\(\int \frac {(b+a x^2) \sqrt [3]{-x+x^3}}{x^2} \, dx\) [1955]

Optimal result

Integrand size = 22, antiderivative size = 138 \[ \int \frac {\left (b+a x^2\right ) \sqrt [3]{-x+x^3}}{x^2} \, dx=\frac {\left (-3 b+a x^2\right ) \sqrt [3]{-x+x^3}}{2 x}+\frac {1}{6} \left (\sqrt {3} a-3 \sqrt {3} b\right ) \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x+x^3}}\right )+\frac {1}{6} (a-3 b) \log \left (-x+\sqrt [3]{-x+x^3}\right )+\frac {1}{12} (-a+3 b) \log \left (x^2+x \sqrt [3]{-x+x^3}+\left (-x+x^3\right )^{2/3}\right ) \]

[Out]

1/2*(a*x^2-3*b)*(x^3-x)^(1/3)/x+1/6*(3^(1/2)*a-3*3^(1/2)*b)*arctan(3^(1/2)*x/(x+2*(x^3-x)^(1/3)))+1/6*(a-3*b)*
ln(-x+(x^3-x)^(1/3))+1/12*(-a+3*b)*ln(x^2+x*(x^3-x)^(1/3)+(x^3-x)^(2/3))

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2063, 2029, 2057, 335, 281, 337} \[ \int \frac {\left (b+a x^2\right ) \sqrt [3]{-x+x^3}}{x^2} \, dx=\frac {x^{2/3} \left (x^2-1\right )^{2/3} (a-3 b) \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} \left (x^3-x\right )^{2/3}}+\frac {1}{2} x \sqrt [3]{x^3-x} (a-3 b)+\frac {x^{2/3} \left (x^2-1\right )^{2/3} (a-3 b) \log \left (x^{2/3}-\sqrt [3]{x^2-1}\right )}{4 \left (x^3-x\right )^{2/3}}+\frac {3 b \left (x^3-x\right )^{4/3}}{2 x^2} \]

[In]

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

[Out]

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

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

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 2029

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[x*((a*x^j + b*x^n)^p/(n*p + 1)), x] + Dist[a
*(n - j)*(p/(n*p + 1)), Int[x^j*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] &&  !IntegerQ[p] && LtQ[0,
j, n] && GtQ[p, 0] && NeQ[n*p + 1, 0]

Rule 2057

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[c^IntPart[m]*(c*x)^FracPa
rt[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(FracPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p])), Int[x^(m
+ j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && PosQ[n
- j]

