {2} (-a)^{17/8}} \] Output:

-9/4/a^2/x^(1/2)+1/4/a/x^(1/2)/(c*x^4+a)-9/32*c^(1/8)*arctan(-1+2^(1/2)*c^ 
(1/8)*x^(1/2)/(-a)^(1/8))*2^(1/2)/(-a)^(17/8)-9/32*c^(1/8)*arctan(1+2^(1/2 
)*c^(1/8)*x^(1/2)/(-a)^(1/8))*2^(1/2)/(-a)^(17/8)-9/16*c^(1/8)*arctan(c^(1 
/8)*x^(1/2)/(-a)^(1/8))/(-a)^(17/8)+9/16*c^(1/8)*arctanh(c^(1/8)*x^(1/2)/( 
-a)^(1/8))/(-a)^(17/8)+9/32*c^(1/8)*arctanh(2^(1/2)*(-a)^(1/8)*c^(1/8)*x^( 
1/2)/((-a)^(1/4)+c^(1/4)*x))*2^(1/2)/(-a)^(17/8)
 

Mathematica [A] (verified)

Time = 1.24 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x^{3/2} \left (a+c x^4\right )^2} \, dx=\frac {-\frac {8 \sqrt [8]{a} \left (8 a+9 c x^4\right )}{\sqrt {x} \left (a+c x^4\right )}+9 \sqrt {2+\sqrt {2}} \sqrt [8]{c} \arctan \left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )+9 \sqrt {2-\sqrt {2}} \sqrt [8]{c} \arctan \left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )+9 \sqrt {2+\sqrt {2}} \sqrt [8]{c} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )+9 \sqrt {2-\sqrt {2}} \sqrt [8]{c} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [8]{c} \sqrt {-\left (\left (-2+\sqrt {2}\right ) x\right )}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{32 a^{17/8}} \] Input:

Integrate[1/(x^(3/2)*(a + c*x^4)^2),x]
 

Output:

((-8*a^(1/8)*(8*a + 9*c*x^4))/(Sqrt[x]*(a + c*x^4)) + 9*Sqrt[2 + Sqrt[2]]* 
c^(1/8)*ArcTan[(Sqrt[1 - 1/Sqrt[2]]*(a^(1/4) - c^(1/4)*x))/(a^(1/8)*c^(1/8 
)*Sqrt[x])] + 9*Sqrt[2 - Sqrt[2]]*c^(1/8)*ArcTan[(Sqrt[1 + 1/Sqrt[2]]*(a^( 
1/4) - c^(1/4)*x))/(a^(1/8)*c^(1/8)*Sqrt[x])] + 9*Sqrt[2 + Sqrt[2]]*c^(1/8 
)*ArcTanh[(Sqrt[2 + Sqrt[2]]*a^(1/8)*c^(1/8)*Sqrt[x])/(a^(1/4) + c^(1/4)*x 
)] + 9*Sqrt[2 - Sqrt[2]]*c^(1/8)*ArcTanh[(a^(1/8)*c^(1/8)*Sqrt[-((-2 + Sqr 
t[2])*x)])/(a^(1/4) + c^(1/4)*x)])/(32*a^(17/8))
 

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.41, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {819, 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.

\(\displaystyle \int \frac {1}{x^{3/2} \left (a+c x^4\right )^2} \, dx\)

\(\Big \downarrow \) 819

\(\displaystyle \frac {9 \int \frac {1}{x^{3/2} \left (c x^4+a\right )}dx}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+c x^4\right )}\)

\(\Big \downarrow \) 847

\(\displaystyle \frac {9 \left (-\frac {c \int \frac {x^{5/2}}{c x^4+a}dx}{a}-\frac {2}{a \sqrt {x}}\right )}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+c x^4\right )}\)

\(\Big \downarrow \) 851

\(\displaystyle \frac {9 \left (-\frac {2 c \int \frac {x^3}{c x^4+a}d\sqrt {x}}{a}-\frac {2}{a \sqrt {x}}\right )}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+c x^4\right )}\)

