\(\int \frac {x}{(x^2 (-a+x))^{2/3} (a+(-1+d) x)} \, dx\) [2046]

Optimal result

Integrand size = 24, antiderivative size = 147 \[ \int \frac {x}{\left (x^2 (-a+x)\right )^{2/3} (a+(-1+d) x)} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} x}{\sqrt [3]{d} x+2 \sqrt [3]{-a x^2+x^3}}\right )}{a d^{2/3}}-\frac {\log \left (-\sqrt [3]{d} x+\sqrt [3]{-a x^2+x^3}\right )}{a d^{2/3}}+\frac {\log \left (d^{2/3} x^2+\sqrt [3]{d} x \sqrt [3]{-a x^2+x^3}+\left (-a x^2+x^3\right )^{2/3}\right )}{2 a d^{2/3}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.31, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6851, 93} \[ \int \frac {x}{\left (x^2 (-a+x)\right )^{2/3} (a+(-1+d) x)} \, dx=-\frac {\sqrt {3} x^{4/3} (x-a)^{2/3} \arctan \left (\frac {2 \sqrt [3]{d} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-a}}+\frac {1}{\sqrt {3}}\right )}{a d^{2/3} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}+\frac {x^{4/3} (x-a)^{2/3} \log (a-(1-d) x)}{2 a d^{2/3} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}-\frac {3 x^{4/3} (x-a)^{2/3} \log \left (\sqrt [3]{d} \sqrt [3]{x}-\sqrt [3]{x-a}\right )}{2 a d^{2/3} \left (-\left (x^2 (a-x)\right )\right )^{2/3}} \]

[In]

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

[Out]

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

Rule 93

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

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

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.06 \[ \int \frac {x}{\left (x^2 (-a+x)\right )^{2/3} (a+(-1+d) x)} \, dx=\frac {x^{4/3} (-a+x)^{2/3} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{d} \sqrt [3]{x}+2 \sqrt [3]{-a+x}}\right )-2 \log \left (-\sqrt [3]{d} \sqrt [3]{x}+\sqrt [3]{-a+x}\right )+\log \left (d^{2/3} x^{2/3}+\sqrt [3]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )\right )}{2 a d^{2/3} \left (x^2 (-a+x)\right )^{2/3}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.78

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{3}} x +2 \left (-\left (a -x \right ) x^{2}\right )^{\frac {1}{3}}\right )}{3 d^{\frac {1}{3}} x}\right )+\ln \left (\frac {d^{\frac {2}{3}} x^{2}+d^{\frac {1}{3}} \left (-\left (a -x \right ) x^{2}\right )^{\frac {1}{3}} x +\left (-\left (a -x \right ) x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-d^{\frac {1}{3}} x +\left (-\left (a -x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )}{2 a \,d^{\frac {2}{3}}}\)	\(115\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.26 \[ \int \frac {x}{\left (x^2 (-a+x)\right )^{2/3} (a+(-1+d) x)} \, dx=\frac {2 \, \sqrt {3} d \sqrt {-\left (-d^{2}\right )^{\frac {1}{3}}} \arctan \left (-\frac {\sqrt {3} {\left (\left (-d^{2}\right )^{\frac {1}{3}} d x - 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} \left (-d^{2}\right )^{\frac {2}{3}}\right )} \sqrt {-\left (-d^{2}\right )^{\frac {1}{3}}}}{3 \, d^{2} x}\right ) - 2 \, \left (-d^{2}\right )^{\frac {2}{3}} \log \left (-\frac {\left (-d^{2}\right )^{\frac {2}{3}} x - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d}{x}\right ) + \left (-d^{2}\right )^{\frac {2}{3}} \log \left (-\frac {\left (-d^{2}\right )^{\frac {1}{3}} d x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} \left (-d^{2}\right )^{\frac {2}{3}} x - {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d}{x^{2}}\right )}{2 \, a d^{2}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.67 \[ \int \frac {x}{\left (x^2 (-a+x)\right )^{2/3} (a+(-1+d) x)} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (d^{\frac {1}{3}} + 2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}\right )}}{3 \, d^{\frac {1}{3}}}\right )}{a d^{\frac {2}{3}}} + \frac {\log \left (d^{\frac {2}{3}} + d^{\frac {1}{3}} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}}\right )}{2 \, a d^{\frac {2}{3}}} - \frac {\log \left ({\left | -d^{\frac {1}{3}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} \right |}\right )}{a d^{\frac {2}{3}}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\left (x^2 (-a+x)\right )^{2/3} (a+(-1+d) x)} \, dx=\int \frac {x}{\left (a+x\,\left (d-1\right )\right )\,{\left (-x^2\,\left (a-x\right )\right )}^{2/3}} \,d x \]

[In]

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

[Out]

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