Integrand size = 26, antiderivative size = 304 \[ \int \frac {x^2}{\left (-b+a x^2\right ) \sqrt [3]{-x+x^3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x+x^3}}\right )}{2 a}+\frac {\sqrt {3} \sqrt [3]{b} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a-b} x}{\sqrt [3]{a-b} x-2 \sqrt [3]{b} \sqrt [3]{-x+x^3}}\right )}{2 a \sqrt [3]{a-b}}-\frac {\log \left (-x+\sqrt [3]{-x+x^3}\right )}{2 a}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a-b} x+\sqrt [3]{b} \sqrt [3]{-x+x^3}\right )}{2 a \sqrt [3]{a-b}}+\frac {\log \left (x^2+x \sqrt [3]{-x+x^3}+\left (-x+x^3\right )^{2/3}\right )}{4 a}+\frac {\sqrt [3]{b} \log \left ((a-b)^{2/3} x^2-\sqrt [3]{a-b} \sqrt [3]{b} x \sqrt [3]{-x+x^3}+b^{2/3} \left (-x+x^3\right )^{2/3}\right )}{4 a \sqrt [3]{a-b}} \]
1/2*3^(1/2)*arctan(3^(1/2)*x/(x+2*(x^3-x)^(1/3)))/a+1/2*3^(1/2)*b^(1/3)*ar ctan(3^(1/2)*(a-b)^(1/3)*x/((a-b)^(1/3)*x-2*b^(1/3)*(x^3-x)^(1/3)))/a/(a-b )^(1/3)-1/2*ln(-x+(x^3-x)^(1/3))/a-1/2*b^(1/3)*ln((a-b)^(1/3)*x+b^(1/3)*(x ^3-x)^(1/3))/a/(a-b)^(1/3)+1/4*ln(x^2+x*(x^3-x)^(1/3)+(x^3-x)^(2/3))/a+1/4 *b^(1/3)*ln((a-b)^(2/3)*x^2-(a-b)^(1/3)*b^(1/3)*x*(x^3-x)^(1/3)+b^(2/3)*(x ^3-x)^(2/3))/a/(a-b)^(1/3)
Time = 8.34 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.10 \[ \int \frac {x^2}{\left (-b+a x^2\right ) \sqrt [3]{-x+x^3}} \, dx=\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \left (2 \sqrt {3} \sqrt [3]{a-b} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{-1+x^2}}\right )+2 \sqrt {3} \sqrt [3]{b} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a-b} x^{2/3}}{\sqrt [3]{a-b} x^{2/3}-2 \sqrt [3]{b} \sqrt [3]{-1+x^2}}\right )-2 \sqrt [3]{a-b} \log \left (a \left (-x^{2/3}+\sqrt [3]{-1+x^2}\right )\right )-2 \sqrt [3]{b} \log \left (\sqrt [3]{a-b} x^{2/3}+\sqrt [3]{b} \sqrt [3]{-1+x^2}\right )+\sqrt [3]{a-b} \log \left (x^{4/3}+x^{2/3} \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right )+\sqrt [3]{b} \log \left ((a-b)^{2/3} x^{4/3}-\sqrt [3]{a-b} \sqrt [3]{b} x^{2/3} \sqrt [3]{-1+x^2}+b^{2/3} \left (-1+x^2\right )^{2/3}\right )\right )}{4 a \sqrt [3]{a-b} \sqrt [3]{x \left (-1+x^2\right )}} \]
(x^(1/3)*(-1 + x^2)^(1/3)*(2*Sqrt[3]*(a - b)^(1/3)*ArcTan[(Sqrt[3]*x^(2/3) )/(x^(2/3) + 2*(-1 + x^2)^(1/3))] + 2*Sqrt[3]*b^(1/3)*ArcTan[(Sqrt[3]*(a - b)^(1/3)*x^(2/3))/((a - b)^(1/3)*x^(2/3) - 2*b^(1/3)*(-1 + x^2)^(1/3))] - 2*(a - b)^(1/3)*Log[a*(-x^(2/3) + (-1 + x^2)^(1/3))] - 2*b^(1/3)*Log[(a - b)^(1/3)*x^(2/3) + b^(1/3)*(-1 + x^2)^(1/3)] + (a - b)^(1/3)*Log[x^(4/3) + x^(2/3)*(-1 + x^2)^(1/3) + (-1 + x^2)^(2/3)] + b^(1/3)*Log[(a - b)^(2/3) *x^(4/3) - (a - b)^(1/3)*b^(1/3)*x^(2/3)*(-1 + x^2)^(1/3) + b^(2/3)*(-1 + x^2)^(2/3)]))/(4*a*(a - b)^(1/3)*(x*(-1 + x^2))^(1/3))
Time = 0.36 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.74, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1948, 25, 368, 965, 983, 769, 901}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^2}{\sqrt [3]{x^3-x} \left (a x^2-b\right )} \, dx\) |
| \(\Big \downarrow \) 1948 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \int -\frac {x^{5/3}}{\sqrt [3]{x^2-1} \left (b-a x^2\right )}dx}{\sqrt [3]{x^3-x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \int \frac {x^{5/3}}{\sqrt [3]{x^2-1} \left (b-a x^2\right )}dx}{\sqrt [3]{x^3-x}}\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^2-1} \int \frac {x^{7/3}}{\sqrt [3]{x^2-1} \left (b-a x^2\right )}d\sqrt [3]{x}}{\sqrt [3]{x^3-x}}\) |
