\(\int \frac {x (3+7 x^4)}{\sqrt [3]{1+x^4} (-4+x^3+x^7)} \, dx\) [2463]

Optimal result

Integrand size = 28, antiderivative size = 200 \[ \int \frac {x \left (3+7 x^4\right )}{\sqrt [3]{1+x^4} \left (-4+x^3+x^7\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {3 \sqrt {3} x \sqrt [3]{1+x^4}-6 x^2 \sqrt [3]{1+x^4}}{-6 2^{2/3}+4\ 2^{2/3} \sqrt {3} x-3 x \sqrt [3]{1+x^4}+2 \sqrt {3} x^2 \sqrt [3]{1+x^4}}\right )}{2^{2/3}}+\sqrt [3]{2} \text {arctanh}\left (1-\sqrt [3]{2} x \sqrt [3]{1+x^4}\right )-\frac {\text {arctanh}\left (\frac {2 \sqrt [3]{2}+2^{2/3} x \sqrt [3]{1+x^4}}{2 \sqrt [3]{2}+2^{2/3} x \sqrt [3]{1+x^4}+2 x^2 \left (1+x^4\right )^{2/3}}\right )}{2^{2/3}} \] Output:

-1/2*3^(1/2)*arctan((3*3^(1/2)*x*(x^4+1)^(1/3)-6*x^2*(x^4+1)^(1/3))/(-6*2^ 
(2/3)+4*2^(2/3)*x*3^(1/2)-3*x*(x^4+1)^(1/3)+2*3^(1/2)*x^2*(x^4+1)^(1/3)))* 
2^(1/3)-2^(1/3)*arctanh(-1+2^(1/3)*x*(x^4+1)^(1/3))-1/2*arctanh((2*2^(1/3) 
+2^(2/3)*x*(x^4+1)^(1/3))/(2*2^(1/3)+2^(2/3)*x*(x^4+1)^(1/3)+2*x^2*(x^4+1) 
^(2/3)))*2^(1/3)
                                                                                    
                                                                                    
 

Mathematica [F]

\[ \int \frac {x \left (3+7 x^4\right )}{\sqrt [3]{1+x^4} \left (-4+x^3+x^7\right )} \, dx=\int \frac {x \left (3+7 x^4\right )}{\sqrt [3]{1+x^4} \left (-4+x^3+x^7\right )} \, dx \] Input:

Integrate[(x*(3 + 7*x^4))/((1 + x^4)^(1/3)*(-4 + x^3 + x^7)),x]
 

Output:

Integrate[(x*(3 + 7*x^4))/((1 + x^4)^(1/3)*(-4 + x^3 + x^7)), x]
 

Rubi [F]

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.

\(\displaystyle \int \frac {x \left (7 x^4+3\right )}{\sqrt [3]{x^4+1} \left (x^7+x^3-4\right )} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {3 x}{\sqrt [3]{x^4+1} \left (x^7+x^3-4\right )}+\frac {7 x^5}{\sqrt [3]{x^4+1} \left (x^7+x^3-4\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 3 \int \frac {x}{\sqrt [3]{x^4+1} \left (x^7+x^3-4\right )}dx+7 \int \frac {x^5}{\sqrt [3]{x^4+1} \left (x^7+x^3-4\right )}dx\)

Input:

Int[(x*(3 + 7*x^4))/((1 + x^4)^(1/3)*(-4 + x^3 + x^7)),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

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

Time = 43.81 (sec) , antiderivative size = 690, normalized size of antiderivative = 3.45

method	result	size
trager	\(\text {Expression too large to display}\)	\(690\)

Input:

int(x*(7*x^4+3)/(x^4+1)^(1/3)/(x^7+x^3-4),x,method=_RETURNVERBOSE)
 

Output:

