3.74 \(\int \frac {x}{(4-d x^3) \sqrt {-1+d x^3}} \, dx\)

Optimal. Leaf size=157 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [3]{2} \sqrt [3]{d} x+1}{\sqrt {d x^3-1}}\right )}{3\ 2^{2/3} d^{2/3}}-\frac {\tan ^{-1}\left (\sqrt {d x^3-1}\right )}{9\ 2^{2/3} d^{2/3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt {d x^3-1}}\right )}{3\ 2^{2/3} \sqrt {3} d^{2/3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d x^3-1}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3} d^{2/3}} \]

[Out]

-1/6*arctan((1+2^(1/3)*d^(1/3)*x)/(d*x^3-1)^(1/2))*2^(1/3)/d^(2/3)-1/18*arctan((d*x^3-1)^(1/2))*2^(1/3)/d^(2/3
)-1/18*arctanh((1-2^(1/3)*d^(1/3)*x)*3^(1/2)/(d*x^3-1)^(1/2))*2^(1/3)/d^(2/3)*3^(1/2)-1/18*arctanh(1/3*(d*x^3-
1)^(1/2)*3^(1/2))*2^(1/3)/d^(2/3)*3^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {485} \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [3]{2} \sqrt [3]{d} x+1}{\sqrt {d x^3-1}}\right )}{3\ 2^{2/3} d^{2/3}}-\frac {\tan ^{-1}\left (\sqrt {d x^3-1}\right )}{9\ 2^{2/3} d^{2/3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt {d x^3-1}}\right )}{3\ 2^{2/3} \sqrt {3} d^{2/3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d x^3-1}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3} d^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[x/((4 - d*x^3)*Sqrt[-1 + d*x^3]),x]

[Out]

-ArcTan[(1 + 2^(1/3)*d^(1/3)*x)/Sqrt[-1 + d*x^3]]/(3*2^(2/3)*d^(2/3)) - ArcTan[Sqrt[-1 + d*x^3]]/(9*2^(2/3)*d^
(2/3)) - ArcTanh[(Sqrt[3]*(1 - 2^(1/3)*d^(1/3)*x))/Sqrt[-1 + d*x^3]]/(3*2^(2/3)*Sqrt[3]*d^(2/3)) - ArcTanh[Sqr
t[-1 + d*x^3]/Sqrt[3]]/(3*2^(2/3)*Sqrt[3]*d^(2/3))

Rule 485

Int[(x_)/(((a_) + (b_.)*(x_)^3)*Sqrt[(c_) + (d_.)*(x_)^3]), x_Symbol] :> With[{q = Rt[d/c, 3]}, -Simp[(q*ArcTa
n[Sqrt[c + d*x^3]/Rt[-c, 2]])/(9*2^(2/3)*b*Rt[-c, 2]), x] + (-Simp[(q*ArcTan[(Rt[-c, 2]*(1 - 2^(1/3)*q*x))/Sqr
t[c + d*x^3]])/(3*2^(2/3)*b*Rt[-c, 2]), x] - Simp[(q*ArcTanh[Sqrt[c + d*x^3]/(Sqrt[3]*Rt[-c, 2])])/(3*2^(2/3)*
Sqrt[3]*b*Rt[-c, 2]), x] - Simp[(q*ArcTanh[(Sqrt[3]*Rt[-c, 2]*(1 + 2^(1/3)*q*x))/Sqrt[c + d*x^3]])/(3*2^(2/3)*
Sqrt[3]*b*Rt[-c, 2]), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[4*b*c - a*d, 0] && NegQ[c]

Rubi steps

\begin {align*} \int \frac {x}{\left (4-d x^3\right ) \sqrt {-1+d x^3}} \, dx &=-\frac {\tan ^{-1}\left (\frac {1+\sqrt [3]{2} \sqrt [3]{d} x}{\sqrt {-1+d x^3}}\right )}{3\ 2^{2/3} d^{2/3}}-\frac {\tan ^{-1}\left (\sqrt {-1+d x^3}\right )}{9\ 2^{2/3} d^{2/3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt {-1+d x^3}}\right )}{3\ 2^{2/3} \sqrt {3} d^{2/3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {-1+d x^3}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3} d^{2/3}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 54, normalized size = 0.34 \[ \frac {x^2 \sqrt {1-d x^3} F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};d x^3,\frac {d x^3}{4}\right )}{8 \sqrt {d x^3-1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x/((4 - d*x^3)*Sqrt[-1 + d*x^3]),x]

