3.28.18 \(\int \frac {x}{\sqrt [3]{x^2 (-a+x)} (-a^2+2 a x+(-1+d) x^2)} \, dx\) [2718]

Optimal. Leaf size=249 \[ -\frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{d} x^2}{\sqrt [3]{d} x^2+2 \left (-a x^2+x^3\right )^{2/3}}\right )}{2 a d^{2/3}}-\frac {\log \left (-\sqrt [6]{d} x+\sqrt [3]{-a x^2+x^3}\right )}{2 a d^{2/3}}-\frac {\log \left (\sqrt [6]{d} x+\sqrt [3]{-a x^2+x^3}\right )}{2 a d^{2/3}}+\frac {\log \left (\sqrt [3]{d} x^2-\sqrt [6]{d} x \sqrt [3]{-a x^2+x^3}+\left (-a x^2+x^3\right )^{2/3}\right )}{4 a d^{2/3}}+\frac {\log \left (\sqrt [3]{d} x^2+\sqrt [6]{d} x \sqrt [3]{-a x^2+x^3}+\left (-a x^2+x^3\right )^{2/3}\right )}{4 a d^{2/3}} \]

[Out]

-1/2*3^(1/2)*arctan(3^(1/2)*d^(1/3)*x^2/(d^(1/3)*x^2+2*(-a*x^2+x^3)^(2/3)))/a/d^(2/3)-1/2*ln(-d^(1/6)*x+(-a*x^
2+x^3)^(1/3))/a/d^(2/3)-1/2*ln(d^(1/6)*x+(-a*x^2+x^3)^(1/3))/a/d^(2/3)+1/4*ln(d^(1/3)*x^2-d^(1/6)*x*(-a*x^2+x^
3)^(1/3)+(-a*x^2+x^3)^(2/3))/a/d^(2/3)+1/4*ln(d^(1/3)*x^2+d^(1/6)*x*(-a*x^2+x^3)^(1/3)+(-a*x^2+x^3)^(2/3))/a/d
^(2/3)

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Rubi [A]
time = 0.54, antiderivative size = 397, normalized size of antiderivative = 1.59, number of steps used = 11, number of rules used = 6, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {6851, 925, 129, 494, 245, 384} \begin {gather*} \frac {\sqrt {3} x^{2/3} \sqrt [3]{x-a} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [3]{x}}{\sqrt [3]{x-a}}}{\sqrt {3}}\right )}{2 a d^{2/3} \sqrt [3]{-\left (x^2 (a-x)\right )}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{x-a} \text {ArcTan}\left (\frac {\frac {2 \sqrt [6]{d} \sqrt [3]{x}}{\sqrt [3]{x-a}}+1}{\sqrt {3}}\right )}{2 a d^{2/3} \sqrt [3]{-\left (x^2 (a-x)\right )}}+\frac {x^{2/3} \sqrt [3]{x-a} \log \left (a \left (1-\sqrt {d}\right )-(1-d) x\right )}{4 a d^{2/3} \sqrt [3]{-\left (x^2 (a-x)\right )}}+\frac {x^{2/3} \sqrt [3]{x-a} \log \left (a \left (\sqrt {d}+1\right )-(1-d) x\right )}{4 a d^{2/3} \sqrt [3]{-\left (x^2 (a-x)\right )}}-\frac {3 x^{2/3} \sqrt [3]{x-a} \log \left (\sqrt [6]{d} \sqrt [3]{x}-\sqrt [3]{x-a}\right )}{4 a d^{2/3} \sqrt [3]{-\left (x^2 (a-x)\right )}}-\frac {3 x^{2/3} \sqrt [3]{x-a} \log \left (\sqrt [3]{x-a}+\sqrt [6]{d} \sqrt [3]{x}\right )}{4 a d^{2/3} \sqrt [3]{-\left (x^2 (a-x)\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/((x^2*(-a + x))^(1/3)*(-a^2 + 2*a*x + (-1 + d)*x^2)),x]

