Integrand size = 100, antiderivative size = 279 \[ \int \frac {1+(-2+3 k) x-\left (k+4 k^2\right ) x^2+3 k^2 x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(1+5 b k) x-\left (1+10 b k^2\right ) x^2+10 b k^3 x^3-5 b k^4 x^4+b k^5 x^5\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x+(-1-k) x^2+k x^3}}{2 \sqrt [3]{b}-4 \sqrt [3]{b} k x+2 \sqrt [3]{b} k^2 x^2+\sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{\sqrt [3]{b}}+\frac {\log \left (-\sqrt [3]{b}+2 \sqrt [3]{b} k x-\sqrt [3]{b} k^2 x^2+\sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{\sqrt [3]{b}}-\frac {\log \left (b^{2/3}-4 b^{2/3} k x+6 b^{2/3} k^2 x^2-4 b^{2/3} k^3 x^3+b^{2/3} k^4 x^4+\left (\sqrt [3]{b}-2 \sqrt [3]{b} k x+\sqrt [3]{b} k^2 x^2\right ) \sqrt [3]{x+(-1-k) x^2+k x^3}+\left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{2 \sqrt [3]{b}} \]
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)*k*x +2*b^(1/3)*k^2*x^2+(x+(-1-k)*x^2+k*x^3)^(1/3)))/b^(1/3)+ln(-b^(1/3)+2*b^(1 /3)*k*x-b^(1/3)*k^2*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)*k*x+6*b^(2/3)*k^2*x^2-4*b^(2/3)*k^3*x^3+b^(2/3)*k^4*x^4+(b^(1/3 )-2*b^(1/3)*k*x+b^(1/3)*k^2*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)
\[ \int \frac {1+(-2+3 k) x-\left (k+4 k^2\right ) x^2+3 k^2 x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(1+5 b k) x-\left (1+10 b k^2\right ) x^2+10 b k^3 x^3-5 b k^4 x^4+b k^5 x^5\right )} \, dx=\int \frac {1+(-2+3 k) x-\left (k+4 k^2\right ) x^2+3 k^2 x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(1+5 b k) x-\left (1+10 b k^2\right ) x^2+10 b k^3 x^3-5 b k^4 x^4+b k^5 x^5\right )} \, dx \]
Integrate[(1 + (-2 + 3*k)*x - (k + 4*k^2)*x^2 + 3*k^2*x^3)/(((1 - x)*x*(1 - k*x))^(1/3)*(-b + (1 + 5*b*k)*x - (1 + 10*b*k^2)*x^2 + 10*b*k^3*x^3 - 5* b*k^4*x^4 + b*k^5*x^5)),x]
Integrate[(1 + (-2 + 3*k)*x - (k + 4*k^2)*x^2 + 3*k^2*x^3)/(((1 - x)*x*(1 - k*x))^(1/3)*(-b + (1 + 5*b*k)*x - (1 + 10*b*k^2)*x^2 + 10*b*k^3*x^3 - 5* b*k^4*x^4 + b*k^5*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.
