3.108 \(\int \frac{(1+\sqrt{3}) \sqrt [3]{a}+\sqrt [3]{b} x}{((1-\sqrt{3}) \sqrt [3]{a}+\sqrt [3]{b} x) \sqrt{-a-b x^3}} \, dx\)
Optimal. Leaf size=72 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{-a-b x^3}}\right )}{\sqrt{2 \sqrt{3}-3} \sqrt [6]{a} \sqrt [3]{b}} \]
[Out]
(-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)*b^(1/3))
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Rubi [A] time = 0.168736, antiderivative size = 72, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 2, integrand size = 61, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033, Rules used =
{2140, 203} \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{-a-b x^3}}\right )}{\sqrt{2 \sqrt{3}-3} \sqrt [6]{a} \sqrt [3]{b}} \]
Antiderivative was successfully verified.
[In]
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]
[Out]
(-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)*b^(1/3))
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]
Rule 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rubi steps
\begin{align*} \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{\left (2 \sqrt [3]{a}\right ) \operatorname{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 [C] time = 0.530354, size = 325, normalized size = 4.51 \[ \frac{2 \sqrt{\frac{\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (\frac{4 (-1)^{5/6} \left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1} \Pi \left (\frac{2 \sqrt{3}}{-3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )}{\left ((1+2 i) \sqrt{3}-3 i\right ) \sqrt [3]{b}}-\frac{\left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\sqrt [6]{-1}-\frac{i \sqrt [3]{b} x}{\sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt{\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}}\right )}{\sqrt{-a-b x^3}} \]
Warning: Unable to verify antiderivative.
[In]
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]
[Out]
(2*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(-((((-1)^(1/3)*a^(1/3) - b^(1/3)*x)*Sqrt[(-1)^(1/6)
- (I*b^(1/3)*x)/a^(1/3)]*EllipticF[ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]],
(-1)^(1/3)])/(3^(1/4)*b^(1/3)*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))])) + (4*(-1)^(
5/6)*(1 + (-1)^(1/3))*a^(1/3)*Sqrt[1 - (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(2*Sqrt[3])/(-3
*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)
])/((-3*I + (1 + 2*I)*Sqrt[3])*b^(1/3))))/Sqrt[-a - b*x^3]
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Maple [F] time = 0.073, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \sqrt [3]{b}x+\sqrt [3]{a} \left ( 1+\sqrt{3} \right ) \right ) \left ( \sqrt [3]{b}x+\sqrt [3]{a} \left ( 1-\sqrt{3} \right ) \right ) ^{-1}{\frac{1}{\sqrt{-b{x}^{3}-a}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
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)
[Out]
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)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
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="maxim
a")
[Out]
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)
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Fricas [B] time = 6.39838, size = 3348, normalized size = 46.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
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="frica
s")
[Out]
[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 + 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 + 984
96*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 - 492
8*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) + 4*sqrt
(1/3)*((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 - 1979
2*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))*sqrt(-b*x^3 - a)*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 - 4
928*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))*sqrt(-b*x^3 - a)*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))*sqrt(-b*x^3 - a)*b^(1/3))*sqr
t(-(2*sqrt(3) + 3)/(a*b^(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))/(b^8*x^24 + 80*a*b^7*x^21 + 23
68*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)*a^(1/3)*sqrt((2*sqrt(3) + 3)/(a*b^(2/3)))*arctan(1/2*sqrt(1/3)*(sqrt(-b*x^3 -
a)*a^(1/3)*b*x^2 + 2*sqrt(-b*x^3 - a)*(sqrt(3)*x - 2*x)*a^(2/3)*b^(2/3) + 2*sqrt(-b*x^3 - a)*(sqrt(3)*a - a)*
b^(1/3))*sqrt((2*sqrt(3) + 3)/(a*b^(2/3)))/(b*x^3 + a))]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
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)
[Out]
Integral((a**(1/3) + sqrt(3)*a**(1/3) + b**(1/3)*x)/(sqrt(-a - b*x**3)*(-sqrt(3)*a**(1/3) + a**(1/3) + b**(1/3
)*x)), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
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"
)
[Out]
sage0*x