3.299 \(\int \frac{\left (c+d x^3\right )^{3/2}}{x \left (8 c-d x^3\right )} \, dx\)

Optimal. Leaf size=73 \[ -\frac{2}{3} \sqrt{c+d x^3}+\frac{9}{4} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )-\frac{1}{12} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right ) \]

[Out]

(-2*Sqrt[c + d*x^3])/3 + (9*Sqrt[c]*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/4 - (S
qrt[c]*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/12

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Rubi [A]  time = 0.274042, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{2}{3} \sqrt{c+d x^3}+\frac{9}{4} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )-\frac{1}{12} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[c + d*x^3])/3 + (9*Sqrt[c]*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/4 - (S
qrt[c]*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/12

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Rubi in Sympy [A]  time = 35.6595, size = 63, normalized size = 0.86 \[ \frac{9 \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{4} - \frac{\sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{12} - \frac{2 \sqrt{c + d x^{3}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*sqrt(c)*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/4 - sqrt(c)*atanh(sqrt(c + d*x**3)
/sqrt(c))/12 - 2*sqrt(c + d*x**3)/3

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Mathematica [C]  time = 0.292338, size = 319, normalized size = 4.37 \[ \frac{2 \left (\frac{240 c^2 d x^3 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{\left (8 c-d x^3\right ) \left (d x^3 \left (F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+16 c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )}+\frac{5 c^2 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}{\left (d x^3-8 c\right ) \left (5 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )+16 c F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )-c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )\right )}-3 \left (c+d x^3\right )\right )}{9 \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*(-3*(c + d*x^3) + (240*c^2*d*x^3*AppellF1[1, 1/2, 1, 2, -((d*x^3)/c), (d*x^3)
/(8*c)])/((8*c - d*x^3)*(16*c*AppellF1[1, 1/2, 1, 2, -((d*x^3)/c), (d*x^3)/(8*c)
] + d*x^3*(AppellF1[2, 1/2, 2, 3, -((d*x^3)/c), (d*x^3)/(8*c)] - 4*AppellF1[2, 3
/2, 1, 3, -((d*x^3)/c), (d*x^3)/(8*c)]))) + (5*c^2*d*x^3*AppellF1[3/2, 1/2, 1, 5
/2, -(c/(d*x^3)), (8*c)/(d*x^3)])/((-8*c + d*x^3)*(5*d*x^3*AppellF1[3/2, 1/2, 1,
5/2, -(c/(d*x^3)), (8*c)/(d*x^3)] + 16*c*AppellF1[5/2, 1/2, 2, 7/2, -(c/(d*x^3)
), (8*c)/(d*x^3)] - c*AppellF1[5/2, 3/2, 1, 7/2, -(c/(d*x^3)), (8*c)/(d*x^3)])))
)/(9*Sqrt[c + d*x^3])

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Maple [C]  time = 0.03, size = 500, normalized size = 6.9 \[{\frac{1}{8\,c} \left ({\frac{2\,d{x}^{3}}{9}\sqrt{d{x}^{3}+c}}+{\frac{8\,c}{9}\sqrt{d{x}^{3}+c}}-{\frac{2}{3}{c}^{{\frac{3}{2}}}{\it Artanh} \left ({1\sqrt{d{x}^{3}+c}{\frac{1}{\sqrt{c}}}} \right ) } \right ) }-{\frac{d}{8\,c} \left ({\frac{2\,{x}^{3}}{9}\sqrt{d{x}^{3}+c}}+{\frac{56\,c}{9\,d}\sqrt{d{x}^{3}+c}}+{\frac{3\,ic\sqrt{2}}{{d}^{3}}\sum _{{\it \_alpha}={\it RootOf} \left ( d{{\it \_Z}}^{3}-8\,c \right ) }{1\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},-{\frac{1}{18\,cd} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8/c*(2/9*d*x^3*(d*x^3+c)^(1/2)+8/9*c*(d*x^3+c)^(1/2)-2/3*c^(3/2)*arctanh((d*x^
3+c)^(1/2)/c^(1/2)))-1/8*d/c*(2/9*x^3*(d*x^3+c)^(1/2)+56/9*c*(d*x^3+c)^(1/2)/d+3
*I*c/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)*_alp
ha*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),-1/18/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-8*c)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}}}{{\left (d x^{3} - 8 \, c\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(d*x^3 + c)^(3/2)/((d*x^3 - 8*c)*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.26347, size = 1, normalized size = 0.01 \[ \left [\frac{9}{8} \, \sqrt{c} \log \left (\frac{d x^{3} + 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) + \frac{1}{24} \, \sqrt{c} \log \left (\frac{d x^{3} - 2 \, \sqrt{d x^{3} + c} \sqrt{c} + 2 \, c}{x^{3}}\right ) - \frac{2}{3} \, \sqrt{d x^{3} + c}, -\frac{1}{12} \, \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right ) + \frac{9}{4} \, \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right ) - \frac{2}{3} \, \sqrt{d x^{3} + c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(d*x^3 + c)^(3/2)/((d*x^3 - 8*c)*x),x, algorithm="fricas")

[Out]

[9/8*sqrt(c)*log((d*x^3 + 6*sqrt(d*x^3 + c)*sqrt(c) + 10*c)/(d*x^3 - 8*c)) + 1/2
4*sqrt(c)*log((d*x^3 - 2*sqrt(d*x^3 + c)*sqrt(c) + 2*c)/x^3) - 2/3*sqrt(d*x^3 +
c), -1/12*sqrt(-c)*arctan(sqrt(d*x^3 + c)/sqrt(-c)) + 9/4*sqrt(-c)*arctan(1/3*sq
rt(d*x^3 + c)/sqrt(-c)) - 2/3*sqrt(d*x^3 + c)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.221476, size = 82, normalized size = 1.12 \[ \frac{c \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{12 \, \sqrt{-c}} - \frac{9 \, c \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{4 \, \sqrt{-c}} - \frac{2}{3} \, \sqrt{d x^{3} + c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(d*x^3 + c)^(3/2)/((d*x^3 - 8*c)*x),x, algorithm="giac")

[Out]

1/12*c*arctan(sqrt(d*x^3 + c)/sqrt(-c))/sqrt(-c) - 9/4*c*arctan(1/3*sqrt(d*x^3 +
c)/sqrt(-c))/sqrt(-c) - 2/3*sqrt(d*x^3 + c)