3.43 \(\int \frac{1}{x \sqrt [3]{(-1+x) (q-2 q x+x^2)}} \, dx\)
Optimal. Leaf size=118 \[ -\frac{3 \log \left (\sqrt [3]{(x-1) \left (-2 q x+q+x^2\right )}-\sqrt [3]{q} (x-1)\right )}{4 \sqrt [3]{q}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{q} (x-1)}{\sqrt{3} \sqrt [3]{(x-1) \left (-2 q x+q+x^2\right )}}+\frac{1}{\sqrt{3}}\right )}{2 \sqrt [3]{q}}+\frac{\log (1-x)}{4 \sqrt [3]{q}}+\frac{\log (x)}{2 \sqrt [3]{q}} \]
[Out]
(Sqrt[3]*ArcTan[1/Sqrt[3] + (2*q^(1/3)*(-1 + x))/(Sqrt[3]*((-1 + x)*(q - 2*q*x + x^2))^(1/3))])/(2*q^(1/3)) +
Log[1 - x]/(4*q^(1/3)) + Log[x]/(2*q^(1/3)) - (3*Log[-(q^(1/3)*(-1 + x)) + ((-1 + x)*(q - 2*q*x + x^2))^(1/3)]
)/(4*q^(1/3))
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Rubi [F] time = 21.9369, antiderivative size = 0, normalized size of antiderivative = 0.,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {}
\[ \int \frac{1}{x \sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}} \, dx \]
Verification is Not applicable to the result.
[In]
Int[1/(x*((-1 + x)*(q - 2*q*x + x^2))^(1/3)),x]
[Out]
((-1 - 2*q - (1 - 5*q + 4*q^2 + (1 + 6*q - 15*q^2 + 8*q^3 + 3*Sqrt[3]*Sqrt[-((-1 + q)^3*q)])^(2/3))/(1 + 6*q -
15*q^2 + 8*q^3 + 3*Sqrt[3]*Sqrt[-((-1 + q)^3*q)])^(1/3) + 3*x)^(1/3)*(-1 + 5*q - 4*q^2 + ((1 - 4*q)^2*(1 - q)
^2)/(1 + 6*q - 15*q^2 + 8*q^3 + 3*Sqrt[3]*Sqrt[(1 - q)^3*q])^(2/3) + (1 + 6*q - 15*q^2 + 8*q^3 + 3*Sqrt[3]*Sqr
t[(1 - q)^3*q])^(2/3) + (3*(1 - 5*q + 4*q^2 + (1 + 6*q - 15*q^2 + 8*q^3 + 3*Sqrt[3]*Sqrt[(1 - q)^3*q])^(2/3))*
((-1 - 2*q)/3 + x))/(1 + 6*q - 15*q^2 + 8*q^3 + 3*Sqrt[3]*Sqrt[(1 - q)^3*q])^(1/3) + 9*((-1 - 2*q)/3 + x)^2)^(
1/3)*Defer[Subst][Defer[Int][1/(((1 + 2*q)/3 + x)*(-(1 - 5*q + 4*q^2 + (1 + 6*q - 15*q^2 + 8*q^3 + 3*Sqrt[3]*S
qrt[(1 - q)^3*q])^(2/3))/(3*(1 + 6*q - 15*q^2 + 8*q^3 + 3*Sqrt[3]*Sqrt[(1 - q)^3*q])^(1/3)) + x)^(1/3)*((-1 +
5*q - 4*q^2 + ((1 - 4*q)^2*(1 - q)^2)/(1 + 6*q - 15*q^2 + 8*q^3 + 3*Sqrt[3]*Sqrt[(1 - q)^3*q])^(2/3) + (1 + 6*
q - 15*q^2 + 8*q^3 + 3*Sqrt[3]*Sqrt[(1 - q)^3*q])^(2/3))/9 + ((1 - 5*q + 4*q^2 + (1 + 6*q - 15*q^2 + 8*q^3 + 3
*Sqrt[3]*Sqrt[(1 - q)^3*q])^(2/3))*x)/(3*(1 + 6*q - 15*q^2 + 8*q^3 + 3*Sqrt[3]*Sqrt[(1 - q)^3*q])^(1/3)) + x^2
)^(1/3)), x], x, (-1 - 2*q)/3 + x])/(3*(-q + 3*q*x + (-1 - 2*q)*x^2 + x^3)^(1/3))
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\left (\frac{1}{3} (1+2 q)+x\right ) \sqrt [3]{-\frac{2}{27} (1-q)^2 (1+8 q)-\frac{1}{3} (1-4 q) (1-q) x+x^3}} \, dx,x,\frac{1}{3} (-1-2 q)+x\right )\\ &=\frac{\left (\sqrt [3]{-1-2 q-\frac{1-5 q+4 q^2+\left (1+6 q-15 q^2+8 q^3+3 \sqrt{3} \sqrt{-(-1+q)^3 q}\right )^{2/3}}{\sqrt [3]{1+6 q-15 q^2+8 q^3+3 \sqrt{3} \sqrt{-(-1+q)^3 q}}}+3 x} \sqrt [3]{-1+5 q-4 q^2+\frac{(1-4 q)^2 (1-q)^2}{\left (1+6 