Integrand size = 81, antiderivative size = 257 \[ \int \frac {1+(3-2 k) x-(4+k) x^2+3 k x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(1+5 b) x-(10 b+k) x^2+10 b x^3-5 b x^4+b x^5\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x+(-1-k) x^2+k x^3}}{2 \sqrt [3]{b}-4 \sqrt [3]{b} x+2 \sqrt [3]{b} x^2+\sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{\sqrt [3]{b}}+\frac {\log \left (-\sqrt [3]{b}+2 \sqrt [3]{b} x-\sqrt [3]{b} x^2+\sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{\sqrt [3]{b}}-\frac {\log \left (b^{2/3}-4 b^{2/3} x+6 b^{2/3} x^2-4 b^{2/3} x^3+b^{2/3} x^4+\left (\sqrt [3]{b}-2 \sqrt [3]{b} x+\sqrt [3]{b} x^2\right ) \sqrt [3]{x+(-1-k) x^2+k x^3}+\left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{2 \sqrt [3]{b}} \] Output:
3^(1/2)*arctan(3^(1/2)*(x+(-1-k)*x^2+k*x^3)^(1/3)/(2*b^(1/3)-4*b^(1/3)*x+2 *b^(1/3)*x^2+(x+(-1-k)*x^2+k*x^3)^(1/3)))/b^(1/3)+ln(-b^(1/3)+2*b^(1/3)*x- b^(1/3)*x^2+(x+(-1-k)*x^2+k*x^3)^(1/3))/b^(1/3)-1/2*ln(b^(2/3)-4*b^(2/3)*x +6*b^(2/3)*x^2-4*b^(2/3)*x^3+b^(2/3)*x^4+(b^(1/3)-2*b^(1/3)*x+b^(1/3)*x^2) *(x+(-1-k)*x^2+k*x^3)^(1/3)+(x+(-1-k)*x^2+k*x^3)^(2/3))/b^(1/3)
\[ \int \frac {1+(3-2 k) x-(4+k) x^2+3 k x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(1+5 b) x-(10 b+k) x^2+10 b x^3-5 b x^4+b x^5\right )} \, dx=\int \frac {1+(3-2 k) x-(4+k) x^2+3 k x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(1+5 b) x-(10 b+k) x^2+10 b x^3-5 b x^4+b x^5\right )} \, dx \] Input:
Integrate[(1 + (3 - 2*k)*x - (4 + k)*x^2 + 3*k*x^3)/(((1 - x)*x*(1 - k*x)) ^(1/3)*(-b + (1 + 5*b)*x - (10*b + k)*x^2 + 10*b*x^3 - 5*b*x^4 + b*x^5)),x ]
Output:
Integrate[(1 + (3 - 2*k)*x - (4 + k)*x^2 + 3*k*x^3)/(((1 - x)*x*(1 - k*x)) ^(1/3)*(-b + (1 + 5*b)*x - (10*b + k)*x^2 + 10*b*x^3 - 5*b*x^4 + b*x^5)), 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.
| \(\displaystyle \int \frac {3 k x^3-(k+4) x^2+(3-2 k) x+1}{\sqrt [3]{(1-x) x (1-k x)} \left (-x^2 (10 b+k)+b x^5-5 b x^4+10 b x^3+(5 b+1) x-b\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int -\frac {3 k x^3-(k+4) x^2+(3-2 k) x+1}{\sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \left (-b x^5+5 b x^4-10 b x^3+(10 b+k) x^2-(5 b+1) x+b\right )}dx}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {3 k x^3-(k+4) x^2+(3-2 k) x+1}{\sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \left (-b x^5+5 b x^4-10 b x^3+(10 b+k) x^2-(5 b+1) x+b\right )}dx}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {\sqrt [3]{x} \left (3 k x^3-(k+4) x^2+(3-2 k) x+1\right )}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b x^5+5 b x^4-10 b x^3+(10 b+k) x^2-(5 b+1) x+b\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {(1-x) \sqrt [3]{x} \left (-3 k x^2+2 (2-k) x+1\right )}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b x^5+5 b x^4-10 b x^3+(10 b+k) x^2-(5 b+1) x+b\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 1395 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \frac {(1-x)^{2/3} \sqrt [3]{x} \left (-3 k x^2+2 (2-k) x+1\right )}{\sqrt [3]{1-k x} \left (-b x^5+5 b x^4-10 b x^3+(10 b+k) x^2-(5 b+1) x+b\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \left (\frac {3 k (1-x)^{2/3} x^{7/3}}{\sqrt [3]{1-k x} \left (b x^5-5 b x^4+10 b x^3-10 b \left (\frac {k}{10 b}+1\right ) x^2+(5 b+1) x-b\right )}+\frac {2 (2-k) (1-x)^{2/3} x^{4/3}}{\sqrt [3]{1-k x} \left (-b x^5+5 b x^4-10 b x^3+10 b \left (\frac {k}{10 b}+1\right ) x^2-(5 b+1) x+b\right )}+\frac {(1-x)^{2/3} \sqrt [3]{x}}{\sqrt [3]{1-k x} \left (-b x^5+5 b x^4-10 b x^3+10 b \left (\frac {k}{10 b}+1\right ) x^2-(5 b+1) x+b\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \left (\int \frac {(1-x)^{2/3} \sqrt [3]{x}}{\sqrt [3]{1-k x} \left (-b (x-1)^5+k x^2-x\right )}d\sqrt [3]{x}+3 k \int \frac {(1-x)^{2/3} x^{7/3}}{\sqrt [3]{1-k x} \left (b (x-1)^5-k x^2+x\right )}d\sqrt [3]{x}+2 (2-k) \int \frac {(1-x)^{2/3} x^{4/3}}{\sqrt [3]{1-k x} \left (-b (x-1)^5+k x^2-x\right )}d\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}}\) |
Input:
Int[(1 + (3 - 2*k)*x - (4 + k)*x^2 + 3*k*x^3)/(((1 - x)*x*(1 - k*x))^(1/3) *(-b + (1 + 5*b)*x - (10*b + k)*x^2 + 10*b*x^3 - 5*b*x^4 + b*x^5)),x]
Output:
$Aborted
\[\int \frac {1+\left (3-2 k \right ) x -\left (4+k \right ) x^{2}+3 k \,x^{3}}{\left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {1}{3}} \left (-b +\left (1+5 b \right ) x -\left (10 b +k \right ) x^{2}+10 b \,x^{3}-5 b \,x^{4}+b \,x^{5}\right )}d x\]
Input:
int((1+(3-2*k)*x-(4+k)*x^2+3*k*x^3)/((1-x)*x*(-k*x+1))^(1/3)/(-b+(1+5*b)*x -(10*b+k)*x^2+10*b*x^3-5*b*x^4+b*x^5),x)
Output:
int((1+(3-2*k)*x-(4+k)*x^2+3*k*x^3)/((1-x)*x*(-k*x+1))^(1/3)/(-b+(1+5*b)*x -(10*b+k)*x^2+10*b*x^3-5*b*x^4+b*x^5),x)
Timed out. \[ \int \frac {1+(3-2 k) x-(4+k) x^2+3 k x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(1+5 b) x-(10 b+k) x^2+10 b x^3-5 b x^4+b x^5\right )} \, dx=\text {Timed out} \] Input:
integrate((1+(3-2*k)*x-(4+k)*x^2+3*k*x^3)/((1-x)*x*(-k*x+1))^(1/3)/(-b+(1+ 5*b)*x-(10*b+k)*x^2+10*b*x^3-5*b*x^4+b*x^5),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1+(3-2 k) x-(4+k) x^2+3 k x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(1+5 b) x-(10 b+k) x^2+10 b x^3-5 b x^4+b x^5\right )} \, dx=\int \frac {\left (x - 1\right ) \left (3 k x^{2} + 2 k x - 4 x - 1\right )}{\sqrt [3]{x \left (x - 1\right ) \left (k x - 1\right )} \left (b x^{5} - 5 b x^{4} + 10 b x^{3} - 10 b x^{2} + 5 b x - b - k x^{2} + x\right )}\, dx \] Input:
integrate((1+(3-2*k)*x-(4+k)*x**2+3*k*x**3)/((1-x)*x*(-k*x+1))**(1/3)/(-b+ (1+5*b)*x-(10*b+k)*x**2+10*b*x**3-5*b*x**4+b*x**5),x)
Output:
Integral((x - 1)*(3*k*x**2 + 2*k*x - 4*x - 1)/((x*(x - 1)*(k*x - 1))**(1/3 )*(b*x**5 - 5*b*x**4 + 10*b*x**3 - 10*b*x**2 + 5*b*x - b - k*x**2 + x)), x )
\[ \int \frac {1+(3-2 k) x-(4+k) x^2+3 k x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(1+5 b) x-(10 b+k) x^2+10 b x^3-5 b x^4+b x^5\right )} \, dx=\int { \frac {3 \, k x^{3} - {\left (k + 4\right )} x^{2} - {\left (2 \, k - 3\right )} x + 1}{{\left (b x^{5} - 5 \, b x^{4} + 10 \, b x^{3} - {\left (10 \, b + k\right )} x^{2} + {\left (5 \, b + 1\right )} x - b\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}}} \,d x } \] Input:
integrate((1+(3-2*k)*x-(4+k)*x^2+3*k*x^3)/((1-x)*x*(-k*x+1))^(1/3)/(-b+(1+ 5*b)*x-(10*b+k)*x^2+10*b*x^3-5*b*x^4+b*x^5),x, algorithm="maxima")
Output:
integrate((3*k*x^3 - (k + 4)*x^2 - (2*k - 3)*x + 1)/((b*x^5 - 5*b*x^4 + 10 *b*x^3 - (10*b + k)*x^2 + (5*b + 1)*x - b)*((k*x - 1)*(x - 1)*x)^(1/3)), x )
