Integrand size = 58, antiderivative size = 69 \[ \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {a+b x^3}}\right )}{\sqrt {-3+2 \sqrt {3}} \sqrt [6]{a} \sqrt [3]{b}} \] Output:
-2*arctanh((-3+2*3^(1/2))^(1/2)*a^(1/6)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/2 ))/(-3+2*3^(1/2))^(1/2)/a^(1/6)/b^(1/3)
Time = 7.47 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.22 \[ \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {1+\frac {2}{\sqrt {3}}} \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [6]{a} \sqrt {a+b x^3}}\right )}{\sqrt {-3+2 \sqrt {3}} \sqrt [6]{a} \sqrt [3]{b}} \] Input:
Integrate[((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)/(((1 - Sqrt[3])*a^(1/3) + b^ (1/3)*x)*Sqrt[a + b*x^3]),x]
Output:
(-2*ArcTanh[(Sqrt[1 + 2/Sqrt[3]]*(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^ 2))/(a^(1/6)*Sqrt[a + b*x^3])])/(Sqrt[-3 + 2*Sqrt[3]]*a^(1/6)*b^(1/3))
Time = 0.56 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2565, 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.
| \(\displaystyle \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx\) |
| \(\Big \downarrow \) 2565 |
| \(\displaystyle -\frac {2 \sqrt [3]{a} \int \frac {1}{\frac {\left (3-2 \sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{b x^3+a}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt {b x^3+a}}}{\sqrt [3]{b}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {2 \text {arctanh}\left (\frac {\sqrt {2 \sqrt {3}-3} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {a+b x^3}}\right )}{\sqrt {2 \sqrt {3}-3} \sqrt [6]{a} \sqrt [3]{b}}\) |
Input:
Int[((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)/(((1 - Sqrt[3])*a^(1/3) + b^(1/3)* x)*Sqrt[a + b*x^3]),x]
Output:
(-2*ArcTanh[(Sqrt[-3 + 2*Sqrt[3]]*a^(1/6)*(a^(1/3) + b^(1/3)*x))/Sqrt[a + b*x^3]])/(Sqrt[-3 + 2*Sqrt[3]]*a^(1/6)*b^(1/3))
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])
Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ Symbol] :> With[{k = Simplify[(d*e + 2*c*f)/(c*f)]}, Simp[(1 + k)*(e/d) S ubst[Int[1/(1 + (3 + 2*k)*a*x^2), x], x, (1 + (1 + k)*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^2*c ^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0] && EqQ[6*a*d^4*e - c*f*(b*c^3 - 22*a*d^ 3), 0]