| \(\Big \downarrow \) 965 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^2-1} \int \frac {x}{\sqrt [3]{x-1} (b-a x)}dx^{2/3}}{2 \sqrt [3]{x^3-x}}\) |
| \(\Big \downarrow \) 983 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^2-1} \left (\frac {b \int \frac {1}{\sqrt [3]{x-1} (b-a x)}dx^{2/3}}{a}-\frac {\int \frac {1}{\sqrt [3]{x-1}}dx^{2/3}}{a}\right )}{2 \sqrt [3]{x^3-x}}\) |
| \(\Big \downarrow \) 769 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^2-1} \left (\frac {b \int \frac {1}{\sqrt [3]{x-1} (b-a x)}dx^{2/3}}{a}-\frac {\frac {\arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (\sqrt [3]{x-1}-x^{2/3}\right )}{a}\right )}{2 \sqrt [3]{x^3-x}}\) |
| \(\Big \downarrow \) 901 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^2-1} \left (\frac {b \left (-\frac {\arctan \left (\frac {1-\frac {2 x^{2/3} \sqrt [3]{a-b}}{\sqrt [3]{b} \sqrt [3]{x-1}}}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3} \sqrt [3]{a-b}}+\frac {\log \left (-\frac {x^{2/3} \sqrt [3]{a-b}}{\sqrt [3]{b}}-\sqrt [3]{x-1}\right )}{2 b^{2/3} \sqrt [3]{a-b}}-\frac {\log (b-a x)}{6 b^{2/3} \sqrt [3]{a-b}}\right )}{a}-\frac {\frac {\arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (\sqrt [3]{x-1}-x^{2/3}\right )}{a}\right )}{2 \sqrt [3]{x^3-x}}\) |
(-3*x^(1/3)*(-1 + x^2)^(1/3)*(-((ArcTan[(1 + (2*x^(2/3))/(-1 + x)^(1/3))/S qrt[3]]/Sqrt[3] - Log[(-1 + x)^(1/3) - x^(2/3)]/2)/a) + (b*(-(ArcTan[(1 - (2*(a - b)^(1/3)*x^(2/3))/(b^(1/3)*(-1 + x)^(1/3)))/Sqrt[3]]/(Sqrt[3]*(a - b)^(1/3)*b^(2/3))) + Log[-(-1 + x)^(1/3) - ((a - b)^(1/3)*x^(2/3))/b^(1/3 )]/(2*(a - b)^(1/3)*b^(2/3)) - Log[b - a*x]/(6*(a - b)^(1/3)*b^(2/3))))/a) )/(2*(-x + x^3)^(1/3))
3.29.65.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*2)/e^2))^p*(c + d*(x^(k*2)/e^2))^q, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[m ] && IntegerQ[p]
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]]/(Sqrt[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]
Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> Wit h[{q = Rt[(b*c - a*d)/c, 3]}, Simp[ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/S qrt[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]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /; Free Q[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]
Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_)^(n_))^(q_.))/((a_) + (b_.)*(x_)^( n_)), x_Symbol] :> Simp[e^n/b Int[(e*x)^(m - n)*(c + d*x^n)^q, x], x] - S imp[a*(e^n/b) Int[(e*x)^(m - n)*((c + d*x^n)^q/(a + b*x^n)), x], x] /; Fr eeQ[{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]
Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Simp[e^IntPart[m]*(e*x)^FracPart[m]*( (a*x^j + b*x^(j + n))^FracPart[p]/(x^(FracPart[m] + j*FracPart[p])*(a + b*x ^n)^FracPart[p])) Int[x^(m + j*p)*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, j, m, n, p, q}, x] && EqQ[jn, j + n] && !IntegerQ[p] && NeQ[b*c - a*d, 0] && !(EqQ[n, 1] && EqQ[j, 1])
Time = 1.00 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.80
| method | result | size |
