3.313 \(\int \frac{1}{x^4 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx\)

Optimal. Leaf size=81 \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{288 c^{5/2}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{32 c^{5/2}}-\frac{\sqrt{c+d x^3}}{24 c^2 x^3} \]

[Out]

-Sqrt[c + d*x^3]/(24*c^2*x^3) + (d*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(288*c^
(5/2)) + (d*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(32*c^(5/2))

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Rubi [A]  time = 0.268044, antiderivative size = 81, 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 \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{288 c^{5/2}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{32 c^{5/2}}-\frac{\sqrt{c+d x^3}}{24 c^2 x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(8*c - d*x^3)*Sqrt[c + d*x^3]),x]

[Out]

-Sqrt[c + d*x^3]/(24*c^2*x^3) + (d*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(288*c^
(5/2)) + (d*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(32*c^(5/2))

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Rubi in Sympy [A]  time = 32.6038, size = 70, normalized size = 0.86 \[ - \frac{\sqrt{c + d x^{3}}}{24 c^{2} x^{3}} + \frac{d \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{288 c^{\frac{5}{2}}} + \frac{d \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{32 c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c + d*x**3)/(24*c**2*x**3) + d*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/(288*c*
*(5/2)) + d*atanh(sqrt(c + d*x**3)/sqrt(c))/(32*c**(5/2))

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Mathematica [C]  time = 0.341684, size = 326, normalized size = 4.02 \[ \frac{\frac{8 c d^2 x^6 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{\left (8 c-d x^3\right ) \left (d x^3 \left (F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+16 c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )}+\frac{10 c d^2 x^6 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}{\left (8 c-d x^3\right ) \left (5 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )+16 c F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )-c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )\right )}-c-d x^3}{24 c^2 x^3 \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^4*(8*c - d*x^3)*Sqrt[c + d*x^3]),x]

[Out]

(-c - d*x^3 + (8*c*d^2*x^6*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)]))) + (10*c*d^2*x^6*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)])))/(24*c^2*
x^3*Sqrt[c + d*x^3])

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Maple [C]  time = 0.015, size = 477, normalized size = 5.9 \[{\frac{1}{8\,c} \left ( -{\frac{1}{3\,c{x}^{3}}\sqrt{d{x}^{3}+c}}+{\frac{d}{3}{\it Artanh} \left ({1\sqrt{d{x}^{3}+c}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}} \right ) }-{\frac{d}{96}{\it Artanh} \left ({1\sqrt{d{x}^{3}+c}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{5}{2}}}}-{\frac{{\frac{i}{1728}}\sqrt{2}}{d{c}^{3}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}d-8\,c \right ) }{1\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},-{\frac{1}{18\,cd} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(-d*x^3+8*c)/(d*x^3+c)^(1/2),x)

[Out]

1/8/c*(-1/3*(d*x^3+c)^(1/2)/c/x^3+1/3*d*arctanh((d*x^3+c)^(1/2)/c^(1/2))/c^(3/2)
)-1/96*d*arctanh((d*x^3+c)^(1/2)/c^(1/2))/c^(5/2)-1/1728*I/d/c^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),-1/18/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. \[ -\int \frac{1}{\sqrt{d x^{3} + c}{\left (d x^{3} - 8 \, c\right )} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)*x^4),x, algorithm="maxima")

[Out]

-integrate(1/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)*x^4), x)

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Fricas [A]  time = 0.253168, size = 1, normalized size = 0.01 \[ \left [\frac{d x^{3} \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) + 9 \, d x^{3} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right ) - 24 \, \sqrt{d x^{3} + c} \sqrt{c}}{576 \, c^{\frac{5}{2}} x^{3}}, -\frac{d x^{3} \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) + 9 \, d x^{3} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) + 12 \, \sqrt{d x^{3} + c} \sqrt{-c}}{288 \, \sqrt{-c} c^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)*x^4),x, algorithm="fricas")

[Out]

[1/576*(d*x^3*log(((d*x^3 + 10*c)*sqrt(c) + 6*sqrt(d*x^3 + c)*c)/(d*x^3 - 8*c))
+ 9*d*x^3*log(((d*x^3 + 2*c)*sqrt(c) + 2*sqrt(d*x^3 + c)*c)/x^3) - 24*sqrt(d*x^3
+ c)*sqrt(c))/(c^(5/2)*x^3), -1/288*(d*x^3*arctan(3*c/(sqrt(d*x^3 + c)*sqrt(-c)
)) + 9*d*x^3*arctan(c/(sqrt(d*x^3 + c)*sqrt(-c))) + 12*sqrt(d*x^3 + c)*sqrt(-c))
/(sqrt(-c)*c^2*x^3)]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221023, size = 103, normalized size = 1.27 \[ -\frac{1}{288} \, d{\left (\frac{9 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{2}} + \frac{\arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} c^{2}} + \frac{12 \, \sqrt{d x^{3} + c}}{c^{2} d x^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)*x^4),x, algorithm="giac")

[Out]

-1/288*d*(9*arctan(sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^2) + arctan(1/3*sqrt(d*
x^3 + c)/sqrt(-c))/(sqrt(-c)*c^2) + 12*sqrt(d*x^3 + c)/(c^2*d*x^3))