\[\int \frac {\left (1+\sqrt {3}\right ) a^{\frac {1}{3}}+b^{\frac {1}{3}} x}{\left (\left (1-\sqrt {3}\right ) a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \sqrt {b \,x^{3}+a}}d x\]
Input:
int(((1+3^(1/2))*a^(1/3)+b^(1/3)*x)/((1-3^(1/2))*a^(1/3)+b^(1/3)*x)/(b*x^3 +a)^(1/2),x)
Output:
int(((1+3^(1/2))*a^(1/3)+b^(1/3)*x)/((1-3^(1/2))*a^(1/3)+b^(1/3)*x)/(b*x^3 +a)^(1/2),x)
Time = 0.97 (sec) , antiderivative size = 1240, normalized size of antiderivative = 17.97 \[ \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx=\text {Too large to display} \] Input:
integrate(((1+3^(1/2))*a^(1/3)+b^(1/3)*x)/((1-3^(1/2))*a^(1/3)+b^(1/3)*x)/ (b*x^3+a)^(1/2),x, algorithm="fricas")
Output:
[1/2*sqrt(1/3)*a^(1/3)*sqrt((2*sqrt(3) + 3)/(a*b^(2/3)))*log((b^8*x^24 - 1 840*a*b^7*x^21 + 67264*a^2*b^6*x^18 - 58624*a^3*b^5*x^15 + 504064*a^4*b^4* x^12 + 2140160*a^5*b^3*x^9 + 3100672*a^6*b^2*x^6 + 1089536*a^7*b*x^3 + 286 72*a^8 + 4*sqrt(1/3)*sqrt(b*x^3 + a)*((3*b^7*x^22 - 2688*a*b^6*x^19 + 5695 2*a^2*b^5*x^16 - 93504*a^3*b^4*x^13 - 63552*a^4*b^3*x^10 - 377856*a^5*b^2* x^7 - 314880*a^6*b*x^4 - 24576*a^7*x - 2*sqrt(3)*(b^7*x^22 - 764*a*b^6*x^1 9 + 16860*a^2*b^5*x^16 - 19792*a^3*b^4*x^13 + 42368*a^4*b^3*x^10 + 104448* a^5*b^2*x^7 + 90880*a^6*b*x^4 + 7168*a^7*x))*a^(2/3)*b^(2/3) + 6*(81*a*b^7 *x^20 - 4752*a^2*b^6*x^17 + 14472*a^3*b^5*x^14 - 24192*a^4*b^4*x^11 - 3974 4*a^5*b^3*x^8 - 69120*a^6*b^2*x^5 - 13824*a^7*b*x^2 - sqrt(3)*(47*a*b^7*x^ 20 - 2724*a^2*b^6*x^17 + 8976*a^3*b^5*x^14 - 4928*a^4*b^4*x^11 + 32448*a^5 *b^3*x^8 + 37632*a^6*b^2*x^5 + 8192*a^7*b*x^2))*a^(1/3) - 2*(30*a*b^7*x^21 - 5010*a^2*b^6*x^18 + 44640*a^3*b^5*x^15 - 21360*a^4*b^4*x^12 + 79872*a^5 *b^3*x^9 + 233856*a^6*b^2*x^6 + 86016*a^7*b*x^3 + 3072*a^8 - sqrt(3)*(17*a *b^7*x^21 - 2920*a^2*b^6*x^18 + 24864*a^3*b^5*x^15 - 26576*a^4*b^4*x^12 - 56000*a^5*b^3*x^9 - 115968*a^6*b^2*x^6 - 56320*a^7*b*x^3 - 1024*a^8))*b^(1 /3))*sqrt((2*sqrt(3) + 3)/(a*b^(2/3))) + 32*(9*b^7*x^22 - 846*a*b^6*x^19 + 4617*a^2*b^5*x^16 + 5472*a^3*b^4*x^13 + 43776*a^4*b^3*x^10 + 98496*a^5*b^ 2*x^7 + 59328*a^6*b*x^4 + 4608*a^7*x - sqrt(3)*(5*b^7*x^22 - 505*a*b^6*x^1 9 + 2130*a^2*b^5*x^16 - 4928*a^3*b^4*x^13 - 28688*a^4*b^3*x^10 - 53760*...
\[ \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx=\int \frac {\sqrt [3]{a} + \sqrt {3} \sqrt [3]{a} + \sqrt [3]{b} x}{\sqrt {a + b x^{3}} \left (- \sqrt {3} \sqrt [3]{a} + \sqrt [3]{a} + \sqrt [3]{b} x\right )}\, dx \] Input:
