\(\int \frac {x^{3/2}}{(a+c x^4)^3} \, dx\) [156]

Optimal result

Integrand size = 15, antiderivative size = 270 \[ \int \frac {x^{3/2}}{\left (a+c x^4\right )^3} \, dx=\frac {x^{5/2}}{8 a \left (a+c x^4\right )^2}+\frac {11 x^{5/2}}{64 a^2 \left (a+c x^4\right )}-\frac {33 \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{19/8} c^{5/8}}+\frac {33 \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{19/8} c^{5/8}}-\frac {33 \arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{19/8} c^{5/8}}-\frac {33 \text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{19/8} c^{5/8}}+\frac {33 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{-a}+\sqrt [4]{c} x}\right )}{256 \sqrt {2} (-a)^{19/8} c^{5/8}} \] Output:

1/8*x^(5/2)/a/(c*x^4+a)^2+11/64*x^(5/2)/a^2/(c*x^4+a)+33/512*arctan(-1+2^( 
1/2)*c^(1/8)*x^(1/2)/(-a)^(1/8))*2^(1/2)/(-a)^(19/8)/c^(5/8)+33/512*arctan 
(1+2^(1/2)*c^(1/8)*x^(1/2)/(-a)^(1/8))*2^(1/2)/(-a)^(19/8)/c^(5/8)-33/256* 
arctan(c^(1/8)*x^(1/2)/(-a)^(1/8))/(-a)^(19/8)/c^(5/8)-33/256*arctanh(c^(1 
/8)*x^(1/2)/(-a)^(1/8))/(-a)^(19/8)/c^(5/8)+33/512*arctanh(2^(1/2)*(-a)^(1 
/8)*c^(1/8)*x^(1/2)/((-a)^(1/4)+c^(1/4)*x))*2^(1/2)/(-a)^(19/8)/c^(5/8)
 

Mathematica [A] (verified)

Time = 1.01 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.06 \[ \int \frac {x^{3/2}}{\left (a+c x^4\right )^3} \, dx=\frac {\frac {8 a^{3/8} x^{5/2} \left (19 a+11 c x^4\right )}{\left (a+c x^4\right )^2}+\frac {33 \sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )}{c^{5/8}}-\frac {33 \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )}{c^{5/8}}-\frac {33 \sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{c^{5/8}}+\frac {33 \sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [8]{c} \sqrt {-\left (\left (-2+\sqrt {2}\right ) x\right )}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{c^{5/8}}}{512 a^{19/8}} \] Input:

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

Output:

((8*a^(3/8)*x^(5/2)*(19*a + 11*c*x^4))/(a + c*x^4)^2 + (33*Sqrt[2 - Sqrt[2 
]]*ArcTan[(Sqrt[1 - 1/Sqrt[2]]*(a^(1/4) - c^(1/4)*x))/(a^(1/8)*c^(1/8)*Sqr 
t[x])])/c^(5/8) - (33*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^(5/8) - (33*Sqrt[2 - Sqrt[2 
]]*ArcTanh[(Sqrt[2 + Sqrt[2]]*a^(1/8)*c^(1/8)*Sqrt[x])/(a^(1/4) + c^(1/4)* 
x)])/c^(5/8) + (33*Sqrt[2 + Sqrt[2]]*ArcTanh[(a^(1/8)*c^(1/8)*Sqrt[-((-2 + 
 Sqrt[2])*x)])/(a^(1/4) + c^(1/4)*x)])/c^(5/8))/(512*a^(19/8))
 

Rubi [A] (verified)

Time = 1.05 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.42, 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, 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.

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

\(\Big \downarrow \) 819

\(\displaystyle \frac {11 \int \frac {x^{3/2}}{\left (c x^4+a\right )^2}dx}{16 a}+\frac {x^{5/2}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 819

\(\displaystyle \frac {11 \left (\frac {3 \int \frac {x^{3/2}}{c x^4+a}dx}{8 a}+\frac {x^{5/2}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {x^{5/2}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 851

\(\displaystyle \frac {11 \left (\frac {3 \int \frac {x^2}{c x^4+a}d\sqrt {x}}{4 a}+\frac {x^{5/2}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {x^{5/2}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 830

