Integrand size = 38, antiderivative size = 43 \[ \int \frac {-x+4 x^6}{\left (1+x^5\right ) \left (a-x+a x^5\right ) \sqrt {x+x^6}} \, dx=\frac {2 \sqrt {x+x^6}}{1+x^5}-2 \sqrt {a} \text {arctanh}\left (\frac {x}{\sqrt {a} \sqrt {x+x^6}}\right ) \] Output:
2*(x^6+x)^(1/2)/(x^5+1)-2*a^(1/2)*arctanh(x/a^(1/2)/(x^6+x)^(1/2))
\[ \int \frac {-x+4 x^6}{\left (1+x^5\right ) \left (a-x+a x^5\right ) \sqrt {x+x^6}} \, dx=\int \frac {-x+4 x^6}{\left (1+x^5\right ) \left (a-x+a x^5\right ) \sqrt {x+x^6}} \, dx \] Input:
Integrate[(-x + 4*x^6)/((1 + x^5)*(a - x + a*x^5)*Sqrt[x + x^6]),x]
Output:
Integrate[(-x + 4*x^6)/((1 + x^5)*(a - x + a*x^5)*Sqrt[x + x^6]), x]
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.
| \(\displaystyle \int \frac {4 x^6-x}{\left (x^5+1\right ) \sqrt {x^6+x} \left (a x^5+a-x\right )} \, dx\) |
| \(\Big \downarrow \) 2027 |
| \(\displaystyle \int \frac {x \left (4 x^5-1\right )}{\left (x^5+1\right ) \sqrt {x^6+x} \left (a x^5+a-x\right )}dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {x^5+1} \int -\frac {\sqrt {x} \left (1-4 x^5\right )}{\left (x^5+1\right )^{3/2} \left (a x^5-x+a\right )}dx}{\sqrt {x^6+x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {x^5+1} \int \frac {\sqrt {x} \left (1-4 x^5\right )}{\left (x^5+1\right )^{3/2} \left (a x^5-x+a\right )}dx}{\sqrt {x^6+x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^5+1} \int \frac {x \left (1-4 x^5\right )}{\left (x^5+1\right )^{3/2} \left (a x^5-x+a\right )}d\sqrt {x}}{\sqrt {x^6+x}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^5+1} \int \left (\frac {(5 a-4 x) x}{a \left (x^5+1\right )^{3/2} \left (a x^5-x+a\right )}-\frac {4 x}{a \left (x^5+1\right )^{3/2}}\right )d\sqrt {x}}{\sqrt {x^6+x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^5+1} \left (5 \int \frac {x}{\left (x^5+1\right )^{3/2} \left (a x^5-x+a\right )}d\sqrt {x}-\frac {4 \int \frac {x^2}{\left (x^5+1\right )^{3/2} \left (a x^5-x+a\right )}d\sqrt {x}}{a}-\frac {8 x^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{10},\frac {1}{2},\frac {13}{10},-x^5\right )}{15 a}-\frac {4 x^{3/2}}{5 a \sqrt {x^5+1}}\right )}{\sqrt {x^6+x}}\) |
Input:
Int[(-x + 4*x^6)/((1 + x^5)*(a - x + a*x^5)*Sqrt[x + x^6]),x]
Output:
$Aborted
Time = 0.60 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.91
| method | result | size |
| pseudoelliptic | \(\frac {-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {a}\, \sqrt {x^{6}+x}}{x}\right ) \sqrt {x^{6}+x}+2 x}{\sqrt {x^{6}+x}}\) | \(39\) |
Input:
int((4*x^6-x)/(x^5+1)/(a*x^5+a-x)/(x^6+x)^(1/2),x,method=_RETURNVERBOSE)
Output:
2*(-a^(1/2)*arctanh(a^(1/2)*(x^6+x)^(1/2)/x)*(x^6+x)^(1/2)+x)/(x^6+x)^(1/2 )
