Integrand size = 32, antiderivative size = 277 \[ \int \frac {x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx=\frac {4 \arctan \left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt {-a+b x^3}}\right )}{9 \sqrt [6]{a} b^{2/3}}-\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {-a+b x^3}} \]
4/9*arctan(1/3*(a^(1/3)-b^(1/3)*x)^2/a^(1/6)/(b*x^3-a)^(1/2))/a^(1/6)/b^(2 /3)-2/9*(a^(1/3)-b^(1/3)*x)*EllipticF((-b^(1/3)*x+a^(1/3)*(1+3^(1/2)))/(-b ^(1/3)*x+a^(1/3)*(1-3^(1/2))),2*I-I*3^(1/2))*((a^(2/3)+a^(1/3)*b^(1/3)*x+b ^(2/3)*x^2)/(-b^(1/3)*x+a^(1/3)*(1-3^(1/2)))^2)^(1/2)*(1/2*6^(1/2)-1/2*2^( 1/2))*3^(3/4)/b^(2/3)/(b*x^3-a)^(1/2)/(-a^(1/3)*(a^(1/3)-b^(1/3)*x)/(-b^(1 /3)*x+a^(1/3)*(1-3^(1/2)))^2)^(1/2)
Result contains complex when optimal does not.
Time = 10.47 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.34 \[ \int \frac {x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx=\frac {2 \sqrt {\frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (\left (-2+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {\sqrt [3]{-1} \left (\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right ),\sqrt [3]{-1}\right )+\frac {2 \sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a} \sqrt {\frac {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}+\frac {b^{2/3} x^2}{a^{2/3}}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {3}}{3 i+\sqrt {3}},\arcsin \left (\sqrt {\frac {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right ),\sqrt [3]{-1}\right )}{\sqrt {3}}\right )}{\left (-2+\sqrt [3]{-1}\right ) b^{2/3} \sqrt {\frac {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {-a+b x^3}} \]
(2*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*((-2 + (-1)^(1/3 ))*((-1)^(1/3)*a^(1/3) + b^(1/3)*x)*Sqrt[((-1)^(1/3)*(a^(1/3) + (-1)^(1/3) *b^(1/3)*x))/((1 + (-1)^(1/3))*a^(1/3))]*EllipticF[ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)] + (2*(-1)^ (1/3)*(1 + (-1)^(1/3))*a^(1/3)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[1 + (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3 )]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[Sqrt[(a^(1/3) - (-1)^(2/ 3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/Sqrt[3]))/((-2 + (-1)^(1/3))*b^(2/3)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3) )*a^(1/3))]*Sqrt[-a + b*x^3])
Time = 0.60 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2564, 27, 760, 2563, 216}
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.
| \(\displaystyle \int \frac {x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {b x^3-a}} \, dx\) |
| \(\Big \downarrow \) 2564 |
| \(\displaystyle \frac {\int \frac {1}{\sqrt {b x^3-a}}dx}{3 \sqrt [3]{b}}-\frac {\int \frac {2 \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\sqrt [3]{b} x+2 \sqrt [3]{a}\right ) \sqrt {b x^3-a}}dx}{3 \sqrt [3]{b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {1}{\sqrt {b x^3-a}}dx}{3 \sqrt [3]{b}}-\frac {2 \int \frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\left (\sqrt [3]{b} x+2 \sqrt [3]{a}\right ) \sqrt {b x^3-a}}dx}{3 \sqrt [3]{b}}\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle -\frac {2 \int \frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\left (\sqrt [3]{b} x+2 \sqrt [3]{a}\right ) \sqrt {b x^3-a}}dx}{3 \sqrt [3]{b}}-\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {b x^3-a}}\) |
| \(\Big \downarrow \) 2563 |
| \(\displaystyle \frac {4 \sqrt [3]{a} \int \frac {1}{\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^4}{\sqrt [3]{a} \left (b x^3-a\right )}+9}d\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{a^{2/3} \sqrt {b x^3-a}}}{3 b^{2/3}}-\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {b x^3-a}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {4 \arctan \left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt {b x^3-a}}\right )}{9 \sqrt [6]{a} b^{2/3}}-\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {b x^3-a}}\) |
(4*ArcTan[(a^(1/3) - b^(1/3)*x)^2/(3*a^(1/6)*Sqrt[-a + b*x^3])])/(9*a^(1/6 )*b^(2/3)) - (2*Sqrt[2 - Sqrt[3]]*(a^(1/3) - b^(1/3)*x)*Sqrt[(a^(2/3) + a^ (1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)^2]*Elli pticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - b^(1/3)*x)/((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)], -7 + 4*Sqrt[3]])/(3*3^(1/4)*b^(2/3)*Sqrt[-((a^(1/3)*(a^(1/3) - b^(1/3)*x))/((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)^2)]*Sqrt[-a + b*x^3])
3.1.98.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s *x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) *s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x ] && NegQ[a]
Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ Symbol] :> Simp[-2*(e/d) Subst[Int[1/(9 - a*x^2), x], x, (1 + f*(x/e))^2/ Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] & & EqQ[b*c^3 + 8*a*d^3, 0] && EqQ[2*d*e + c*f, 0]
Int[((e_.) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x _Symbol] :> Simp[(2*d*e + c*f)/(3*c*d) Int[1/Sqrt[a + b*x^3], x], x] + Si mp[(d*e - c*f)/(3*c*d) Int[(c - 2*d*x)/((c + d*x)*Sqrt[a + b*x^3]), x], x ] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && (EqQ[b*c^3 - 4*a* d^3, 0] || EqQ[b*c^3 + 8*a*d^3, 0]) && NeQ[2*d*e + c*f, 0]
\[\int \frac {x}{\left (2 a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \sqrt {b \,x^{3}-a}}d x\]
Timed out. \[ \int \frac {x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx=\text {Timed out} \]
\[ \int \frac {x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx=\int \frac {x}{\left (2 \sqrt [3]{a} + \sqrt [3]{b} x\right ) \sqrt {- a + b x^{3}}}\, dx \]
\[ \int \frac {x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx=\int { \frac {x}{\sqrt {b x^{3} - a} {\left (b^{\frac {1}{3}} x + 2 \, a^{\frac {1}{3}}\right )}} \,d x } \]
Timed out. \[ \int \frac {x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx=\int \frac {x}{\left (b^{1/3}\,x+2\,a^{1/3}\right )\,\sqrt {b\,x^3-a}} \,d x \]