\(\displaystyle \frac {11 \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt {c}}-\frac {\int \frac {1}{\sqrt {-a}-\sqrt {c} x^2}d\sqrt {x}}{2 \sqrt {c}}\right )}{4 a}+\frac {x^{5/2}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {x^{5/2}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {11 \left (\frac {3 \left (\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}}{2 \sqrt {c}}-\frac {\int \frac {1}{\sqrt {-a}-\sqrt {c} x^2}d\sqrt {x}}{2 \sqrt {c}}\right )}{4 a}+\frac {x^{5/2}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {x^{5/2}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {11 \left (\frac {3 \left (\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\int \frac {1}{\sqrt [4]{c} x+\sqrt [4]{-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}}{2 \sqrt {c}}\right )}{4 a}+\frac {x^{5/2}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {x^{5/2}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {11 \left (\frac {3 \left (\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {c}}\right )}{4 a}+\frac {x^{5/2}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {x^{5/2}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {11 \left (\frac {3 \left (\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}}{2 \sqrt {c}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {c}}\right )}{4 a}+\frac {x^{5/2}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {x^{5/2}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {11 \left (\frac {3 \left (\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [4]{c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {c}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {c}}\right )}{4 a}+\frac {x^{5/2}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {x^{5/2}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {11 \left (\frac {3 \left (\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {c}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {c}}\right )}{4 a}+\frac {x^{5/2}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {x^{5/2}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {11 \left (\frac {3 \left (\frac {\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {c}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {c}}\right )}{4 a}+\frac {x^{5/2}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {x^{5/2}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {11 \left (\frac {3 \left (\frac {\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{c} \left (x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}\right )}{\sqrt [8]{c} \left (x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {c}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {c}}\right )}{4 a}+\frac {x^{5/2}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {x^{5/2}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {11 \left (\frac {3 \left (\frac {\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{c} \left (x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}\right )}{\sqrt [8]{c} \left (x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {c}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {c}}\right )}{4 a}+\frac {x^{5/2}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {x^{5/2}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {11 \left (\frac {3 \left (\frac {\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}}{x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [8]{-a} \sqrt [4]{c}}}{2 \sqrt [4]{-a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {c}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {c}}\right )}{4 a}+\frac {x^{5/2}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {x^{5/2}}{8 a \left (a+c x^4\right )^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {11 \left (\frac {3 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{-a}}}{2 \sqrt {c}}-\frac {\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} \sqrt [8]{c}}}{2 \sqrt {c}}\right )}{4 a}+\frac {x^{5/2}}{4 a \left (a+c x^4\right )}\right )}{16 a}+\frac {x^{5/2}}{8 a \left (a+c x^4\right )^2}\)

Input:

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

Output:

x^(5/2)/(8*a*(a + c*x^4)^2) + (11*(x^(5/2)/(4*a*(a + c*x^4)) + (3*(-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[c] + ((-(ArcTan[1 - 
(Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(Sqrt[2]*(-a)^(1/8)*c^(1/8))) + ArcT 
an[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[c])))/(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 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 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 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 = 62, normalized size of antiderivative = 0.23

method	result	size
derivativedivides	\(\frac {\frac {19 x^{\frac {5}{2}}}{64 a}+\frac {11 c \,x^{\frac {13}{2}}}{64 a^{2}}}{\left (c \,x^{4}+a \right )^{2}}+\frac {33 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{512 a^{2} c}\)	\(62\)
default	\(\frac {\frac {19 x^{\frac {5}{2}}}{64 a}+\frac {11 c \,x^{\frac {13}{2}}}{64 a^{2}}}{\left (c \,x^{4}+a \right )^{2}}+\frac {33 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{512 a^{2} c}\)	\(62\)

Input:

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

Output:

2*(19/128/a*x^(5/2)+11/128/a^2*c*x^(13/2))/(c*x^4+a)^2+33/512/a^2/c*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 = 551, normalized size of antiderivative = 2.04 \[ \int \frac {x^{3/2}}{\left (a+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/1024*(33*sqrt(2)*(-(I + 1)*a^2*c^2*x^8 - (2*I + 2)*a^3*c*x^4 - (I + 1)* 
a^4)*(-1/(a^19*c^5))^(1/8)*log((1/2*I + 1/2)*sqrt(2)*a^12*c^3*(-1/(a^19*c^ 
5))^(5/8) + sqrt(x)) + 33*sqrt(2)*((I - 1)*a^2*c^2*x^8 + (2*I - 2)*a^3*c*x 
^4 + (I - 1)*a^4)*(-1/(a^19*c^5))^(1/8)*log(-(1/2*I - 1/2)*sqrt(2)*a^12*c^ 
3*(-1/(a^19*c^5))^(5/8) + sqrt(x)) + 33*sqrt(2)*(-(I - 1)*a^2*c^2*x^8 - (2 
*I - 2)*a^3*c*x^4 - (I - 1)*a^4)*(-1/(a^19*c^5))^(1/8)*log((1/2*I - 1/2)*s 
qrt(2)*a^12*c^3*(-1/(a^19*c^5))^(5/8) + sqrt(x)) + 33*sqrt(2)*((I + 1)*a^2 
*c^2*x^8 + (2*I + 2)*a^3*c*x^4 + (I + 1)*a^4)*(-1/(a^19*c^5))^(1/8)*log(-( 
1/2*I + 1/2)*sqrt(2)*a^12*c^3*(-1/(a^19*c^5))^(5/8) + sqrt(x)) + 66*(a^2*c 
^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^19*c^5))^(1/8)*log(a^12*c^3*(-1/(a^19*c 
^5))^(5/8) + sqrt(x)) + 66*(I*a^2*c^2*x^8 + 2*I*a^3*c*x^4 + I*a^4)*(-1/(a^ 
19*c^5))^(1/8)*log(I*a^12*c^3*(-1/(a^19*c^5))^(5/8) + sqrt(x)) + 66*(-I*a^ 
2*c^2*x^8 - 2*I*a^3*c*x^4 - I*a^4)*(-1/(a^19*c^5))^(1/8)*log(-I*a^12*c^3*( 
-1/(a^19*c^5))^(5/8) + sqrt(x)) - 66*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1 
/(a^19*c^5))^(1/8)*log(-a^12*c^3*(-1/(a^19*c^5))^(5/8) + sqrt(x)) - 16*(11 
*c*x^6 + 19*a*x^2)*sqrt(x))/(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