[Out]

(x^2*Sqrt[1 - d*x^3]*AppellF1[2/3, 1/2, 1, 5/3, d*x^3, (d*x^3)/4])/(8*Sqrt[-1 + d*x^3])

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fricas [B]  time = 1.81, size = 1666, normalized size = 10.61 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-d*x^3+4)/(d*x^3-1)^(1/2),x, algorithm="fricas")

[Out]

-1/9*sqrt(3)*(1/432)^(1/6)*(d^(-4))^(1/6)*arctan(1/3*(3*(sqrt(3)*sqrt(1/3)*d^2*sqrt(d^(-4))*x + 2*sqrt(3)*(1/4
32)^(1/6)*d*(d^(-4))^(1/6)*x^2 - 24*sqrt(3)*(1/432)^(5/6)*(d^4*x^3 - 4*d^3)*(d^(-4))^(5/6))*sqrt(d*x^3 - 1) +
(2*sqrt(3)*(1/2)^(1/3)*(d^2*x^3 - d)*(d^(-4))^(1/3) + sqrt(3)*(d*x^4 - x) + 3*(sqrt(3)*sqrt(1/3)*d^2*sqrt(d^(-
4))*x + 2*sqrt(3)*(1/432)^(1/6)*d*(d^(-4))^(1/6)*x^2 + 24*sqrt(3)*(1/432)^(5/6)*(d^4*x^3 + 2*d^3)*(d^(-4))^(5/
6))*sqrt(d*x^3 - 1))*sqrt((d^3*x^9 - 60*d^2*x^6 - 24*(1/2)^(2/3)*(d^5*x^7 - 5*d^4*x^4 + 4*d^3*x)*(d^(-4))^(2/3
) + 12*(1/2)^(1/3)*(d^4*x^8 + 7*d^3*x^5 - 8*d^2*x^2)*(d^(-4))^(1/3) + 12*(648*(1/432)^(5/6)*d^5*(d^(-4))^(5/6)
*x^5 - sqrt(1/3)*(d^4*x^6 + 16*d^3*x^3 - 8*d^2)*sqrt(d^(-4)) - (1/432)^(1/6)*(d^3*x^7 - 2*d^2*x^4 - 8*d*x)*(d^
(-4))^(1/6))*sqrt(d*x^3 - 1) + 32)/(d^3*x^9 - 12*d^2*x^6 + 48*d*x^3 - 64)))/(d*x^4 - x)) - 1/9*sqrt(3)*(1/432)
^(1/6)*(d^(-4))^(1/6)*arctan(1/3*(3*(sqrt(3)*sqrt(1/3)*d^2*sqrt(d^(-4))*x + 2*sqrt(3)*(1/432)^(1/6)*d*(d^(-4))
^(1/6)*x^2 - 24*sqrt(3)*(1/432)^(5/6)*(d^4*x^3 - 4*d^3)*(d^(-4))^(5/6))*sqrt(d*x^3 - 1) - (2*sqrt(3)*(1/2)^(1/
3)*(d^2*x^3 - d)*(d^(-4))^(1/3) + sqrt(3)*(d*x^4 - x) - 3*(sqrt(3)*sqrt(1/3)*d^2*sqrt(d^(-4))*x + 2*sqrt(3)*(1
/432)^(1/6)*d*(d^(-4))^(1/6)*x^2 + 24*sqrt(3)*(1/432)^(5/6)*(d^4*x^3 + 2*d^3)*(d^(-4))^(5/6))*sqrt(d*x^3 - 1))
*sqrt((d^3*x^9 - 60*d^2*x^6 - 24*(1/2)^(2/3)*(d^5*x^7 - 5*d^4*x^4 + 4*d^3*x)*(d^(-4))^(2/3) + 12*(1/2)^(1/3)*(
d^4*x^8 + 7*d^3*x^5 - 8*d^2*x^2)*(d^(-4))^(1/3) - 12*(648*(1/432)^(5/6)*d^5*(d^(-4))^(5/6)*x^5 - sqrt(1/3)*(d^
4*x^6 + 16*d^3*x^3 - 8*d^2)*sqrt(d^(-4)) - (1/432)^(1/6)*(d^3*x^7 - 2*d^2*x^4 - 8*d*x)*(d^(-4))^(1/6))*sqrt(d*
