3.109 \(\int \frac {1+\sqrt {3}+\sqrt [3]{\frac {b}{a}} x}{(1-\sqrt {3}+\sqrt [3]{\frac {b}{a}} x) \sqrt {a+b x^3}} \, dx\)
Optimal. Leaf size=73 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {2 \sqrt {3}-3} \sqrt {a} \left (x \sqrt [3]{\frac {b}{a}}+1\right )}{\sqrt {a+b x^3}}\right )}{\sqrt {2 \sqrt {3}-3} \sqrt {a} \sqrt [3]{\frac {b}{a}}} \]
[Out]
-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)
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Rubi [A] time = 0.20, antiderivative size = 73, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 2, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used =
{2140, 206} \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {2 \sqrt {3}-3} \sqrt {a} \left (x \sqrt [3]{\frac {b}{a}}+1\right )}{\sqrt {a+b x^3}}\right )}{\sqrt {2 \sqrt {3}-3} \sqrt {a} \sqrt [3]{\frac {b}{a}}} \]
Antiderivative was successfully verified.
[In]
Int[(1 + Sqrt[3] + (b/a)^(1/3)*x)/((1 - Sqrt[3] + (b/a)^(1/3)*x)*Sqrt[a + b*x^3]),x]
[Out]
(-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))
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 2140
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*} \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 \operatorname {Subst}\left (\int \frac {1}{1+\left (3-2 \sqrt {3}\right ) a x^2} \, dx,x,\frac {1+\sqrt [3]{\frac {b}{a}} x}{\sqrt {a+b x^3}}\right )}{\sqrt [3]{\frac {b}{a}}}\\ &=-\frac {2 \tanh ^{-1}\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}}}\\ \end {align*}
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Mathematica [C] time = 1.28, size = 663, normalized size = 9.08 \[ \frac {x \left (-\frac {3 \left (10496 \sqrt {3} a^3 F_1\left (\frac {1}{3};\frac {1}{2},1;\frac {4}{3};-\frac {b x^3}{a},\frac {b x^3}{6 \sqrt {3} a-10 a}\right )-18176 a^3 F_1\left (\frac {1}{3};\frac {1}{2},1;\frac {4}{3};-\frac {b x^3}{a},\frac {b x^3}{6 \sqrt {3} a-10 a}\right )+b x^3 \left (2 \left (3 \sqrt {3}-5\right ) a-b x^3\right ) \sqrt {\frac {b x^3}{a}+1} F_1\left (\frac {4}{3};\frac {1}{2},1;\frac {7}{3};-\frac {b x^3}{a},\frac {b x^3}{6 \sqrt {3} a-10 a}\right ) \left (3 b x^3 \left (F_1\left (\frac {4}{3};\frac {1}{2},2;\frac {7}{3};-\frac {b x^3}{a},-\frac {b x^3}{10 a-6 \sqrt {3} a}\right )+\left (5-3 \sqrt {3}\right ) F_1\left (\frac {4}{3};\frac {3}{2},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {b x^3}{10 a-6 \sqrt {3} a}\right )\right )+8 \left (3 \sqrt {3}-5\right ) a F_1\left (\frac {1}{3};\frac {1}{2},1;\frac {4}{3};-\frac {b x^3}{a},-\frac {b x^3}{10 a-6 \sqrt {3} a}\right )\right )\right )}{a \left (2 \left (3 \sqrt {3}-5\right ) a-b x^3\right ) \left (3 b x^3 \left (F_1\left (\frac {4}{3};\frac {1}{2},2;\frac {7}{3};-\frac {b x^3}{a},-\frac {b x^3}{10 a-6 \sqrt {3} a}\right )+\left (5-3 \sqrt {3}\right ) F_1\left (\frac {4}{3};\frac {3}{2},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {b x^3}{10 a-6 \sqrt {3} a}\right )\right )+8 \left (3 \sqrt {3}-5\right ) a F_1\left (\frac {1}{3};\frac {1}{2},1;\frac {4}{3};-\frac {b x^3}{a},-\frac {b x^3}{10 a-6 \sqrt {3} a}\right )\right )}+12 \left (\sqrt {3}-3\right ) x \sqrt [3]{\frac {b}{a}} \sqrt {\frac {b x^3}{a}+1} F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};-\frac {b x^3}{a},\frac {b x^3}{6 \sqrt {3} a-10 a}\right )-8 x^2 \left (\frac {b}{a}\right )^{2/3} \sqrt {\frac {3 b x^3}{a}+3} F_1\left (1;\frac {1}{2},1;2;-\frac {b x^3}{a},\frac {b x^3}{6 \sqrt {3} a-10 a}\right )\right )}{24 \left (3 \sqrt {3}-5\right ) \sqrt {a+b x^3}} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(1 + Sqrt[3] + (b/a)^(1/3)*x)/((1 - Sqrt[3] + (b/a)^(1/3)*x)*Sqrt[a + b*x^3]),x]
[Out]
(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)] + 1
0496*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*Sq
