\(\int \frac {1}{\sqrt {x} (a+c x^4)^3} \, dx\) [158]

Optimal result

Integrand size = 15, antiderivative size = 270 \[ \int \frac {1}{\sqrt {x} \left (a+c x^4\right )^3} \, dx=\frac {\sqrt {x}}{8 a \left (a+c x^4\right )^2}+\frac {15 \sqrt {x}}{64 a^2 \left (a+c x^4\right )}+\frac {105 \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{23/8} \sqrt [8]{c}}-\frac {105 \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{23/8} \sqrt [8]{c}}-\frac {105 \arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}}-\frac {105 \text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}}-\frac {105 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{-a}+\sqrt [4]{c} x}\right )}{256 \sqrt {2} (-a)^{23/8} \sqrt [8]{c}} \] Output:

1/8*x^(1/2)/a/(c*x^4+a)^2+15/64*x^(1/2)/a^2/(c*x^4+a)-105/512*arctan(-1+2^ 
(1/2)*c^(1/8)*x^(1/2)/(-a)^(1/8))*2^(1/2)/(-a)^(23/8)/c^(1/8)-105/512*arct 
an(1+2^(1/2)*c^(1/8)*x^(1/2)/(-a)^(1/8))*2^(1/2)/(-a)^(23/8)/c^(1/8)-105/2 
56*arctan(c^(1/8)*x^(1/2)/(-a)^(1/8))/(-a)^(23/8)/c^(1/8)-105/256*arctanh( 
c^(1/8)*x^(1/2)/(-a)^(1/8))/(-a)^(23/8)/c^(1/8)-105/512*arctanh(2^(1/2)*(- 
a)^(1/8)*c^(1/8)*x^(1/2)/((-a)^(1/4)+c^(1/4)*x))*2^(1/2)/(-a)^(23/8)/c^(1/ 
8)
 

Mathematica [A] (verified)

Time = 1.02 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\sqrt {x} \left (a+c x^4\right )^3} \, dx=\frac {\frac {8 a^{7/8} \sqrt {x} \left (23 a+15 c x^4\right )}{\left (a+c x^4\right )^2}-\frac {105 \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )}{\sqrt [8]{c}}-\frac {105 \sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )}{\sqrt [8]{c}}+\frac {105 \sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{\sqrt [8]{c}}+\frac {105 \sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [8]{c} \sqrt {-\left (\left (-2+\sqrt {2}\right ) x\right )}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{\sqrt [8]{c}}}{512 a^{23/8}} \] Input:

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

Output:

((8*a^(7/8)*Sqrt[x]*(23*a + 15*c*x^4))/(a + c*x^4)^2 - (105*Sqrt[2 + Sqrt[ 
2]]*ArcTan[(Sqrt[1 - 1/Sqrt[2]]*(a^(1/4) - c^(1/4)*x))/(a^(1/8)*c^(1/8)*Sq 
rt[x])])/c^(1/8) - (105*Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[1 + 1/Sqrt[2]]*(a^( 
1/4) - c^(1/4)*x))/(a^(1/8)*c^(1/8)*Sqrt[x])])/c^(1/8) + (105*Sqrt[2 + Sqr 
t[2]]*ArcTanh[(Sqrt[2 + Sqrt[2]]*a^(1/8)*c^(1/8)*Sqrt[x])/(a^(1/4) + c^(1/ 
4)*x)])/c^(1/8) + (105*Sqrt[2 - Sqrt[2]]*ArcTanh[(a^(1/8)*c^(1/8)*Sqrt[-(( 
-2 + Sqrt[2])*x)])/(a^(1/4) + c^(1/4)*x)])/c^(1/8))/(512*a^(23/8))
 

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.44, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {819, 819, 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.

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

\(\Big \downarrow \) 819

\(\displaystyle \frac {15 \int \frac {1}{\sqrt {x} \left (c x^4+a\right )^2}dx}{16 a}+\frac {\sqrt {x}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 819

\(\displaystyle \frac {15 \left (\frac {7 \int \frac {1}{\sqrt {x} \left (c x^4+a\right )}dx}{8 a}+\frac {\sqrt {x}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {\sqrt {x}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 851

\(\displaystyle \frac {15 \left (\frac {7 \int \frac {1}{c x^4+a}d\sqrt {x}}{4 a}+\frac {\sqrt {x}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {\sqrt {x}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 758

