3.74 \(\int \frac{x}{\left (8-d x^3\right ) \sqrt{1+d x^3}} \, dx\)

Optimal. Leaf size=103 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (\sqrt [3]{d} x+1\right )}{\sqrt{d x^3+1}}\right )}{6 \sqrt{3} d^{2/3}}+\frac{\tanh ^{-1}\left (\frac{\left (\sqrt [3]{d} x+1\right )^2}{3 \sqrt{d x^3+1}}\right )}{18 d^{2/3}}-\frac{\tanh ^{-1}\left (\frac{1}{3} \sqrt{d x^3+1}\right )}{18 d^{2/3}} \]

[Out]

-ArcTan[(Sqrt[3]*(1 + d^(1/3)*x))/Sqrt[1 + d*x^3]]/(6*Sqrt[3]*d^(2/3)) + ArcTanh
[(1 + d^(1/3)*x)^2/(3*Sqrt[1 + d*x^3])]/(18*d^(2/3)) - ArcTanh[Sqrt[1 + d*x^3]/3
]/(18*d^(2/3))

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Rubi [A]  time = 0.544631, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (\sqrt [3]{d} x+1\right )}{\sqrt{d x^3+1}}\right )}{6 \sqrt{3} d^{2/3}}+\frac{\tanh ^{-1}\left (\frac{\left (\sqrt [3]{d} x+1\right )^2}{3 \sqrt{d x^3+1}}\right )}{18 d^{2/3}}-\frac{\tanh ^{-1}\left (\frac{1}{3} \sqrt{d x^3+1}\right )}{18 d^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[x/((8 - d*x^3)*Sqrt[1 + d*x^3]),x]

[Out]

-ArcTan[(Sqrt[3]*(1 + d^(1/3)*x))/Sqrt[1 + d*x^3]]/(6*Sqrt[3]*d^(2/3)) + ArcTanh
[(1 + d^(1/3)*x)^2/(3*Sqrt[1 + d*x^3])]/(18*d^(2/3)) - ArcTanh[Sqrt[1 + d*x^3]/3
]/(18*d^(2/3))

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Rubi in Sympy [A]  time = 3.31384, size = 22, normalized size = 0.21 \[ \frac{x^{2} \operatorname{appellf_{1}}{\left (\frac{2}{3},\frac{1}{2},1,\frac{5}{3},- d x^{3},\frac{d x^{3}}{8} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(-d*x**3+8)/(d*x**3+1)**(1/2),x)

[Out]

x**2*appellf1(2/3, 1/2, 1, 5/3, -d*x**3, d*x**3/8)/16

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Mathematica [C]  time = 0.18094, size = 139, normalized size = 1.35 \[ -\frac{20 x^2 F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-d x^3,\frac{d x^3}{8}\right )}{\left (d x^3-8\right ) \sqrt{d x^3+1} \left (3 d x^3 \left (F_1\left (\frac{5}{3};\frac{1}{2},2;\frac{8}{3};-d x^3,\frac{d x^3}{8}\right )-4 F_1\left (\frac{5}{3};\frac{3}{2},1;\frac{8}{3};-d x^3,\frac{d x^3}{8}\right )\right )+40 F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-d x^3,\frac{d x^3}{8}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x/((8 - d*x^3)*Sqrt[1 + d*x^3]),x]

[Out]

(-20*x^2*AppellF1[2/3, 1/2, 1, 5/3, -(d*x^3), (d*x^3)/8])/((-8 + d*x^3)*Sqrt[1 +
d*x^3]*(40*AppellF1[2/3, 1/2, 1, 5/3, -(d*x^3), (d*x^3)/8] + 3*d*x^3*(AppellF1[
5/3, 1/2, 2, 8/3, -(d*x^3), (d*x^3)/8] - 4*AppellF1[5/3, 3/2, 1, 8/3, -(d*x^3),
(d*x^3)/8])))

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Maple [C]  time = 0.197, size = 383, normalized size = 3.7 \[{\frac{-{\frac{i}{27}}\sqrt{2}}{{d}^{3}}\sum _{{\it \_alpha}={\it RootOf} \left ( d{{\it \_Z}}^{3}-8 \right ) }{\frac{1}{{\it \_alpha}}\sqrt [3]{-{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-{d}^{2}}+\sqrt [3]{-{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-{d}^{2}}+i\sqrt{3}\sqrt [3]{-{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-{d}^{2}}+\sqrt [3]{-{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-{d}^{2}}}}}} \left ( i\sqrt [3]{-{d}^{2}}{\it \_alpha}\,\sqrt{3}d-i\sqrt{3} \left ( -{d}^{2} \right ) ^{{\frac{2}{3}}}+2\,{{\it \_alpha}}^{2}{d}^{2}-\sqrt [3]{-{d}^{2}}{\it \_alpha}\,d- \left ( -{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-{d}^{2}}}}}}},-{\frac{1}{18\,d} \left ( 2\,i\sqrt [3]{-{d}^{2}}{{\it \_alpha}}^{2}\sqrt{3}d-i \left ( -{d}^{2} \right ) ^{{\frac{2}{3}}}{\it \_alpha}\,\sqrt{3}-3\, \left ( -{d}^{2} \right ) ^{2/3}{\it \_alpha}+i\sqrt{3}d-3\,d \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+1}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-d*x^3+8)/(d*x^3+1)^(1/2),x)

[Out]

