\(\int \frac {e+f x}{(c+d x) \sqrt {-1+x^3}} \, dx\) [212]

Optimal result

Integrand size = 22, antiderivative size = 475 \[ \int \frac {e+f x}{(c+d x) \sqrt {-1+x^3}} \, dx=-\frac {(d e-c f) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \text {arctanh}\left (\frac {\sqrt {c^2-c d+d^2} \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}}}{\sqrt {d} \sqrt {c+d} \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}}\right )}{\sqrt {d} \sqrt {c+d} \sqrt {c^2-c d+d^2} \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {2 \sqrt {2-\sqrt {3}} \left (e+f+\sqrt {3} f\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \left (c+d+\sqrt {3} d\right ) \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (d e-c f) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (c+d+\sqrt {3} d\right )^2}{\left (c+d-\sqrt {3} d\right )^2},\arcsin \left (\frac {1-\sqrt {3}-x}{1+\sqrt {3}-x}\right ),-7-4 \sqrt {3}\right )}{\left (c^2+2 c d-2 d^2\right ) \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \] Output:

-(-c*f+d*e)*(1-x)*((x^2+x+1)/(1+3^(1/2)-x)^2)^(1/2)*arctanh((c^2-c*d+d^2)^ 
(1/2)*((1-x)/(1+3^(1/2)-x)^2)^(1/2)/d^(1/2)/(c+d)^(1/2)/((x^2+x+1)/(1+3^(1 
/2)-x)^2)^(1/2))/d^(1/2)/(c+d)^(1/2)/(c^2-c*d+d^2)^(1/2)/((1-x)/(1+3^(1/2) 
-x)^2)^(1/2)/(x^3-1)^(1/2)-2/3*(1/2*6^(1/2)-1/2*2^(1/2))*(e+f+3^(1/2)*f)*( 
1-x)*((x^2+x+1)/(1-3^(1/2)-x)^2)^(1/2)*EllipticF((1+3^(1/2)-x)/(1-3^(1/2)- 
x),2*I-I*3^(1/2))*3^(3/4)/(c+d+3^(1/2)*d)/(-(1-x)/(1-3^(1/2)-x)^2)^(1/2)/( 
x^3-1)^(1/2)-4*3^(1/4)*(1/2*6^(1/2)+1/2*2^(1/2))*(-c*f+d*e)*(1-x)*((x^2+x+ 
1)/(1+3^(1/2)-x)^2)^(1/2)*EllipticPi((1-3^(1/2)-x)/(1+3^(1/2)-x),(c+d+3^(1 
/2)*d)^2/(c+d-3^(1/2)*d)^2,I*3^(1/2)+2*I)/(c^2+2*c*d-2*d^2)/((1-x)/(1+3^(1 
/2)-x)^2)^(1/2)/(x^3-1)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 10.86 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.49 \[ \int \frac {e+f x}{(c+d x) \sqrt {-1+x^3}} \, dx=\frac {2 \sqrt {\frac {1-x}{1+\sqrt [3]{-1}}} \left (\frac {3 f \left (\sqrt [3]{-1}+x\right ) \sqrt {\frac {\sqrt [3]{-1}+(-1)^{2/3} x}{1+\sqrt [3]{-1}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac {\sqrt [3]{-1} \sqrt {3} \left (1+\sqrt [3]{-1}\right ) (-d e+c f) \sqrt {1+x+x^2} \operatorname {EllipticPi}\left (\frac {i \sqrt {3} d}{-c+\sqrt [3]{-1} d},\arcsin \left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{-c+\sqrt [3]{-1} d}\right )}{3 d \sqrt {-1+x^3}} \] Input:

Integrate[(e + f*x)/((c + d*x)*Sqrt[-1 + x^3]),x]
 

Output:

(2*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*((3*f*((-1)^(1/3) + x)*Sqrt[((-1)^(1/3) 
+ (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 - (-1)^(2/3)*x) 
/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3)) 
] + ((-1)^(1/3)*Sqrt[3]*(1 + (-1)^(1/3))*(-(d*e) + c*f)*Sqrt[1 + x + x^2]* 
EllipticPi[(I*Sqrt[3]*d)/(-c + (-1)^(1/3)*d), ArcSin[Sqrt[(1 - (-1)^(2/3)* 
x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(-c + (-1)^(1/3)*d)))/(3*d*Sqrt[-1 + x 
^3])
 

Rubi [A] (warning: unable to verify)

Time = 1.99 (sec) , antiderivative size = 463, normalized size of antiderivative = 0.97, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2569, 760, 2567, 2538, 412, 435, 104, 221}

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.