x^3 - 1) + 32)/(d^3*x^9 - 12*d^2*x^6 + 48*d*x^3 - 64)))/(d*x^4 - x)) + 1/18*(1/432)^(1/6)*(d^(-4))^(1/6)*log((
d^3*x^9 + 66*d^2*x^6 - 72*d*x^3 + 48*(1/2)^(2/3)*(d^5*x^7 + d^4*x^4 - 2*d^3*x)*(d^(-4))^(2/3) + 12*(1/2)^(1/3)
*(d^4*x^8 + 7*d^3*x^5 - 8*d^2*x^2)*(d^(-4))^(1/3) + 6*(1296*(1/432)^(5/6)*d^5*(d^(-4))^(5/6)*x^5 + sqrt(1/3)*(
5*d^4*x^6 + 20*d^3*x^3 - 16*d^2)*sqrt(d^(-4)) + 2*(1/432)^(1/6)*(d^3*x^7 + 16*d^2*x^4 - 8*d*x)*(d^(-4))^(1/6))
*sqrt(d*x^3 - 1) + 32)/(d^3*x^9 - 12*d^2*x^6 + 48*d*x^3 - 64)) - 1/18*(1/432)^(1/6)*(d^(-4))^(1/6)*log((d^3*x^
9 + 66*d^2*x^6 - 72*d*x^3 + 48*(1/2)^(2/3)*(d^5*x^7 + d^4*x^4 - 2*d^3*x)*(d^(-4))^(2/3) + 12*(1/2)^(1/3)*(d^4*
x^8 + 7*d^3*x^5 - 8*d^2*x^2)*(d^(-4))^(1/3) - 6*(1296*(1/432)^(5/6)*d^5*(d^(-4))^(5/6)*x^5 + sqrt(1/3)*(5*d^4*
x^6 + 20*d^3*x^3 - 16*d^2)*sqrt(d^(-4)) + 2*(1/432)^(1/6)*(d^3*x^7 + 16*d^2*x^4 - 8*d*x)*(d^(-4))^(1/6))*sqrt(
d*x^3 - 1) + 32)/(d^3*x^9 - 12*d^2*x^6 + 48*d*x^3 - 64)) - 1/36*(1/432)^(1/6)*(d^(-4))^(1/6)*log((d^3*x^9 - 60
*d^2*x^6 - 24*(1/2)^(2/3)*(d^5*x^7 - 5*d^4*x^4 + 4*d^3*x)*(d^(-4))^(2/3) + 12*(1/2)^(1/3)*(d^4*x^8 + 7*d^3*x^5
- 8*d^2*x^2)*(d^(-4))^(1/3) + 12*(648*(1/432)^(5/6)*d^5*(d^(-4))^(5/6)*x^5 - sqrt(1/3)*(d^4*x^6 + 16*d^3*x^3
- 8*d^2)*sqrt(d^(-4)) - (1/432)^(1/6)*(d^3*x^7 - 2*d^2*x^4 - 8*d*x)*(d^(-4))^(1/6))*sqrt(d*x^3 - 1) + 32)/(d^3
*x^9 - 12*d^2*x^6 + 48*d*x^3 - 64)) + 1/36*(1/432)^(1/6)*(d^(-4))^(1/6)*log((d^3*x^9 - 60*d^2*x^6 - 24*(1/2)^(
2/3)*(d^5*x^7 - 5*d^4*x^4 + 4*d^3*x)*(d^(-4))^(2/3) + 12*(1/2)^(1/3)*(d^4*x^8 + 7*d^3*x^5 - 8*d^2*x^2)*(d^(-4)
)^(1/3) - 12*(648*(1/432)^(5/6)*d^5*(d^(-4))^(5/6)*x^5 - sqrt(1/3)*(d^4*x^6 + 16*d^3*x^3 - 8*d^2)*sqrt(d^(-4))
- (1/432)^(1/6)*(d^3*x^7 - 2*d^2*x^4 - 8*d*x)*(d^(-4))^(1/6))*sqrt(d*x^3 - 1) + 32)/(d^3*x^9 - 12*d^2*x^6 + 4
8*d*x^3 - 64))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {x}{\sqrt {d x^{3} - 1} {\left (d x^{3} - 4\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-d*x^3+4)/(d*x^3-1)^(1/2),x, algorithm="giac")

[Out]

integrate(-x/(sqrt(d*x^3 - 1)*(d*x^3 - 4)), x)