[Out]

(Sqrt[3]*x^(2/3)*(-a + x)^(1/3)*ArcTan[(1 - (2*d^(1/6)*x^(1/3))/(-a + x)^(1/3))/Sqrt[3]])/(2*a*d^(2/3)*(-((a -
x)*x^2))^(1/3)) + (Sqrt[3]*x^(2/3)*(-a + x)^(1/3)*ArcTan[(1 + (2*d^(1/6)*x^(1/3))/(-a + x)^(1/3))/Sqrt[3]])/(
2*a*d^(2/3)*(-((a - x)*x^2))^(1/3)) + (x^(2/3)*(-a + x)^(1/3)*Log[a*(1 - Sqrt[d]) - (1 - d)*x])/(4*a*d^(2/3)*(
-((a - x)*x^2))^(1/3)) + (x^(2/3)*(-a + x)^(1/3)*Log[a*(1 + Sqrt[d]) - (1 - d)*x])/(4*a*d^(2/3)*(-((a - x)*x^2
))^(1/3)) - (3*x^(2/3)*(-a + x)^(1/3)*Log[d^(1/6)*x^(1/3) - (-a + x)^(1/3)])/(4*a*d^(2/3)*(-((a - x)*x^2))^(1/
3)) - (3*x^(2/3)*(-a + x)^(1/3)*Log[d^(1/6)*x^(1/3) + (-a + x)^(1/3)])/(4*a*d^(2/3)*(-((a - x)*x^2))^(1/3))

Rule 129

Int[((e_.)*(x_))^(p_)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> With[{k = Denominator[p]
}, Dist[k/e, Subst[Int[x^(k*(p + 1) - 1)*(a + b*(x^k/e))^m*(c + d*(x^k/e))^n, x], x, (e*x)^(1/k)], x]] /; Free
Q[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && FractionQ[p] && IntegerQ[m]

Rule 245

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 494

Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_)^(n_))^(q_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Dist[e^n/b, Int[
(e*x)^(m - n)*(c + d*x^n)^q, x], x] - Dist[a*(e^n/b), Int[(e*x)^(m - n)*((c + d*x^n)^q/(a + b*x^n)), x], x] /;
FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1] && IntBinomialQ[a, b
, c, d, e, m, n, -1, q, x]

Rule 925

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n, 1/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x
] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[m] &&  !IntegerQ[n]

Rule 6851

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a*v^m*w^n)^FracPart[p]/(v^(m*Fr
acPart[p])*w^(n*FracPart[p]))), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rubi steps

