Integrand size = 26, antiderivative size = 201 \[ \int \frac {x \sqrt [3]{a+b x^3}}{a d-b d x^3} \, dx=\frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3} d}-\frac {\sqrt [3]{2} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3} d}+\frac {\log \left (a d-b d x^3\right )}{3\ 2^{2/3} b^{2/3} d}+\frac {\log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2 b^{2/3} d}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2^{2/3} b^{2/3} d} \]
1/6*ln(-b*d*x^3+a*d)*2^(1/3)/b^(2/3)/d+1/2*ln(b^(1/3)*x-(b*x^3+a)^(1/3))/b ^(2/3)/d-1/2*ln(2^(1/3)*b^(1/3)*x-(b*x^3+a)^(1/3))*2^(1/3)/b^(2/3)/d+1/3*a rctan(1/3*(1+2*b^(1/3)*x/(b*x^3+a)^(1/3))*3^(1/2))/b^(2/3)/d*3^(1/2)-1/3*2 ^(1/3)*arctan(1/3*(1+2*2^(1/3)*b^(1/3)*x/(b*x^3+a)^(1/3))*3^(1/2))/b^(2/3) /d*3^(1/2)
Time = 0.70 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.32 \[ \int \frac {x \sqrt [3]{a+b x^3}}{a d-b d x^3} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )-2 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2^{2/3} \sqrt [3]{a+b x^3}}\right )+2 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )-2 \sqrt [3]{2} \log \left (-2 \sqrt [3]{b} x+2^{2/3} \sqrt [3]{a+b x^3}\right )-\log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )+\sqrt [3]{2} \log \left (2 b^{2/3} x^2+2^{2/3} \sqrt [3]{b} x \sqrt [3]{a+b x^3}+\sqrt [3]{2} \left (a+b x^3\right )^{2/3}\right )}{6 b^{2/3} d} \]
(2*Sqrt[3]*ArcTan[(Sqrt[3]*b^(1/3)*x)/(b^(1/3)*x + 2*(a + b*x^3)^(1/3))] - 2*2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*b^(1/3)*x)/(b^(1/3)*x + 2^(2/3)*(a + b* x^3)^(1/3))] + 2*Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)] - 2*2^(1/3)*Log[-2* b^(1/3)*x + 2^(2/3)*(a + b*x^3)^(1/3)] - Log[b^(2/3)*x^2 + b^(1/3)*x*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)] + 2^(1/3)*Log[2*b^(2/3)*x^2 + 2^(2/3)*b^ (1/3)*x*(a + b*x^3)^(1/3) + 2^(1/3)*(a + b*x^3)^(2/3)])/(6*b^(2/3)*d)
Time = 0.29 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {984, 27, 853, 992}
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.
| \(\displaystyle \int \frac {x \sqrt [3]{a+b x^3}}{a d-b d x^3} \, dx\) |
| \(\Big \downarrow \) 984 |
| \(\displaystyle 2 a \int \frac {x}{d \left (a-b x^3\right ) \left (b x^3+a\right )^{2/3}}dx-\frac {\int \frac {x}{\left (b x^3+a\right )^{2/3}}dx}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 a \int \frac {x}{\left (a-b x^3\right ) \left (b x^3+a\right )^{2/3}}dx}{d}-\frac {\int \frac {x}{\left (b x^3+a\right )^{2/3}}dx}{d}\) |
| \(\Big \downarrow \) 853 |
| \(\displaystyle \frac {2 a \int \frac {x}{\left (a-b x^3\right ) \left (b x^3+a\right )^{2/3}}dx}{d}-\frac {-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3}}-\frac {\log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2 b^{2/3}}}{d}\) |
| \(\Big \downarrow \) 992 |
| \(\displaystyle \frac {2 a \left (-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} a b^{2/3}}+\frac {\log \left (a-b x^3\right )}{6\ 2^{2/3} a b^{2/3}}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2\ 2^{2/3} a b^{2/3}}\right )}{d}-\frac {-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3}}-\frac {\log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2 b^{2/3}}}{d}\) |
-((-(ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*b^(2/3 ))) - Log[b^(1/3)*x - (a + b*x^3)^(1/3)]/(2*b^(2/3)))/d) + (2*a*(-(ArcTan[ (1 + (2*2^(1/3)*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(2^(2/3)*Sqrt[3]*a* b^(2/3))) + Log[a - b*x^3]/(6*2^(2/3)*a*b^(2/3)) - Log[2^(1/3)*b^(1/3)*x - (a + b*x^3)^(1/3)]/(2*2^(2/3)*a*b^(2/3))))/d
3.6.76.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Sim p[-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]
Int[((x_)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol ] :> Simp[b/d Int[x*(a + b*x^n)^(p - 1), x], x] - Simp[(b*c - a*d)/d In t[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]
Int[(x_)/(((a_) + (b_.)*(x_)^3)^(2/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[-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]
Time = 4.68 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.09
| method | result | size |
| pseudoelliptic | \(\frac {2 \,2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{3 b^{\frac {1}{3}} x}\right )-2 \,2^{\frac {1}{3}} \ln \left (\frac {-2^{\frac {1}{3}} b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )+2^{\frac {1}{3}} \ln \left (\frac {2^{\frac {2}{3}} b^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right )+2 \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{6 d \,b^{\frac {2}{3}}}\) | \(219\) |
1/6*(2*2^(1/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2^(2/3)*(b*x^3+a)^(1/3)+b^(1/3) *x)/b^(1/3)/x)-2*2^(1/3)*ln((-2^(1/3)*b^(1/3)*x+(b*x^3+a)^(1/3))/x)+2^(1/3 )*ln((2^(2/3)*b^(2/3)*x^2+2^(1/3)*b^(1/3)*(b*x^3+a)^(1/3)*x+(b*x^3+a)^(2/3 ))/x^2)-2*3^(1/2)*arctan(1/3*3^(1/2)*(b^(1/3)*x+2*(b*x^3+a)^(1/3))/b^(1/3) /x)+2*ln((-b^(1/3)*x+(b*x^3+a)^(1/3))/x)-ln((b^(2/3)*x^2+b^(1/3)*(b*x^3+a) ^(1/3)*x+(b*x^3+a)^(2/3))/x^2))/d/b^(2/3)
Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (156) = 312\).
