3.3.48 \(\int \frac {x}{\sqrt {-a-b x^3} (-2 (5+3 \sqrt {3}) a-b x^3)} \, dx\)

Optimal. Leaf size=330 \[ \frac {\left (2-\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \sqrt [6]{a} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {-a-b x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac {\left (2-\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {-a-b x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac {\left (2-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {-a-b x^3}}\right )}{2 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}}-\frac {\left (2-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt {-a-b x^3}}{\sqrt {2} 3^{3/4} \sqrt {a}}\right )}{3 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}} \]

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Rubi [A]  time = 0.07, antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {488} \begin {gather*} \frac {\left (2-\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \sqrt [6]{a} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {-a-b x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac {\left (2-\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {-a-b x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac {\left (2-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {-a-b x^3}}\right )}{2 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}}-\frac {\left (2-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt {-a-b x^3}}{\sqrt {2} 3^{3/4} \sqrt {a}}\right )}{3 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(Sqrt[-a - b*x^3]*(-2*(5 + 3*Sqrt[3])*a - b*x^3)),x]

[Out]

((2 - Sqrt[3])*ArcTan[(3^(1/4)*a^(1/6)*((1 + Sqrt[3])*a^(1/3) - 2*b^(1/3)*x))/(Sqrt[2]*Sqrt[-a - b*x^3])])/(3*
Sqrt[2]*3^(1/4)*a^(5/6)*b^(2/3)) + ((2 - Sqrt[3])*ArcTan[(3^(1/4)*(1 - Sqrt[3])*a^(1/6)*(a^(1/3) + b^(1/3)*x))
/(Sqrt[2]*Sqrt[-a - b*x^3])])/(6*Sqrt[2]*3^(1/4)*a^(5/6)*b^(2/3)) + ((2 - Sqrt[3])*ArcTanh[(3^(1/4)*(1 + Sqrt[
3])*a^(1/6)*(a^(1/3) + b^(1/3)*x))/(Sqrt[2]*Sqrt[-a - b*x^3])])/(2*Sqrt[2]*3^(3/4)*a^(5/6)*b^(2/3)) - ((2 - Sq
rt[3])*ArcTanh[((1 - Sqrt[3])*Sqrt[-a - b*x^3])/(Sqrt[2]*3^(3/4)*Sqrt[a])])/(3*Sqrt[2]*3^(3/4)*a^(5/6)*b^(2/3)
)

Rule 488

Int[(x_)/(Sqrt[(a_) + (b_.)*(x_)^3]*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[b/a, 3], r = Simplify[(b
*c - 10*a*d)/(6*a*d)]}, Simp[(q*(2 - r)*ArcTanh[((1 - r)*Sqrt[a + b*x^3])/(Sqrt[2]*Rt[-a, 2]*r^(3/2))])/(3*Sqr
t[2]*Rt[-a, 2]*d*r^(3/2)), x] + (-Simp[(q*(2 - r)*ArcTanh[(Rt[-a, 2]*Sqrt[r]*(1 + r)*(1 + q*x))/(Sqrt[2]*Sqrt[
a + b*x^3])])/(2*Sqrt[2]*Rt[-a, 2]*d*r^(3/2)), x] - Simp[(q*(2 - r)*ArcTan[(Rt[-a, 2]*Sqrt[r]*(1 + r - 2*q*x))
/(Sqrt[2]*Sqrt[a + b*x^3])])/(3*Sqrt[2]*Rt[-a, 2]*d*Sqrt[r]), x] - Simp[(q*(2 - r)*ArcTan[(Rt[-a, 2]*(1 - r)*S
qrt[r]*(1 + q*x))/(Sqrt[2]*Sqrt[a + b*x^3])])/(6*Sqrt[2]*Rt[-a, 2]*d*Sqrt[r]), x])] /; FreeQ[{a, b, c, d}, x]
&& NeQ[b*c - a*d, 0] && EqQ[b^2*c^2 - 20*a*b*c*d - 8*a^2*d^2, 0] && NegQ[a]

Rubi steps

\begin {align*} \int \frac {x}{\sqrt {-a-b x^3} \left (-2 \left (5+3 \sqrt {3}\right ) a-b x^3\right )} \, dx &=\frac {\left (2-\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \sqrt [6]{a} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {-a-b x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac {\left (2-\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {-a-b x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac {\left (2-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {-a-b x^3}}\right )}{2 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}}-\frac {\left (2-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt {-a-b x^3}}{\sqrt {2} 3^{3/4} \sqrt {a}}\right )}{3 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}}\\ \end {align*}

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Mathematica [C]  time = 0.11, size = 87, normalized size = 0.26 \begin {gather*} -\frac {x^2 \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 )}{\left (12 \sqrt {3} a+20 a\right ) \sqrt {-a-b x^3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[x/(Sqrt[-a - b*x^3]*(-2*(5 + 3*Sqrt[3])*a - b*x^3)),x]

[Out]

-((x^2*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))])/((20*a +
12*Sqrt[3]*a)*Sqrt[-a - b*x^3]))

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IntegrateAlgebraic [F]  time = 31.17, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {-a-b x^3} \left (-2 \left (5+3 \sqrt {3}\right ) a-b x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x/(Sqrt[-a - b*x^3]*(-2*(5 + 3*Sqrt[3])*a - b*x^3)),x]

[Out]

Defer[IntegrateAlgebraic][x/(Sqrt[-a - b*x^3]*(-2*(5 + 3*Sqrt[3])*a - b*x^3)), x]