\[ \int \frac {1+(3-2 k) x-(4+k) x^2+3 k x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(1+5 b) x-(10 b+k) x^2+10 b x^3-5 b x^4+b x^5\right )} \, dx=\int { \frac {3 \, k x^{3} - {\left (k + 4\right )} x^{2} - {\left (2 \, k - 3\right )} x + 1}{{\left (b x^{5} - 5 \, b x^{4} + 10 \, b x^{3} - {\left (10 \, b + k\right )} x^{2} + {\left (5 \, b + 1\right )} x - b\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}}} \,d x } \] Input:
integrate((1+(3-2*k)*x-(4+k)*x^2+3*k*x^3)/((1-x)*x*(-k*x+1))^(1/3)/(-b+(1+ 5*b)*x-(10*b+k)*x^2+10*b*x^3-5*b*x^4+b*x^5),x, algorithm="giac")
Output:
integrate((3*k*x^3 - (k + 4)*x^2 - (2*k - 3)*x + 1)/((b*x^5 - 5*b*x^4 + 10 *b*x^3 - (10*b + k)*x^2 + (5*b + 1)*x - b)*((k*x - 1)*(x - 1)*x)^(1/3)), x )
Timed out. \[ \int \frac {1+(3-2 k) x-(4+k) x^2+3 k x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(1+5 b) x-(10 b+k) x^2+10 b x^3-5 b x^4+b x^5\right )} \, dx=-\int -\frac {-3\,k\,x^3+\left (k+4\right )\,x^2+\left (2\,k-3\right )\,x-1}{{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{1/3}\,\left (-b\,x^5+5\,b\,x^4-10\,b\,x^3+\left (10\,b+k\right )\,x^2+\left (-5\,b-1\right )\,x+b\right )} \,d x \] Input:
int((x*(2*k - 3) + x^2*(k + 4) - 3*k*x^3 - 1)/((x*(k*x - 1)*(x - 1))^(1/3) *(b - 10*b*x^3 + 5*b*x^4 - b*x^5 + x^2*(10*b + k) - x*(5*b + 1))),x)
Output:
-int(-(x*(2*k - 3) + x^2*(k + 4) - 3*k*x^3 - 1)/((x*(k*x - 1)*(x - 1))^(1/ 3)*(b - 10*b*x^3 + 5*b*x^4 - b*x^5 + x^2*(10*b + k) - x*(5*b + 1))), x)
\[ \int \frac {1+(3-2 k) x-(4+k) x^2+3 k x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(1+5 b) x-(10 b+k) x^2+10 b x^3-5 b x^4+b x^5\right )} \, dx =\text {Too large to display} \] Input:
int((1+(3-2*k)*x-(4+k)*x^2+3*k*x^3)/((1-x)*x*(-k*x+1))^(1/3)/(-b+(1+5*b)*x -(10*b+k)*x^2+10*b*x^3-5*b*x^4+b*x^5),x)
Output:
3*int(x**3/(x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*b*x**5 - 5*x**(1/3)*(k* x**2 - k*x - x + 1)**(1/3)*b*x**4 + 10*x**(1/3)*(k*x**2 - k*x - x + 1)**(1 /3)*b*x**3 - 10*x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*b*x**2 + 5*x**(1/3) *(k*x**2 - k*x - x + 1)**(1/3)*b*x - x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3 )*b - x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*k*x**2 + x**(1/3)*(k*x**2 - k *x - x + 1)**(1/3)*x),x)*k - int(x**2/(x**(1/3)*(k*x**2 - k*x - x + 1)**(1 /3)*b*x**5 - 5*x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*b*x**4 + 10*x**(1/3) *(k*x**2 - k*x - x + 1)**(1/3)*b*x**3 - 10*x**(1/3)*(k*x**2 - k*x - x + 1) **(1/3)*b*x**2 + 5*x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*b*x - x**(1/3)*( k*x**2 - k*x - x + 1)**(1/3)*b - x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*k* x**2 + x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*x),x)*k - 4*int(x**2/(x**(1/ 3)*(k*x**2 - k*x - x + 1)**(1/3)*b*x**5 - 5*x**(1/3)*(k*x**2 - k*x - x + 1 )**(1/3)*b*x**4 + 10*x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*b*x**3 - 10*x* *(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*b*x**2 + 5*x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*b*x - x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*b - x**(1/3)*(k* x**2 - k*x - x + 1)**(1/3)*k*x**2 + x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3) *x),x) - 2*int(x/(x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*b*x**5 - 5*x**(1/ 3)*(k*x**2 - k*x - x + 1)**(1/3)*b*x**4 + 10*x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*b*x**3 - 10*x**(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*b*x**2 + 5*x* *(1/3)*(k*x**2 - k*x - x + 1)**(1/3)*b*x - x**(1/3)*(k*x**2 - k*x - x +...