Integrand size = 44, antiderivative size = 336 \[ \int \frac {e+f x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=\frac {\left (\sqrt [3]{b} e+\left (1-\sqrt {3}\right ) \sqrt [3]{a} f\right ) \text {arctanh}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt {a-b x^3}}\right )}{\sqrt {3 \left (-3+2 \sqrt {3}\right )} \sqrt {a} b^{2/3}}+\frac {\sqrt {2+\sqrt {3}} \left (\sqrt [3]{b} e+\left (1+\sqrt {3}\right ) \sqrt [3]{a} f\right ) \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt [3]{a} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {a-b x^3}} \]
arctanh(a^(1/6)*(a^(1/3)-b^(1/3)*x)*(-3+2*3^(1/2))^(1/2)/(-b*x^3+a)^(1/2)) *(b^(1/3)*e+a^(1/3)*f*(1-3^(1/2)))/b^(2/3)/a^(1/2)/(-9+6*3^(1/2))^(1/2)+1/ 3*(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))),I*3^(1/2)+2*I)*(b^(1/3)*e+a^(1/3)*f*(1+3^(1/2)))*( 1/2*6^(1/2)+1/2*2^(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)*3^(1/4)/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 = 11.26 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.39 \[ \int \frac {e+f x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=-\frac {4 \sqrt {\frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (\frac {1}{2} f \left (i \left (-3+(2+i) \sqrt {3}\right ) \sqrt [3]{a}+\left (3-(2-i) \sqrt {3}\right ) \sqrt [3]{b} x\right ) \sqrt {\frac {\left (-i+\sqrt {3}\right ) \sqrt [3]{a}+\left (i+\sqrt {3}\right ) \sqrt [3]{b} x}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {i \left (2 \sqrt [3]{a}+\left (1-i \sqrt {3}\right ) \sqrt [3]{b} x\right )}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}}\right ),\frac {1}{2} \left (1+i \sqrt {3}\right )\right )-i \left (\sqrt [3]{b} e-\left (-1+\sqrt {3}\right ) \sqrt [3]{a} f\right ) \sqrt {-\frac {i \left (2 \sqrt [3]{a}+\left (1-i \sqrt {3}\right ) \sqrt [3]{b} x\right )}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}} \sqrt {1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}+\frac {b^{2/3} x^2}{a^{2/3}}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {3}}{-3 i+(1+2 i) \sqrt {3}},\arcsin \left (\sqrt {-\frac {i \left (2 \sqrt [3]{a}+\left (1-i \sqrt {3}\right ) \sqrt [3]{b} x\right )}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}}\right ),\frac {1}{2} \left (1+i \sqrt {3}\right )\right )\right )}{\left (3-(2-i) \sqrt {3}\right ) b^{2/3} \sqrt {\frac {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {a-b x^3}} \]
(-4*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*((f*(I*(-3 + (2 + I)*Sqrt[3])*a^(1/3) + (3 - (2 - 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))]*Ellipt icF[ArcSin[Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sq rt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2])/2 - I*(b^(1/3)*e - (-1 + Sqrt[3])*a ^(1/3)*f)*Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqr t[3])*a^(1/3))]*Sqrt[1 + (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*Elli pticPi[(2*Sqrt[3])/(-3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[((-I)*(2*a^(1/3 ) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[ 3])/2]))/((3 - (2 - I)*Sqrt[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 = 0.78 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2566, 27, 759, 2565, 219}
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.