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maple [C]  time = 0.22, size = 240, normalized size = 1.53 \[ -\frac {i \sqrt {-\frac {i \left (2 x +\frac {1}{d^{\frac {1}{3}}}+\frac {i \sqrt {3}}{d^{\frac {1}{3}}}\right ) d^{\frac {1}{3}}}{2}}\, \sqrt {\frac {x -\frac {1}{d^{\frac {1}{3}}}}{-\frac {3}{d^{\frac {1}{3}}}-\frac {i \sqrt {3}}{d^{\frac {1}{3}}}}}\, \sqrt {i \left (2 x +\frac {1}{d^{\frac {1}{3}}}-\frac {i \sqrt {3}}{d^{\frac {1}{3}}}\right ) d^{\frac {1}{3}}}\, \left (-2 \RootOf \left (d \,\textit {\_Z}^{3}-4\right )^{2} d +i \sqrt {3}\, \RootOf \left (d \,\textit {\_Z}^{3}-4\right ) d^{\frac {2}{3}}+\RootOf \left (d \,\textit {\_Z}^{3}-4\right ) d^{\frac {2}{3}}-i \sqrt {3}\, d^{\frac {1}{3}}+d^{\frac {1}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {-i \left (x +\frac {1}{2 d^{\frac {1}{3}}}+\frac {i \sqrt {3}}{2 d^{\frac {1}{3}}}\right ) \sqrt {3}\, d^{\frac {1}{3}}}}{3}, \frac {i \sqrt {3}\, \RootOf \left (d \,\textit {\_Z}^{3}-4\right )^{2} d^{\frac {2}{3}}}{3}-\frac {i \sqrt {3}\, \RootOf \left (d \,\textit {\_Z}^{3}-4\right ) d^{\frac {1}{3}}}{6}+\frac {\RootOf \left (d \,\textit {\_Z}^{3}-4\right ) d^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}}{6}-\frac {1}{2}, \sqrt {-\frac {i \sqrt {3}}{\left (-\frac {3}{2 d^{\frac {1}{3}}}-\frac {i \sqrt {3}}{2 d^{\frac {1}{3}}}\right ) d^{\frac {1}{3}}}}\right )}{9 \sqrt {d \,x^{3}-1}\, \RootOf \left (d \,\textit {\_Z}^{3}-4\right ) d^{\frac {4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(-d*x^3+4)/(d*x^3-1)^(1/2),x)

[Out]

-1/9*I*2^(1/2)*sum(1/_alpha/d^(4/3)*(-1/2*I*(2*x+1/d^(1/3)+I*3^(1/2)/d^(1/3))*d^(1/3))^(1/2)*((x-1/d^(1/3))/(-
3/d^(1/3)-I*3^(1/2)/d^(1/3)))^(1/2)*(1/2*I*(2*x+1/d^(1/3)-I*3^(1/2)/d^(1/3))*d^(1/3))^(1/2)/(d*x^3-1)^(1/2)*(-
2*_alpha^2*d+I*3^(1/2)*_alpha*d^(2/3)+_alpha*d^(2/3)-I*3^(1/2)*d^(1/3)+d^(1/3))*EllipticPi(1/3*3^(1/2)*(-I*(x+
1/2/d^(1/3)+1/2*I*3^(1/2)/d^(1/3))*3^(1/2)*d^(1/3))^(1/2),1/3*I*_alpha^2*d^(2/3)*3^(1/2)-1/6*I*_alpha*d^(1/3)*
3^(1/2)+1/2*_alpha*d^(1/3)-1/6*I*3^(1/2)-1/2,(-I*3^(1/2)/d^(1/3)/(-3/2/d^(1/3)-1/2*I*3^(1/2)/d^(1/3)))^(1/2)),
_alpha=RootOf(_Z^3*d-4))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {x}{\sqrt {d x^{3} - 1} {\left (d x^{3} - 4\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-d*x^3+4)/(d*x^3-1)^(1/2),x, algorithm="maxima")

[Out]

-integrate(x/(sqrt(d*x^3 - 1)*(d*x^3 - 4)), x)

