3.329 \(\int \frac{1}{x \left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=76 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{324 c^{5/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{12 c^{5/2}}+\frac{2}{27 c^2 \sqrt{c+d x^3}} \]

[Out]

2/(27*c^2*Sqrt[c + d*x^3]) + ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])]/(324*c^(5/2))
- ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]]/(12*c^(5/2))

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Rubi [A]  time = 0.254474, antiderivative size = 76, 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 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{324 c^{5/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{12 c^{5/2}}+\frac{2}{27 c^2 \sqrt{c+d x^3}} \]

Antiderivative was successfully verified.

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

[Out]

2/(27*c^2*Sqrt[c + d*x^3]) + ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])]/(324*c^(5/2))
- ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]]/(12*c^(5/2))

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Rubi in Sympy [A]  time = 31.3079, size = 65, normalized size = 0.86 \[ \frac{2}{27 c^{2} \sqrt{c + d x^{3}}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{324 c^{\frac{5}{2}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{12 c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(27*c**2*sqrt(c + d*x**3)) + atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/(324*c**(5/2)
) - atanh(sqrt(c + d*x**3)/sqrt(c))/(12*c**(5/2))

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Mathematica [C]  time = 0.271127, size = 310, normalized size = 4.08 \[ \frac{2 \left (-\frac{8 c d x^3 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{\left (8 c-d x^3\right ) \left (d x^3 \left (F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+16 c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )}+\frac{15 c d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}{\left (d x^3-8 c\right ) \left (5 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )+16 c F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )-c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )\right )}+1\right )}{27 c^2 \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*(1 - (8*c*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*(Appell
F1[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)]))) + (15*c*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)]))))/(27*c^2*Sqrt[c +
d*x^3])

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Maple [C]  time = 0.036, size = 485, normalized size = 6.4 \[{\frac{1}{8\,c} \left ({\frac{2}{3\,c}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{c}{d}} \right ) d}}}}-{\frac{2}{3}{\it Artanh} \left ({1\sqrt{d{x}^{3}+c}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}} \right ) }-{\frac{d}{8\,c} \left ({\frac{2}{27\,cd}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{c}{d}} \right ) d}}}}+{\frac{{\frac{i}{243}}\sqrt{2}}{{d}^{3}{c}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ( d{{\it \_Z}}^{3}-8\,c \right ) }{1\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},-{\frac{1}{18\,cd} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8/c*(2/3/c/((x^3+c/d)*d)^(1/2)-2/3*arctanh((d*x^3+c)^(1/2)/c^(1/2))/c^(3/2))-1
/8*d/c*(2/27/d/c/((x^3+c/d)*d)^(1/2)+1/243*I/d^3/c^2*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),-1/18/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=RootO
f(_Z^3*d-8*c)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{1}{{\left (d x^{3} + c\right )}^{\frac{3}{2}}{\left (d x^{3} - 8 \, c\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.238704, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{d x^{3} + c} \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) + 27 \, \sqrt{d x^{3} + c} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right ) + 48 \, \sqrt{c}}{648 \, \sqrt{d x^{3} + c} c^{\frac{5}{2}}}, -\frac{\sqrt{d x^{3} + c} \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 27 \, \sqrt{d x^{3} + c} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 24 \, \sqrt{-c}}{324 \, \sqrt{d x^{3} + c} \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)*x),x, algorithm="fricas")

[Out]

[1/648*(sqrt(d*x^3 + c)*log(((d*x^3 + 10*c)*sqrt(c) + 6*sqrt(d*x^3 + c)*c)/(d*x^
3 - 8*c)) + 27*sqrt(d*x^3 + c)*log(((d*x^3 + 2*c)*sqrt(c) - 2*sqrt(d*x^3 + c)*c)
/x^3) + 48*sqrt(c))/(sqrt(d*x^3 + c)*c^(5/2)), -1/324*(sqrt(d*x^3 + c)*arctan(3*
c/(sqrt(d*x^3 + c)*sqrt(-c))) - 27*sqrt(d*x^3 + c)*arctan(c/(sqrt(d*x^3 + c)*sqr
t(-c))) - 24*sqrt(-c))/(sqrt(d*x^3 + c)*sqrt(-c)*c^2)]

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Sympy [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/x/(-d*x**3+8*c)/(d*x**3+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216823, size = 92, normalized size = 1.21 \[ \frac{\arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{12 \, \sqrt{-c} c^{2}} - \frac{\arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{324 \, \sqrt{-c} c^{2}} + \frac{2}{27 \, \sqrt{d x^{3} + c} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)*x),x, algorithm="giac")

[Out]

1/12*arctan(sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^2) - 1/324*arctan(1/3*sqrt(d*x
^3 + c)/sqrt(-c))/(sqrt(-c)*c^2) + 2/27/(sqrt(d*x^3 + c)*c^2)