1/2*RootOf(_Z^3-2)*ln((-RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2 
)*RootOf(_Z^3-2)^3*x^7-RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2) 
^2*RootOf(_Z^3-2)^2*x^7+3*(x^4+1)^(2/3)*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^ 
3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^2-RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf 
(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^3-RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf( 
_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^3-3*(x^4+1)^(1/3)*RootOf(_Z^3-2)^2*x- 
6*(x^4+1)^(1/3)*RootOf(_Z^3-2)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2) 
+4*_Z^2)*x+4*RootOf(_Z^3-2)+4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+ 
4*_Z^2))/(x^7+x^3-4))+RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)* 
ln(-(-RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3 
*x^7-4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2 
)^2*x^7-RootOf(_Z^3-2)*x^7-4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4 
*_Z^2)*x^7+6*(x^4+1)^(2/3)*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+2*_Z*R 
ootOf(_Z^3-2)+4*_Z^2)*x^2-RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z 
^2)*RootOf(_Z^3-2)^3*x^3-4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_ 
Z^2)^2*RootOf(_Z^3-2)^2*x^3+6*x^2*(x^4+1)^(2/3)-6*(x^4+1)^(1/3)*RootOf(_Z^ 
3-2)^2*x-RootOf(_Z^3-2)*x^3-4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+ 
4*_Z^2)*x^3-4*RootOf(_Z^3-2)-16*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2 
)+4*_Z^2))/(x^7+x^3-4))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 344 vs. \(2 (157) = 314\).

Time = 45.51 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.72 \[ \int \frac {x \left (3+7 x^4\right )}{\sqrt [3]{1+x^4} \left (-4+x^3+x^7\right )} \, dx=-\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (6 \cdot 4^{\frac {2}{3}} {\left (x^{16} + 2 \, x^{12} + 4 \, x^{9} + x^{8} + 4 \, x^{5} - 32 \, x^{2}\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{21} + 3 \, x^{17} + 60 \, x^{14} + 3 \, x^{13} + 120 \, x^{10} + x^{9} + 192 \, x^{7} + 60 \, x^{6} + 192 \, x^{3} - 64\right )} + 24 \, {\left (x^{15} + 2 \, x^{11} + 28 \, x^{8} + x^{7} + 28 \, x^{4} + 16 \, x\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (x^{21} + 3 \, x^{17} - 12 \, x^{14} + 3 \, x^{13} - 24 \, x^{10} + x^{9} - 384 \, x^{7} - 12 \, x^{6} - 384 \, x^{3} - 64\right )}}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \log \left (-\frac {12 \, {\left (x^{4} + 1\right )}^{\frac {2}{3}} x^{2} - 4^{\frac {2}{3}} {\left (x^{7} + x^{3} - 4\right )} - 12 \cdot 4^{\frac {1}{3}} {\left (x^{4} + 1\right )}^{\frac {1}{3}} x}{x^{7} + x^{3} - 4}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \log \left (\frac {3 \cdot 4^{\frac {2}{3}} {\left (x^{9} + x^{5} + 8 \, x^{2}\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{14} + 2 \, x^{10} + 28 \, x^{7} + x^{6} + 28 \, x^{3} + 16\right )} + 24 \, {\left (x^{8} + x^{4} + 2 \, x\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{x^{14} + 2 \, x^{10} - 8 \, x^{7} + x^{6} - 8 \, x^{3} + 16}\right ) \] Input:

integrate(x*(7*x^4+3)/(x^4+1)^(1/3)/(x^7+x^3-4),x, algorithm="fricas")
 

Output:

-1/6*4^(1/6)*sqrt(3)*arctan(1/6*4^(1/6)*sqrt(3)*(6*4^(2/3)*(x^16 + 2*x^12 
+ 4*x^9 + x^8 + 4*x^5 - 32*x^2)*(x^4 + 1)^(2/3) + 4^(1/3)*(x^21 + 3*x^17 + 
 60*x^14 + 3*x^13 + 120*x^10 + x^9 + 192*x^7 + 60*x^6 + 192*x^3 - 64) + 24 
*(x^15 + 2*x^11 + 28*x^8 + x^7 + 28*x^4 + 16*x)*(x^4 + 1)^(1/3))/(x^21 + 3 
*x^17 - 12*x^14 + 3*x^13 - 24*x^10 + x^9 - 384*x^7 - 12*x^6 - 384*x^3 - 64 
)) + 1/12*4^(2/3)*log(-(12*(x^4 + 1)^(2/3)*x^2 - 4^(2/3)*(x^7 + x^3 - 4) - 
 12*4^(1/3)*(x^4 + 1)^(1/3)*x)/(x^7 + x^3 - 4)) - 1/24*4^(2/3)*log((3*4^(2 
/3)*(x^9 + x^5 + 8*x^2)*(x^4 + 1)^(2/3) + 4^(1/3)*(x^14 + 2*x^10 + 28*x^7 
+ x^6 + 28*x^3 + 16) + 24*(x^8 + x^4 + 2*x)*(x^4 + 1)^(1/3))/(x^14 + 2*x^1 
0 - 8*x^7 + x^6 - 8*x^3 + 16))
 

