\(\int \frac {e+f x}{(2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x) \sqrt {-a+b x^3}} \, dx\) [108]

Optimal result

Integrand size = 41, antiderivative size = 333 \[ \int \frac {e+f x}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx=-\frac {2 \left (\sqrt [3]{b} e+2^{2/3} \sqrt [3]{a} f\right ) \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{b} x\right )}{\sqrt {-a+b x^3}}\right )}{3 \sqrt {3} \sqrt {a} b^{2/3}}-\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{2} \sqrt [3]{b} e-\sqrt [3]{a} f\right ) \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt [3]{a} b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {-a+b x^3}} \] Output:

-2/9*(b^(1/3)*e+2^(2/3)*a^(1/3)*f)*arctanh(3^(1/2)*a^(1/6)*(a^(1/3)-2^(1/3 
)*b^(1/3)*x)/(b*x^3-a)^(1/2))*3^(1/2)/a^(1/2)/b^(2/3)-2/9*(1/2*6^(1/2)-1/2 
*2^(1/2))*(2^(1/3)*b^(1/3)*e-a^(1/3)*f)*(a^(1/3)-b^(1/3)*x)*((a^(2/3)+a^(1 
/3)*b^(1/3)*x+b^(2/3)*x^2)/((1-3^(1/2))*a^(1/3)-b^(1/3)*x)^2)^(1/2)*Ellipt 
icF(((1+3^(1/2))*a^(1/3)-b^(1/3)*x)/((1-3^(1/2))*a^(1/3)-b^(1/3)*x),2*I-I* 
3^(1/2))*3^(3/4)/a^(1/3)/b^(2/3)/(-a^(1/3)*(a^(1/3)-b^(1/3)*x)/((1-3^(1/2) 
)*a^(1/3)-b^(1/3)*x)^2)^(1/2)/(b*x^3-a)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 10.65 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.20 \[ \int \frac {e+f x}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx=-\frac {2 \sqrt {\frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (\left (\sqrt [3]{-1}+2^{2/3}\right ) f \left (\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {\sqrt [3]{-1} \left (\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right ),\sqrt [3]{-1}\right )-\frac {\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{b} e+2^{2/3} \sqrt [3]{a} f\right ) \sqrt {\frac {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}+\frac {b^{2/3} x^2}{a^{2/3}}} \operatorname {EllipticPi}\left (\frac {i \sqrt {3}}{\sqrt [3]{-1}+2^{2/3}},\arcsin \left (\sqrt {\frac {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right ),\sqrt [3]{-1}\right )}{\sqrt {3}}\right )}{\left (\sqrt [3]{-1}+2^{2/3}\right ) b^{2/3} \sqrt {\frac {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {-a+b x^3}} \] Input:

Integrate[(e + f*x)/((2^(2/3)*a^(1/3) - b^(1/3)*x)*Sqrt[-a + b*x^3]),x]
 

Output:

(-2*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(((-1)^(1/3) + 
2^(2/3))*f*((-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)] - 
((-1)^(1/3)*(1 + (-1)^(1/3))*(b^(1/3)*e + 2^(2/3)*a^(1/3)*f)*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[(I*Sqrt[3])/((-1)^(1/3) + 2^(2 
/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]))/(((-1)^(1/3) + 2^(2/3))*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])
 

Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {2564, 27, 760, 2562, 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.

\(\displaystyle \int \frac {e+f x}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {b x^3-a}} \, dx\)

\(\Big \downarrow \) 2564

\(\displaystyle \frac {1}{3} \left (\frac {\sqrt [3]{2} e}{\sqrt [3]{a}}-\frac {f}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt {b x^3-a}}dx+\frac {1}{6} \left (\frac {\sqrt [3]{2} e}{\sqrt [3]{a}}+\frac {2 f}{\sqrt [3]{b}}\right ) \int \frac {2^{2/3} \left (\sqrt [3]{2} \sqrt [3]{b} x+\sqrt [3]{a}\right )}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {b x^3-a}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (\frac {\sqrt [3]{2} e}{\sqrt [3]{a}}-\frac {f}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt {b x^3-a}}dx+\frac {\left (\frac {\sqrt [3]{2} e}{\sqrt [3]{a}}+\frac {2 f}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{2} \sqrt [3]{b} x+\sqrt [3]{a}}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {b x^3-a}}dx}{3 \sqrt [3]{2}}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {\left (\frac {\sqrt [3]{2} e}{\sqrt [3]{a}}+\frac {2 f}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{2} \sqrt [3]{b} x+\sqrt [3]{a}}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {b x^3-a}}dx}{3 \sqrt [3]{2}}-\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \left (\frac {\sqrt [3]{2} e}{\sqrt [3]{a}}-\frac {f}{\sqrt [3]{b}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt [3]{b} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {b x^3-a}}\)

\(\Big \downarrow \) 2562

\(\displaystyle -\frac {2^{2/3} \sqrt [3]{a} \left (\frac {\sqrt [3]{2} e}{\sqrt [3]{a}}+\frac {2 f}{\sqrt [3]{b}}\right ) \int \frac {1}{1-\frac {3 \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{b} x\right )^2}{b x^3-a}}d\frac {\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a} \sqrt {b x^3-a}}}{3 \sqrt [3]{b}}-\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \left (\frac {\sqrt [3]{2} e}{\sqrt [3]{a}}-\frac {f}{\sqrt [3]{b}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt [3]{b} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {b x^3-a}}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \left (\frac {\sqrt [3]{2} e}{\sqrt [3]{a}}-\frac {f}{\sqrt [3]{b}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt [3]{b} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {b x^3-a}}-\frac {2^{2/3} \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{b} x\right )}{\sqrt {b x^3-a}}\right ) \left (\frac {\sqrt [3]{2} e}{\sqrt [3]{a}}+\frac {2 f}{\sqrt [3]{b}}\right )}{3 \sqrt {3} \sqrt [6]{a} \sqrt [3]{b}}\)

