\(\int \frac {e+f x}{(c+d x) \sqrt {-1-x^3}} \, dx\) [213]

Optimal result

Integrand size = 24, antiderivative size = 463 \[ \int \frac {e+f x}{(c+d x) \sqrt {-1-x^3}} \, dx=\frac {(d e-c f) (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \arctan \left (\frac {\sqrt {c^2+c d+d^2} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}}}{\sqrt {c-d} \sqrt {d} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}}\right )}{\sqrt {c-d} \sqrt {d} \sqrt {c^2+c d+d^2} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}+\frac {2 \sqrt {2-\sqrt {3}} \left (e-f-\sqrt {3} f\right ) (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \left (c-d-\sqrt {3} d\right ) \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}-\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (d e-c f) (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (c-\left (1+\sqrt {3}\right ) d\right )^2}{\left (c-\left (1-\sqrt {3}\right ) d\right )^2},\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{\left (c^2-2 c d-2 d^2\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 10.72 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.46 \[ \int \frac {e+f x}{(c+d x) \sqrt {-1-x^3}} \, dx=\frac {2 \sqrt {\frac {1+x}{1+\sqrt [3]{-1}}} \left (-\frac {f \left (\sqrt [3]{-1}-x\right ) \sqrt {\frac {\sqrt [3]{-1}-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1+(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{\sqrt {\frac {1+(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac {i (-d e+c f) \sqrt {1-x+x^2} \operatorname {EllipticPi}\left (\frac {i \sqrt {3} d}{c+\sqrt [3]{-1} d},\arcsin \left (\sqrt {\frac {1+(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{c+\sqrt [3]{-1} d}\right )}{d \sqrt {-1-x^3}} \] Input:

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

Output:

(2*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*(-((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)) 
]) + (I*(-(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)))/(d*Sqrt[-1 - x^3])
 

Rubi [A] (warning: unable to verify)

Time = 2.28 (sec) , antiderivative size = 460, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2569, 760, 2567, 25, 2538, 412, 435, 104, 218}

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

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

\(\Big \downarrow \) 760

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

\(\Big \downarrow \) 2567

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2538

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

\(\Big \downarrow \) 412

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

\(\Big \downarrow \) 435

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[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 = 0.34 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.57

method	result	size
default	\(-\frac {2 i f \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 d \sqrt {-x^{3}-1}}+\frac {2 i \left (c f -d e \right ) \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}+\frac {c}{d}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 d^{2} \sqrt {-x^{3}-1}\, \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}+\frac {c}{d}\right )}\)	\(265\)
elliptic	\(-\frac {2 i f \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 d \sqrt {-x^{3}-1}}+\frac {2 i \left (c f -d e \right ) \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}+\frac {c}{d}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 d^{2} \sqrt {-x^{3}-1}\, \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}+\frac {c}{d}\right )}\)	\(265\)

Input:

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

Output:

-2/3*I*f/d*3^(1/2)*(I*(x-1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*((x+1)/(3/2+1/2 
*I*3^(1/2)))^(1/2)*(-I*(x-1/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(-x^3-1)^(1/2) 
*EllipticF(1/3*3^(1/2)*(I*(x-1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),(I*3^(1/2)/ 
(3/2+1/2*I*3^(1/2)))^(1/2))+2/3*I*(c*f-d*e)/d^2*3^(1/2)*(I*(x-1/2-1/2*I*3^ 
(1/2))*3^(1/2))^(1/2)*((x+1)/(3/2+1/2*I*3^(1/2)))^(1/2)*(-I*(x-1/2+1/2*I*3 
^(1/2))*3^(1/2))^(1/2)/(-x^3-1)^(1/2)/(1/2+1/2*I*3^(1/2)+c/d)*EllipticPi(1 
/3*3^(1/2)*(I*(x-1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),I*3^(1/2)/(1/2+1/2*I*3^ 
(1/2)+c/d),(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 = 21.73 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.84 \[ \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 {x^3+1}\,\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}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\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^3-1}\,\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 {x^3+1}\,\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}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\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\,\sqrt {-x^3-1}\,\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*f*((3^(1/2)*1i)/2 + 3/2)*(x^3 + 1)^(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)*(((3^(1 
/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticF(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*(- x^3 - 1)^(1/2)*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i 
)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)) - 
(2*((3^(1/2)*1i)/2 + 3/2)*(x^3 + 1)^(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)*(((3^(1/2)*1i)/2 - x + 1/2)/((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*(- x^3 - 1)^(1/ 
2)*(c/d - 1)*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) 
- ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(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)