Time = 0.27 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.56 \[ \int \frac {x \sqrt [3]{a+b x^3}}{a d-b d x^3} \, dx=-\frac {2 \, \sqrt {3} 2^{\frac {1}{3}} b^{2} \left (-\frac {1}{b^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} b \left (-\frac {1}{b^{2}}\right )^{\frac {2}{3}} + \sqrt {3} x}{3 \, x}\right ) - 2 \cdot 2^{\frac {1}{3}} b^{2} \left (-\frac {1}{b^{2}}\right )^{\frac {1}{3}} \log \left (\frac {2^{\frac {1}{3}} b x \left (-\frac {1}{b^{2}}\right )^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + 2^{\frac {1}{3}} b^{2} \left (-\frac {1}{b^{2}}\right )^{\frac {1}{3}} \log \left (\frac {2^{\frac {2}{3}} b^{2} x^{2} \left (-\frac {1}{b^{2}}\right )^{\frac {2}{3}} - 2^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x \left (-\frac {1}{b^{2}}\right )^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 2 \, \sqrt {3} {\left (b^{2}\right )}^{\frac {1}{6}} b \arctan \left (\frac {{\left (\sqrt {3} {\left (b^{2}\right )}^{\frac {1}{3}} b x + 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b^{2}\right )}^{\frac {2}{3}}\right )} {\left (b^{2}\right )}^{\frac {1}{6}}}{3 \, b^{2} x}\right ) - 2 \, {\left (b^{2}\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (b^{2}\right )}^{\frac {2}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}} b}{x}\right ) + {\left (b^{2}\right )}^{\frac {2}{3}} \log \left (\frac {{\left (b^{2}\right )}^{\frac {1}{3}} b x^{2} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b^{2}\right )}^{\frac {2}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}} b}{x^{2}}\right )}{6 \, b^{2} d} \]
-1/6*(2*sqrt(3)*2^(1/3)*b^2*(-1/b^2)^(1/3)*arctan(1/3*(sqrt(3)*2^(2/3)*(b* x^3 + a)^(1/3)*b*(-1/b^2)^(2/3) + sqrt(3)*x)/x) - 2*2^(1/3)*b^2*(-1/b^2)^( 1/3)*log((2^(1/3)*b*x*(-1/b^2)^(1/3) + (b*x^3 + a)^(1/3))/x) + 2^(1/3)*b^2 *(-1/b^2)^(1/3)*log((2^(2/3)*b^2*x^2*(-1/b^2)^(2/3) - 2^(1/3)*(b*x^3 + a)^ (1/3)*b*x*(-1/b^2)^(1/3) + (b*x^3 + a)^(2/3))/x^2) + 2*sqrt(3)*(b^2)^(1/6) *b*arctan(1/3*(sqrt(3)*(b^2)^(1/3)*b*x + 2*sqrt(3)*(b*x^3 + a)^(1/3)*(b^2) ^(2/3))*(b^2)^(1/6)/(b^2*x)) - 2*(b^2)^(2/3)*log(-((b^2)^(2/3)*x - (b*x^3 + a)^(1/3)*b)/x) + (b^2)^(2/3)*log(((b^2)^(1/3)*b*x^2 + (b*x^3 + a)^(1/3)* (b^2)^(2/3)*x + (b*x^3 + a)^(2/3)*b)/x^2))/(b^2*d)
\[ \int \frac {x \sqrt [3]{a+b x^3}}{a d-b d x^3} \, dx=- \frac {\int \frac {x \sqrt [3]{a + b x^{3}}}{- a + b x^{3}}\, dx}{d} \]
\[ \int \frac {x \sqrt [3]{a+b x^3}}{a d-b d x^3} \, dx=\int { -\frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} x}{b d x^{3} - a d} \,d x } \]
\[ \int \frac {x \sqrt [3]{a+b x^3}}{a d-b d x^3} \, dx=\int { -\frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} x}{b d x^{3} - a d} \,d x } \]
Timed out. \[ \int \frac {x \sqrt [3]{a+b x^3}}{a d-b d x^3} \, dx=\int \frac {x\,{\left (b\,x^3+a\right )}^{1/3}}{a\,d-b\,d\,x^3} \,d x \]