| \(\displaystyle \int \frac {e+f x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx\) |
| \(\Big \downarrow \) 2566 |
| \(\displaystyle -\frac {\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a} f+\sqrt [3]{b} e\right ) \int -\frac {6 a b \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}}dx}{12 \sqrt {3} a^{4/3} b^{4/3}}-\frac {\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} f+\sqrt [3]{b} e\right ) \int \frac {1}{\sqrt {a-b x^3}}dx}{2 \sqrt {3} \sqrt [3]{a} \sqrt [3]{b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a} f+\sqrt [3]{b} e\right ) \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}}dx}{2 \sqrt {3} \sqrt [3]{a} \sqrt [3]{b}}-\frac {\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} f+\sqrt [3]{b} e\right ) \int \frac {1}{\sqrt {a-b x^3}}dx}{2 \sqrt {3} \sqrt [3]{a} \sqrt [3]{b}}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a} f+\sqrt [3]{b} e\right ) \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}}dx}{2 \sqrt {3} \sqrt [3]{a} \sqrt [3]{b}}+\frac {\sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} f+\sqrt [3]{b} e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt [3]{a} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {a-b x^3}}\) |
| \(\Big \downarrow \) 2565 |
| \(\displaystyle \frac {\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a} f+\sqrt [3]{b} e\right ) \int \frac {1}{\frac {\left (3-2 \sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{a-b x^3}+1}d\frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt [3]{a} \sqrt {a-b x^3}}}{\sqrt {3} b^{2/3}}+\frac {\sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} f+\sqrt [3]{b} e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt [3]{a} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {a-b x^3}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} f+\sqrt [3]{b} e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt [3]{a} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {a-b x^3}}+\frac {\text {arctanh}\left (\frac {\sqrt {2 \sqrt {3}-3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt {a-b x^3}}\right ) \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a} f+\sqrt [3]{b} e\right )}{\sqrt {3 \left (2 \sqrt {3}-3\right )} \sqrt {a} b^{2/3}}\) |
((b^(1/3)*e + (1 - Sqrt[3])*a^(1/3)*f)*ArcTanh[(Sqrt[-3 + 2*Sqrt[3]]*a^(1/ 6)*(a^(1/3) - b^(1/3)*x))/Sqrt[a - b*x^3]])/(Sqrt[3*(-3 + 2*Sqrt[3])]*Sqrt [a]*b^(2/3)) + (Sqrt[2 + Sqrt[3]]*(b^(1/3)*e + (1 + Sqrt[3])*a^(1/3)*f)*(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 - 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/ 4)*a^(1/3)*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])
3.2.32.3.1 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]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[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] & & PosQ[a]
Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ Symbol] :> With[{k = Simplify[(d*e + 2*c*f)/(c*f)]}, Simp[(1 + k)*(e/d) S ubst[Int[1/(1 + (3 + 2*k)*a*x^2), x], x, (1 + (1 + k)*d*(x/c))/Sqrt[a + b*x ^3]], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b^2*c ^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0] && EqQ[6*a*d^4*e - c*f*(b*c^3 - 22*a*d^ 3), 0]
Int[((e_.) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x _Symbol] :> Simp[-(6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3))/(c*d*(b*c^3 - 28*a*d ^3)) Int[1/Sqrt[a + b*x^3], x], x] + Simp[(d*e - c*f)/(c*d*(b*c^3 - 28*a* d^3)) Int[(c*(b*c^3 - 22*a*d^3) + 6*a*d^4*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^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0] && NeQ[6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3) , 0]
\[\int \frac {f x +e}{\left (-b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )\right ) \sqrt {-b \,x^{3}+a}}d x\]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 26.95 (sec) , antiderivative size = 7063, normalized size of antiderivative = 21.02 \[ \int \frac {e+f x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=\text {Too large to display} \]
\[ \int \frac {e+f x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=- \int \frac {e}{- \sqrt [3]{a} \sqrt {a - b x^{3}} + \sqrt {3} \sqrt [3]{a} \sqrt {a - b x^{3}} + \sqrt [3]{b} x \sqrt {a - b x^{3}}}\, dx - \int \frac {f x}{- \sqrt [3]{a} \sqrt {a - b x^{3}} + \sqrt {3} \sqrt [3]{a} \sqrt {a - b x^{3}} + \sqrt [3]{b} x \sqrt {a - b x^{3}}}\, dx \]
-Integral(e/(-a**(1/3)*sqrt(a - b*x**3) + sqrt(3)*a**(1/3)*sqrt(a - b*x**3 ) + b**(1/3)*x*sqrt(a - b*x**3)), x) - Integral(f*x/(-a**(1/3)*sqrt(a - b* x**3) + sqrt(3)*a**(1/3)*sqrt(a - b*x**3) + b**(1/3)*x*sqrt(a - b*x**3)), x)
\[ \int \frac {e+f x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=\int { -\frac {f x + e}{\sqrt {-b x^{3} + a} {\left (b^{\frac {1}{3}} x + a^{\frac {1}{3}} {\left (\sqrt {3} - 1\right )}\right )}} \,d x } \]
Timed out. \[ \int \frac {e+f x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {e+f x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=\int -\frac {e+f\,x}{\sqrt {a-b\,x^3}\,\left (b^{1/3}\,x+a^{1/3}\,\left (\sqrt {3}-1\right )\right )} \,d x \]