Rule 2063

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Sim
p[c*e^(j - 1)*(e*x)^(m - j + 1)*((a*x^j + b*x^(j + n))^(p + 1)/(a*(m + j*p + 1))), x] + Dist[(a*d*(m + j*p + 1
) - b*c*(m + n + p*(j + n) + 1))/(a*e^n*(m + j*p + 1)), Int[(e*x)^(m + n)*(a*x^j + b*x^(j + n))^p, x], x] /; F
reeQ[{a, b, c, d, e, j, p}, x] && EqQ[jn, j + n] &&  !IntegerQ[p] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && (LtQ[m
+ j*p, -1] || (IntegersQ[m - 1/2, p - 1/2] && LtQ[p, 0] && LtQ[m, (-n)*p - 1])) && (GtQ[e, 0] || IntegersQ[j,
n]) && NeQ[m + j*p + 1, 0] && NeQ[m - n + j*p + 1, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {3 b \left (-x+x^3\right )^{4/3}}{2 x^2}+(a-3 b) \int \sqrt [3]{-x+x^3} \, dx \\ & = \frac {1}{2} (a-3 b) x \sqrt [3]{-x+x^3}+\frac {3 b \left (-x+x^3\right )^{4/3}}{2 x^2}+\frac {1}{3} (-a+3 b) \int \frac {x}{\left (-x+x^3\right )^{2/3}} \, dx \\ & = \frac {1}{2} (a-3 b) x \sqrt [3]{-x+x^3}+\frac {3 b \left (-x+x^3\right )^{4/3}}{2 x^2}+\frac {\left ((-a+3 b) x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (-1+x^2\right )^{2/3}} \, dx}{3 \left (-x+x^3\right )^{2/3}} \\ & = \frac {1}{2} (a-3 b) x \sqrt [3]{-x+x^3}+\frac {3 b \left (-x+x^3\right )^{4/3}}{2 x^2}+\frac {\left ((-a+3 b) x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\left (-x+x^3\right )^{2/3}} \\ & = \frac {1}{2} (a-3 b) x \sqrt [3]{-x+x^3}+\frac {3 b \left (-x+x^3\right )^{4/3}}{2 x^2}+\frac {\left ((-a+3 b) x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{2 \left (-x+x^3\right )^{2/3}} \\ & = \frac {1}{2} (a-3 b) x \sqrt [3]{-x+x^3}+\frac {3 b \left (-x+x^3\right )^{4/3}}{2 x^2}+\frac {(a-3 b) x^{2/3} \left (-1+x^2\right )^{2/3} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt {3} \left (-x+x^3\right )^{2/3}}+\frac {(a-3 b) x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (x^{2/3}-\sqrt [3]{-1+x^2}\right )}{4 \left (-x+x^3\right )^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.07 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.51 \[ \int \frac {\left (b+a x^2\right ) \sqrt [3]{-x+x^3}}{x^2} \, dx=\frac {\left (-1+x^2\right )^{2/3} \left (-18 b \sqrt [3]{-1+x^2}+6 a x^2 \sqrt [3]{-1+x^2}+2 \sqrt {3} (a-3 b) x^{2/3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{-1+x^2}}\right )+2 (a-3 b) x^{2/3} \log \left (-x^{2/3}+\sqrt [3]{-1+x^2}\right )-a x^{2/3} \log \left (x^{4/3}+x^{2/3} \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right )+3 b x^{2/3} \log \left (x^{4/3}+x^{2/3} \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right )\right )}{12 \left (x \left (-1+x^2\right )\right )^{2/3}} \]

[In]

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

[Out]

((-1 + x^2)^(2/3)*(-18*b*(-1 + x^2)^(1/3) + 6*a*x^2*(-1 + x^2)^(1/3) + 2*Sqrt[3]*(a - 3*b)*x^(2/3)*ArcTan[(Sqr
t[3]*x^(2/3))/(x^(2/3) + 2*(-1 + x^2)^(1/3))] + 2*(a - 3*b)*x^(2/3)*Log[-x^(2/3) + (-1 + x^2)^(1/3)] - a*x^(2/
3)*Log[x^(4/3) + x^(2/3)*(-1 + x^2)^(1/3) + (-1 + x^2)^(2/3)] + 3*b*x^(2/3)*Log[x^(4/3) + x^(2/3)*(-1 + x^2)^(
1/3) + (-1 + x^2)^(2/3)]))/(12*(x*(-1 + x^2))^(2/3))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.92 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.49

