3.325 \(\int \frac{x^{11}}{\left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=90 \[ \frac{1024 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{81 d^4}+\frac{2 c^2}{27 d^4 \sqrt{c+d x^3}}-\frac{4 c \sqrt{c+d x^3}}{d^4}-\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^4} \]

[Out]

(2*c^2)/(27*d^4*Sqrt[c + d*x^3]) - (4*c*Sqrt[c + d*x^3])/d^4 - (2*(c + d*x^3)^(3
/2))/(9*d^4) + (1024*c^(3/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(81*d^4)

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Rubi [A]  time = 0.266037, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{1024 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{81 d^4}+\frac{2 c^2}{27 d^4 \sqrt{c+d x^3}}-\frac{4 c \sqrt{c+d x^3}}{d^4}-\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^4} \]

Antiderivative was successfully verified.

[In]  Int[x^11/((8*c - d*x^3)*(c + d*x^3)^(3/2)),x]

[Out]

(2*c^2)/(27*d^4*Sqrt[c + d*x^3]) - (4*c*Sqrt[c + d*x^3])/d^4 - (2*(c + d*x^3)^(3
/2))/(9*d^4) + (1024*c^(3/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(81*d^4)

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Rubi in Sympy [A]  time = 31.0834, size = 83, normalized size = 0.92 \[ \frac{1024 c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{81 d^{4}} + \frac{2 c^{2}}{27 d^{4} \sqrt{c + d x^{3}}} - \frac{4 c \sqrt{c + d x^{3}}}{d^{4}} - \frac{2 \left (c + d x^{3}\right )^{\frac{3}{2}}}{9 d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(-d*x**3+8*c)/(d*x**3+c)**(3/2),x)

[Out]

1024*c**(3/2)*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/(81*d**4) + 2*c**2/(27*d**4*sq
rt(c + d*x**3)) - 4*c*sqrt(c + d*x**3)/d**4 - 2*(c + d*x**3)**(3/2)/(9*d**4)

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Mathematica [A]  time = 0.181066, size = 70, normalized size = 0.78 \[ \frac{2 \left (512 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )-\frac{3 \left (56 c^2+60 c d x^3+3 d^2 x^6\right )}{\sqrt{c+d x^3}}\right )}{81 d^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/((8*c - d*x^3)*(c + d*x^3)^(3/2)),x]

[Out]

(2*((-3*(56*c^2 + 60*c*d*x^3 + 3*d^2*x^6))/Sqrt[c + d*x^3] + 512*c^(3/2)*ArcTanh
[Sqrt[c + d*x^3]/(3*Sqrt[c])]))/(81*d^4)

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Maple [C]  time = 0.084, size = 560, normalized size = 6.2 \[ -{\frac{1}{d} \left ( -{\frac{2\,{c}^{2}}{3\,{d}^{3}}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{c}{d}} \right ) d}}}}+{\frac{2\,{x}^{3}}{9\,{d}^{2}}\sqrt{d{x}^{3}+c}}-{\frac{10\,c}{9\,{d}^{3}}\sqrt{d{x}^{3}+c}} \right ) }-8\,{\frac{c}{{d}^{2}} \left ( 2/3\,{\frac{c}{{d}^{2}}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{c}{d}} \right ) d}}}}+2/3\,{\frac{\sqrt{d{x}^{3}+c}}{{d}^{2}}} \right ) }+{\frac{128\,{c}^{2}}{3\,{d}^{4}}{\frac{1}{\sqrt{d{x}^{3}+c}}}}-512\,{\frac{{c}^{3}}{{d}^{3}} \left ({\frac{2}{27\,cd}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{c}{d}} \right ) d}}}}+{\frac{{\frac{i}{243}}\sqrt{2}}{{d}^{3}{c}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ( d{{\it \_Z}}^{3}-8\,c \right ) }{\frac{\sqrt [3]{-c{d}^{2}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{2/3}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{2/3} \right ) }{\sqrt{d{x}^{3}+c}}\sqrt{{\frac{i/2d}{\sqrt [3]{-c{d}^{2}}} \left ( 2\,x+{\frac{-i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}\sqrt{{\frac{d}{-3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}}} \left ( x-{\frac{\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}\sqrt{{\frac{-i/2d}{\sqrt [3]{-c{d}^{2}}} \left ( 2\,x+{\frac{i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}{\it EllipticPi} \left ( 1/3\,\sqrt{3}\sqrt{{\frac{id\sqrt{3}}{\sqrt [3]{-c{d}^{2}}} \left ( x+1/2\,{\frac{\sqrt [3]{-c{d}^{2}}}{d}}-{\frac{i/2\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d}} \right ) }},-1/18\,{\frac{2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd}{cd}},\sqrt{{\frac{i\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d} \left ( -3/2\,{\frac{\sqrt [3]{-c{d}^{2}}}{d}}+{\frac{i/2\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d}} \right ) ^{-1}}} \right ) }} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(-d*x^3+8*c)/(d*x^3+c)^(3/2),x)

[Out]

-1/d*(-2/3/d^3*c^2/((x^3+c/d)*d)^(1/2)+2/9/d^2*x^3*(d*x^3+c)^(1/2)-10/9*c*(d*x^3
+c)^(1/2)/d^3)-8*c/d^2*(2/3/d^2*c/((x^3+c/d)*d)^(1/2)+2/3*(d*x^3+c)^(1/2)/d^2)+1
28/3*c^2/d^4/(d*x^3+c)^(1/2)-512*c^3/d^3*(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=RootOf(_Z^3*d-8*c)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^11/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237121, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (9 \, d^{2} x^{6} + 180 \, c d x^{3} - 256 \, \sqrt{d x^{3} + c} c^{\frac{3}{2}} \log \left (\frac{d x^{3} + 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) + 168 \, c^{2}\right )}}{81 \, \sqrt{d x^{3} + c} d^{4}}, -\frac{2 \,{\left (9 \, d^{2} x^{6} + 180 \, c d x^{3} - 512 \, \sqrt{d x^{3} + c} \sqrt{-c} c \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right ) + 168 \, c^{2}\right )}}{81 \, \sqrt{d x^{3} + c} d^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^11/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)),x, algorithm="fricas")

[Out]

[-2/81*(9*d^2*x^6 + 180*c*d*x^3 - 256*sqrt(d*x^3 + c)*c^(3/2)*log((d*x^3 + 6*sqr
t(d*x^3 + c)*sqrt(c) + 10*c)/(d*x^3 - 8*c)) + 168*c^2)/(sqrt(d*x^3 + c)*d^4), -2
/81*(9*d^2*x^6 + 180*c*d*x^3 - 512*sqrt(d*x^3 + c)*sqrt(-c)*c*arctan(1/3*sqrt(d*
x^3 + c)/sqrt(-c)) + 168*c^2)/(sqrt(d*x^3 + c)*d^4)]

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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(x**11/(-d*x**3+8*c)/(d*x**3+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219514, size = 111, normalized size = 1.23 \[ -\frac{1024 \, c^{2} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{81 \, \sqrt{-c} d^{4}} + \frac{2 \, c^{2}}{27 \, \sqrt{d x^{3} + c} d^{4}} - \frac{2 \,{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} d^{8} + 18 \, \sqrt{d x^{3} + c} c d^{8}\right )}}{9 \, d^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^11/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)),x, algorithm="giac")

[Out]

-1024/81*c^2*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*d^4) + 2/27*c^2/(sqr
t(d*x^3 + c)*d^4) - 2/9*((d*x^3 + c)^(3/2)*d^8 + 18*sqrt(d*x^3 + c)*c*d^8)/d^12