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

Optimal. Leaf size=88 \[ -\frac{d \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{8 \sqrt{3} c^{3/2}}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{24 c^{3/2}}-\frac{\sqrt{c+d x^3}}{12 c x^3} \]

[Out]

-Sqrt[c + d*x^3]/(12*c*x^3) - (d*ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])])/(8*S
qrt[3]*c^(3/2)) - (d*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(24*c^(3/2))

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Rubi [A]  time = 0.294534, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{d \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{8 \sqrt{3} c^{3/2}}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{24 c^{3/2}}-\frac{\sqrt{c+d x^3}}{12 c x^3} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[c + d*x^3]/(12*c*x^3) - (d*ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])])/(8*S
qrt[3]*c^(3/2)) - (d*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(24*c^(3/2))

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Rubi in Sympy [A]  time = 29.3211, size = 80, normalized size = 0.91 \[ - \frac{\sqrt{c + d x^{3}}}{12 c x^{3}} - \frac{\sqrt{3} d \operatorname{atan}{\left (\frac{\sqrt{3} \sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{24 c^{\frac{3}{2}}} - \frac{d \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{24 c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c + d*x**3)/(12*c*x**3) - sqrt(3)*d*atan(sqrt(3)*sqrt(c + d*x**3)/(3*sqrt(
c)))/(24*c**(3/2)) - d*atanh(sqrt(c + d*x**3)/sqrt(c))/(24*c**(3/2))

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Mathematica [C]  time = 0.78221, size = 319, normalized size = 3.62 \[ \frac{\frac{12 d^2 x^6 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )}{\left (4 c+d x^3\right ) \left (d x^3 \left (F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )+2 F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )\right )-8 c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )\right )}+\frac{10 d^2 x^6 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{4 c}{d x^3}\right )}{\left (4 c+d x^3\right ) \left (c \left (8 F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{4 c}{d x^3}\right )+F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{4 c}{d x^3}\right )\right )-5 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{4 c}{d x^3}\right )\right )}-\frac{3 d x^3}{c}-3}{36 x^3 \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-3 - (3*d*x^3)/c + (12*d^2*x^6*AppellF1[1, 1/2, 1, 2, -((d*x^3)/c), -(d*x^3)/(4
*c)])/((4*c + d*x^3)*(-8*c*AppellF1[1, 1/2, 1, 2, -((d*x^3)/c), -(d*x^3)/(4*c)]
+ d*x^3*(AppellF1[2, 1/2, 2, 3, -((d*x^3)/c), -(d*x^3)/(4*c)] + 2*AppellF1[2, 3/
2, 1, 3, -((d*x^3)/c), -(d*x^3)/(4*c)]))) + (10*d^2*x^6*AppellF1[3/2, 1/2, 1, 5/
2, -(c/(d*x^3)), (-4*c)/(d*x^3)])/((4*c + d*x^3)*(-5*d*x^3*AppellF1[3/2, 1/2, 1,
5/2, -(c/(d*x^3)), (-4*c)/(d*x^3)] + c*(8*AppellF1[5/2, 1/2, 2, 7/2, -(c/(d*x^3
)), (-4*c)/(d*x^3)] + AppellF1[5/2, 3/2, 1, 7/2, -(c/(d*x^3)), (-4*c)/(d*x^3)]))
))/(36*x^3*Sqrt[c + d*x^3])

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Maple [C]  time = 0.034, size = 511, normalized size = 5.8 \[{\frac{1}{4\,c} \left ( -{\frac{1}{3\,{x}^{3}}\sqrt{d{x}^{3}+c}}-{\frac{d}{3}{\it Artanh} \left ({1\sqrt{d{x}^{3}+c}{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}} \right ) }-{\frac{d}{16\,{c}^{2}} \left ({\frac{2}{3}\sqrt{d{x}^{3}+c}}-{\frac{2}{3}{\it Artanh} \left ({1\sqrt{d{x}^{3}+c}{\frac{1}{\sqrt{c}}}} \right ) \sqrt{c}} \right ) }+{\frac{{d}^{2}}{16\,{c}^{2}} \left ({\frac{2}{3\,d}\sqrt{d{x}^{3}+c}}+{\frac{{\frac{i}{3}}\sqrt{2}}{{d}^{3}}\sum _{{\it \_alpha}={\it RootOf} \left ( d{{\it \_Z}}^{3}+4\,c \right ) }{1\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},{\frac{1}{6\,cd} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4/c*(-1/3*(d*x^3+c)^(1/2)/x^3-1/3*d*arctanh((d*x^3+c)^(1/2)/c^(1/2))/c^(1/2))-
1/16*d/c^2*(2/3*(d*x^3+c)^(1/2)-2/3*arctanh((d*x^3+c)^(1/2)/c^(1/2))*c^(1/2))+1/
16*d^2/c^2*(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. \[ \int \frac{\sqrt{d x^{3} + c}}{{\left (d x^{3} + 4 \, c\right )} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.284966, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{3}{\left (\sqrt{3} \sqrt{c} d x^{3} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right ) + 6 \, \sqrt{c} d x^{3} \arctan \left (\frac{\sqrt{3} \sqrt{c}}{\sqrt{d x^{3} + c}}\right ) - 4 \, \sqrt{3} \sqrt{d x^{3} + c} c\right )}}{144 \, c^{2} x^{3}}, -\frac{\sqrt{3}{\left (2 \, \sqrt{3} \sqrt{-c} d x^{3} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) + 3 \, \sqrt{-c} d x^{3} \log \left (\frac{\sqrt{3}{\left (d x^{3} - 2 \, c\right )} \sqrt{-c} - 6 \, \sqrt{d x^{3} + c} c}{d x^{3} + 4 \, c}\right ) + 4 \, \sqrt{3} \sqrt{d x^{3} + c} c\right )}}{144 \, c^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/144*sqrt(3)*(sqrt(3)*sqrt(c)*d*x^3*log(((d*x^3 + 2*c)*sqrt(c) - 2*sqrt(d*x^3
+ c)*c)/x^3) + 6*sqrt(c)*d*x^3*arctan(sqrt(3)*sqrt(c)/sqrt(d*x^3 + c)) - 4*sqrt(
3)*sqrt(d*x^3 + c)*c)/(c^2*x^3), -1/144*sqrt(3)*(2*sqrt(3)*sqrt(-c)*d*x^3*arctan
(c/(sqrt(d*x^3 + c)*sqrt(-c))) + 3*sqrt(-c)*d*x^3*log((sqrt(3)*(d*x^3 - 2*c)*sqr
t(-c) - 6*sqrt(d*x^3 + c)*c)/(d*x^3 + 4*c)) + 4*sqrt(3)*sqrt(d*x^3 + c)*c)/(c^2*
x^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.219369, size = 101, normalized size = 1.15 \[ -\frac{1}{24} \, d{\left (\frac{\sqrt{3} \arctan \left (\frac{\sqrt{3} \sqrt{d x^{3} + c}}{3 \, \sqrt{c}}\right )}{c^{\frac{3}{2}}} - \frac{\arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c} + \frac{2 \, \sqrt{d x^{3} + c}}{c d x^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*d*(sqrt(3)*arctan(1/3*sqrt(3)*sqrt(d*x^3 + c)/sqrt(c))/c^(3/2) - arctan(sq
rt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c) + 2*sqrt(d*x^3 + c)/(c*d*x^3))