Integrand size = 52, antiderivative size = 73 \[ \int \frac {1+\sqrt {3}+\sqrt [3]{\frac {b}{a}} x}{\left (1-\sqrt {3}+\sqrt [3]{\frac {b}{a}} x\right ) \sqrt {a+b x^3}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {a} \left (1+\sqrt [3]{\frac {b}{a}} x\right )}{\sqrt {a+b x^3}}\right )}{\sqrt {-3+2 \sqrt {3}} \sqrt {a} \sqrt [3]{\frac {b}{a}}} \]
-2*arctanh((1+(b/a)^(1/3)*x)*a^(1/2)*(-3+2*3^(1/2))^(1/2)/(b*x^3+a)^(1/2)) /(b/a)^(1/3)/a^(1/2)/(-3+2*3^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 10.97 (sec) , antiderivative size = 663, normalized size of antiderivative = 9.08 \[ \int \frac {1+\sqrt {3}+\sqrt [3]{\frac {b}{a}} x}{\left (1-\sqrt {3}+\sqrt [3]{\frac {b}{a}} x\right ) \sqrt {a+b x^3}} \, dx=\frac {x \left (12 \left (-3+\sqrt {3}\right ) \sqrt [3]{\frac {b}{a}} x \sqrt {1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {b x^3}{a},\frac {b x^3}{-10 a+6 \sqrt {3} a}\right )-8 \left (\frac {b}{a}\right )^{2/3} x^2 \sqrt {3+\frac {3 b x^3}{a}} \operatorname {AppellF1}\left (1,\frac {1}{2},1,2,-\frac {b x^3}{a},\frac {b x^3}{-10 a+6 \sqrt {3} a}\right )-\frac {3 \left (-18176 a^3 \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},\frac {b x^3}{-10 a+6 \sqrt {3} a}\right )+10496 \sqrt {3} a^3 \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},\frac {b x^3}{-10 a+6 \sqrt {3} a}\right )+b x^3 \left (2 \left (-5+3 \sqrt {3}\right ) a-b x^3\right ) \sqrt {1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},1,\frac {7}{3},-\frac {b x^3}{a},\frac {b x^3}{-10 a+6 \sqrt {3} a}\right ) \left (8 \left (-5+3 \sqrt {3}\right ) a \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {b x^3}{10 a-6 \sqrt {3} a}\right )+3 b x^3 \left (\operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {b x^3}{10 a-6 \sqrt {3} a}\right )+\left (5-3 \sqrt {3}\right ) \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {b x^3}{10 a-6 \sqrt {3} a}\right )\right )\right )\right )}{a \left (2 \left (-5+3 \sqrt {3}\right ) a-b x^3\right ) \left (8 \left (-5+3 \sqrt {3}\right ) a \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {b x^3}{10 a-6 \sqrt {3} a}\right )+3 b x^3 \left (\operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {b x^3}{10 a-6 \sqrt {3} a}\right )+\left (5-3 \sqrt {3}\right ) \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {b x^3}{10 a-6 \sqrt {3} a}\right )\right )\right )}\right )}{24 \left (-5+3 \sqrt {3}\right ) \sqrt {a+b x^3}} \]
