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]

1/4*ln(1-x)/q^(1/3)+1/2*ln(x)/q^(1/3)-3/4*ln(-q^(1/3)*(-1+x)+((-1+x)*(-2*q*x+x^2+q))^(1/3))/q^(1/3)+1/2*arctan
(1/3*3^(1/2)+2/3*q^(1/3)*(-1+x)/((-1+x)*(-2*q*x+x^2+q))^(1/3)*3^(1/2))*3^(1/2)/q^(1/3)

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Rubi [F]  time = 21.94, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [C]  time = 0.20, 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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fricas [B]  time = 52.74, size = 1496, normalized size = 12.68 \[ \text {result too large to display} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-{\left (2 \, q x - x^{2} - q\right )} {\left (x - 1\right )}\right )^{\frac {1}{3}} x}\,{d x} \]

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)

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\left (x -1\right ) \left (-2 q x +x^{2}+q \right )\right )^{\frac {1}{3}} x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/((x-1)*(-2*q*x+x^2+q))^(1/3),x)

[Out]

int(1/x/((x-1)*(-2*q*x+x^2+q))^(1/3),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-{\left (2 \, q x - x^{2} - q\right )} {\left (x - 1\right )}\right )^{\frac {1}{3}} x}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x\,{\left (\left (x-1\right )\,\left (x^2-2\,q\,x+q\right )\right )}^{1/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*((x - 1)*(q - 2*q*x + x^2))^(1/3)),x)

[Out]

int(1/(x*((x - 1)*(q - 2*q*x + x^2))^(1/3)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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