3.2.0 ∫ (1−√ _ 3) 3 √_a+ 3 √_bx _________________________________ ((1+√ _ 3) 3 √_a+ 3 √_bx)√____

Integrand size = 58, antiderivative size = 69 \[ \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {a+b x^3}}\right )}{\sqrt {3+2 \sqrt {3}} \sqrt [6]{a} \sqrt [3]{b}} \]

-2*arctan(a^(1/6)*(a^(1/3)+b^(1/3)*x)*(3+2*3^(1/2))^(1/2)/(b*x^3+a)^(1/2))/a^(1/6)/b^(1/3)/(3+2*3^(1/2))^(1/2)

Rubi [A] (veriﬁed)

Time = 0.13 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2165, 209} \[ \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {a+b x^3}}\right )}{\sqrt {3+2 \sqrt {3}} \sqrt [6]{a} \sqrt [3]{b}} \]

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]

(-2*ArcTan[(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)*

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;

FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[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)]}, Dist[(1 + k)*(e/d), Subst[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (2 \sqrt [3]{a}\right ) \text {Subst}\left (\int \frac {1}{1+\left (3+2 \sqrt {3}\right ) a x^2} \, dx,x,\frac {1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {a+b x^3}}\right )}{\sqrt [3]{b}} \\ & = -\frac {2 \tan ^{-1}\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}} \\ \end{align*}

Mathematica [A] (veriﬁed)

Time = 7.32 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.22 \[ \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx=\frac {2 \arctan \left (\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\sqrt {3+2 \sqrt {3}} \sqrt [6]{a} \sqrt {a+b x^3}}\right )}{\sqrt {3+2 \sqrt {3}} \sqrt [6]{a} \sqrt [3]{b}} \]

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]

(2*ArcTan[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(Sqrt[3 + 2*Sqrt[3]]*a^(1/6)*Sqrt[a + b*x^3])])/(Sqrt[3

Maple [F]

\[\int \frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right ) \sqrt {b \,x^{3}+a}}d x\]

int((b^(1/3)*x+a^(1/3)*(1-3^(1/2)))/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))/(b*x^3+a)^(1/2),x)

Fricas [A] (veriﬁcation not implemented)

Time = 1.05 (sec) , antiderivative size = 1236, normalized size of antiderivative = 17.91 \[ \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx=\text {Too large to display} \]

integrate((b^(1/3)*x+a^(1/3)*(1-3^(1/2)))/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))/(b*x^3+a)^(1/2),x, algorithm="fricas

[1/2*sqrt(1/3)*a^(1/3)*sqrt(-(2*sqrt(3) - 3)/(a*b^(2/3)))*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)*sqrt(b*x^3 + a)*((3*b^7*x^22 - 2688*a*b^6*x^19 + 56952*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^19 + 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

- 39744*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^19 + 2130*a^2*b^5*x^16

- 4928*a^3*b^4*x^13 - 28688*a^4*b^3*x^10 - 53760*a^5*b^2*x^7 - 35200*a^6*b*x^4 - 2560*a^7*x))*a^(2/3)*b^(1/3)

- 8*(3*b^7*x^23 - 1077*a*b^6*x^20 + 13320*a^2*b^5*x^17 - 19200*a^3*b^4*x^14 - 111360*a^4*b^3*x^11 - 345024*a^5

*b^2*x^8 - 328704*a^6*b*x^5 - 61440*a^7*x^2 + 2*sqrt(3)*(b^7*x^23 - 299*a*b^6*x^20 + 4260*a^2*b^5*x^17 + 1520*

a^3*b^4*x^14 + 26720*a^4*b^3*x^11 + 105024*a^5*b^2*x^8 + 93184*a^6*b*x^5 + 17920*a^7*x^2))*a^(1/3)*b^(2/3) - 3

2*sqrt(3)*(35*a*b^7*x^21 - 1141*a^2*b^6*x^18 + 2544*a^3*b^5*x^15 + 6760*a^4*b^4*x^12 + 39520*a^5*b^3*x^9 + 556

80*a^6*b^2*x^6 + 19712*a^7*b*x^3 + 512*a^8))/(b^8*x^24 + 80*a*b^7*x^21 + 2368*a^2*b^6*x^18 + 30080*a^3*b^5*x^1

5 + 121984*a^4*b^4*x^12 - 240640*a^5*b^3*x^9 + 151552*a^6*b^2*x^6 - 40960*a^7*b*x^3 + 4096*a^8)), -sqrt(1/3)*a

^(1/3)*sqrt((2*sqrt(3) - 3)/(a*b^(2/3)))*arctan(-1/2*sqrt(1/3)*(a^(1/3)*b*x^2 - 2*(sqrt(3)*x + 2*x)*a^(2/3)*b^

(2/3) - 2*(sqrt(3)*a + a)*b^(1/3))*sqrt((2*sqrt(3) - 3)/(a*b^(2/3)))/sqrt(b*x^3 + a))]

Sympy [F]

\[ \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx=\int \frac {- \sqrt {3} \sqrt [3]{a} + \sqrt [3]{a} + \sqrt [3]{b} x}{\sqrt {a + b x^{3}} \left (\sqrt [3]{a} + \sqrt {3} \sqrt [3]{a} + \sqrt [3]{b} x\right )}\, dx \]

integrate((b**(1/3)*x+a**(1/3)*(1-3**(1/2)))/(b**(1/3)*x+a**(1/3)*(1+3**(1/2)))/(b*x**3+a)**(1/2),x)

Integral((-sqrt(3)*a**(1/3) + a**(1/3) + b**(1/3)*x)/(sqrt(a + b*x**3)*(a**(1/3) + sqrt(3)*a**(1/3) + b**(1/3)

Maxima [F]

\[ \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx=\int { \frac {b^{\frac {1}{3}} x - a^{\frac {1}{3}} {\left (\sqrt {3} - 1\right )}}{\sqrt {b x^{3} + a} {\left (b^{\frac {1}{3}} x + a^{\frac {1}{3}} {\left (\sqrt {3} + 1\right )}\right )}} \,d x } \]

integrate((b^(1/3)*x+a^(1/3)*(1-3^(1/2)))/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))/(b*x^3+a)^(1/2),x, algorithm="maxima

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)

Giac [F(-1)]

Timed out. \[ \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx=\text {Timed out} \]

integrate((b^(1/3)*x+a^(1/3)*(1-3^(1/2)))/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))/(b*x^3+a)^(1/2),x, algorithm="giac")

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx=\text {Hanged} \]

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)

\(\int \frac {(1-\sqrt {3}) \sqrt [3]{a}+\sqrt [3]{b} x}{((1+\sqrt {3}) \sqrt [3]{a}+\sqrt [3]{b} x) \sqrt {a+b x^3}} \, dx\) [117]

Optimal result