3.452 \(\int \frac{1}{\sqrt [3]{x} (8 c-d x) \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=143 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt{c+d x}}\right )}{2 \sqrt{3} c^{5/6} d^{2/3}}+\frac{\tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x}}\right )}{6 c^{5/6} d^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x}}{3 \sqrt{c}}\right )}{6 c^{5/6} d^{2/3}} \]

[Out]

-ArcTan[(Sqrt[3]*c^(1/6)*(c^(1/3) + d^(1/3)*x^(1/3)))/Sqrt[c + d*x]]/(2*Sqrt[3]*
c^(5/6)*d^(2/3)) + ArcTanh[(c^(1/3) + d^(1/3)*x^(1/3))^2/(3*c^(1/6)*Sqrt[c + d*x
])]/(6*c^(5/6)*d^(2/3)) - ArcTanh[Sqrt[c + d*x]/(3*Sqrt[c])]/(6*c^(5/6)*d^(2/3))

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Rubi [A]  time = 0.876482, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt{c+d x}}\right )}{2 \sqrt{3} c^{5/6} d^{2/3}}+\frac{\tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x}}\right )}{6 c^{5/6} d^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x}}{3 \sqrt{c}}\right )}{6 c^{5/6} d^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

-ArcTan[(Sqrt[3]*c^(1/6)*(c^(1/3) + d^(1/3)*x^(1/3)))/Sqrt[c + d*x]]/(2*Sqrt[3]*
c^(5/6)*d^(2/3)) + ArcTanh[(c^(1/3) + d^(1/3)*x^(1/3))^2/(3*c^(1/6)*Sqrt[c + d*x
])]/(6*c^(5/6)*d^(2/3)) - ArcTanh[Sqrt[c + d*x]/(3*Sqrt[c])]/(6*c^(5/6)*d^(2/3))

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Rubi in Sympy [A]  time = 10.6358, size = 48, normalized size = 0.34 \[ \frac{3 x^{\frac{2}{3}} \sqrt{c + d x} \operatorname{appellf_{1}}{\left (\frac{2}{3},\frac{1}{2},1,\frac{5}{3},- \frac{d x}{c},\frac{d x}{8 c} \right )}}{16 c^{2} \sqrt{1 + \frac{d x}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.236398, size = 148, normalized size = 1.03 \[ \frac{60 c x^{2/3} F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x}{c},\frac{d x}{8 c}\right )}{(8 c-d x) \sqrt{c+d x} \left (40 c F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x}{c},\frac{d x}{8 c}\right )+3 d x \left (F_1\left (\frac{5}{3};\frac{1}{2},2;\frac{8}{3};-\frac{d x}{c},\frac{d x}{8 c}\right )-4 F_1\left (\frac{5}{3};\frac{3}{2},1;\frac{8}{3};-\frac{d x}{c},\frac{d x}{8 c}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \[ \int{\frac{1}{-dx+8\,c}{\frac{1}{\sqrt [3]{x}}}{\frac{1}{\sqrt{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(1/x^(1/3)/(-d*x+8*c)/(d*x+c)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{1}{- 8 c \sqrt [3]{x} \sqrt{c + d x} + d x^{\frac{4}{3}} \sqrt{c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(1/(-8*c*x**(1/3)*sqrt(c + d*x) + d*x**(4/3)*sqrt(c + d*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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