Sympy [F]

\[ \int \frac {x \left (3+7 x^4\right )}{\sqrt [3]{1+x^4} \left (-4+x^3+x^7\right )} \, dx=\int \frac {x \left (7 x^{4} + 3\right )}{\sqrt [3]{x^{4} + 1} \left (x^{7} + x^{3} - 4\right )}\, dx \] Input:

integrate(x*(7*x**4+3)/(x**4+1)**(1/3)/(x**7+x**3-4),x)
 

Output:

Integral(x*(7*x**4 + 3)/((x**4 + 1)**(1/3)*(x**7 + x**3 - 4)), x)
 

Maxima [F]

\[ \int \frac {x \left (3+7 x^4\right )}{\sqrt [3]{1+x^4} \left (-4+x^3+x^7\right )} \, dx=\int { \frac {{\left (7 \, x^{4} + 3\right )} x}{{\left (x^{7} + x^{3} - 4\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate(x*(7*x^4+3)/(x^4+1)^(1/3)/(x^7+x^3-4),x, algorithm="maxima")
 

Output:

integrate((7*x^4 + 3)*x/((x^7 + x^3 - 4)*(x^4 + 1)^(1/3)), x)
 

Giac [F]

\[ \int \frac {x \left (3+7 x^4\right )}{\sqrt [3]{1+x^4} \left (-4+x^3+x^7\right )} \, dx=\int { \frac {{\left (7 \, x^{4} + 3\right )} x}{{\left (x^{7} + x^{3} - 4\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate(x*(7*x^4+3)/(x^4+1)^(1/3)/(x^7+x^3-4),x, algorithm="giac")
 

Output:

integrate((7*x^4 + 3)*x/((x^7 + x^3 - 4)*(x^4 + 1)^(1/3)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x \left (3+7 x^4\right )}{\sqrt [3]{1+x^4} \left (-4+x^3+x^7\right )} \, dx=\int \frac {x\,\left (7\,x^4+3\right )}{{\left (x^4+1\right )}^{1/3}\,\left (x^7+x^3-4\right )} \,d x \] Input:

int((x*(7*x^4 + 3))/((x^4 + 1)^(1/3)*(x^3 + x^7 - 4)),x)
 

Output:

int((x*(7*x^4 + 3))/((x^4 + 1)^(1/3)*(x^3 + x^7 - 4)), x)
 

Reduce [F]

\[ \int \frac {x \left (3+7 x^4\right )}{\sqrt [3]{1+x^4} \left (-4+x^3+x^7\right )} \, dx=7 \left (\int \frac {x^{5}}{\left (x^{4}+1\right )^{\frac {1}{3}} x^{7}+\left (x^{4}+1\right )^{\frac {1}{3}} x^{3}-4 \left (x^{4}+1\right )^{\frac {1}{3}}}d x \right )+3 \left (\int \frac {x}{\left (x^{4}+1\right )^{\frac {1}{3}} x^{7}+\left (x^{4}+1\right )^{\frac {1}{3}} x^{3}-4 \left (x^{4}+1\right )^{\frac {1}{3}}}d x \right ) \] Input:

int(x*(7*x^4+3)/(x^4+1)^(1/3)/(x^7+x^3-4),x)
 

Output:

7*int(x**5/((x**4 + 1)**(1/3)*x**7 + (x**4 + 1)**(1/3)*x**3 - 4*(x**4 + 1) 
**(1/3)),x) + 3*int(x/((x**4 + 1)**(1/3)*x**7 + (x**4 + 1)**(1/3)*x**3 - 4 
*(x**4 + 1)**(1/3)),x)