| \(\displaystyle \int \frac {3 k^2 x^3-\left (4 k^2+k\right ) x^2+(3 k-2) x+1}{\sqrt [3]{(1-x) x (1-k x)} \left (b k^5 x^5-5 b k^4 x^4+10 b k^3 x^3-x^2 \left (10 b k^2+1\right )+x (5 b k+1)-b\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int -\frac {3 k^2 x^3-k (4 k+1) x^2-(2-3 k) x+1}{\sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \left (-b k^5 x^5+5 b k^4 x^4-10 b k^3 x^3+\left (10 b k^2+1\right ) x^2-(5 b k+1) x+b\right )}dx}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {3 k^2 x^3-k (4 k+1) x^2-(2-3 k) x+1}{\sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \left (-b k^5 x^5+5 b k^4 x^4-10 b k^3 x^3+\left (10 b k^2+1\right ) x^2-(5 b k+1) x+b\right )}dx}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {\sqrt [3]{x} \left (3 k^2 x^3-k (4 k+1) x^2-(2-3 k) x+1\right )}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b k^5 x^5+5 b k^4 x^4-10 b k^3 x^3+\left (10 b k^2+1\right ) x^2-(5 b k+1) x+b\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \left (\frac {3 k^2 x^{10/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b k^5 x^5+5 b k^4 x^4-10 b k^3 x^3+\left (10 b k^2+1\right ) x^2-(5 b k+1) x+b\right )}+\frac {(-4 k-1) k x^{7/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b k^5 x^5+5 b k^4 x^4-10 b k^3 x^3+\left (10 b k^2+1\right ) x^2-(5 b k+1) x+b\right )}+\frac {(3 k-2) x^{4/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b k^5 x^5+5 b k^4 x^4-10 b k^3 x^3+\left (10 b k^2+1\right ) x^2-(5 b k+1) x+b\right )}+\frac {\sqrt [3]{x}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b k^5 x^5+5 b k^4 x^4-10 b k^3 x^3+\left (10 b k^2+1\right ) x^2-(5 b k+1) x+b\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \left (3 k^2 \int \frac {x^{10/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b (k x-1)^5+x^2-x\right )}d\sqrt [3]{x}+\int \frac {\sqrt [3]{x}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b (k x-1)^5+x^2-x\right )}d\sqrt [3]{x}-(4 k+1) k \int \frac {x^{7/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b (k x-1)^5+x^2-x\right )}d\sqrt [3]{x}-(2-3 k) \int \frac {x^{4/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b (k x-1)^5+x^2-x\right )}d\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}}\) |
Int[(1 + (-2 + 3*k)*x - (k + 4*k^2)*x^2 + 3*k^2*x^3)/(((1 - x)*x*(1 - k*x) )^(1/3)*(-b + (1 + 5*b*k)*x - (1 + 10*b*k^2)*x^2 + 10*b*k^3*x^3 - 5*b*k^4* x^4 + b*k^5*x^5)),x]
3.29.21.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
\[\int \frac {1+\left (-2+3 k \right ) x -\left (4 k^{2}+k \right ) x^{2}+3 k^{2} x^{3}}{\left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {1}{3}} \left (-b +\left (5 b k +1\right ) x -\left (10 b \,k^{2}+1\right ) x^{2}+10 b \,k^{3} x^{3}-5 b \,k^{4} x^{4}+b \,k^{5} x^{5}\right )}d x\]
int((1+(-2+3*k)*x-(4*k^2+k)*x^2+3*k^2*x^3)/((1-x)*x*(-k*x+1))^(1/3)/(-b+(5 *b*k+1)*x-(10*b*k^2+1)*x^2+10*b*k^3*x^3-5*b*k^4*x^4+b*k^5*x^5),x)
int((1+(-2+3*k)*x-(4*k^2+k)*x^2+3*k^2*x^3)/((1-x)*x*(-k*x+1))^(1/3)/(-b+(5 *b*k+1)*x-(10*b*k^2+1)*x^2+10*b*k^3*x^3-5*b*k^4*x^4+b*k^5*x^5),x)
