3.27.84 \(\int \frac {-a x+x^2}{(x^2 (-a+x))^{2/3} (a^2 d-2 a d x+(-1+d) x^2)} \, dx\) [2684]

Optimal. Leaf size=243 \[ -\frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{d} \left (-a x^2+x^3\right )^{2/3}}\right )}{2 a d^{2/3}}+\frac {\log \left (x-\sqrt [6]{d} \sqrt [3]{-a x^2+x^3}\right )}{2 a d^{2/3}}+\frac {\log \left (x+\sqrt [6]{d} \sqrt [3]{-a x^2+x^3}\right )}{2 a d^{2/3}}-\frac {\log \left (x^2-\sqrt [6]{d} x \sqrt [3]{-a x^2+x^3}+\sqrt [3]{d} \left (-a x^2+x^3\right )^{2/3}\right )}{4 a d^{2/3}}-\frac {\log \left (x^2+\sqrt [6]{d} x \sqrt [3]{-a x^2+x^3}+\sqrt [3]{d} \left (-a x^2+x^3\right )^{2/3}\right )}{4 a d^{2/3}} \]

[Out]

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

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Rubi [A]
time = 0.64, antiderivative size = 405, normalized size of antiderivative = 1.67, number of steps used = 12, number of rules used = 7, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1607, 6851, 925, 129, 495, 337, 503} \begin {gather*} \frac {\sqrt {3} x^{4/3} (x-a)^{2/3} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{x}}{\sqrt [6]{d} \sqrt [3]{x-a}}}{\sqrt {3}}\right )}{2 a d^{2/3} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}+\frac {\sqrt {3} x^{4/3} (x-a)^{2/3} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [6]{d} \sqrt [3]{x-a}}+1}{\sqrt {3}}\right )}{2 a d^{2/3} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}-\frac {x^{4/3} (x-a)^{2/3} \log \left (a \left (1-\sqrt {d}\right ) \sqrt {d}-(1-d) x\right )}{4 a d^{2/3} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}-\frac {x^{4/3} (x-a)^{2/3} \log \left (a \sqrt {d} \left (\sqrt {d}+1\right )+(1-d) x\right )}{4 a d^{2/3} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}+\frac {3 x^{4/3} (x-a)^{2/3} \log \left (\sqrt [3]{x}-\sqrt [6]{d} \sqrt [3]{x-a}\right )}{4 a d^{2/3} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}+\frac {3 x^{4/3} (x-a)^{2/3} \log \left (\sqrt [6]{d} \sqrt [3]{x-a}+\sqrt [3]{x}\right )}{4 a d^{2/3} \left (-\left (x^2 (a-x)\right )\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 337

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

Rule 495

Int[((x_)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[b/d, Int[x*(a + b*x^n)^(p
- 1), x], x] - Dist[(b*c - a*d)/d, Int[x*((a + b*x^n)^(p - 1)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d}, x]
&& NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[p, 0] && IntBinomialQ[a, b, c, d, 1, 1, n, p, -1, x]

Rule 503

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

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 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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

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Mathematica [A]
time = 0.01, size = 217, normalized size = 0.89 \begin {gather*} -\frac {x^{4/3} (-a+x)^{2/3} \left (2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{d} (-a+x)^{2/3}}\right )-2 \log \left (\sqrt [3]{x}-\sqrt [6]{d} \sqrt [3]{-a+x}\right )-2 \log \left (\sqrt [3]{x}+\sqrt [6]{d} \sqrt [3]{-a+x}\right )+\log \left (x^{2/3}-\sqrt [6]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )+\log \left (x^{2/3}+\sqrt [6]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )\right )}{4 a d^{2/3} \left (x^2 (-a+x)\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.15, size = 0, normalized size = 0.00 \[\int \frac {-a x +x^{2}}{\left (x^{2} \left (-a +x \right )\right )^{\frac {2}{3}} \left (a^{2} d -2 a d x +\left (-1+d \right ) x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [A]
time = 0.37, size = 167, normalized size = 0.69 \begin {gather*} \frac {2 \, \sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} d \arctan \left (\frac {\sqrt {3} {\left ({\left (d^{2}\right )}^{\frac {1}{3}} x^{2} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d\right )} {\left (d^{2}\right )}^{\frac {1}{6}}}{3 \, d x^{2}}\right ) - {\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}} {\left (d^{2}\right )}^{\frac {1}{3}} d - {\left (a d^{2} - d^{2} x\right )} {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x^{2}}\right ) + 2 \, {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (d^{2}\right )}^{\frac {1}{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((-a*x+x^2)/(x^2*(-a+x))^(2/3)/(a^2*d-2*a*d*x+(-1+d)*x^2),x, algorithm="fricas")

[Out]

1/4*(2*sqrt(3)*(d^2)^(1/6)*d*arctan(1/3*sqrt(3)*((d^2)^(1/3)*x^2 + 2*(-a*x^2 + x^3)^(2/3)*d)*(d^2)^(1/6)/(d*x^
2)) - (d^2)^(2/3)*log(((d^2)^(2/3)*x^2 + (-a*x^2 + x^3)^(2/3)*(d^2)^(1/3)*d - (a*d^2 - d^2*x)*(-a*x^2 + x^3)^(
1/3))/x^2) + 2*(d^2)^(2/3)*log(-((d^2)^(1/3)*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 \left (- a + x\right )}{\left (x^{2} \left (- a + x\right )\right )^{\frac {2}{3}} \left (a^{2} d - 2 a d x + d x^{2} - x^{2}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.43, size = 108, normalized size = 0.44 \begin {gather*} \frac {\sqrt {3} {\left | d \right |}^{\frac {4}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} d^{\frac {1}{3}} {\left (2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + \frac {1}{d^{\frac {1}{3}}}\right )}\right )}{2 \, a d^{2}} - \frac {{\left | d \right |}^{\frac {4}{3}} \log \left ({\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + \frac {{\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}}}{d^{\frac {1}{3}}} + \frac {1}{d^{\frac {2}{3}}}\right )}{4 \, a d^{2}} + \frac {\log \left ({\left | {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} - \frac {1}{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((-a*x+x^2)/(x^2*(-a+x))^(2/3)/(a^2*d-2*a*d*x+(-1+d)*x^2),x, algorithm="giac")

[Out]

1/2*sqrt(3)*abs(d)^(4/3)*arctan(1/3*sqrt(3)*d^(1/3)*(2*(-a/x + 1)^(2/3) + 1/d^(1/3)))/(a*d^2) - 1/4*abs(d)^(4/
3)*log((-a/x + 1)^(4/3) + (-a/x + 1)^(2/3)/d^(1/3) + 1/d^(2/3))/(a*d^2) + 1/2*log(abs((-a/x + 1)^(2/3) - 1/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 {a\,x-x^2}{{\left (-x^2\,\left (a-x\right )\right )}^{2/3}\,\left (d\,a^2-2\,d\,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(-(a*x - x^2)/((-x^2*(a - x))^(2/3)*(a^2*d + x^2*(d - 1) - 2*a*d*x)),x)

[Out]

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

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