| pseudoelliptic | \(-\frac {\left (2 \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-x \right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}-\ln \left (\frac {x^{2}+x \left (x^{3}-x \right )^{\frac {1}{3}}+\left (x^{3}-x \right )^{\frac {2}{3}}}{x^{2}}\right )+2 \ln \left (\frac {-x +\left (x^{3}-x \right )^{\frac {1}{3}}}{x}\right )\right ) \left (\frac {a -b}{b}\right )^{\frac {1}{3}}+2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {a -b}{b}\right )^{\frac {1}{3}} x -2 \left (x^{3}-x \right )^{\frac {1}{3}}\right )}{3 \left (\frac {a -b}{b}\right )^{\frac {1}{3}} x}\right )+2 \ln \left (\frac {\left (\frac {a -b}{b}\right )^{\frac {1}{3}} x +\left (x^{3}-x \right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {\left (\frac {a -b}{b}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a -b}{b}\right )^{\frac {1}{3}} \left (x^{3}-x \right )^{\frac {1}{3}} x +\left (x^{3}-x \right )^{\frac {2}{3}}}{x^{2}}\right )}{4 \left (\frac {a -b}{b}\right )^{\frac {1}{3}} a}\) | \(244\) |
-1/4/((a-b)/b)^(1/3)*((2*arctan(1/3*3^(1/2)/x*(x+2*(x^3-x)^(1/3)))*3^(1/2) -ln((x^2+x*(x^3-x)^(1/3)+(x^3-x)^(2/3))/x^2)+2*ln((-x+(x^3-x)^(1/3))/x))*( (a-b)/b)^(1/3)+2*3^(1/2)*arctan(1/3*3^(1/2)*(((a-b)/b)^(1/3)*x-2*(x^3-x)^( 1/3))/((a-b)/b)^(1/3)/x)+2*ln((((a-b)/b)^(1/3)*x+(x^3-x)^(1/3))/x)-ln((((a -b)/b)^(2/3)*x^2-((a-b)/b)^(1/3)*(x^3-x)^(1/3)*x+(x^3-x)^(2/3))/x^2))/a
Timed out. \[ \int \frac {x^2}{\left (-b+a x^2\right ) \sqrt [3]{-x+x^3}} \, dx=\text {Timed out} \]
\[ \int \frac {x^2}{\left (-b+a x^2\right ) \sqrt [3]{-x+x^3}} \, dx=\int \frac {x^{2}}{\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )} \left (a x^{2} - b\right )}\, dx \]
\[ \int \frac {x^2}{\left (-b+a x^2\right ) \sqrt [3]{-x+x^3}} \, dx=\int { \frac {x^{2}}{{\left (a x^{2} - b\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}}} \,d x } \]
Time = 0.32 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.89 \[ \int \frac {x^2}{\left (-b+a x^2\right ) \sqrt [3]{-x+x^3}} \, dx=-\frac {b \left (-\frac {a - b}{b}\right )^{\frac {2}{3}} \log \left ({\left | -\left (-\frac {a - b}{b}\right )^{\frac {1}{3}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} \right |}\right )}{2 \, {\left (a^{2} - a b\right )}} - \frac {3 \, {\left (-a b^{2} + b^{3}\right )}^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (\left (-\frac {a - b}{b}\right )^{\frac {1}{3}} + 2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a - b}{b}\right )^{\frac {1}{3}}}\right )}{2 \, {\left (\sqrt {3} a^{2} b - \sqrt {3} a b^{2}\right )}} + \frac {{\left (-a b^{2} + b^{3}\right )}^{\frac {2}{3}} \log \left (\left (-\frac {a - b}{b}\right )^{\frac {2}{3}} + \left (-\frac {a - b}{b}\right )^{\frac {1}{3}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}}\right )}{4 \, {\left (a^{2} b - a b^{2}\right )}} - \frac {\sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right )}{2 \, a} + \frac {\log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}{4 \, a} - \frac {\log \left ({\left | {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right )}{2 \, a} \]
-1/2*b*(-(a - b)/b)^(2/3)*log(abs(-(-(a - b)/b)^(1/3) + (-1/x^2 + 1)^(1/3) ))/(a^2 - a*b) - 3/2*(-a*b^2 + b^3)^(2/3)*arctan(1/3*sqrt(3)*((-(a - b)/b) ^(1/3) + 2*(-1/x^2 + 1)^(1/3))/(-(a - b)/b)^(1/3))/(sqrt(3)*a^2*b - sqrt(3 )*a*b^2) + 1/4*(-a*b^2 + b^3)^(2/3)*log((-(a - b)/b)^(2/3) + (-(a - b)/b)^ (1/3)*(-1/x^2 + 1)^(1/3) + (-1/x^2 + 1)^(2/3))/(a^2*b - a*b^2) - 1/2*sqrt( 3)*arctan(1/3*sqrt(3)*(2*(-1/x^2 + 1)^(1/3) + 1))/a + 1/4*log((-1/x^2 + 1) ^(2/3) + (-1/x^2 + 1)^(1/3) + 1)/a - 1/2*log(abs((-1/x^2 + 1)^(1/3) - 1))/ a
Timed out. \[ \int \frac {x^2}{\left (-b+a x^2\right ) \sqrt [3]{-x+x^3}} \, dx=-\int \frac {x^2}{{\left (x^3-x\right )}^{1/3}\,\left (b-a\,x^2\right )} \,d x \]