\(\int \frac {1}{\sqrt [3]{x^2 (-a+x)} (a+(-1+d) x)} \, dx\) [2040]

Optimal result

Integrand size = 23, antiderivative size = 146 \[ \int \frac {1}{\sqrt [3]{x^2 (-a+x)} (a+(-1+d) x)} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} x}{\sqrt [3]{d} x+2 \sqrt [3]{-a x^2+x^3}}\right )}{a \sqrt [3]{d}}-\frac {\log \left (-\sqrt [3]{d} x+\sqrt [3]{-a x^2+x^3}\right )}{a \sqrt [3]{d}}+\frac {\log \left (d^{2/3} x^2+\sqrt [3]{d} x \sqrt [3]{-a x^2+x^3}+\left (-a x^2+x^3\right )^{2/3}\right )}{2 a \sqrt [3]{d}} \]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.68, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2106, 2102, 93} \[ \int \frac {1}{\sqrt [3]{x^2 (-a+x)} (a+(-1+d) x)} \, dx=-\frac {\sqrt {3} \left (-a^2 x\right )^{2/3} \sqrt [3]{a^2 (x-a)} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{a^2 (x-a)}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{-a^2 x}}\right )}{a^3 \sqrt [3]{d} \sqrt [3]{x^2 (x-a)}}+\frac {\left (-a^2 x\right )^{2/3} \sqrt [3]{a^2 (x-a)} \log (a+(d-1) x)}{2 a^3 \sqrt [3]{d} \sqrt [3]{x^2 (x-a)}}-\frac {3 \left (-a^2 x\right )^{2/3} \sqrt [3]{a^2 (x-a)} \log \left (-\frac {\sqrt [3]{\frac {2}{3}} \sqrt [3]{a^2 (x-a)}}{\sqrt [3]{d}}-\sqrt [3]{\frac {2}{3}} \sqrt [3]{-a^2 x}\right )}{2 a^3 \sqrt [3]{d} \sqrt [3]{x^2 (x-a)}} \]

[In]

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

[Out]

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

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 2102

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> Dist[(a + b*x + d*x^3)^p/
((3*a - b*x)^p*(3*a + 2*b*x)^(2*p)), Int[(e + f*x)^m*(3*a - b*x)^p*(3*a + 2*b*x)^(2*p), x], x] /; FreeQ[{a, b,
d, e, f, m, p}, x] && EqQ[4*b^3 + 27*a^2*d, 0] &&  !IntegerQ[p]

Rule 2106

Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = C
oeff[P3, x, 2], d = Coeff[P3, x, 3]}, Subst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2
)/(27*d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[{e, f, m, p}, x
] && PolyQ[P3, x, 3]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\left (\frac {1}{3} (3 a+a (-1+d))+(-1+d) x\right ) \sqrt [3]{-\frac {2 a^3}{27}-\frac {a^2 x}{3}+x^3}} \, dx,x,-\frac {a}{3}+x\right ) \\ & = \frac {\left (2^{2/3} \left (-a^2 x\right )^{2/3} \sqrt [3]{a^2 (-a+x)}\right ) \text {Subst}\left (\int \frac {1}{\left (-\frac {2 a^3}{9}-\frac {2 a^2 x}{3}\right )^{2/3} \sqrt [3]{-\frac {2 a^3}{9}+\frac {a^2 x}{3}} \left (\frac {1}{3} (3 a+a (-1+d))+(-1+d) x\right )} \, dx,x,-\frac {a}{3}+x\right )}{3 \sqrt [3]{-a x^2+x^3}} \\ & = -\frac {\sqrt {3} \sqrt [3]{-a^2 (a-x)} \left (-a^2 x\right )^{2/3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-a^2 (a-x)}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{-a^2 x}}\right )}{a^3 \sqrt [3]{d} \sqrt [3]{-a x^2+x^3}}+\frac {\sqrt [3]{-a^2 (a-x)} \left (-a^2 x\right )^{2/3} \log (a-(1-d) x)}{2 a^3 \sqrt [3]{d} \sqrt [3]{-a x^2+x^3}}-\frac {3 \sqrt [3]{-a^2 (a-x)} \left (-a^2 x\right )^{2/3} \log \left (\sqrt [3]{-a^2 (a-x)}+\sqrt [3]{d} \sqrt [3]{-a^2 x}\right )}{2 a^3 \sqrt [3]{d} \sqrt [3]{-a x^2+x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\sqrt [3]{x^2 (-a+x)} (a+(-1+d) x)} \, dx=\frac {x^{2/3} \sqrt [3]{-a+x} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{d} \sqrt [3]{x}+2 \sqrt [3]{-a+x}}\right )-2 \log \left (-\sqrt [3]{d} \sqrt [3]{x}+\sqrt [3]{-a+x}\right )+\log \left (d^{2/3} x^{2/3}+\sqrt [3]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )\right )}{2 a \sqrt [3]{d} \sqrt [3]{x^2 (-a+x)}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.75 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.78