rt[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*(App
ellF1[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*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))])))))/(24*(-5 + 3*Sqrt[3])*Sqrt[a + b*x^3])
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fricas [A] time = 1.18, size = 1273, normalized size = 17.44 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
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")
[Out]
[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 - 5862
4*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 - 9350
4*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^16 - 19792*a^4*b^4*x^13 + 42368*a^5*b^3*x^10 + 104448*a^6*b^2*x^7 + 9
0880*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 + 37632*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 + 3
072*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/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^20 + 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^3*x^11 + 105024*a^6*b^2*x^8 + 93184*a^7*b*x^5 + 17920*a^8*x^2))
*(b/a)^(2/3) + 32*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 + 55680*a^6*b^2*x^6 + 19712*a^7*b*x^3 + 512*a^8) + 32*(9*a*b^7*x^22 - 846*a^2*b^6*x^19 + 4617*a^3*b
^5*x^16 + 5472*a^4*b^4*x^13 + 43776*a^5*b^3*x^10 + 98496*a^6*b^2*x^7 + 59328*a^7*b*x^4 + 4608*a^8*x - sqrt(3)*
(5*a*b^7*x^22 - 505*a^2*b^6*x^19 + 2130*a^3*b^5*x^16 - 4928*a^4*b^4*x^13 - 28688*a^5*b^3*x^10 - 53760*a^6*b^2*
x^7 - 35200*a^7*b*x^4 - 2560*a^8*x))*(b/a)^(1/3))/(b^8*x^24 + 80*a*b^7*x^21 + 2368*a^2*b^6*x^18 + 30080*a^3*b^
5*x^15 + 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)*sqrt(-(2*sqrt(3) + 3)*(b/a)^(1/3)/b)*arctan(1/2*sqrt(1/3)*(b*x^2 + 2*(sqrt(3)*a*x - 2*a*x)*(b/a)^(2/3) + 2*
(sqrt(3)*a - a)*(b/a)^(1/3))*sqrt(-(2*sqrt(3) + 3)*(b/a)^(1/3)/b)/sqrt(b*x^3 + a))]
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
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")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const ge
n & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueEvaluation time: 0.41index.cc index_m oper
ator + Error: Bad Argument Value
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maple [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {b}{a}\right )^{\frac {1}{3}} x +1+\sqrt {3}}{\left (\left (\frac {b}{a}\right )^{\frac {1}{3}} x +1-\sqrt {3}\right ) \sqrt {b \,x^{3}+a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((1+(1/a*b)^(1/3)*x+3^(1/2))/(1+(1/a*b)^(1/3)*x-3^(1/2))/(b*x^3+a)^(1/2),x)
[Out]
int((1+(1/a*b)^(1/3)*x+3^(1/2))/(1+(1/a*b)^(1/3)*x-3^(1/2))/(b*x^3+a)^(1/2),x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \sqrt {3} + 1}{\sqrt {b x^{3} + a} {\left (x \left (\frac {b}{a}\right )^{\frac {1}{3}} - \sqrt {3} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
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")
[Out]
integrate((x*(b/a)^(1/3) + sqrt(3) + 1)/(sqrt(b*x^3 + a)*(x*(b/a)^(1/3) - sqrt(3) + 1)), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {3}+x\,{\left (\frac {b}{a}\right )}^{1/3}+1}{\sqrt {b\,x^3+a}\,\left (x\,{\left (\frac {b}{a}\right )}^{1/3}-\sqrt {3}+1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((3^(1/2) + x*(b/a)^(1/3) + 1)/((a + b*x^3)^(1/2)*(x*(b/a)^(1/3) - 3^(1/2) + 1)),x)
[Out]
int((3^(1/2) + x*(b/a)^(1/3) + 1)/((a + b*x^3)^(1/2)*(x*(b/a)^(1/3) - 3^(1/2) + 1)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \sqrt [3]{\frac {b}{a}} + 1 + \sqrt {3}}{\sqrt {a + b x^{3}} \left (x \sqrt [3]{\frac {b}{a}} - \sqrt {3} + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
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)
[Out]
Integral((x*(b/a)**(1/3) + 1 + sqrt(3))/(sqrt(a + b*x**3)*(x*(b/a)**(1/3) - sqrt(3) + 1)), x)
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