1/64*(11*c*x^(13/2) + 19*a*x^(5/2))/(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4) + 33 
*integrate(1/128*x^(3/2)/(a^2*c*x^4 + a^3), x)
                                                                                    
                                                                                    
 

Giac [B] (verification not implemented)

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

Time = 0.27 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.75 \[ \int \frac {x^{3/2}}{\left (a+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-33/256*(a/c)^(5/8)*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)) - 33/256*(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^3*sqrt(2*sqrt(2) + 4)) + 33/256*(a/c)^(5/8)*arctan((sqr 
t(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)) + 33/256*(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^3*sqrt(-2*s 
qrt(2) + 4)) - 33/512*(a/c)^(5/8)*log(sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8 
) + x + (a/c)^(1/4))/(a^3*sqrt(2*sqrt(2) + 4)) + 33/512*(a/c)^(5/8)*log(-s 
qrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^3*sqrt(2*sqrt(2 
) + 4)) + 33/512*(a/c)^(5/8)*log(sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + 
x + (a/c)^(1/4))/(a^3*sqrt(-2*sqrt(2) + 4)) - 33/512*(a/c)^(5/8)*log(-sqrt 
(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^3*sqrt(-2*sqrt(2) 
 + 4)) + 1/64*(11*c*x^(13/2) + 19*a*x^(5/2))/((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 \[ \int \frac {x^{3/2}}{\left (a+c x^4\right )^3} \, dx=\frac {\frac {19\,x^{5/2}}{64\,a}+\frac {11\,c\,x^{13/2}}{64\,a^2}}{a^2+2\,a\,c\,x^4+c^2\,x^8}-\frac {33\,\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{256\,{\left (-a\right )}^{19/8}\,c^{5/8}}+\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,33{}\mathrm {i}}{256\,{\left (-a\right )}^{19/8}\,c^{5/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (\frac {33}{512}+\frac {33}{512}{}\mathrm {i}\right )}{{\left (-a\right )}^{19/8}\,c^{5/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (\frac {33}{512}-\frac {33}{512}{}\mathrm {i}\right )}{{\left (-a\right )}^{19/8}\,c^{5/8}} \] Input:

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

Output:

((19*x^(5/2))/(64*a) + (11*c*x^(13/2))/(64*a^2))/(a^2 + c^2*x^8 + 2*a*c*x^ 
4) - (33*atan((c^(1/8)*x^(1/2))/(-a)^(1/8)))/(256*(-a)^(19/8)*c^(5/8)) + ( 
atan((c^(1/8)*x^(1/2)*1i)/(-a)^(1/8))*33i)/(256*(-a)^(19/8)*c^(5/8)) + (2^ 
(1/2)*atan((2^(1/2)*c^(1/8)*x^(1/2)*(1/2 - 1i/2))/(-a)^(1/8))*(33/512 + 33 
i/512))/((-a)^(19/8)*c^(5/8)) + (2^(1/2)*atan((2^(1/2)*c^(1/8)*x^(1/2)*(1/ 
2 + 1i/2))/(-a)^(1/8))*(33/512 - 33i/512))/((-a)^(19/8)*c^(5/8))
 

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 2367, normalized size of antiderivative = 8.77 \[ \int \frac {x^{3/2}}{\left (a+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(66*c**(3/8)*a**(5/8)*sqrt(sqrt(2) + 2)*sqrt(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**2 + 132*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)*s 
qrt(sqrt(2) + 2)))*a*c*x**4 + 66*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)))*c**2*x**8 - 66*c**(3/8)*a**(5/8)*sqrt(s 
qrt(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 - 132*c**(3/8)*a**(5/8)*s 
qrt(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 - 66*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)))*c**2*x**8 - 66*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)))* 
a**2 - 132*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(sq 
rt(2) + 2)))*a*c*x**4 - 66*c**(3/8)*a**(5/8)*sqrt(sqrt(2) + 2)*sqrt(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)))*c**2*x**8 + 66*c**(3/8)*a**(5/8)*sqrt(sqrt...