q-15 q^2+8 q^3+3 \sqrt{3} \sqrt{(1-q)^3 q}\right )^{2/3}}+\left (1+6 q-15 q^2+8 q^3+3 \sqrt{3} \sqrt{(1-q)^3 q}\right )^{2/3}+9 \left (\frac{1}{3} (-1-2 q)+x\right )^2+\frac{\left (1-5 q+4 q^2+\left (1+6 q-15 q^2+8 q^3+3 \sqrt{3} \sqrt{(1-q)^3 q}\right )^{2/3}\right ) (-1-2 q+3 x)}{\sqrt [3]{1+6 q-15 q^2+8 q^3+3 \sqrt{3} \sqrt{(1-q)^3 q}}}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\frac{1}{3} (1+2 q)+x\right ) \sqrt [3]{-\frac{1-5 q+4 q^2+\left (1+6 q-15 q^2+8 q^3+3 \sqrt{3} \sqrt{(1-q)^3 q}\right )^{2/3}}{3 \sqrt [3]{1+6 q-15 q^2+8 q^3+3 \sqrt{3} \sqrt{(1-q)^3 q}}}+x} \sqrt [3]{\frac{1}{9} \left (-1+5 q-4 q^2+\frac{(1-4 q)^2 (1-q)^2}{\left (1+6 q-15 q^2+8 q^3+3 \sqrt{3} \sqrt{(1-q)^3 q}\right )^{2/3}}+\left (1+6 q-15 q^2+8 q^3+3 \sqrt{3} \sqrt{(1-q)^3 q}\right )^{2/3}\right )+\frac{\left (1-5 q+4 q^2+\left (1+6 q-15 q^2+8 q^3+3 \sqrt{3} \sqrt{(1-q)^3 q}\right )^{2/3}\right ) x}{3 \sqrt [3]{1+6 q-15 q^2+8 q^3+3 \sqrt{3} \sqrt{(1-q)^3 q}}}+x^2}} \, dx,x,\frac{1}{3} (-1-2 q)+x\right )}{3 \sqrt [3]{-q+3 q x-(1+2 q) x^2+x^3}}\\ \end{align*}
Mathematica [C] time = 0.19341, size = 55, normalized size = 0.47 \[ \frac{3 \left ((x-1) \left (-2 q x+q+x^2\right )\right )^{2/3} \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{x^2-2 q x+q}{q (x-1)^2}\right )}{4 q (x-1)^2} \]
Antiderivative was successfully verified.
[In]
Integrate[1/(x*((-1 + x)*(q - 2*q*x + x^2))^(1/3)),x]
[Out]
(3*((-1 + x)*(q - 2*q*x + x^2))^(2/3)*Hypergeometric2F1[2/3, 1, 5/3, (q - 2*q*x + x^2)/(q*(-1 + x)^2)])/(4*q*(
-1 + x)^2)
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Maple [F] time = 0.014, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}{\frac{1}{\sqrt [3]{ \left ( -1+x \right ) \left ( -2\,qx+{x}^{2}+q \right ) }}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x/((-1+x)*(-2*q*x+x^2+q))^(1/3),x)
[Out]
int(1/x/((-1+x)*(-2*q*x+x^2+q))^(1/3),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (-{\left (2 \, q x - x^{2} - q\right )}{\left (x - 1\right )}\right )^{\frac{1}{3}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/((-1+x)*(-2*q*x+x^2+q))^(1/3),x, algorithm="maxima")
[Out]
integrate(1/((-(2*q*x - x^2 - q)*(x - 1))^(1/3)*x), x)
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Fricas [B] time = 167.475, size = 3509, normalized size = 29.74 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/((-1+x)*(-2*q*x+x^2+q))^(1/3),x, algorithm="fricas")
[Out]
[1/12*(sqrt(3)*q*sqrt((-q)^(1/3)/q)*log(-((q^3 - 30*q^2 - 51*q - 1)*x^6 + 54*(q^3 + 6*q^2 + 2*q)*x^5 - 27*(17*
q^3 + 26*q^2 + 2*q)*x^4 + 486*q^3*x + 540*(2*q^3 + q^2)*x^3 - 81*q^3 - 135*(8*q^3 + q^2)*x^2 + 9*((2*q^2 - q -
1)*x^4 - 6*(q^2 - q)*x^3 + 3*(q^2 - q)*x^2)*(-(2*q + 1)*x^2 + x^3 + 3*q*x - q)^(2/3)*(-q)^(1/3) + 9*((q^2 + 7