\(\displaystyle \int \frac {e+f x}{\sqrt {x^3-1} (c+d x)} \, dx\)

\(\Big \downarrow \) 2569

\(\displaystyle \frac {\left (e+\sqrt {3} f+f\right ) \int \frac {1}{\sqrt {x^3-1}}dx}{c+\sqrt {3} d+d}+\frac {(d e-c f) \int \frac {-x+\sqrt {3}+1}{(c+d x) \sqrt {x^3-1}}dx}{c+\sqrt {3} d+d}\)

\(\Big \downarrow \) 760

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

\(\Big \downarrow \) 2567

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

\(\Big \downarrow \) 2538

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

\(\Big \downarrow \) 412

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

\(\Big \downarrow \) 435

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 221

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

Input:

Int[(e + f*x)/((c + d*x)*Sqrt[-1 + x^3]),x]
 

Output:

(-2*Sqrt[2 - Sqrt[3]]*(e + f + Sqrt[3]*f)*(1 - x)*Sqrt[(1 + x + x^2)/(1 - 
Sqrt[3] - x)^2]*EllipticF[ArcSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 
+ 4*Sqrt[3]])/(3^(1/4)*(c + d + Sqrt[3]*d)*Sqrt[-((1 - x)/(1 - Sqrt[3] - x 
)^2)]*Sqrt[-1 + x^3]) + (4*3^(1/4)*Sqrt[2 - Sqrt[3]]*(d*e - c*f)*(1 - x)*S 
qrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*(((c + d + Sqrt[3]*d)*ArcTanh[(Sqrt 
[2 - Sqrt[3]]*Sqrt[c^2 - c*d + d^2]*(1 - Sqrt[3] - x))/(3^(1/4)*Sqrt[d]*Sq 
rt[c + d]*(1 + Sqrt[3] - x))])/(4*3^(1/4)*Sqrt[2 - Sqrt[3]]*Sqrt[d]*Sqrt[c 
 + d]*Sqrt[c^2 - c*d + d^2]) - EllipticPi[(c + d + Sqrt[3]*d)^2/(c + d - S 
qrt[3]*d)^2, ArcSin[(1 - Sqrt[3] - x)/(1 + Sqrt[3] - x)], -7 - 4*Sqrt[3]]/ 
(Sqrt[7 - 4*Sqrt[3]]*(c + d - Sqrt[3]*d))))/((c + d + Sqrt[3]*d)*Sqrt[(1 - 
 x)/(1 + Sqrt[3] - x)^2]*Sqrt[-1 + x^3])
 

Defintions of rubi rules used

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* 
(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, 
 f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 435

Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^(q_.)*(( 
e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2) 
*(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^2], x] /; FreeQ[{a, b, c, d, 
 e, f, p, q, r}, x] && IntegerQ[(m - 1)/2]
 

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 2538

Int[1/(((a_) + (b_.)*(x_))*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_) 
^2]), x_Symbol] :> Simp[a   Int[1/((a^2 - b^2*x^2)*Sqrt[c + d*x^2]*Sqrt[e + 
 f*x^2]), x], x] - Simp[b   Int[x/((a^2 - b^2*x^2)*Sqrt[c + d*x^2]*Sqrt[e + 
 f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x]
 

rule 2567

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ 
Symbol] :> With[{q = Simplify[(1 + Sqrt[3])*(f/e)]}, Simp[4*3^(1/4)*Sqrt[2 
- Sqrt[3]]*f*(1 + q*x)*(Sqrt[(1 - q*x + q^2*x^2)/(1 + Sqrt[3] + q*x)^2]/(q* 
Sqrt[a + b*x^3]*Sqrt[(1 + q*x)/(1 + Sqrt[3] + q*x)^2]))   Subst[Int[1/(((1 
- Sqrt[3])*d - c*q + ((1 + Sqrt[3])*d - c*q)*x)*Sqrt[1 - x^2]*Sqrt[7 - 4*Sq 
rt[3] + x^2]), x], x, (-1 + Sqrt[3] - q*x)/(1 + Sqrt[3] + q*x)], x]] /; Fre 
eQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b*e^3 - 2*(5 + 3*Sqrt 
[3])*a*f^3, 0] && NeQ[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
 