method	result	size
pseudoelliptic	\(-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{3}} x +2 \left (-\left (a -x \right ) x^{2}\right )^{\frac {1}{3}}\right )}{3 d^{\frac {1}{3}} x}\right )+\ln \left (\frac {-d^{\frac {1}{3}} x +\left (-\left (a -x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {d^{\frac {2}{3}} x^{2}+d^{\frac {1}{3}} \left (-\left (a -x \right ) x^{2}\right )^{\frac {1}{3}} x +\left (-\left (a -x \right ) x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}}{d^{\frac {1}{3}} a}\)	\(114\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.58 \[ \int \frac {1}{\sqrt [3]{x^2 (-a+x)} (a+(-1+d) x)} \, dx=\left [\frac {\sqrt {3} d \sqrt {\frac {\left (-d\right )^{\frac {1}{3}}}{d}} \log \left (-\frac {{\left (d + 2\right )} x^{2} - 2 \, a x - 3 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} \left (-d\right )^{\frac {2}{3}} x - \sqrt {3} {\left (\left (-d\right )^{\frac {1}{3}} d x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d x + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} \left (-d\right )^{\frac {2}{3}}\right )} \sqrt {\frac {\left (-d\right )^{\frac {1}{3}}}{d}}}{{\left (d - 1\right )} x^{2} + a x}\right ) - 2 \, \left (-d\right )^{\frac {2}{3}} \log \left (\frac {\left (-d\right )^{\frac {1}{3}} x + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x}\right ) + \left (-d\right )^{\frac {2}{3}} \log \left (\frac {\left (-d\right )^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} \left (-d\right )^{\frac {1}{3}} x + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2 \, a d}, -\frac {2 \, \sqrt {3} d \sqrt {-\frac {\left (-d\right )^{\frac {1}{3}}}{d}} \arctan \left (-\frac {\sqrt {3} {\left (\left (-d\right )^{\frac {1}{3}} x - 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-d\right )^{\frac {1}{3}}}{d}}}{3 \, x}\right ) + 2 \, \left (-d\right )^{\frac {2}{3}} \log \left (\frac {\left (-d\right )^{\frac {1}{3}} x + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x}\right ) - \left (-d\right )^{\frac {2}{3}} \log \left (\frac {\left (-d\right )^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} \left (-d\right )^{\frac {1}{3}} x + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2 \, a d}\right ] \]

[In]

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

[Out]

[1/2*(sqrt(3)*d*sqrt((-d)^(1/3)/d)*log(-((d + 2)*x^2 - 2*a*x - 3*(-a*x^2 + x^3)^(1/3)*(-d)^(2/3)*x - sqrt(3)*(
(-d)^(1/3)*d*x^2 - (-a*x^2 + x^3)^(1/3)*d*x + 2*(-a*x^2 + x^3)^(2/3)*(-d)^(2/3))*sqrt((-d)^(1/3)/d))/((d - 1)*
x^2 + a*x)) - 2*(-d)^(2/3)*log(((-d)^(1/3)*x + (-a*x^2 + x^3)^(1/3))/x) + (-d)^(2/3)*log(((-d)^(2/3)*x^2 - (-a
*x^2 + x^3)^(1/3)*(-d)^(1/3)*x + (-a*x^2 + x^3)^(2/3))/x^2))/(a*d), -1/2*(2*sqrt(3)*d*sqrt(-(-d)^(1/3)/d)*arct
an(-1/3*sqrt(3)*((-d)^(1/3)*x - 2*(-a*x^2 + x^3)^(1/3))*sqrt(-(-d)^(1/3)/d)/x) + 2*(-d)^(2/3)*log(((-d)^(1/3)*
x + (-a*x^2 + x^3)^(1/3))/x) - (-d)^(2/3)*log(((-d)^(2/3)*x^2 - (-a*x^2 + x^3)^(1/3)*(-d)^(1/3)*x + (-a*x^2 +
x^3)^(2/3))/x^2))/(a*d)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\sqrt [3]{x^2 (-a+x)} (a+(-1+d) x)} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (d^{\frac {1}{3}} + 2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}\right )}}{3 \, d^{\frac {1}{3}}}\right )}{a d^{\frac {1}{3}}} + \frac {\log \left (d^{\frac {2}{3}} + d^{\frac {1}{3}} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}}\right )}{2 \, a d^{\frac {1}{3}}} - \frac {\log \left ({\left | -d^{\frac {1}{3}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} \right |}\right )}{a d^{\frac {1}{3}}} \]

[In]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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