\(\Big \downarrow \) 830

\(\displaystyle \frac {9 \left (-\frac {2 c \left (\frac {\int \frac {x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt {c}}-\frac {\int \frac {x}{\sqrt {-a}-\sqrt {c} x^2}d\sqrt {x}}{2 \sqrt {c}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+c x^4\right )}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {9 \left (-\frac {2 c \left (\frac {\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\int \frac {x}{\sqrt {-a}-\sqrt {c} x^2}d\sqrt {x}}{2 \sqrt {c}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+c x^4\right )}\)

\(\Big \downarrow \) 827

\(\displaystyle \frac {9 \left (-\frac {2 c \left (\frac {\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\int \frac {1}{\sqrt [4]{c} x+\sqrt [4]{-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+c x^4\right )}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {9 \left (-\frac {2 c \left (\frac {\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+c x^4\right )}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {9 \left (-\frac {2 c \left (\frac {\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+c x^4\right )}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {9 \left (-\frac {2 c \left (\frac {\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [4]{c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+c x^4\right )}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {9 \left (-\frac {2 c \left (\frac {\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+c x^4\right )}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {9 \left (-\frac {2 c \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+c x^4\right )}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {9 \left (-\frac {2 c \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{c} \left (x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}\right )}{\sqrt [8]{c} \left (x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+c x^4\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {9 \left (-\frac {2 c \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{c} \left (x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}\right )}{\sqrt [8]{c} \left (x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+c x^4\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {9 \left (-\frac {2 c \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}}{x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [8]{-a} \sqrt [4]{c}}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+c x^4\right )}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {9 \left (-\frac {2 c \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+c x^4\right )}\)

Input:

Int[1/(x^(3/2)*(a + c*x^4)^2),x]
 

Output:

1/(4*a*Sqrt[x]*(a + c*x^4)) + (9*(-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)*Sqr 
t[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)]/(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*c^( 
1/4)) - (-1/2*Log[(-a)^(1/4) - Sqrt[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))/(8*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 819

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( 
c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 
1) + 1)/(a*n*(p + 1))   Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a 
, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p 
, x]
 

rule 826

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]]))
 

rule 827

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]
 

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 = 0.51 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.21

method	result	size
risch	\(-\frac {2}{a^{2} \sqrt {x}}-\frac {c \,x^{\frac {7}{2}}}{4 a^{2} \left (c \,x^{4}+a \right )}-\frac {9 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}}\right )}{32 a^{2}}\)	\(56\)
derivativedivides	\(-\frac {2 c \left (\frac {x^{\frac {7}{2}}}{8 c \,x^{4}+8 a}+\frac {9 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}}\right )}{64 c}\right )}{a^{2}}-\frac {2}{a^{2} \sqrt {x}}\)	\(59\)
default	\(-\frac {2 c \left (\frac {x^{\frac {7}{2}}}{8 c \,x^{4}+8 a}+\frac {9 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}}\right )}{64 c}\right )}{a^{2}}-\frac {2}{a^{2} \sqrt {x}}\)	\(59\)

Input:

int(1/x^(3/2)/(c*x^4+a)^2,x,method=_RETURNVERBOSE)
 

Output:

-2/a^2/x^(1/2)-1/4/a^2*c*x^(7/2)/(c*x^4+a)-9/32/a^2*sum(1/_R*ln(x^(1/2)-_R 
),_R=RootOf(_Z^8*c+a))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.65 \[ \int \frac {1}{x^{3/2} \left (a+c x^4\right )^2} \, dx =\text {Too large to display} \] Input:

integrate(1/x^(3/2)/(c*x^4+a)^2,x, algorithm="fricas")
 

Output:

