Integrand size = 32, antiderivative size = 241 \[ \int \frac {1-\sqrt {3}+\sqrt [3]{\frac {b}{a}} x}{\sqrt {a+b x^3}} \, dx=\frac {2 \left (\frac {b}{a}\right )^{2/3} \sqrt {a+b x^3}}{b \left (1+\sqrt {3}+\sqrt [3]{\frac {b}{a}} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt [3]{\frac {b}{a}} x\right ) \sqrt {\frac {1-\sqrt [3]{\frac {b}{a}} x+\left (\frac {b}{a}\right )^{2/3} x^2}{\left (1+\sqrt {3}+\sqrt [3]{\frac {b}{a}} x\right )^2}} E\left (\arcsin \left (\frac {1-\sqrt {3}+\sqrt [3]{\frac {b}{a}} x}{1+\sqrt {3}+\sqrt [3]{\frac {b}{a}} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{\frac {b}{a}} \sqrt {\frac {1+\sqrt [3]{\frac {b}{a}} x}{\left (1+\sqrt {3}+\sqrt [3]{\frac {b}{a}} x\right )^2}} \sqrt {a+b x^3}} \] Output:
2*(b/a)^(2/3)*(b*x^3+a)^(1/2)/b/(1+3^(1/2)+(b/a)^(1/3)*x)-3^(1/4)*(1/2*6^( 1/2)-1/2*2^(1/2))*(1+(b/a)^(1/3)*x)*((1-(b/a)^(1/3)*x+(b/a)^(2/3)*x^2)/(1+ 3^(1/2)+(b/a)^(1/3)*x)^2)^(1/2)*EllipticE((1-3^(1/2)+(b/a)^(1/3)*x)/(1+3^( 1/2)+(b/a)^(1/3)*x),I*3^(1/2)+2*I)/(b/a)^(1/3)/((1+(b/a)^(1/3)*x)/(1+3^(1/ 2)+(b/a)^(1/3)*x)^2)^(1/2)/(b*x^3+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.04 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.37 \[ \int \frac {1-\sqrt {3}+\sqrt [3]{\frac {b}{a}} x}{\sqrt {a+b x^3}} \, dx=\frac {x \sqrt {1+\frac {b x^3}{a}} \left (-2 \left (-1+\sqrt {3}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a}\right )+\sqrt [3]{\frac {b}{a}} x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )\right )}{2 \sqrt {a+b x^3}} \] Input:
Integrate[(1 - Sqrt[3] + (b/a)^(1/3)*x)/Sqrt[a + b*x^3],x]
Output:
(x*Sqrt[1 + (b*x^3)/a]*(-2*(-1 + Sqrt[3])*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + (b/a)^(1/3)*x*Hypergeometric2F1[1/2, 2/3, 5/3, -((b*x^3)/ a)]))/(2*Sqrt[a + b*x^3])
Time = 0.56 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {2416}
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.
\(\displaystyle \int \frac {x \sqrt [3]{\frac {b}{a}}-\sqrt {3}+1}{\sqrt {a+b x^3}} \, dx\) |
\(\Big \downarrow \) 2416 |
\(\displaystyle \frac {2 \left (\frac {b}{a}\right )^{2/3} \sqrt {a+b x^3}}{b \left (x \sqrt [3]{\frac {b}{a}}+\sqrt {3}+1\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (x \sqrt [3]{\frac {b}{a}}+1\right ) \sqrt {\frac {x^2 \left (\frac {b}{a}\right )^{2/3}-x \sqrt [3]{\frac {b}{a}}+1}{\left (x \sqrt [3]{\frac {b}{a}}+\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{\frac {b}{a}} x-\sqrt {3}+1}{\sqrt [3]{\frac {b}{a}} x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{\frac {b}{a}} \sqrt {\frac {x \sqrt [3]{\frac {b}{a}}+1}{\left (x \sqrt [3]{\frac {b}{a}}+\sqrt {3}+1\right )^2}} \sqrt {a+b x^3}}\) |
Input:
Int[(1 - Sqrt[3] + (b/a)^(1/3)*x)/Sqrt[a + b*x^3],x]
Output:
(2*(b/a)^(2/3)*Sqrt[a + b*x^3])/(b*(1 + Sqrt[3] + (b/a)^(1/3)*x)) - (3^(1/ 4)*Sqrt[2 - Sqrt[3]]*(1 + (b/a)^(1/3)*x)*Sqrt[(1 - (b/a)^(1/3)*x + (b/a)^( 2/3)*x^2)/(1 + Sqrt[3] + (b/a)^(1/3)*x)^2]*EllipticE[ArcSin[(1 - Sqrt[3] + (b/a)^(1/3)*x)/(1 + Sqrt[3] + (b/a)^(1/3)*x)], -7 - 4*Sqrt[3]])/((b/a)^(1 /3)*Sqrt[(1 + (b/a)^(1/3)*x)/(1 + Sqrt[3] + (b/a)^(1/3)*x)^2]*Sqrt[a + b*x ^3])
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Simplify[(1 - Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 - Sqrt[3])*(d/c) ]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x] - S imp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( (1 + Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt [3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) *s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1003 vs. \(2 (196 ) = 392\).