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mupad [B]  time = 15.03, size = 331, normalized size = 2.11 \[ \frac {\sqrt {3}\,{314928}^{1/3}\,\ln \left (\frac {\left (54\,\sqrt {d\,x^3-1}+54\,\sqrt {3}-54\,2^{1/3}\,\sqrt {3}\,d^{1/3}\,x\right )\,{\left (\sqrt {d\,x^3-1}-\sqrt {3}+2^{1/3}\,\sqrt {3}\,d^{1/3}\,x\right )}^3}{{\left (2^{2/3}-d^{1/3}\,x\right )}^6}\right )}{2916\,d^{2/3}}+\frac {\sqrt {3}\,{314928}^{1/3}\,\ln \left (\frac {{\left (2\,\sqrt {d\,x^3-1}+2\,\sqrt {3}+2^{1/3}\,\sqrt {3}\,d^{1/3}\,x+2^{1/3}\,d^{1/3}\,x\,3{}\mathrm {i}\right )}^3\,\left (108\,\sqrt {3}-108\,\sqrt {d\,x^3-1}+54\,2^{1/3}\,\sqrt {3}\,d^{1/3}\,x+2^{1/3}\,d^{1/3}\,x\,162{}\mathrm {i}\right )}{{\left (2^{2/3}+2\,d^{1/3}\,x-2^{2/3}\,\sqrt {3}\,1{}\mathrm {i}\right )}^6}\right )\,\sqrt {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}{2916\,d^{2/3}}+\frac {\sqrt {3}\,{314928}^{1/3}\,\ln \left (\frac {{\left (2\,\sqrt {d\,x^3-1}-2\,\sqrt {3}-2^{1/3}\,\sqrt {3}\,d^{1/3}\,x+2^{1/3}\,d^{1/3}\,x\,3{}\mathrm {i}\right )}^3\,\left (108\,\sqrt {d\,x^3-1}+108\,\sqrt {3}+54\,2^{1/3}\,\sqrt {3}\,d^{1/3}\,x-2^{1/3}\,d^{1/3}\,x\,162{}\mathrm {i}\right )}{{\left (2^{2/3}+2\,d^{1/3}\,x+2^{2/3}\,\sqrt {3}\,1{}\mathrm {i}\right )}^6}\right )\,\sqrt {\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\,1{}\mathrm {i}}{2916\,d^{2/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x/((d*x^3 - 1)^(1/2)*(d*x^3 - 4)),x)

[Out]

(3^(1/2)*314928^(1/3)*log(((54*(d*x^3 - 1)^(1/2) + 54*3^(1/2) - 54*2^(1/3)*3^(1/2)*d^(1/3)*x)*((d*x^3 - 1)^(1/
2) - 3^(1/2) + 2^(1/3)*3^(1/2)*d^(1/3)*x)^3)/(2^(2/3) - d^(1/3)*x)^6))/(2916*d^(2/3)) + (3^(1/2)*314928^(1/3)*
log(((2*(d*x^3 - 1)^(1/2) + 2*3^(1/2) + 2^(1/3)*d^(1/3)*x*3i + 2^(1/3)*3^(1/2)*d^(1/3)*x)^3*(108*3^(1/2) - 108
*(d*x^3 - 1)^(1/2) + 2^(1/3)*d^(1/3)*x*162i + 54*2^(1/3)*3^(1/2)*d^(1/3)*x))/(2^(2/3) - 2^(2/3)*3^(1/2)*1i + 2
*d^(1/3)*x)^6)*((3^(1/2)*1i)/2 - 1/2)^(1/2))/(2916*d^(2/3)) + (3^(1/2)*314928^(1/3)*log(((2*(d*x^3 - 1)^(1/2)
- 2*3^(1/2) + 2^(1/3)*d^(1/3)*x*3i - 2^(1/3)*3^(1/2)*d^(1/3)*x)^3*(108*(d*x^3 - 1)^(1/2) + 108*3^(1/2) - 2^(1/
3)*d^(1/3)*x*162i + 54*2^(1/3)*3^(1/2)*d^(1/3)*x))/(2^(2/3)*3^(1/2)*1i + 2^(2/3) + 2*d^(1/3)*x)^6)*((3^(1/2)*1
i)/2 + 1/2)^(1/2)*1i)/(2916*d^(2/3))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {x}{d x^{3} \sqrt {d x^{3} - 1} - 4 \sqrt {d x^{3} - 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-d*x**3+4)/(d*x**3-1)**(1/2),x)

[Out]

-Integral(x/(d*x**3*sqrt(d*x**3 - 1) - 4*sqrt(d*x**3 - 1)), x)

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