Input:

Int[(e + f*x)/((2^(2/3)*a^(1/3) - b^(1/3)*x)*Sqrt[-a + b*x^3]),x]
 

Output:

-1/3*(2^(2/3)*((2^(1/3)*e)/a^(1/3) + (2*f)/b^(1/3))*ArcTanh[(Sqrt[3]*a^(1/ 
6)*(a^(1/3) - 2^(1/3)*b^(1/3)*x))/Sqrt[-a + b*x^3]])/(Sqrt[3]*a^(1/6)*b^(1 
/3)) - (2*Sqrt[2 - Sqrt[3]]*((2^(1/3)*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 + 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^(1/ 
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])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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])
 

rule 760

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]
 

rule 2562

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ 
Symbol] :> Simp[2*(e/d)   Subst[Int[1/(1 + 3*a*x^2), x], x, (1 + 2*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*c^3 - 4*a*d^3, 0] && EqQ[2*d*e + c*f, 0]
 

rule 2564

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]
 

Maple [F]

\[\int \frac {f x +e}{\left (2^{\frac {2}{3}} a^{\frac {1}{3}}-b^{\frac {1}{3}} x \right ) \sqrt {b \,x^{3}-a}}d x\]

Input:

int((f*x+e)/(2^(2/3)*a^(1/3)-b^(1/3)*x)/(b*x^3-a)^(1/2),x)
 

Output:

int((f*x+e)/(2^(2/3)*a^(1/3)-b^(1/3)*x)/(b*x^3-a)^(1/2),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {e+f x}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx=\text {Timed out} \] Input:

integrate((f*x+e)/(2^(2/3)*a^(1/3)-b^(1/3)*x)/(b*x^3-a)^(1/2),x, algorithm 
="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {e+f x}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx=- \int \frac {e}{- 2^{\frac {2}{3}} \sqrt [3]{a} \sqrt {- a + b x^{3}} + \sqrt [3]{b} x \sqrt {- a + b x^{3}}}\, dx - \int \frac {f x}{- 2^{\frac {2}{3}} \sqrt [3]{a} \sqrt {- a + b x^{3}} + \sqrt [3]{b} x \sqrt {- a + b x^{3}}}\, dx \] Input:

integrate((f*x+e)/(2**(2/3)*a**(1/3)-b**(1/3)*x)/(b*x**3-a)**(1/2),x)
 

Output:

-Integral(e/(-2**(2/3)*a**(1/3)*sqrt(-a + b*x**3) + b**(1/3)*x*sqrt(-a + b 
*x**3)), x) - Integral(f*x/(-2**(2/3)*a**(1/3)*sqrt(-a + b*x**3) + b**(1/3 
)*x*sqrt(-a + b*x**3)), x)
 

Maxima [F]

\[ \int \frac {e+f x}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx=\int { -\frac {f x + e}{\sqrt {b x^{3} - a} {\left (b^{\frac {1}{3}} x - 2^{\frac {2}{3}} a^{\frac {1}{3}}\right )}} \,d x } \] Input:

integrate((f*x+e)/(2^(2/3)*a^(1/3)-b^(1/3)*x)/(b*x^3-a)^(1/2),x, algorithm 
="maxima")
 

Output:

-integrate((f*x + e)/(sqrt(b*x^3 - a)*(b^(1/3)*x - 2^(2/3)*a^(1/3))), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {e+f x}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx=\text {Timed out} \] Input:

integrate((f*x+e)/(2^(2/3)*a^(1/3)-b^(1/3)*x)/(b*x^3-a)^(1/2),x, algorithm 
="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e+f x}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx=\int \frac {e+f\,x}{\sqrt {b\,x^3-a}\,\left (2^{2/3}\,a^{1/3}-b^{1/3}\,x\right )} \,d x \] Input:

int((e + f*x)/((b*x^3 - a)^(1/2)*(2^(2/3)*a^(1/3) - b^(1/3)*x)),x)
 

Output:

int((e + f*x)/((b*x^3 - a)^(1/2)*(2^(2/3)*a^(1/3) - b^(1/3)*x)), x)
 

Reduce [F]

\[ \int \frac {e+f x}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx=\left (\int \frac {x}{a^{\frac {1}{3}} \sqrt {b \,x^{3}-a}\, 2^{\frac {2}{3}}-b^{\frac {1}{3}} \sqrt {b \,x^{3}-a}\, x}d x \right ) f +\left (\int \frac {1}{a^{\frac {1}{3}} \sqrt {b \,x^{3}-a}\, 2^{\frac {2}{3}}-b^{\frac {1}{3}} \sqrt {b \,x^{3}-a}\, x}d x \right ) e \] Input:

int((f*x+e)/(2^(2/3)*a^(1/3)-b^(1/3)*x)/(b*x^3-a)^(1/2),x)
 

Output:

int(x/(a**(1/3)*sqrt( - a + b*x**3)*2**(2/3) - b**(1/3)*sqrt( - a + b*x**3 
)*x),x)*f + int(1/(a**(1/3)*sqrt( - a + b*x**3)*2**(2/3) - b**(1/3)*sqrt( 
- a + b*x**3)*x),x)*e