integrate(((1+3**(1/2))*a**(1/3)+b**(1/3)*x)/((1-3**(1/2))*a**(1/3)+b**(1/ 3)*x)/(b*x**3+a)**(1/2),x)
Output:
Integral((a**(1/3) + sqrt(3)*a**(1/3) + b**(1/3)*x)/(sqrt(a + b*x**3)*(-sq rt(3)*a**(1/3) + a**(1/3) + b**(1/3)*x)), x)
\[ \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx=\int { \frac {b^{\frac {1}{3}} x + a^{\frac {1}{3}} {\left (\sqrt {3} + 1\right )}}{\sqrt {b x^{3} + a} {\left (b^{\frac {1}{3}} x - a^{\frac {1}{3}} {\left (\sqrt {3} - 1\right )}\right )}} \,d x } \] Input:
integrate(((1+3^(1/2))*a^(1/3)+b^(1/3)*x)/((1-3^(1/2))*a^(1/3)+b^(1/3)*x)/ (b*x^3+a)^(1/2),x, algorithm="maxima")
Output:
integrate((b^(1/3)*x + a^(1/3)*(sqrt(3) + 1))/(sqrt(b*x^3 + a)*(b^(1/3)*x - a^(1/3)*(sqrt(3) - 1))), x)
Timed out. \[ \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx=\text {Timed out} \] Input:
integrate(((1+3^(1/2))*a^(1/3)+b^(1/3)*x)/((1-3^(1/2))*a^(1/3)+b^(1/3)*x)/ (b*x^3+a)^(1/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx=\text {Hanged} \] Input:
int((b^(1/3)*x + a^(1/3)*(3^(1/2) + 1))/((a + b*x^3)^(1/2)*(b^(1/3)*x - a^ (1/3)*(3^(1/2) - 1))),x)
Output:
\text{Hanged}
\[ \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx=6 b^{\frac {2}{3}} a^{\frac {2}{3}} \sqrt {3}\, \left (\int \frac {\sqrt {b \,x^{3}+a}\, x^{2}}{4 a^{\frac {7}{3}}+8 a^{\frac {4}{3}} b \,x^{3}+4 a^{\frac {1}{3}} b^{2} x^{6}-8 b^{\frac {1}{3}} a^{2} x -7 b^{\frac {4}{3}} a \,x^{4}+b^{\frac {7}{3}} x^{7}}d x \right )+6 b^{\frac {2}{3}} a^{\frac {2}{3}} \left (\int \frac {\sqrt {b \,x^{3}+a}\, x^{2}}{4 a^{\frac {7}{3}}+8 a^{\frac {4}{3}} b \,x^{3}+4 a^{\frac {1}{3}} b^{2} x^{6}-8 b^{\frac {1}{3}} a^{2} x -7 b^{\frac {4}{3}} a \,x^{4}+b^{\frac {7}{3}} x^{7}}d x \right )-4 a^{\frac {4}{3}} \sqrt {3}\, \left (\int \frac {\sqrt {b \,x^{3}+a}}{4 a^{\frac {7}{3}}+8 a^{\frac {4}{3}} b \,x^{3}+4 a^{\frac {1}{3}} b^{2} x^{6}-8 b^{\frac {1}{3}} a^{2} x -7 b^{\frac {4}{3}} a \,x^{4}+b^{\frac {7}{3}} x^{7}}d x \right )+2 a^{\frac {1}{3}} \sqrt {3}\, \left (\int \frac {\sqrt {b \,x^{3}+a}\, x^{3}}{4 a^{\frac {7}{3}}+8 a^{\frac {4}{3}} b \,x^{3}+4 a^{\frac {1}{3}} b^{2} x^{6}-8 b^{\frac {1}{3}} a^{2} x -7 b^{\frac {4}{3}} a \,x^{4}+b^{\frac {7}{3}} x^{7}}d x \right ) b -8 a^{\frac {4}{3}} \left (\int \frac {\sqrt {b \,x^{3}+a}}{4 a^{\frac {7}{3}}+8 a^{\frac {4}{3}} b \,x^{3}+4 a^{\frac {1}{3}} b^{2} x^{6}-8 b^{\frac {1}{3}} a^{2} x -7 b^{\frac {4}{3}} a \,x^{4}+b^{\frac {7}{3}} x^{7}}d x \right )+4 a^{\frac {1}{3}} \left (\int \frac {\sqrt {b \,x^{3}+a}\, x^{3}}{4 a^{\frac {7}{3}}+8 a^{\frac {4}{3}} b \,x^{3}+4 a^{\frac {1}{3}} b^{2} x^{6}-8 b^{\frac {1}{3}} a^{2} x -7 b^{\frac {4}{3}} a \,x^{4}+b^{\frac {7}{3}} x^{7}}d x \right ) b +b^{\frac {4}{3}} \left (\int \frac {\sqrt {b \,x^{3}+a}\, x^{4}}{4 a^{\frac {7}{3}}+8 a^{\frac {4}{3}} b \,x^{3}+4 a^{\frac {1}{3}} b^{2} x^{6}-8 b^{\frac {1}{3}} a^{2} x -7 b^{\frac {4}{3}} a \,x^{4}+b^{\frac {7}{3}} x^{7}}d x \right )+4 b^{\frac {1}{3}} \left (\int \frac {\sqrt {b \,x^{3}+a}\, x}{4 a^{\frac {7}{3}}+8 a^{\frac {4}{3}} b \,x^{3}+4 a^{\frac {1}{3}} b^{2} x^{6}-8 b^{\frac {1}{3}} a^{2} x -7 b^{\frac {4}{3}} a \,x^{4}+b^{\frac {7}{3}} x^{7}}d x \right ) a \] Input:
int(((1+3^(1/2))*a^(1/3)+b^(1/3)*x)/((1-3^(1/2))*a^(1/3)+b^(1/3)*x)/(b*x^3 +a)^(1/2),x)
Output:
6*b**(2/3)*a**(2/3)*sqrt(3)*int((sqrt(a + b*x**3)*x**2)/(4*a**(1/3)*a**2 + 8*a**(1/3)*a*b*x**3 + 4*a**(1/3)*b**2*x**6 - 8*b**(1/3)*a**2*x - 7*b**(1/ 3)*a*b*x**4 + b**(1/3)*b**2*x**7),x) + 6*b**(2/3)*a**(2/3)*int((sqrt(a + b *x**3)*x**2)/(4*a**(1/3)*a**2 + 8*a**(1/3)*a*b*x**3 + 4*a**(1/3)*b**2*x**6 - 8*b**(1/3)*a**2*x - 7*b**(1/3)*a*b*x**4 + b**(1/3)*b**2*x**7),x) - 4*a* *(1/3)*sqrt(3)*int(sqrt(a + b*x**3)/(4*a**(1/3)*a**2 + 8*a**(1/3)*a*b*x**3 + 4*a**(1/3)*b**2*x**6 - 8*b**(1/3)*a**2*x - 7*b**(1/3)*a*b*x**4 + b**(1/ 3)*b**2*x**7),x)*a + 2*a**(1/3)*sqrt(3)*int((sqrt(a + b*x**3)*x**3)/(4*a** (1/3)*a**2 + 8*a**(1/3)*a*b*x**3 + 4*a**(1/3)*b**2*x**6 - 8*b**(1/3)*a**2* x - 7*b**(1/3)*a*b*x**4 + b**(1/3)*b**2*x**7),x)*b - 8*a**(1/3)*int(sqrt(a + b*x**3)/(4*a**(1/3)*a**2 + 8*a**(1/3)*a*b*x**3 + 4*a**(1/3)*b**2*x**6 - 8*b**(1/3)*a**2*x - 7*b**(1/3)*a*b*x**4 + b**(1/3)*b**2*x**7),x)*a + 4*a* *(1/3)*int((sqrt(a + b*x**3)*x**3)/(4*a**(1/3)*a**2 + 8*a**(1/3)*a*b*x**3 + 4*a**(1/3)*b**2*x**6 - 8*b**(1/3)*a**2*x - 7*b**(1/3)*a*b*x**4 + b**(1/3 )*b**2*x**7),x)*b + b**(1/3)*int((sqrt(a + b*x**3)*x**4)/(4*a**(1/3)*a**2 + 8*a**(1/3)*a*b*x**3 + 4*a**(1/3)*b**2*x**6 - 8*b**(1/3)*a**2*x - 7*b**(1 /3)*a*b*x**4 + b**(1/3)*b**2*x**7),x)*b + 4*b**(1/3)*int((sqrt(a + b*x**3) *x)/(4*a**(1/3)*a**2 + 8*a**(1/3)*a*b*x**3 + 4*a**(1/3)*b**2*x**6 - 8*b**( 1/3)*a**2*x - 7*b**(1/3)*a*b*x**4 + b**(1/3)*b**2*x**7),x)*a