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-b*x^3-2*a*(5+3*3^(1/2)))/(-b*x^3-a)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-b*x^3-2*a*(5+3*3^(1/2)))/(-b*x^3-a)^(1/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:index.cc index_m
operator + Error: Bad Argument Value

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maple [C]  time = 0.36, size = 541, normalized size = 1.64 \begin {gather*} \frac {i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (2 x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}-i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right ) b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {\left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right ) b}{-3 \left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i \left (2 x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right ) b}{2 \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (4 \sqrt {3}\, \RootOf \left (\textit {\_Z}^{3} b +6 \sqrt {3}\, a +10 a \right )^{2} b^{2}-6 \RootOf \left (\textit {\_Z}^{3} b +6 \sqrt {3}\, a +10 a \right )^{2} b^{2}-3 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \RootOf \left (\textit {\_Z}^{3} b +6 \sqrt {3}\, a +10 a \right ) b -2 \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3} b +6 \sqrt {3}\, a +10 a \right ) b +6 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3} b +6 \sqrt {3}\, a +10 a \right ) b +3 \left (-a \,b^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3} b +6 \sqrt {3}\, a +10 a \right ) b +3 i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}-2 \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}}-6 i \left (-a \,b^{2}\right )^{\frac {2}{3}}+3 \left (-a \,b^{2}\right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {2 i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3} b +6 \sqrt {3}\, a +10 a \right )^{2} b -4 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3} b +6 \sqrt {3}\, a +10 a \right )^{2} b +i \sqrt {3}\, a b +2 \sqrt {3}\, a b -2 i a b -3 a b -i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3} b +6 \sqrt {3}\, a +10 a \right )+2 \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3} b +6 \sqrt {3}\, a +10 a \right )+2 i \left (-a \,b^{2}\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3} b +6 \sqrt {3}\, a +10 a \right )-3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3} b +6 \sqrt {3}\, a +10 a \right )}{6 a b}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) b}}\right )}{27 a \,b^{3} \sqrt {-b \,x^{3}-a}\, \RootOf \left (\textit {\_Z}^{3} b +6 \sqrt {3}\, a +10 a \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(-b*x^3-2*a*(5+3*3^(1/2)))/(-b*x^3-a)^(1/2),x)

[Out]

1/27*I/b^3/a*2^(1/2)*sum(1/_alpha*(-a*b^2)^(1/3)*(1/2*I*(2*x+((-a*b^2)^(1/3)-I*3^(1/2)*(-a*b^2)^(1/3))/b)/(-a*
b^2)^(1/3)*b)^(1/2)*((x-(-a*b^2)^(1/3)/b)/(-3*(-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1/3))*b)^(1/2)*(-1/2*I*(2*x+(
(-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1/3))/b)/(-a*b^2)^(1/3)*b)^(1/2)/(-b*x^3-a)^(1/2)*(4*3^(1/2)*_alpha^2*b^2-6
*_alpha^2*b^2-3*I*(-a*b^2)^(1/3)*3^(1/2)*_alpha*b-2*3^(1/2)*(-a*b^2)^(1/3)*_alpha*b+6*I*(-a*b^2)^(1/3)*_alpha*
b+3*(-a*b^2)^(1/3)*_alpha*b+3*I*(-a*b^2)^(2/3)*3^(1/2)-2*3^(1/2)*(-a*b^2)^(2/3)-6*I*(-a*b^2)^(2/3)+3*(-a*b^2)^
(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2*(-a*b^2)^(1/3)/b-1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)*3^(1/2)/(-a*b^2)^(1/3
)*b)^(1/2),-1/6*(2*I*3^(1/2)*(-a*b^2)^(1/3)*_alpha^2*b-4*I*(-a*b^2)^(1/3)*_alpha^2*b+I*3^(1/2)*a*b+2*3^(1/2)*a
*b-2*I*a*b-3*a*b-I*3^(1/2)*(-a*b^2)^(2/3)*_alpha+2*3^(1/2)*(-a*b^2)^(2/3)*_alpha+2*I*(-a*b^2)^(2/3)*_alpha-3*(
-a*b^2)^(2/3)*_alpha)/a/b,(I*3^(1/2)*(-a*b^2)^(1/3)/(-3/2*(-a*b^2)^(1/3)/b+1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)/b)^
(1/2)),_alpha=RootOf(_Z^3*b+6*3^(1/2)*a+10*a))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {x}{{\left (b x^{3} + 2 \, a {\left (3 \, \sqrt {3} + 5\right )}\right )} \sqrt {-b x^{3} - a}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-b*x^3-2*a*(5+3*3^(1/2)))/(-b*x^3-a)^(1/2),x, algorithm="maxima")

[Out]

-integrate(x/((b*x^3 + 2*a*(3*sqrt(3) + 5))*sqrt(-b*x^3 - a)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {x}{\sqrt {-b\,x^3-a}\,\left (b\,x^3+2\,a\,\left (3\,\sqrt {3}+5\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x/((- a - b*x^3)^(1/2)*(b*x^3 + 2*a*(3*3^(1/2) + 5))),x)

[Out]

int(-x/((- a - b*x^3)^(1/2)*(b*x^3 + 2*a*(3*3^(1/2) + 5))), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x}{10 a \sqrt {- a - b x^{3}} + 6 \sqrt {3} a \sqrt {- a - b x^{3}} + b x^{3} \sqrt {- a - b x^{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-b*x**3-2*a*(5+3*3**(1/2)))/(-b*x**3-a)**(1/2),x)

[Out]

-Integral(x/(10*a*sqrt(-a - b*x**3) + 6*sqrt(3)*a*sqrt(-a - b*x**3) + b*x**3*sqrt(-a - b*x**3)), x)

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