Time = 0.58 (sec) , antiderivative size = 1004, normalized size of antiderivative = 4.17
Input:
int((1-3^(1/2)+(b/a)^(1/3)*x)/(b*x^3+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
2*I/b*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^( 1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))/(-3/2/b*(-a* b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-a*b^2)^(1 /3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)/(b*x^3 +a)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b *(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/ 3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))-2/3*I*3^ (1/2)/b*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2) ^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))/(-3/2/b*(- a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-a*b^2)^ (1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)/(b*x ^3+a)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2) /b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^( 1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))-2/3*I* (b/a)^(1/3)*3^(1/2)/b*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1 /2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/ 3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*(x+1 /2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/ 3))^(1/2)/(b*x^3+a)^(1/2)*((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2) ^(1/3))*EllipticE(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/...
Time = 0.08 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.22 \[ \int \frac {1-\sqrt {3}+\sqrt [3]{\frac {b}{a}} x}{\sqrt {a+b x^3}} \, dx=-\frac {2 \, {\left (\sqrt {b} {\left (\sqrt {3} - 1\right )} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + \sqrt {b} \left (\frac {b}{a}\right )^{\frac {1}{3}} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right )\right )}}{b} \] Input:
integrate((1-3^(1/2)+(b/a)^(1/3)*x)/(b*x^3+a)^(1/2),x, algorithm="fricas")
Output:
-2*(sqrt(b)*(sqrt(3) - 1)*weierstrassPInverse(0, -4*a/b, x) + sqrt(b)*(b/a )^(1/3)*weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)))/b
Time = 1.44 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.51 \[ \int \frac {1-\sqrt {3}+\sqrt [3]{\frac {b}{a}} x}{\sqrt {a+b x^3}} \, dx=\frac {x^{2} \sqrt [3]{\frac {b}{a}} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {5}{3}\right )} - \frac {\sqrt {3} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {4}{3}\right )} + \frac {x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {4}{3}\right )} \] Input:
integrate((1-3**(1/2)+(b/a)**(1/3)*x)/(b*x**3+a)**(1/2),x)
Output:
x**2*(b/a)**(1/3)*gamma(2/3)*hyper((1/2, 2/3), (5/3,), b*x**3*exp_polar(I* pi)/a)/(3*sqrt(a)*gamma(5/3)) - sqrt(3)*x*gamma(1/3)*hyper((1/3, 1/2), (4/ 3,), b*x**3*exp_polar(I*pi)/a)/(3*sqrt(a)*gamma(4/3)) + x*gamma(1/3)*hyper ((1/3, 1/2), (4/3,), b*x**3*exp_polar(I*pi)/a)/(3*sqrt(a)*gamma(4/3))
\[ \int \frac {1-\sqrt {3}+\sqrt [3]{\frac {b}{a}} x}{\sqrt {a+b x^3}} \, dx=\int { \frac {x \left (\frac {b}{a}\right )^{\frac {1}{3}} - \sqrt {3} + 1}{\sqrt {b x^{3} + a}} \,d x } \] Input:
integrate((1-3^(1/2)+(b/a)^(1/3)*x)/(b*x^3+a)^(1/2),x, algorithm="maxima")
Output:
integrate((x*(b/a)^(1/3) - sqrt(3) + 1)/sqrt(b*x^3 + a), x)
\[ \int \frac {1-\sqrt {3}+\sqrt [3]{\frac {b}{a}} x}{\sqrt {a+b x^3}} \, dx=\int { \frac {x \left (\frac {b}{a}\right )^{\frac {1}{3}} - \sqrt {3} + 1}{\sqrt {b x^{3} + a}} \,d x } \] Input:
integrate((1-3^(1/2)+(b/a)^(1/3)*x)/(b*x^3+a)^(1/2),x, algorithm="giac")
Output:
integrate((x*(b/a)^(1/3) - sqrt(3) + 1)/sqrt(b*x^3 + a), x)
Timed out. \[ \int \frac {1-\sqrt {3}+\sqrt [3]{\frac {b}{a}} x}{\sqrt {a+b x^3}} \, dx=\int \frac {x\,{\left (\frac {b}{a}\right )}^{1/3}-\sqrt {3}+1}{\sqrt {b\,x^3+a}} \,d x \] Input:
int((x*(b/a)^(1/3) - 3^(1/2) + 1)/(a + b*x^3)^(1/2),x)
Output:
int((x*(b/a)^(1/3) - 3^(1/2) + 1)/(a + b*x^3)^(1/2), x)
\[ \int \frac {1-\sqrt {3}+\sqrt [3]{\frac {b}{a}} x}{\sqrt {a+b x^3}} \, dx=\frac {-a^{\frac {1}{3}} \sqrt {3}\, \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{3}+a}d x \right )+a^{\frac {1}{3}} \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{3}+a}d x \right )+b^{\frac {1}{3}} \left (\int \frac {\sqrt {b \,x^{3}+a}\, x}{b \,x^{3}+a}d x \right )}{a^{\frac {1}{3}}} \] Input:
int((1-3^(1/2)+(b/a)^(1/3)*x)/(b*x^3+a)^(1/2),x)
Output:
( - a**(1/3)*sqrt(3)*int(sqrt(a + b*x**3)/(a + b*x**3),x) + a**(1/3)*int(s qrt(a + b*x**3)/(a + b*x**3),x) + b**(1/3)*int((sqrt(a + b*x**3)*x)/(a + b *x**3),x))/a**(1/3)