Time = 0.19 (sec) , antiderivative size = 181, normalized size of antiderivative = 4.21 \[ \int \frac {-x+4 x^6}{\left (1+x^5\right ) \left (a-x+a x^5\right ) \sqrt {x+x^6}} \, dx=\left [\frac {{\left (x^{5} + 1\right )} \sqrt {a} \log \left (-\frac {a^{2} x^{10} + 2 \, a^{2} x^{5} + 6 \, a x^{6} - 4 \, {\left (a x^{5} + a + x\right )} \sqrt {x^{6} + x} \sqrt {a} + a^{2} + 6 \, a x + x^{2}}{a^{2} x^{10} + 2 \, a^{2} x^{5} - 2 \, a x^{6} + a^{2} - 2 \, a x + x^{2}}\right ) + 4 \, \sqrt {x^{6} + x}}{2 \, {\left (x^{5} + 1\right )}}, \frac {{\left (x^{5} + 1\right )} \sqrt {-a} \arctan \left (\frac {{\left (a x^{5} + a + x\right )} \sqrt {x^{6} + x} \sqrt {-a}}{2 \, {\left (a x^{6} + a x\right )}}\right ) + 2 \, \sqrt {x^{6} + x}}{x^{5} + 1}\right ] \] Input:
integrate((4*x^6-x)/(x^5+1)/(a*x^5+a-x)/(x^6+x)^(1/2),x, algorithm="fricas ")
Output:
[1/2*((x^5 + 1)*sqrt(a)*log(-(a^2*x^10 + 2*a^2*x^5 + 6*a*x^6 - 4*(a*x^5 + a + x)*sqrt(x^6 + x)*sqrt(a) + a^2 + 6*a*x + x^2)/(a^2*x^10 + 2*a^2*x^5 - 2*a*x^6 + a^2 - 2*a*x + x^2)) + 4*sqrt(x^6 + x))/(x^5 + 1), ((x^5 + 1)*sqr t(-a)*arctan(1/2*(a*x^5 + a + x)*sqrt(x^6 + x)*sqrt(-a)/(a*x^6 + a*x)) + 2 *sqrt(x^6 + x))/(x^5 + 1)]
Timed out. \[ \int \frac {-x+4 x^6}{\left (1+x^5\right ) \left (a-x+a x^5\right ) \sqrt {x+x^6}} \, dx=\text {Timed out} \] Input:
integrate((4*x**6-x)/(x**5+1)/(a*x**5+a-x)/(x**6+x)**(1/2),x)
Output:
Timed out
\[ \int \frac {-x+4 x^6}{\left (1+x^5\right ) \left (a-x+a x^5\right ) \sqrt {x+x^6}} \, dx=\int { \frac {4 \, x^{6} - x}{{\left (a x^{5} + a - x\right )} \sqrt {x^{6} + x} {\left (x^{5} + 1\right )}} \,d x } \] Input:
integrate((4*x^6-x)/(x^5+1)/(a*x^5+a-x)/(x^6+x)^(1/2),x, algorithm="maxima ")
Output:
integrate((4*x^6 - x)/((a*x^5 + a - x)*sqrt(x^6 + x)*(x^5 + 1)), x)
Timed out. \[ \int \frac {-x+4 x^6}{\left (1+x^5\right ) \left (a-x+a x^5\right ) \sqrt {x+x^6}} \, dx=\text {Timed out} \] Input:
integrate((4*x^6-x)/(x^5+1)/(a*x^5+a-x)/(x^6+x)^(1/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {-x+4 x^6}{\left (1+x^5\right ) \left (a-x+a x^5\right ) \sqrt {x+x^6}} \, dx=\int -\frac {x-4\,x^6}{\left (x^5+1\right )\,\sqrt {x^6+x}\,\left (a\,x^5-x+a\right )} \,d x \] Input:
int(-(x - 4*x^6)/((x^5 + 1)*(x + x^6)^(1/2)*(a - x + a*x^5)),x)
Output:
int(-(x - 4*x^6)/((x^5 + 1)*(x + x^6)^(1/2)*(a - x + a*x^5)), x)
\[ \int \frac {-x+4 x^6}{\left (1+x^5\right ) \left (a-x+a x^5\right ) \sqrt {x+x^6}} \, dx=4 \left (\int \frac {\sqrt {x}\, \sqrt {x^{5}+1}\, x^{5}}{a \,x^{15}+3 a \,x^{10}-x^{11}+3 a \,x^{5}-2 x^{6}+a -x}d x \right )-\left (\int \frac {\sqrt {x}\, \sqrt {x^{5}+1}}{a \,x^{15}+3 a \,x^{10}-x^{11}+3 a \,x^{5}-2 x^{6}+a -x}d x \right ) \] Input:
int((4*x^6-x)/(x^5+1)/(a*x^5+a-x)/(x^6+x)^(1/2),x)
Output:
4*int((sqrt(x)*sqrt(x**5 + 1)*x**5)/(a*x**15 + 3*a*x**10 + 3*a*x**5 + a - x**11 - 2*x**6 - x),x) - int((sqrt(x)*sqrt(x**5 + 1))/(a*x**15 + 3*a*x**10 + 3*a*x**5 + a - x**11 - 2*x**6 - x),x)