3.451 \(\int \frac{1}{\sqrt [3]{x} \sqrt{c+d x} (4 c+d x)} \, dx\)

Optimal. Leaf size=199 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt{c+d x}}\right )}{2^{2/3} \sqrt{3} c^{5/6} d^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{3} \sqrt{c}}\right )}{2^{2/3} \sqrt{3} c^{5/6} d^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt{c+d x}}\right )}{2^{2/3} c^{5/6} d^{2/3}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{3\ 2^{2/3} c^{5/6} d^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.263475, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt{c+d x}}\right )}{2^{2/3} \sqrt{3} c^{5/6} d^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{3} \sqrt{c}}\right )}{2^{2/3} \sqrt{3} c^{5/6} d^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt{c+d x}}\right )}{2^{2/3} c^{5/6} d^{2/3}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{3\ 2^{2/3} c^{5/6} d^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.034, size = 49, normalized size = 0.25 \[ \frac{3 x^{\frac{2}{3}} \sqrt{c + d x} \operatorname{appellf_{1}}{\left (\frac{2}{3},\frac{1}{2},1,\frac{5}{3},- \frac{d x}{c},- \frac{d x}{4 c} \right )}}{8 c^{2} \sqrt{1 + \frac{d x}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(1/3)/(d*x+4*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/(4*c))/(8*c**2*
sqrt(1 + d*x/c))

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Mathematica [C]  time = 0.232704, size = 147, normalized size = 0.74 \[ \frac{30 c x^{2/3} F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x}{c},-\frac{d x}{4 c}\right )}{\sqrt{c+d x} (4 c+d x) \left (20 c F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x}{c},-\frac{d x}{4 c}\right )-3 d x \left (F_1\left (\frac{5}{3};\frac{1}{2},2;\frac{8}{3};-\frac{d x}{c},-\frac{d x}{4 c}\right )+2 F_1\left (\frac{5}{3};\frac{3}{2},1;\frac{8}{3};-\frac{d x}{c},-\frac{d x}{4 c}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int{\frac{1}{dx+4\,c}{\frac{1}{\sqrt [3]{x}}}{\frac{1}{\sqrt{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(1/3)/(d*x+4*c)/(d*x+c)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d x + 4 \, c\right )} \sqrt{d x + c} x^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((d*x + 4*c)*sqrt(d*x + c)*x^(1/3)),x, algorithm="maxima")

[Out]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt [3]{x} \sqrt{c + d x} \left (4 c + d x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(1/3)/(d*x+4*c)/(d*x+c)**(1/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d x + 4 \, c\right )} \sqrt{d x + c} x^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((d*x + 4*c)*sqrt(d*x + c)*x^(1/3)),x, algorithm="giac")

[Out]

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