(x*(12*(-3 + Sqrt[3])*(b/a)^(1/3)*x*Sqrt[1 + (b*x^3)/a]*AppellF1[2/3, 1/2, 1, 5/3, -((b*x^3)/a), (b*x^3)/(-10*a + 6*Sqrt[3]*a)] - 8*(b/a)^(2/3)*x^2* Sqrt[3 + (3*b*x^3)/a]*AppellF1[1, 1/2, 1, 2, -((b*x^3)/a), (b*x^3)/(-10*a + 6*Sqrt[3]*a)] - (3*(-18176*a^3*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), (b*x^3)/(-10*a + 6*Sqrt[3]*a)] + 10496*Sqrt[3]*a^3*AppellF1[1/3, 1/2, 1, 4 /3, -((b*x^3)/a), (b*x^3)/(-10*a + 6*Sqrt[3]*a)] + b*x^3*(2*(-5 + 3*Sqrt[3 ])*a - b*x^3)*Sqrt[1 + (b*x^3)/a]*AppellF1[4/3, 1/2, 1, 7/3, -((b*x^3)/a), (b*x^3)/(-10*a + 6*Sqrt[3]*a)]*(8*(-5 + 3*Sqrt[3])*a*AppellF1[1/3, 1/2, 1 , 4/3, -((b*x^3)/a), -((b*x^3)/(10*a - 6*Sqrt[3]*a))] + 3*b*x^3*(AppellF1[ 4/3, 1/2, 2, 7/3, -((b*x^3)/a), -((b*x^3)/(10*a - 6*Sqrt[3]*a))] + (5 - 3* Sqrt[3])*AppellF1[4/3, 3/2, 1, 7/3, -((b*x^3)/a), -((b*x^3)/(10*a - 6*Sqrt [3]*a))]))))/(a*(2*(-5 + 3*Sqrt[3])*a - b*x^3)*(8*(-5 + 3*Sqrt[3])*a*Appel lF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((b*x^3)/(10*a - 6*Sqrt[3]*a))] + 3*b *x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((b*x^3)/a), -((b*x^3)/(10*a - 6*Sqrt[3] *a))] + (5 - 3*Sqrt[3])*AppellF1[4/3, 3/2, 1, 7/3, -((b*x^3)/a), -((b*x^3) /(10*a - 6*Sqrt[3]*a))])))))/(24*(-5 + 3*Sqrt[3])*Sqrt[a + b*x^3])
Time = 0.35 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, 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.
| \(\displaystyle \int \frac {x \sqrt [3]{\frac {b}{a}}+\sqrt {3}+1}{\left (x \sqrt [3]{\frac {b}{a}}-\sqrt {3}+1\right ) \sqrt {a+b x^3}} \, dx\) |
| \(\Big \downarrow \) 2565 |
| \(\displaystyle -\frac {2 \int \frac {1}{\frac {\left (3-2 \sqrt {3}\right ) a \left (\sqrt [3]{\frac {b}{a}} x+1\right )^2}{b x^3+a}+1}d\frac {\sqrt [3]{\frac {b}{a}} x+1}{\sqrt {b x^3+a}}}{\sqrt [3]{\frac {b}{a}}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {2 \text {arctanh}\left (\frac {\sqrt {2 \sqrt {3}-3} \sqrt {a} \left (x \sqrt [3]{\frac {b}{a}}+1\right )}{\sqrt {a+b x^3}}\right )}{\sqrt {2 \sqrt {3}-3} \sqrt {a} \sqrt [3]{\frac {b}{a}}}\) |
(-2*ArcTanh[(Sqrt[-3 + 2*Sqrt[3]]*Sqrt[a]*(1 + (b/a)^(1/3)*x))/Sqrt[a + b* x^3]])/(Sqrt[-3 + 2*Sqrt[3]]*Sqrt[a]*(b/a)^(1/3))
3.2.9.3.1 Defintions of rubi rules used
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]
\[\int \frac {1+\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\sqrt {3}}{\left (1+\left (\frac {b}{a}\right )^{\frac {1}{3}} x -\sqrt {3}\right ) \sqrt {b \,x^{3}+a}}d x\]
Time = 0.72 (sec) , antiderivative size = 1273, normalized size of antiderivative = 17.44 \[ \int \frac {1+\sqrt {3}+\sqrt [3]{\frac {b}{a}} x}{\left (1-\sqrt {3}+\sqrt [3]{\frac {b}{a}} x\right ) \sqrt {a+b x^3}} \, dx=\text {Too large to display} \]
integrate((1+(b/a)^(1/3)*x+3^(1/2))/(1+(b/a)^(1/3)*x-3^(1/2))/(b*x^3+a)^(1 /2),x, algorithm="fricas")