-1/64*(9*sqrt(2)*(-(I - 1)*a^2*c*x^5 - (I - 1)*a^3*x)*(-c/a^17)^(1/8)*log( 
(4782969/2*I + 4782969/2)*sqrt(2)*a^15*(-c/a^17)^(7/8) + 4782969*c*sqrt(x) 
) + 9*sqrt(2)*((I + 1)*a^2*c*x^5 + (I + 1)*a^3*x)*(-c/a^17)^(1/8)*log(-(47 
82969/2*I - 4782969/2)*sqrt(2)*a^15*(-c/a^17)^(7/8) + 4782969*c*sqrt(x)) + 
 9*sqrt(2)*(-(I + 1)*a^2*c*x^5 - (I + 1)*a^3*x)*(-c/a^17)^(1/8)*log((47829 
69/2*I - 4782969/2)*sqrt(2)*a^15*(-c/a^17)^(7/8) + 4782969*c*sqrt(x)) + 9* 
sqrt(2)*((I - 1)*a^2*c*x^5 + (I - 1)*a^3*x)*(-c/a^17)^(1/8)*log(-(4782969/ 
2*I + 4782969/2)*sqrt(2)*a^15*(-c/a^17)^(7/8) + 4782969*c*sqrt(x)) + 18*(a 
^2*c*x^5 + a^3*x)*(-c/a^17)^(1/8)*log(4782969*a^15*(-c/a^17)^(7/8) + 47829 
69*c*sqrt(x)) + 18*(-I*a^2*c*x^5 - I*a^3*x)*(-c/a^17)^(1/8)*log(4782969*I* 
a^15*(-c/a^17)^(7/8) + 4782969*c*sqrt(x)) + 18*(I*a^2*c*x^5 + I*a^3*x)*(-c 
/a^17)^(1/8)*log(-4782969*I*a^15*(-c/a^17)^(7/8) + 4782969*c*sqrt(x)) - 18 
*(a^2*c*x^5 + a^3*x)*(-c/a^17)^(1/8)*log(-4782969*a^15*(-c/a^17)^(7/8) + 4 
782969*c*sqrt(x)) + 16*(9*c*x^4 + 8*a)*sqrt(x))/(a^2*c*x^5 + a^3*x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^{3/2} \left (a+c x^4\right )^2} \, dx=\text {Timed out} \] Input:

integrate(1/x**(3/2)/(c*x**4+a)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{x^{3/2} \left (a+c x^4\right )^2} \, dx=\int { \frac {1}{{\left (c x^{4} + a\right )}^{2} x^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/x^(3/2)/(c*x^4+a)^2,x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

-9*c*integrate(1/8*x^(5/2)/(a^2*c*x^4 + a^3), x) - 1/4*(9*c*x^(7/2) + 8*a/ 
sqrt(x))/(a^2*c*x^4 + a^3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 481 vs. \(2 (176) = 352\).

Time = 0.24 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.84 \[ \int \frac {1}{x^{3/2} \left (a+c x^4\right )^2} \, dx =\text {Too large to display} \] Input:

integrate(1/x^(3/2)/(c*x^4+a)^2,x, algorithm="giac")
 

Output:

-9/16*c*(a/c)^(7/8)*arctan((sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + 2*sqrt(x))/(s 
qrt(sqrt(2) + 2)*(a/c)^(1/8)))/(a^3*sqrt(-2*sqrt(2) + 4)) - 9/16*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^3*sqrt(-2*sqrt(2) + 4)) - 9/16*c*(a/c)^(7/8)*arctan((s 
qrt(sqrt(2) + 2)*(a/c)^(1/8) + 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)) 
)/(a^3*sqrt(2*sqrt(2) + 4)) - 9/16*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^3*sqrt(2*s 
qrt(2) + 4)) + 9/32*c*(a/c)^(7/8)*log(sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8 
) + x + (a/c)^(1/4))/(a^3*sqrt(-2*sqrt(2) + 4)) - 9/32*c*(a/c)^(7/8)*log(- 
sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^3*sqrt(-2*sqrt 
(2) + 4)) + 9/32*c*(a/c)^(7/8)*log(sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) 
+ x + (a/c)^(1/4))/(a^3*sqrt(2*sqrt(2) + 4)) - 9/32*c*(a/c)^(7/8)*log(-sqr 
t(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^3*sqrt(2*sqrt(2) 
 + 4)) - 1/4*(9*c*x^4 + 8*a)/((c*x^(9/2) + a*sqrt(x))*a^2)
 