\begin {align*} \int \frac {x}{\sqrt [3]{x^2 (-a+x)} \left (-a^2+2 a x+(-1+d) x^2\right )} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-a+x}\right ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{-a+x} \left (-a^2+2 a x+(-1+d) x^2\right )} \, dx}{\sqrt [3]{x^2 (-a+x)}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{-a+x}\right ) \int \left (\frac {(-1+d) \sqrt [3]{x}}{a \sqrt {d} \sqrt [3]{-a+x} \left (2 a-2 a \sqrt {d}-2 (1-d) x\right )}+\frac {(-1+d) \sqrt [3]{x}}{a \sqrt {d} \sqrt [3]{-a+x} \left (-2 a-2 a \sqrt {d}+2 (1-d) x\right )}\right ) \, dx}{\sqrt [3]{x^2 (-a+x)}}\\ &=-\frac {\left ((1-d) x^{2/3} \sqrt [3]{-a+x}\right ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{-a+x} \left (2 a-2 a \sqrt {d}-2 (1-d) x\right )} \, dx}{a \sqrt {d} \sqrt [3]{x^2 (-a+x)}}-\frac {\left ((1-d) x^{2/3} \sqrt [3]{-a+x}\right ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{-a+x} \left (-2 a-2 a \sqrt {d}+2 (1-d) x\right )} \, dx}{a \sqrt {d} \sqrt [3]{x^2 (-a+x)}}\\ &=-\frac {\left ((1-d) x^{2/3} \sqrt [3]{-a+x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-a+x} \left (-2 a-2 a \sqrt {d}+2 (1-d) x\right )} \, dx}{\left (1-\sqrt {d}\right ) \sqrt {d} \sqrt [3]{x^2 (-a+x)}}-\frac {\left ((1-d) x^{2/3} \sqrt [3]{-a+x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-a+x} \left (2 a-2 a \sqrt {d}-2 (1-d) x\right )} \, dx}{\left (1+\sqrt {d}\right ) \sqrt {d} \sqrt [3]{x^2 (-a+x)}}\\ &=-\frac {\sqrt {3} x^{2/3} \sqrt [3]{-a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-a+x}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{x}}\right )}{2 a d^{2/3} \sqrt [3]{-\left ((a-x) x^2\right )}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{-a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-a+x}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{x}}\right )}{2 a d^{2/3} \sqrt [3]{-\left ((a-x) x^2\right )}}+\frac {x^{2/3} \sqrt [3]{-a+x} \log \left (2 a \left (1-\sqrt {d}\right )-2 (1-d) x\right )}{4 a d^{2/3} \sqrt [3]{-\left ((a-x) x^2\right )}}+\frac {x^{2/3} \sqrt [3]{-a+x} \log \left (-2 a \left (1+\sqrt {d}\right )+2 (1-d) x\right )}{4 a d^{2/3} \sqrt [3]{-\left ((a-x) x^2\right )}}-\frac {3 x^{2/3} \sqrt [3]{-a+x} \log \left (-\sqrt [3]{x}-\frac {\sqrt [3]{-a+x}}{\sqrt [6]{d}}\right )}{4 a d^{2/3} \sqrt [3]{-\left ((a-x) x^2\right )}}-\frac {3 x^{2/3} \sqrt [3]{-a+x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-a+x}}{\sqrt [6]{d}}\right )}{4 a d^{2/3} \sqrt [3]{-\left ((a-x) x^2\right )}}\\ \end {align*}

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Mathematica [A]
time = 0.32, size = 223, normalized size = 0.90 \begin {gather*} \frac {x^{2/3} \sqrt [3]{-a+x} \left (-2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{d} x^{2/3}}{\sqrt [3]{d} x^{2/3}+2 (-a+x)^{2/3}}\right )-2 \log \left (-\sqrt [6]{d} \sqrt [3]{x}+\sqrt [3]{-a+x}\right )-2 \log \left (\sqrt [6]{d} \sqrt [3]{x}+\sqrt [3]{-a+x}\right )+\log \left (\sqrt [3]{d} x^{2/3}-\sqrt [6]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )+\log \left (\sqrt [3]{d} x^{2/3}+\sqrt [6]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )\right )}{4 a d^{2/3} \sqrt [3]{x^2 (-a+x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/((x^2*(-a + x))^(1/3)*(-a^2 + 2*a*x + (-1 + d)*x^2)),x]

[Out]

(x^(2/3)*(-a + x)^(1/3)*(-2*Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*x^(2/3))/(d^(1/3)*x^(2/3) + 2*(-a + x)^(2/3))] - 2
*Log[-(d^(1/6)*x^(1/3)) + (-a + x)^(1/3)] - 2*Log[d^(1/6)*x^(1/3) + (-a + x)^(1/3)] + Log[d^(1/3)*x^(2/3) - d^
(1/6)*x^(1/3)*(-a + x)^(1/3) + (-a + x)^(2/3)] + Log[d^(1/3)*x^(2/3) + d^(1/6)*x^(1/3)*(-a + x)^(1/3) + (-a +
x)^(2/3)]))/(4*a*d^(2/3)*(x^2*(-a + x))^(1/3))