*q + 1)*x^5 - (19*q^2 + 25*q + 1)*x^4 + 9*(7*q^2 + 3*q)*x^3 + 45*q^2*x - 9*(9*q^2 + q)*x^2 - 9*q^2)*(-(2*q + 1
)*x^2 + x^3 + 3*q*x - q)^(1/3)*(-q)^(2/3) + sqrt(3)*(3*((4*q^2 + 13*q + 1)*x^4 - 6*(7*q^2 + 5*q)*x^3 - 72*q^2*
x + 3*(31*q^2 + 5*q)*x^2 + 18*q^2)*(-(2*q + 1)*x^2 + x^3 + 3*q*x - q)^(2/3)*(-q)^(2/3) + 3*((q^3 - 5*q^2 - 5*q
)*x^5 + 5*(q^3 + 7*q^2 + q)*x^4 - 45*q^3*x - 45*(q^3 + q^2)*x^3 + 9*q^3 + 15*(5*q^3 + q^2)*x^2)*(-(2*q + 1)*x^
2 + x^3 + 3*q*x - q)^(1/3) + ((q^3 + 24*q^2 + 3*q - 1)*x^6 - 54*(q^3 + 2*q^2)*x^5 + 81*(3*q^3 + 2*q^2)*x^4 - 1
62*q^3*x - 108*(4*q^3 + q^2)*x^3 + 27*q^3 + 27*(14*q^3 + q^2)*x^2)*(-q)^(1/3))*sqrt((-q)^(1/3)/q))/x^6) - 2*(-
q)^(2/3)*log(((-q)^(2/3)*(q - 1)*x^2 + 3*(-(2*q + 1)*x^2 + x^3 + 3*q*x - q)^(1/3)*(q*x - q)*(-q)^(1/3) + 3*(-(
2*q + 1)*x^2 + x^3 + 3*q*x - q)^(2/3)*q)/x^2) + (-q)^(2/3)*log((3*((2*q + 1)*x^2 - 6*q*x + 3*q)*(-(2*q + 1)*x^
2 + x^3 + 3*q*x - q)^(2/3)*(-q)^(2/3) + 3*((q^2 + 2*q)*x^3 + 9*q^2*x - (7*q^2 + 2*q)*x^2 - 3*q^2)*(-(2*q + 1)*
x^2 + x^3 + 3*q*x - q)^(1/3) - ((q^2 + 7*q + 1)*x^4 - 18*(q^2 + q)*x^3 - 36*q^2*x + 9*(5*q^2 + q)*x^2 + 9*q^2)
*(-q)^(1/3))/x^4))/q, 1/12*(2*sqrt(3)*q*sqrt(-(-q)^(1/3)/q)*arctan(1/3*sqrt(3)*(6*((2*q^2 - q - 1)*x^4 - 6*(q^
2 - q)*x^3 + 3*(q^2 - q)*x^2)*(-(2*q + 1)*x^2 + x^3 + 3*q*x - q)^(2/3)*(-q)^(2/3) - 6*((q^3 + 7*q^2 + q)*x^5 -
(19*q^3 + 25*q^2 + q)*x^4 + 45*q^3*x + 9*(7*q^3 + 3*q^2)*x^3 - 9*q^3 - 9*(9*q^3 + q^2)*x^2)*(-(2*q + 1)*x^2 +
x^3 + 3*q*x - q)^(1/3) - ((q^3 - 12*q^2 - 15*q - 1)*x^6 + 18*(q^3 + 6*q^2 + 2*q)*x^5 - 9*(17*q^3 + 26*q^2 + 2
*q)*x^4 + 162*q^3*x + 180*(2*q^3 + q^2)*x^3 - 27*q^3 - 45*(8*q^3 + q^2)*x^2)*(-q)^(1/3))*sqrt(-(-q)^(1/3)/q)/(
(q^3 + 24*q^2 + 3*q - 1)*x^6 - 54*(q^3 + 2*q^2)*x^5 + 81*(3*q^3 + 2*q^2)*x^4 - 162*q^3*x - 108*(4*q^3 + q^2)*x
^3 + 27*q^3 + 27*(14*q^3 + q^2)*x^2)) - 2*(-q)^(2/3)*log(((-q)^(2/3)*(q - 1)*x^2 + 3*(-(2*q + 1)*x^2 + x^3 + 3
*q*x - q)^(1/3)*(q*x - q)*(-q)^(1/3) + 3*(-(2*q + 1)*x^2 + x^3 + 3*q*x - q)^(2/3)*q)/x^2) + (-q)^(2/3)*log((3*
((2*q + 1)*x^2 - 6*q*x + 3*q)*(-(2*q + 1)*x^2 + x^3 + 3*q*x - q)^(2/3)*(-q)^(2/3) + 3*((q^2 + 2*q)*x^3 + 9*q^2
*x - (7*q^2 + 2*q)*x^2 - 3*q^2)*(-(2*q + 1)*x^2 + x^3 + 3*q*x - q)^(1/3) - ((q^2 + 7*q + 1)*x^4 - 18*(q^2 + q)
*x^3 - 36*q^2*x + 9*(5*q^2 + q)*x^2 + 9*q^2)*(-q)^(1/3))/x^4))/q]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/((-1+x)*(-2*q*x+x**2+q))**(1/3),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (-{\left (2 \, q x - x^{2} - q\right )}{\left (x - 1\right )}\right )^{\frac{1}{3}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/((-1+x)*(-2*q*x+x^2+q))^(1/3),x, algorithm="giac")
[Out]
integrate(1/((-(2*q*x - x^2 - q)*(x - 1))^(1/3)*x), x)