3.262 \(\int \frac{\sqrt{c+d x^3}}{x \left (4 c+d x^3\right )} \, dx\)

Optimal. Leaf size=65 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{2 \sqrt{3} \sqrt{c}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{6 \sqrt{c}} \]

[Out]

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

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Rubi [A]  time = 0.185134, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{2 \sqrt{3} \sqrt{c}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{6 \sqrt{c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.7859, size = 58, normalized size = 0.89 \[ \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{6 \sqrt{c}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{6 \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(3)*atan(sqrt(3)*sqrt(c + d*x**3)/(3*sqrt(c)))/(6*sqrt(c)) - atanh(sqrt(c +
d*x**3)/sqrt(c))/(6*sqrt(c))

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Mathematica [C]  time = 0.30233, size = 158, normalized size = 2.43 \[ -\frac{2 d x^3 \sqrt{c+d x^3} F_1\left (\frac{1}{2};-\frac{1}{2},1;\frac{3}{2};-\frac{c}{d x^3},-\frac{4 c}{d x^3}\right )}{\left (4 c+d x^3\right ) \left (3 d x^3 F_1\left (\frac{1}{2};-\frac{1}{2},1;\frac{3}{2};-\frac{c}{d x^3},-\frac{4 c}{d x^3}\right )+c \left (F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{4 c}{d x^3}\right )-8 F_1\left (\frac{3}{2};-\frac{1}{2},2;\frac{5}{2};-\frac{c}{d x^3},-\frac{4 c}{d x^3}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-2*d*x^3*Sqrt[c + d*x^3]*AppellF1[1/2, -1/2, 1, 3/2, -(c/(d*x^3)), (-4*c)/(d*x^
3)])/((4*c + d*x^3)*(3*d*x^3*AppellF1[1/2, -1/2, 1, 3/2, -(c/(d*x^3)), (-4*c)/(d
*x^3)] + c*(-8*AppellF1[3/2, -1/2, 2, 5/2, -(c/(d*x^3)), (-4*c)/(d*x^3)] + Appel
lF1[3/2, 1/2, 1, 5/2, -(c/(d*x^3)), (-4*c)/(d*x^3)])))

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Maple [C]  time = 0.029, size = 468, normalized size = 7.2 \[{\frac{1}{4\,c} \left ({\frac{2}{3}\sqrt{d{x}^{3}+c}}-{\frac{2}{3}{\it Artanh} \left ({1\sqrt{d{x}^{3}+c}{\frac{1}{\sqrt{c}}}} \right ) \sqrt{c}} \right ) }-{\frac{d}{4\,c} \left ({\frac{2}{3\,d}\sqrt{d{x}^{3}+c}}+{\frac{{\frac{i}{3}}\sqrt{2}}{{d}^{3}}\sum _{{\it \_alpha}={\it RootOf} \left ( d{{\it \_Z}}^{3}+4\,c \right ) }{1\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},{\frac{1}{6\,cd} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.319425, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{3}{\left (\sqrt{3} \sqrt{c} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right ) - 6 \, \sqrt{c} \arctan \left (\frac{\sqrt{3} \sqrt{c}}{\sqrt{d x^{3} + c}}\right )\right )}}{36 \, c}, -\frac{\sqrt{3}{\left (2 \, \sqrt{3} \sqrt{-c} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) + 3 \, \sqrt{-c} \log \left (\frac{\sqrt{3}{\left (d x^{3} - 2 \, c\right )} \sqrt{-c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} + 4 \, c}\right )\right )}}{36 \, c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/36*sqrt(3)*(sqrt(3)*sqrt(c)*log(((d*x^3 + 2*c)*sqrt(c) - 2*sqrt(d*x^3 + c)*c)
/x^3) - 6*sqrt(c)*arctan(sqrt(3)*sqrt(c)/sqrt(d*x^3 + c)))/c, -1/36*sqrt(3)*(2*s
qrt(3)*sqrt(-c)*arctan(c/(sqrt(d*x^3 + c)*sqrt(-c))) + 3*sqrt(-c)*log((sqrt(3)*(
d*x^3 - 2*c)*sqrt(-c) + 6*sqrt(d*x^3 + c)*c)/(d*x^3 + 4*c)))/c]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.21569, size = 68, normalized size = 1.05 \[ \frac{\sqrt{3} \arctan \left (\frac{\sqrt{3} \sqrt{d x^{3} + c}}{3 \, \sqrt{c}}\right )}{6 \, \sqrt{c}} + \frac{\arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{6 \, \sqrt{-c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*sqrt(3)*arctan(1/3*sqrt(3)*sqrt(d*x^3 + c)/sqrt(c))/sqrt(c) + 1/6*arctan(sqr
t(d*x^3 + c)/sqrt(-c))/sqrt(-c)