Integrand size = 38, antiderivative size = 310 \[ \int \frac {e+f x}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx=\frac {2 \left (\sqrt [3]{b} e+2 \sqrt [3]{a} f\right ) \arctan \left (\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt {-a-b x^3}}\right )}{9 \sqrt {a} b^{2/3}}+\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{b} e-\sqrt [3]{a} f\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt [3]{a} b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {-a-b x^3}} \]
2/9*(b^(1/3)*e+2*a^(1/3)*f)*arctan(1/3*(a^(1/3)+b^(1/3)*x)^2/a^(1/6)/(-b*x ^3-a)^(1/2))/b^(2/3)/a^(1/2)+2/9*(b^(1/3)*e-a^(1/3)*f)*(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)/a^(1/3)/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 = 11.05 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.42 \[ \int \frac {e+f x}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx=\frac {2 \sqrt {\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (\frac {1}{2} f \left (\left (-3-i \sqrt {3}\right ) \sqrt [3]{a}+\left (3-i \sqrt {3}\right ) \sqrt [3]{b} x\right ) \sqrt {\frac {\left (-i+\sqrt {3}\right ) \sqrt [3]{a}-\left (i+\sqrt {3}\right ) \sqrt [3]{b} x}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {-2 i \sqrt [3]{a}+\left (i+\sqrt {3}\right ) \sqrt [3]{b} x}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}}\right ),\frac {1}{2} \left (1+i \sqrt {3}\right )\right )+i \left (\sqrt [3]{b} e+2 \sqrt [3]{a} f\right ) \sqrt {\frac {-2 i \sqrt [3]{a}+\left (i+\sqrt {3}\right ) \sqrt [3]{b} x}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}} \sqrt {1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}+\frac {b^{2/3} x^2}{a^{2/3}}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {3}}{3 i+\sqrt {3}},\arcsin \left (\sqrt {\frac {-2 i \sqrt [3]{a}+\left (i+\sqrt {3}\right ) \sqrt [3]{b} x}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}}\right ),\frac {1}{2} \left (1+i \sqrt {3}\right )\right )\right )}{\left (-2+\sqrt [3]{-1}\right ) b^{2/3} \sqrt {\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {-a-b x^3}} \]
(2*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*((f*((-3 - I*Sqr t[3])*a^(1/3) + (3 - I*Sqrt[3])*b^(1/3)*x)*Sqrt[((-I + Sqrt[3])*a^(1/3) - (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]*EllipticF[ArcSin[Sqrt [((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2])/2 + I*(b^(1/3)*e + 2*a^(1/3)*f)*Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[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[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a ^(1/3))]], (1 + I*Sqrt[3])/2]))/((-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 = 0.62 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, 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.
| \(\displaystyle \int \frac {e+f x}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx\) |
| \(\Big \downarrow \) 2564 |
| \(\displaystyle \frac {1}{3} \left (\frac {e}{\sqrt [3]{a}}-\frac {f}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt {-b x^3-a}}dx+\frac {1}{6} \left (\frac {e}{\sqrt [3]{a}}+\frac {2 f}{\sqrt [3]{b}}\right ) \int \frac {2 \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-b x^3-a}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {e}{\sqrt [3]{a}}-\frac {f}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt {-b x^3-a}}dx+\frac {1}{3} \left (\frac {e}{\sqrt [3]{a}}+\frac {2 f}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-b x^3-a}}dx\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {1}{3} \left (\frac {e}{\sqrt [3]{a}}+\frac {2 f}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-b x^3-a}}dx+\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (\frac {e}{\sqrt [3]{a}}-\frac {f}{\sqrt [3]{b}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt [3]{b} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {-a-b x^3}}\) |
| \(\Big \downarrow \) 2563 |
| \(\displaystyle \frac {2 \sqrt [3]{a} \left (\frac {e}{\sqrt [3]{a}}+\frac {2 f}{\sqrt [3]{b}}\right ) \int \frac {1}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^4}{\sqrt [3]{a} \left (-b x^3-a\right )}+9}d\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{a^{2/3} \sqrt {-b x^3-a}}}{3 \sqrt [3]{b}}+\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (\frac {e}{\sqrt [3]{a}}-\frac {f}{\sqrt [3]{b}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt [3]{b} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {-a-b x^3}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (\frac {e}{\sqrt [3]{a}}-\frac {f}{\sqrt [3]{b}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt [3]{b} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {-a-b x^3}}+\frac {2 \arctan \left (\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt {-a-b x^3}}\right ) \left (\frac {e}{\sqrt [3]{a}}+\frac {2 f}{\sqrt [3]{b}}\right )}{9 \sqrt [6]{a} \sqrt [3]{b}}\) |
(2*(e/a^(1/3) + (2*f)/b^(1/3))*ArcTan[(a^(1/3) + b^(1/3)*x)^2/(3*a^(1/6)*S qrt[-a - b*x^3])])/(9*a^(1/6)*b^(1/3)) + (2*Sqrt[2 - Sqrt[3]]*(e/a^(1/3) - f/b^(1/3))*(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]*EllipticF[ArcSin[((1 + Sqr t[3])*a^(1/3) + b^(1/3)*x)/((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 + 4*Sq rt[3]])/(3*3^(1/4)*b^(1/3)*Sqrt[-((a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 - Sq rt[3])*a^(1/3) + b^(1/3)*x)^2)]*Sqrt[-a - b*x^3])
3.1.90.3.1 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]
\[\int \frac {f x +e}{\left (2 a^{\frac {1}{3}}-b^{\frac {1}{3}} x \right ) \sqrt {-b \,x^{3}-a}}d x\]
Timed out. \[ \int \frac {e+f x}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx=\text {Timed out} \]
\[ \int \frac {e+f x}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx=- \int \frac {e}{- 2 \sqrt [3]{a} \sqrt {- a - b x^{3}} + \sqrt [3]{b} x \sqrt {- a - b x^{3}}}\, dx - \int \frac {f x}{- 2 \sqrt [3]{a} \sqrt {- a - b x^{3}} + \sqrt [3]{b} x \sqrt {- a - b x^{3}}}\, dx \]
-Integral(e/(-2*a**(1/3)*sqrt(-a - b*x**3) + b**(1/3)*x*sqrt(-a - b*x**3)) , x) - Integral(f*x/(-2*a**(1/3)*sqrt(-a - b*x**3) + b**(1/3)*x*sqrt(-a - b*x**3)), x)
\[ \int \frac {e+f x}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx=\int { -\frac {f x + e}{\sqrt {-b x^{3} - a} {\left (b^{\frac {1}{3}} x - 2 \, a^{\frac {1}{3}}\right )}} \,d x } \]
Timed out. \[ \int \frac {e+f x}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {e+f x}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx=\int -\frac {e+f\,x}{\left (b^{1/3}\,x-2\,a^{1/3}\right )\,\sqrt {-b\,x^3-a}} \,d x \]