-1/27*I/d^3*2^(1/2)*sum(1/_alpha*(-d^2)^(1/3)*(1/2*I*d*(2*x+1/d*(-I*3^(1/2)*(-d^
2)^(1/3)+(-d^2)^(1/3)))/(-d^2)^(1/3))^(1/2)*(d*(x-1/d*(-d^2)^(1/3))/(-3*(-d^2)^(
1/3)+I*3^(1/2)*(-d^2)^(1/3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I*3^(1/2)*(-d^2)^(1/3)+(
-d^2)^(1/3)))/(-d^2)^(1/3))^(1/2)/(d*x^3+1)^(1/2)*(I*(-d^2)^(1/3)*_alpha*3^(1/2)
*d-I*3^(1/2)*(-d^2)^(2/3)+2*_alpha^2*d^2-(-d^2)^(1/3)*_alpha*d-(-d^2)^(2/3))*Ell
ipticPi(1/3*3^(1/2)*(I*(x+1/2/d*(-d^2)^(1/3)-1/2*I*3^(1/2)/d*(-d^2)^(1/3))*3^(1/
2)*d/(-d^2)^(1/3))^(1/2),-1/18/d*(2*I*(-d^2)^(1/3)*_alpha^2*3^(1/2)*d-I*(-d^2)^(
2/3)*_alpha*3^(1/2)-3*(-d^2)^(2/3)*_alpha+I*3^(1/2)*d-3*d),(I*3^(1/2)/d*(-d^2)^(
1/3)/(-3/2/d*(-d^2)^(1/3)+1/2*I*3^(1/2)/d*(-d^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z
^3*d-8))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{x}{\sqrt{d x^{3} + 1}{\left (d x^{3} - 8\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x/(sqrt(d*x^3 + 1)*(d*x^3 - 8)),x, algorithm="maxima")

[Out]

-integrate(x/(sqrt(d*x^3 + 1)*(d*x^3 - 8)), x)

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Fricas [A]  time = 0.328065, size = 914, normalized size = 8.87 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x/(sqrt(d*x^3 + 1)*(d*x^3 - 8)),x, algorithm="fricas")

[Out]

1/324*sqrt(3)*(2*sqrt(3)*log((18*d^4*x^8 + 684*d^3*x^5 + 1152*d^2*x^2 + (d^3*x^9
+ 318*d^2*x^6 + 1200*d*x^3 + 640)*(d^2)^(2/3) + 6*(30*d^3*x^5 + 192*d^2*x^2 + (
7*d^2*x^6 + 152*d*x^3 + 64)*(d^2)^(2/3) + (d^3*x^7 + 80*d^2*x^4 + 160*d*x)*(d^2)
^(1/3))*sqrt(d*x^3 + 1) + 18*(5*d^3*x^7 + 64*d^2*x^4 + 32*d*x)*(d^2)^(1/3))/(d^3
*x^9 - 24*d^2*x^6 + 192*d*x^3 - 512)) - sqrt(3)*log(-(18*d^3*x^7 - 936*d^2*x^4 -
1440*d*x - 18*(d^2*x^8 + 20*d*x^5 - 8*x^2)*(d^2)^(2/3) + 6*(d^3*x^7 - 28*d^2*x^
4 - 272*d*x - 24*(d*x^5 + x^2)*(d^2)^(2/3) + 4*(d^2*x^6 + 41*d*x^3 + 40)*(d^2)^(
1/3))*sqrt(d*x^3 + 1) - (d^3*x^9 - 276*d^2*x^6 - 1608*d*x^3 - 1088)*(d^2)^(1/3))
/(d^3*x^9 - 24*d^2*x^6 + 192*d*x^3 - 512)) + 6*arctan(-1/9*(sqrt(3)*(d^3*x^9 + 3
18*d^2*x^6 + 1200*d*x^3 + 640)*(d^2)^(2/3) - 9*sqrt(3)*(5*d^3*x^7 + 64*d^2*x^4 +
32*d*x)*(d^2)^(1/3) - 9*sqrt(3)*(d^4*x^8 + 38*d^3*x^5 + 64*d^2*x^2) + 3*sqrt(d*
x^3 + 1)*(2*sqrt(3)*(7*d^2*x^6 + 152*d*x^3 + 64)*(d^2)^(2/3) - sqrt(3)*(d^3*x^7
+ 80*d^2*x^4 + 160*d*x)*(d^2)^(1/3) - 6*sqrt(3)*(5*d^3*x^5 + 32*d^2*x^2)))/(3*d^
4*x^8 + 114*d^3*x^5 + 192*d^2*x^2 + (30*d^3*x^5 + 192*d^2*x^2 - (d^3*x^7 + 80*d^
2*x^4 + 160*d*x)*(d^2)^(1/3))*sqrt(d*x^3 + 1) - 3*(5*d^3*x^7 + 64*d^2*x^4 + 32*d
*x)*(d^2)^(1/3))))/(d^2)^(1/3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{d x^{3} \sqrt{d x^{3} + 1} - 8 \sqrt{d x^{3} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(-d*x**3+8)/(d*x**3+1)**(1/2),x)

[Out]

-Integral(x/(d*x**3*sqrt(d*x**3 + 1) - 8*sqrt(d*x**3 + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{x}{\sqrt{d x^{3} + 1}{\left (d x^{3} - 8\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x/(sqrt(d*x^3 + 1)*(d*x^3 - 8)),x, algorithm="giac")

[Out]

integrate(-x/(sqrt(d*x^3 + 1)*(d*x^3 - 8)), x)