rule 2569

Int[((e_.) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x 
_Symbol] :> With[{q = Rt[b/a, 3]}, Simp[((1 + Sqrt[3])*f - e*q)/((1 + Sqrt[ 
3])*d - c*q)   Int[1/Sqrt[a + b*x^3], x], x] + Simp[(d*e - c*f)/((1 + Sqrt[ 
3])*d - c*q)   Int[(1 + Sqrt[3] + q*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] && NeQ[b^2*c^6 - 20*a 
*b*c^3*d^3 - 8*a^2*d^6, 0] && NeQ[b^2*e^6 - 20*a*b*e^3*f^3 - 8*a^2*f^6, 0]
 

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.58

method	result	size
default	\(\frac {2 f \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{d \sqrt {x^{3}-1}}-\frac {2 \left (c f -d e \right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{1+\frac {c}{d}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{d^{2} \sqrt {x^{3}-1}\, \left (1+\frac {c}{d}\right )}\)	\(274\)
elliptic	\(\frac {2 f \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{d \sqrt {x^{3}-1}}-\frac {2 \left (c f -d e \right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{1+\frac {c}{d}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{d^{2} \sqrt {x^{3}-1}\, \left (1+\frac {c}{d}\right )}\)	\(274\)

Input:

int((f*x+e)/(d*x+c)/(x^3-1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

2*f/d*(-3/2-1/2*I*3^(1/2))*((x-1)/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2-1/2* 
I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2+1/2*I*3^(1/2))/(3/2+1/2*I*3^ 
(1/2)))^(1/2)/(x^3-1)^(1/2)*EllipticF(((x-1)/(-3/2-1/2*I*3^(1/2)))^(1/2),( 
(3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))-2*(c*f-d*e)/d^2*(-3/2-1/2* 
I*3^(1/2))*((x-1)/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2-1/2*I*3^(1/2))/(3/2- 
1/2*I*3^(1/2)))^(1/2)*((x+1/2+1/2*I*3^(1/2))/(3/2+1/2*I*3^(1/2)))^(1/2)/(x 
^3-1)^(1/2)/(1+c/d)*EllipticPi(((x-1)/(-3/2-1/2*I*3^(1/2)))^(1/2),(3/2+1/2 
*I*3^(1/2))/(1+c/d),((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {e+f x}{(c+d x) \sqrt {-1+x^3}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

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

integrate((f*x+e)/(d*x+c)/(x**3-1)**(1/2),x)
 

Output:

Integral((e + f*x)/(sqrt((x - 1)*(x**2 + x + 1))*(c + d*x)), x)
 

Maxima [F]

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

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

Output:

integrate((f*x + e)/(sqrt(x^3 - 1)*(d*x + c)), x)
 

Giac [F]

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

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

Output:

integrate((f*x + e)/(sqrt(x^3 - 1)*(d*x + c)), x)
 

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 355, normalized size of antiderivative = 0.75 \[ \int \frac {e+f x}{(c+d x) \sqrt {-1+x^3}} \, dx=-\frac {2\,f\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{d\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}}+\frac {2\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (c\,f-d\,e\right )\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {c}{d}+1};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{d^2\,\left (\frac {c}{d}+1\right )\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \] Input:

int((e + f*x)/((x^3 - 1)^(1/2)*(c + d*x)),x)
 

Output:

(2*((3^(1/2)*1i)/2 + 3/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 3 
/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(c*f 
- d*e)*(-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi(((3^(1/2)*1i)/2 
+ 3/2)/(c/d + 1), asin((-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2 
)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)))/(d^2*(c/d + 1)*(((3^(1/2)*1i)/2 - 
1/2)*((3^(1/2)*1i)/2 + 1/2) - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 
1/2) + 1) + x^3)^(1/2)) - (2*f*((3^(1/2)*1i)/2 + 3/2)*(-(x - (3^(1/2)*1i)/ 
2 + 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/ 
2)*1i)/2 + 3/2))^(1/2)*(-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticF(a 
sin((-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^ 
(1/2)*1i)/2 - 3/2)))/(d*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) - x 
*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) + x^3)^(1/2))
 

Reduce [F]

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

int((f*x+e)/(d*x+c)/(x^3-1)^(1/2),x)
 

Output:

int((f*x+e)/(d*x+c)/(x^3-1)^(1/2),x)