Timed out. \[ \int \frac {1+(-2+3 k) x-\left (k+4 k^2\right ) x^2+3 k^2 x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(1+5 b k) x-\left (1+10 b k^2\right ) x^2+10 b k^3 x^3-5 b k^4 x^4+b k^5 x^5\right )} \, dx=\text {Timed out} \]
integrate((1+(-2+3*k)*x-(4*k^2+k)*x^2+3*k^2*x^3)/((1-x)*x*(-k*x+1))^(1/3)/ (-b+(5*b*k+1)*x-(10*b*k^2+1)*x^2+10*b*k^3*x^3-5*b*k^4*x^4+b*k^5*x^5),x, al gorithm="fricas")
\[ \int \frac {1+(-2+3 k) x-\left (k+4 k^2\right ) x^2+3 k^2 x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(1+5 b k) x-\left (1+10 b k^2\right ) x^2+10 b k^3 x^3-5 b k^4 x^4+b k^5 x^5\right )} \, dx=\int \frac {\left (k x - 1\right ) \left (3 k x^{2} - 4 k x + 2 x - 1\right )}{\sqrt [3]{x \left (x - 1\right ) \left (k x - 1\right )} \left (b k^{5} x^{5} - 5 b k^{4} x^{4} + 10 b k^{3} x^{3} - 10 b k^{2} x^{2} + 5 b k x - b - x^{2} + x\right )}\, dx \]
integrate((1+(-2+3*k)*x-(4*k**2+k)*x**2+3*k**2*x**3)/((1-x)*x*(-k*x+1))**( 1/3)/(-b+(5*b*k+1)*x-(10*b*k**2+1)*x**2+10*b*k**3*x**3-5*b*k**4*x**4+b*k** 5*x**5),x)
Integral((k*x - 1)*(3*k*x**2 - 4*k*x + 2*x - 1)/((x*(x - 1)*(k*x - 1))**(1 /3)*(b*k**5*x**5 - 5*b*k**4*x**4 + 10*b*k**3*x**3 - 10*b*k**2*x**2 + 5*b*k *x - b - x**2 + x)), x)
\[ \int \frac {1+(-2+3 k) x-\left (k+4 k^2\right ) x^2+3 k^2 x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(1+5 b k) x-\left (1+10 b k^2\right ) x^2+10 b k^3 x^3-5 b k^4 x^4+b k^5 x^5\right )} \, dx=\int { \frac {3 \, k^{2} x^{3} - {\left (4 \, k^{2} + k\right )} x^{2} + {\left (3 \, k - 2\right )} x + 1}{{\left (b k^{5} x^{5} - 5 \, b k^{4} x^{4} + 10 \, b k^{3} x^{3} - {\left (10 \, b k^{2} + 1\right )} x^{2} + {\left (5 \, b k + 1\right )} x - b\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}}} \,d x } \]
integrate((1+(-2+3*k)*x-(4*k^2+k)*x^2+3*k^2*x^3)/((1-x)*x*(-k*x+1))^(1/3)/ (-b+(5*b*k+1)*x-(10*b*k^2+1)*x^2+10*b*k^3*x^3-5*b*k^4*x^4+b*k^5*x^5),x, al gorithm="maxima")
integrate((3*k^2*x^3 - (4*k^2 + k)*x^2 + (3*k - 2)*x + 1)/((b*k^5*x^5 - 5* b*k^4*x^4 + 10*b*k^3*x^3 - (10*b*k^2 + 1)*x^2 + (5*b*k + 1)*x - b)*((k*x - 1)*(x - 1)*x)^(1/3)), x)
Timed out. \[ \int \frac {1+(-2+3 k) x-\left (k+4 k^2\right ) x^2+3 k^2 x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(1+5 b k) x-\left (1+10 b k^2\right ) x^2+10 b k^3 x^3-5 b k^4 x^4+b k^5 x^5\right )} \, dx=\text {Timed out} \]
integrate((1+(-2+3*k)*x-(4*k^2+k)*x^2+3*k^2*x^3)/((1-x)*x*(-k*x+1))^(1/3)/ (-b+(5*b*k+1)*x-(10*b*k^2+1)*x^2+10*b*k^3*x^3-5*b*k^4*x^4+b*k^5*x^5),x, al gorithm="giac")
Timed out. \[ \int \frac {1+(-2+3 k) x-\left (k+4 k^2\right ) x^2+3 k^2 x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(1+5 b k) x-\left (1+10 b k^2\right ) x^2+10 b k^3 x^3-5 b k^4 x^4+b k^5 x^5\right )} \, dx=-\int \frac {x\,\left (3\,k-2\right )+3\,k^2\,x^3-x^2\,\left (4\,k^2+k\right )+1}{{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{1/3}\,\left (b+x^2\,\left (10\,b\,k^2+1\right )-x\,\left (5\,b\,k+1\right )-10\,b\,k^3\,x^3+5\,b\,k^4\,x^4-b\,k^5\,x^5\right )} \,d x \]
int(-(x*(3*k - 2) + 3*k^2*x^3 - x^2*(k + 4*k^2) + 1)/((x*(k*x - 1)*(x - 1) )^(1/3)*(b + x^2*(10*b*k^2 + 1) - x*(5*b*k + 1) - 10*b*k^3*x^3 + 5*b*k^4*x ^4 - b*k^5*x^5)),x)