method	result	size
meijerg	\(\frac {3 a \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{3}} x^{\frac {4}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{2}\right )}{4 {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{3}}}-\frac {3 b \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, -\frac {1}{3}\right ], \left [\frac {2}{3}\right ], x^{2}\right )}{2 {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{3}} x^{\frac {2}{3}}}\)	\(68\)
pseudoelliptic	\(\frac {3 \left (x^{3}-x \right )^{\frac {1}{3}} \left (a \,x^{2}-3 b \right )+x \left (a -3 b \right ) \left (-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-x \right )^{\frac {1}{3}}\right )}{3 x}\right )+\ln \left (\frac {-x +\left (x^{3}-x \right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {x^{2}+x \left (x^{3}-x \right )^{\frac {1}{3}}+\left (x^{3}-x \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right )}{6 \left (\left (x^{3}-x \right )^{\frac {2}{3}}+x \left (x +\left (x^{3}-x \right )^{\frac {1}{3}}\right )\right ) \left (x -\left (x^{3}-x \right )^{\frac {1}{3}}\right )}\)	\(148\)
trager	\(\frac {\left (a \,x^{2}-3 b \right ) \left (x^{3}-x \right )^{\frac {1}{3}}}{2 x}-\frac {\left (a -3 b \right ) \left (12 \ln \left (22320 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{2}-27072 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}-27072 \left (x^{3}-x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x -25212 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}-89280 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}+5355 \left (x^{3}-x \right )^{\frac {2}{3}}+5355 x \left (x^{3}-x \right )^{\frac {1}{3}}+5510 x^{2}+32028 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-1653\right ) \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-12 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \ln \left (22320 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{2}+27072 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+27072 \left (x^{3}-x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x +28932 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}-89280 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}+7611 \left (x^{3}-x \right )^{\frac {2}{3}}+7611 x \left (x^{3}-x \right )^{\frac {1}{3}}+7766 x^{2}-46908 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-4942\right )+\ln \left (22320 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{2}-27072 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}-27072 \left (x^{3}-x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x -25212 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}-89280 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}+5355 \left (x^{3}-x \right )^{\frac {2}{3}}+5355 x \left (x^{3}-x \right )^{\frac {1}{3}}+5510 x^{2}+32028 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-1653\right )\right )}{6}\)	\(473\)
risch	\(\text {Expression too large to display}\)	\(802\)

[In]

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

[Out]

3/4*a*signum(x^2-1)^(1/3)/(-signum(x^2-1))^(1/3)*x^(4/3)*hypergeom([-1/3,2/3],[5/3],x^2)-3/2*b*signum(x^2-1)^(
1/3)/(-signum(x^2-1))^(1/3)/x^(2/3)*hypergeom([-1/3,-1/3],[2/3],x^2)

Fricas [A] (verification not implemented)

none

Time = 45.93 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.89 \[ \int \frac {\left (b+a x^2\right ) \sqrt [3]{-x+x^3}}{x^2} \, dx=\frac {2 \, \sqrt {3} {\left (a - 3 \, b\right )} x \arctan \left (-\frac {44032959556 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (16754327161 \, x^{2} - 2707204793\right )} - 10524305234 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{81835897185 \, x^{2} - 1102302937}\right ) + {\left (a - 3 \, b\right )} x \log \left (-3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} + 1\right ) + 6 \, {\left (a x^{2} - 3 \, b\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}}}{12 \, x} \]

[In]

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

[Out]

1/12*(2*sqrt(3)*(a - 3*b)*x*arctan(-(44032959556*sqrt(3)*(x^3 - x)^(1/3)*x + sqrt(3)*(16754327161*x^2 - 270720
4793) - 10524305234*sqrt(3)*(x^3 - x)^(2/3))/(81835897185*x^2 - 1102302937)) + (a - 3*b)*x*log(-3*(x^3 - x)^(1
/3)*x + 3*(x^3 - x)^(2/3) + 1) + 6*(a*x^2 - 3*b)*(x^3 - x)^(1/3))/x

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.76 \[ \int \frac {\left (b+a x^2\right ) \sqrt [3]{-x+x^3}}{x^2} \, dx=\frac {1}{2} \, a x^{2} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - \frac {1}{6} \, \sqrt {3} {\left (a - 3 \, b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {1}{12} \, {\left (a - 3 \, b\right )} \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, {\left (a - 3 \, b\right )} \log \left ({\left | {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) - \frac {3}{2} \, b {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} \]

[In]

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

[Out]

1/2*a*x^2*(-1/x^2 + 1)^(1/3) - 1/6*sqrt(3)*(a - 3*b)*arctan(1/3*sqrt(3)*(2*(-1/x^2 + 1)^(1/3) + 1)) - 1/12*(a
- 3*b)*log((-1/x^2 + 1)^(2/3) + (-1/x^2 + 1)^(1/3) + 1) + 1/6*(a - 3*b)*log(abs((-1/x^2 + 1)^(1/3) - 1)) - 3/2
*b*(-1/x^2 + 1)^(1/3)

Mupad [F(-1)]

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

[In]

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

[Out]

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