\(\displaystyle \frac {15 \left (\frac {7 \left (-\frac {\int \frac {1}{\sqrt {-a}-\sqrt {c} x^2}d\sqrt {x}}{2 \sqrt {-a}}-\frac {\int \frac {1}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt {-a}}\right )}{4 a}+\frac {\sqrt {x}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {\sqrt {x}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {15 \left (\frac {7 \left (-\frac {\int \frac {1}{\sqrt {-a}-\sqrt {c} x^2}d\sqrt {x}}{2 \sqrt {-a}}-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{4 a}+\frac {\sqrt {x}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {\sqrt {x}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {15 \left (\frac {7 \left (-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\int \frac {1}{\sqrt [4]{c} x+\sqrt [4]{-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{4 a}+\frac {\sqrt {x}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {\sqrt {x}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {15 \left (\frac {7 \left (-\frac {\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {-a}}-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{4 a}+\frac {\sqrt {x}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {\sqrt {x}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {15 \left (\frac {7 \left (-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {-a}}\right )}{4 a}+\frac {\sqrt {x}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {\sqrt {x}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {15 \left (\frac {7 \left (-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [4]{c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {-a}}\right )}{4 a}+\frac {\sqrt {x}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {\sqrt {x}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {15 \left (\frac {7 \left (-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {-a}}\right )}{4 a}+\frac {\sqrt {x}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {\sqrt {x}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {15 \left (\frac {7 \left (-\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {-a}}\right )}{4 a}+\frac {\sqrt {x}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {\sqrt {x}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {15 \left (\frac {7 \left (-\frac {\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{c} \left (x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}\right )}{\sqrt [8]{c} \left (x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {-a}}\right )}{4 a}+\frac {\sqrt {x}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {\sqrt {x}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {15 \left (\frac {7 \left (-\frac {\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{c} \left (x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}\right )}{\sqrt [8]{c} \left (x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {-a}}\right )}{4 a}+\frac {\sqrt {x}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {\sqrt {x}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {15 \left (\frac {7 \left (-\frac {\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}}{x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [8]{-a} \sqrt [4]{c}}}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {-a}}\right )}{4 a}+\frac {\sqrt {x}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {\sqrt {x}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {15 \left (\frac {7 \left (-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {-a}}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{4 a}+\frac {\sqrt {x}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {\sqrt {x}}{8 a \left (a+c x^4\right )^2}\)

Input:

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

Output:

Sqrt[x]/(8*a*(a + c*x^4)^2) + (15*(Sqrt[x]/(4*a*(a + c*x^4)) + (7*(-1/2*(A 
rcTan[(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))) + Arc 
Tan[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) - 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*( 
-a)^(1/4)))/(2*Sqrt[-a])))/(4*a)))/(16*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 758

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]
 

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 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 = 0.55 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.23

method	result	size
derivativedivides	\(\frac {\frac {23 \sqrt {x}}{64 a}+\frac {15 c \,x^{\frac {9}{2}}}{64 a^{2}}}{\left (c \,x^{4}+a \right )^{2}}+\frac {105 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{7}}\right )}{512 a^{2} c}\)	\(62\)
default	\(\frac {\frac {23 \sqrt {x}}{64 a}+\frac {15 c \,x^{\frac {9}{2}}}{64 a^{2}}}{\left (c \,x^{4}+a \right )^{2}}+\frac {105 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{7}}\right )}{512 a^{2} c}\)	\(62\)

Input:

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

Output:

2*(23/128/a*x^(1/2)+15/128/a^2*c*x^(9/2))/(c*x^4+a)^2+105/512/a^2/c*sum(1/ 
_R^7*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 = 524, normalized size of antiderivative = 1.94 \[ \int \frac {1}{\sqrt {x} \left (a+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/1024*(105*sqrt(2)*(-(I + 1)*a^2*c^2*x^8 - (2*I + 2)*a^3*c*x^4 - (I + 1) 
*a^4)*(-1/(a^23*c))^(1/8)*log((1/2*I + 1/2)*sqrt(2)*a^3*(-1/(a^23*c))^(1/8 
) + sqrt(x)) + 105*sqrt(2)*((I - 1)*a^2*c^2*x^8 + (2*I - 2)*a^3*c*x^4 + (I 
 - 1)*a^4)*(-1/(a^23*c))^(1/8)*log(-(1/2*I - 1/2)*sqrt(2)*a^3*(-1/(a^23*c) 
)^(1/8) + sqrt(x)) + 105*sqrt(2)*(-(I - 1)*a^2*c^2*x^8 - (2*I - 2)*a^3*c*x 
^4 - (I - 1)*a^4)*(-1/(a^23*c))^(1/8)*log((1/2*I - 1/2)*sqrt(2)*a^3*(-1/(a 
^23*c))^(1/8) + sqrt(x)) + 105*sqrt(2)*((I + 1)*a^2*c^2*x^8 + (2*I + 2)*a^ 
3*c*x^4 + (I + 1)*a^4)*(-1/(a^23*c))^(1/8)*log(-(1/2*I + 1/2)*sqrt(2)*a^3* 
(-1/(a^23*c))^(1/8) + sqrt(x)) - 210*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1 
/(a^23*c))^(1/8)*log(a^3*(-1/(a^23*c))^(1/8) + sqrt(x)) + 210*(-I*a^2*c^2* 
x^8 - 2*I*a^3*c*x^4 - I*a^4)*(-1/(a^23*c))^(1/8)*log(I*a^3*(-1/(a^23*c))^( 
1/8) + sqrt(x)) + 210*(I*a^2*c^2*x^8 + 2*I*a^3*c*x^4 + I*a^4)*(-1/(a^23*c) 
)^(1/8)*log(-I*a^3*(-1/(a^23*c))^(1/8) + sqrt(x)) + 210*(a^2*c^2*x^8 + 2*a 
^3*c*x^4 + a^4)*(-1/(a^23*c))^(1/8)*log(-a^3*(-1/(a^23*c))^(1/8) + sqrt(x) 
) - 16*(15*c*x^4 + 23*a)*sqrt(x))/(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-105*c*integrate(1/128*x^(7/2)/(a^3*c*x^4 + a^4), x) + 1/64*(105*c^2*x^(17 
/2) + 225*a*c*x^(9/2) + 128*a^2*sqrt(x))/(a^3*c^2*x^8 + 2*a^4*c*x^4 + a^5)
                                                                                    