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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {x}{\left (x^{2} \left (-a +x \right )\right )^{\frac {1}{3}} \left (-a^{2}+2 a x +\left (-1+d \right ) x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2*(-a+x))^(1/3)/(-a^2+2*a*x+(-1+d)*x^2),x, algorithm="maxima")

[Out]

integrate(x/((-(a - x)*x^2)^(1/3)*((d - 1)*x^2 - a^2 + 2*a*x)), x)

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Fricas [A]
time = 0.39, size = 194, normalized size = 0.78 \begin {gather*} \frac {2 \, \sqrt {3} d \sqrt {-\left (-d^{2}\right )^{\frac {1}{3}}} \arctan \left (-\frac {\sqrt {3} {\left (\left (-d^{2}\right )^{\frac {1}{3}} d x^{2} - 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} \left (-d^{2}\right )^{\frac {2}{3}}\right )} \sqrt {-\left (-d^{2}\right )^{\frac {1}{3}}}}{3 \, d^{2} x^{2}}\right ) + \left (-d^{2}\right )^{\frac {2}{3}} \log \left (-\frac {\left (-d^{2}\right )^{\frac {1}{3}} d x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (a d - d x\right )} - {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} \left (-d^{2}\right )^{\frac {2}{3}}}{x^{2}}\right ) - 2 \, \left (-d^{2}\right )^{\frac {2}{3}} \log \left (-\frac {\left (-d^{2}\right )^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d}{x^{2}}\right )}{4 \, a d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2*(-a+x))^(1/3)/(-a^2+2*a*x+(-1+d)*x^2),x, algorithm="fricas")

[Out]

1/4*(2*sqrt(3)*d*sqrt(-(-d^2)^(1/3))*arctan(-1/3*sqrt(3)*((-d^2)^(1/3)*d*x^2 - 2*(-a*x^2 + x^3)^(2/3)*(-d^2)^(
2/3))*sqrt(-(-d^2)^(1/3))/(d^2*x^2)) + (-d^2)^(2/3)*log(-((-d^2)^(1/3)*d*x^2 + (-a*x^2 + x^3)^(1/3)*(a*d - d*x
) - (-a*x^2 + x^3)^(2/3)*(-d^2)^(2/3))/x^2) - 2*(-d^2)^(2/3)*log(-((-d^2)^(2/3)*x^2 - (-a*x^2 + x^3)^(2/3)*d)/
x^2))/(a*d^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt [3]{x^{2} \left (- a + x\right )} \left (- a^{2} + 2 a x + d x^{2} - x^{2}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x**2*(-a+x))**(1/3)/(-a**2+2*a*x+(-1+d)*x**2),x)

[Out]

Integral(x/((x**2*(-a + x))**(1/3)*(-a**2 + 2*a*x + d*x**2 - x**2)), x)

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Giac [A]
time = 0.44, size = 100, normalized size = 0.40 \begin {gather*} \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + d^{\frac {1}{3}}\right )}}{3 \, d^{\frac {1}{3}}}\right )}{2 \, a d^{\frac {2}{3}}} + \frac {\log \left ({\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + d^{\frac {1}{3}} {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + d^{\frac {2}{3}}\right )}{4 \, a d^{\frac {2}{3}}} - \frac {\log \left ({\left | {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} - d^{\frac {1}{3}} \right |}\right )}{2 \, a d^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2*(-a+x))^(1/3)/(-a^2+2*a*x+(-1+d)*x^2),x, algorithm="giac")

[Out]

1/2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(-a/x + 1)^(2/3) + d^(1/3))/d^(1/3))/(a*d^(2/3)) + 1/4*log((-a/x + 1)^(4/3)
+ d^(1/3)*(-a/x + 1)^(2/3) + d^(2/3))/(a*d^(2/3)) - 1/2*log(abs((-a/x + 1)^(2/3) - d^(1/3)))/(a*d^(2/3))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x}{{\left (-x^2\,\left (a-x\right )\right )}^{1/3}\,\left (-a^2+2\,a\,x+\left (d-1\right )\,x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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