[1/2*sqrt(1/3)*sqrt((2*sqrt(3) + 3)*(b/a)^(1/3)/b)*log((b^8*x^24 - 1840*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 + 28672*a^8 + 4*sqrt(1/3)*(486*a*b^7*x^20 - 28512*a^2*b^6*x^17 + 86832*a^3*b^5*x^14 - 145152*a^4*b^4*x^11 - 238464*a^5*b^3*x^8 - 414720*a^6*b^2*x^5 - 82944*a^7 *b*x^2 + (3*a*b^7*x^22 - 2688*a^2*b^6*x^19 + 56952*a^3*b^5*x^16 - 93504*a^ 4*b^4*x^13 - 63552*a^5*b^3*x^10 - 377856*a^6*b^2*x^7 - 314880*a^7*b*x^4 - 24576*a^8*x - 2*sqrt(3)*(a*b^7*x^22 - 764*a^2*b^6*x^19 + 16860*a^3*b^5*x^1 6 - 19792*a^4*b^4*x^13 + 42368*a^5*b^3*x^10 + 104448*a^6*b^2*x^7 + 90880*a ^7*b*x^4 + 7168*a^8*x))*(b/a)^(2/3) - 6*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 + 376 32*a^6*b^2*x^5 + 8192*a^7*b*x^2) - 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 - 115 968*a^6*b^2*x^6 - 56320*a^7*b*x^3 - 1024*a^8))*(b/a)^(1/3))*sqrt(b*x^3 + a )*sqrt((2*sqrt(3) + 3)*(b/a)^(1/3)/b) - 8*(3*a*b^7*x^23 - 1077*a^2*b^6*x^2 0 + 13320*a^3*b^5*x^17 - 19200*a^4*b^4*x^14 - 111360*a^5*b^3*x^11 - 345024 *a^6*b^2*x^8 - 328704*a^7*b*x^5 - 61440*a^8*x^2 - 2*sqrt(3)*(a*b^7*x^23 - 299*a^2*b^6*x^20 + 4260*a^3*b^5*x^17 + 1520*a^4*b^4*x^14 + 26720*a^5*b^...
\[ \int \frac {1+\sqrt {3}+\sqrt [3]{\frac {b}{a}} x}{\left (1-\sqrt {3}+\sqrt [3]{\frac {b}{a}} x\right ) \sqrt {a+b x^3}} \, dx=\int \frac {x \sqrt [3]{\frac {b}{a}} + 1 + \sqrt {3}}{\sqrt {a + b x^{3}} \left (x \sqrt [3]{\frac {b}{a}} - \sqrt {3} + 1\right )}\, dx \]
Integral((x*(b/a)**(1/3) + 1 + sqrt(3))/(sqrt(a + b*x**3)*(x*(b/a)**(1/3) - sqrt(3) + 1)), x)
\[ \int \frac {1+\sqrt {3}+\sqrt [3]{\frac {b}{a}} x}{\left (1-\sqrt {3}+\sqrt [3]{\frac {b}{a}} x\right ) \sqrt {a+b x^3}} \, dx=\int { \frac {x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \sqrt {3} + 1}{\sqrt {b x^{3} + a} {\left (x \left (\frac {b}{a}\right )^{\frac {1}{3}} - \sqrt {3} + 1\right )}} \,d x } \]
integrate((1+(b/a)^(1/3)*x+3^(1/2))/(1+(b/a)^(1/3)*x-3^(1/2))/(b*x^3+a)^(1 /2),x, algorithm="maxima")
Exception generated. \[ \int \frac {1+\sqrt {3}+\sqrt [3]{\frac {b}{a}} x}{\left (1-\sqrt {3}+\sqrt [3]{\frac {b}{a}} x\right ) \sqrt {a+b x^3}} \, dx=\text {Exception raised: TypeError} \]
integrate((1+(b/a)^(1/3)*x+3^(1/2))/(1+(b/a)^(1/3)*x-3^(1/2))/(b*x^3+a)^(1 /2),x, algorithm="giac")
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(con st gen &
Timed out. \[ \int \frac {1+\sqrt {3}+\sqrt [3]{\frac {b}{a}} x}{\left (1-\sqrt {3}+\sqrt [3]{\frac {b}{a}} x\right ) \sqrt {a+b x^3}} \, dx=\int \frac {\sqrt {3}+x\,{\left (\frac {b}{a}\right )}^{1/3}+1}{\sqrt {b\,x^3+a}\,\left (x\,{\left (\frac {b}{a}\right )}^{1/3}-\sqrt {3}+1\right )} \,d x \]