\(\int \frac {e+f x}{(2 \sqrt [3]{a}+\sqrt [3]{b} x) \sqrt {-a+b x^3}} \, dx\) [135]

Optimal result

Integrand size = 36, antiderivative size = 313 \[ \int \frac {e+f x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx=-\frac {2 \left (\sqrt [3]{b} e-2 \sqrt [3]{a} f\right ) \arctan \left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt {-a+b x^3}}\right )}{9 \sqrt {a} b^{2/3}}-\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{b} e+\sqrt [3]{a} f\right ) \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt [3]{a} b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {-a+b x^3}} \] Output:

-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))/a^(1/2)/b^(2/3)-2/9*(1/2*6^(1/2)-1/2*2^(1/2))*(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)*EllipticF(((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 = 11.54 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.43 \[ \int \frac {e+f x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx=\frac {2 \sqrt {\frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (-\frac {1}{2} i f \sqrt {\frac {\left (-i+\sqrt {3}\right ) \sqrt [3]{a}+\left (i+\sqrt {3}\right ) \sqrt [3]{b} x}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}} \left (\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}-\left (3 i+\sqrt {3}\right ) \sqrt [3]{b} x\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {i \left (2 \sqrt [3]{a}+\left (1-i \sqrt {3}\right ) \sqrt [3]{b} x\right )}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}}\right ),\frac {1}{2} \left (1+i \sqrt {3}\right )\right )-i \left (\sqrt [3]{b} e-2 \sqrt [3]{a} f\right ) \sqrt {-\frac {i \left (2 \sqrt [3]{a}+\left (1-i \sqrt {3}\right ) \sqrt [3]{b} x\right )}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}} \sqrt {1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}+\frac {b^{2/3} x^2}{a^{2/3}}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {3}}{3 i+\sqrt {3}},\arcsin \left (\sqrt {-\frac {i \left (2 \sqrt [3]{a}+\left (1-i \sqrt {3}\right ) \sqrt [3]{b} x\right )}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}}\right ),\frac {1}{2} \left (1+i \sqrt {3}\right )\right )\right )}{\left (-2+\sqrt [3]{-1}\right ) b^{2/3} \sqrt {\frac {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {-a+b x^3}} \] Input:

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

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, 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.

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

\(\Big \downarrow \) 2564

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 760

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

\(\Big \downarrow \) 2563

\(\displaystyle -\frac {2 \sqrt [3]{a} \left (\frac {e}{\sqrt [3]{a}}-\frac {2 f}{\sqrt [3]{b}}\right ) \int \frac {1}{\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^4}{\sqrt [3]{a} \left (b x^3-a\right )}+9}d\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{a^{2/3} \sqrt {b x^3-a}}}{3 \sqrt [3]{b}}-\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \left (\frac {e}{\sqrt [3]{a}}+\frac {f}{\sqrt [3]{b}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt [3]{b} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {b x^3-a}}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \left (\frac {e}{\sqrt [3]{a}}+\frac {f}{\sqrt [3]{b}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt [3]{b} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {b x^3-a}}-\frac {2 \arctan \left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt {b x^3-a}}\right ) \left (\frac {e}{\sqrt [3]{a}}-\frac {2 f}{\sqrt [3]{b}}\right )}{9 \sqrt [6]{a} \sqrt [3]{b}}\)

Input:

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

Output:

(-2*(e/a^(1/3) - (2*f)/b^(1/3))*ArcTan[(a^(1/3) - b^(1/3)*x)^2/(3*a^(1/6)* 
Sqrt[-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 + Sq 
rt[3])*a^(1/3) - b^(1/3)*x)/((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)], -7 + 4*S 
qrt[3]])/(3*3^(1/4)*b^(1/3)*Sqrt[-((a^(1/3)*(a^(1/3) - b^(1/3)*x))/((1 - S 
qrt[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 216

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])
 

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 2563

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]
 

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]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {e+f x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx=-\left (\int \frac {\sqrt {b \,x^{3}-a}}{2 a^{\frac {4}{3}}-2 a^{\frac {1}{3}} b \,x^{3}+b^{\frac {1}{3}} a x -b^{\frac {4}{3}} x^{4}}d x \right ) e -\left (\int \frac {\sqrt {b \,x^{3}-a}\, x}{2 a^{\frac {4}{3}}-2 a^{\frac {1}{3}} b \,x^{3}+b^{\frac {1}{3}} a x -b^{\frac {4}{3}} x^{4}}d x \right ) f \] Input:

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

Output:

 - (int(sqrt( - a + b*x**3)/(2*a**(1/3)*a - 2*a**(1/3)*b*x**3 + b**(1/3)*a 
*x - b**(1/3)*b*x**4),x)*e + int((sqrt( - a + b*x**3)*x)/(2*a**(1/3)*a - 2 
*a**(1/3)*b*x**3 + b**(1/3)*a*x - b**(1/3)*b*x**4),x)*f)