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x^{3/2} \left (a+c x^4\right )^2} \, dx=-\frac {\frac {2}{a}+\frac {9\,c\,x^4}{4\,a^2}}{a\,\sqrt {x}+c\,x^{9/2}}-\frac {9\,{\left (-c\right )}^{1/8}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/8}\,\sqrt {x}}{a^{1/8}}\right )}{16\,a^{17/8}}-\frac {{\left (-c\right )}^{1/8}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{a^{1/8}}\right )\,9{}\mathrm {i}}{16\,a^{17/8}}+\frac {\sqrt {2}\,{\left (-c\right )}^{1/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-c\right )}^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (-\frac {9}{32}+\frac {9}{32}{}\mathrm {i}\right )}{a^{17/8}}+\frac {\sqrt {2}\,{\left (-c\right )}^{1/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-c\right )}^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (-\frac {9}{32}-\frac {9}{32}{}\mathrm {i}\right )}{a^{17/8}} \] Input:

int(1/(x^(3/2)*(a + c*x^4)^2),x)
 

Output:

- (2/a + (9*c*x^4)/(4*a^2))/(a*x^(1/2) + c*x^(9/2)) - (9*(-c)^(1/8)*atan(( 
(-c)^(1/8)*x^(1/2))/a^(1/8)))/(16*a^(17/8)) - ((-c)^(1/8)*atan(((-c)^(1/8) 
*x^(1/2)*1i)/a^(1/8))*9i)/(16*a^(17/8)) - (2^(1/2)*(-c)^(1/8)*atan((2^(1/2 
)*(-c)^(1/8)*x^(1/2)*(1/2 - 1i/2))/a^(1/8))*(9/32 - 9i/32))/a^(17/8) - (2^ 
(1/2)*(-c)^(1/8)*atan((2^(1/2)*(-c)^(1/8)*x^(1/2)*(1/2 + 1i/2))/a^(1/8))*( 
9/32 + 9i/32))/a^(17/8)
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 811, normalized size of antiderivative = 3.11 \[ \int \frac {1}{x^{3/2} \left (a+c x^4\right )^2} \, dx =\text {Too large to display} \] Input:

int(1/x^(3/2)/(c*x^4+a)^2,x)
 

Output:

(18*sqrt(x)*c**(1/8)*a**(7/8)*sqrt(sqrt(2) + 2)*atan((c**(1/8)*a**(1/8)*sq 
rt( - sqrt(2) + 2) - 2*sqrt(x)*c**(1/4))/(c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 
 2)))*a + 18*sqrt(x)*c**(1/8)*a**(7/8)*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)))*c*x**4 - 18*sqrt(x)*c**(1/8)*a**(7/8)*sqrt(sqrt(2) + 2)*ata 
n((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)))*a - 18*sqrt(x)*c**(1/8)*a**(7/8)*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)))*c*x**4 + 18*sqrt(x)*c**(1/8)*a**(7/8 
)*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)))*a + 18*sqrt(x)*c**( 
1/8)*a**(7/8)*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)))*c*x**4 
- 18*sqrt(x)*c**(1/8)*a**(7/8)*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)))*a - 18*sqrt(x)*c**(1/8)*a**(7/8)*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)*s 
qrt( - sqrt(2) + 2)))*c*x**4 - 9*sqrt(x)*c**(1/8)*a**(7/8)*sqrt( - sqrt(2) 
 + 2)*log( - sqrt(x)*c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) + a**(1/4) + c 
**(1/4)*x)*a - 9*sqrt(x)*c**(1/8)*a**(7/8)*sqrt( - sqrt(2) + 2)*log( - ...