                                                                                    
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 472 vs. \(2 (185) = 370\).

Time = 0.23 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.75 \[ \int \frac {1}{\sqrt {x} \left (a+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

105/256*(a/c)^(1/8)*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)) + 105/256*(a/c)^ 
(1/8)*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)) + 105/256*(a/c)^(1/8)*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)) + 105/256*(a/c)^(1/8)*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)) + 105/512*(a/c)^(1/8)*log(sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^ 
(1/8) + x + (a/c)^(1/4))/(a^3*sqrt(-2*sqrt(2) + 4)) - 105/512*(a/c)^(1/8)* 
log(-sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^3*sqrt(-2 
*sqrt(2) + 4)) + 105/512*(a/c)^(1/8)*log(sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^ 
(1/8) + x + (a/c)^(1/4))/(a^3*sqrt(2*sqrt(2) + 4)) - 105/512*(a/c)^(1/8)*l 
og(-sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^3*sqrt(2* 
sqrt(2) + 4)) + 1/64*(15*c*x^(9/2) + 23*a*sqrt(x))/((c*x^4 + a)^2*a^2)
 

Mupad [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.58 \[ \int \frac {1}{\sqrt {x} \left (a+c x^4\right )^3} \, dx=\frac {\frac {23\,\sqrt {x}}{64\,a}+\frac {15\,c\,x^{9/2}}{64\,a^2}}{a^2+2\,a\,c\,x^4+c^2\,x^8}-\frac {105\,\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{256\,{\left (-a\right )}^{23/8}\,c^{1/8}}+\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,105{}\mathrm {i}}{256\,{\left (-a\right )}^{23/8}\,c^{1/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {105}{512}-\frac {105}{512}{}\mathrm {i}\right )}{{\left (-a\right )}^{23/8}\,c^{1/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {105}{512}+\frac {105}{512}{}\mathrm {i}\right )}{{\left (-a\right )}^{23/8}\,c^{1/8}} \] Input:

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

Output:

((23*x^(1/2))/(64*a) + (15*c*x^(9/2))/(64*a^2))/(a^2 + c^2*x^8 + 2*a*c*x^4 
) - (105*atan((c^(1/8)*x^(1/2))/(-a)^(1/8)))/(256*(-a)^(23/8)*c^(1/8)) + ( 
atan((c^(1/8)*x^(1/2)*1i)/(-a)^(1/8))*105i)/(256*(-a)^(23/8)*c^(1/8)) - (2 
^(1/2)*atan((2^(1/2)*c^(1/8)*x^(1/2)*(1/2 - 1i/2))/(-a)^(1/8))*(105/512 + 
105i/512))/((-a)^(23/8)*c^(1/8)) - (2^(1/2)*atan((2^(1/2)*c^(1/8)*x^(1/2)* 
(1/2 + 1i/2))/(-a)^(1/8))*(105/512 - 105i/512))/((-a)^(23/8)*c^(1/8))
 

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 1182, normalized size of antiderivative = 4.38 \[ \int \frac {1}{\sqrt {x} \left (a+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 210*c**(7/8)*a**(1/8)*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**2 - 420*c**(7/8)*a**(1/8)*sqrt(sqrt(2) + 2)*atan((c**(1/8)*a**(1/8)*s 
qrt( - sqrt(2) + 2) - 2*sqrt(x)*c**(1/4))/(c**(1/8)*a**(1/8)*sqrt(sqrt(2) 
+ 2)))*a*c*x**4 - 210*c**(7/8)*a**(1/8)*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**2*x**8 + 210*c**(7/8)*a**(1/8)*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**2 + 420*c**(7/8)*a**(1/8)*sqrt(sqrt(2) + 2)*a 
tan((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*c*x**4 + 210*c**(7/8)*a**(1/8)*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**2*x**8 - 210*c**(7/8)*a**(1/8)* 
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**2 - 420*c**(7/8)*a 
**(1/8)*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*c*x**4 - 21 
0*c**(7/8)*a**(1/8)*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 
**2*x**8 + 210*c**(7/8)*a**(1/8)*sqrt( - sqrt(2) + 2)*atan((c**(1/8)*a*...