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

Optimal. Leaf size=90 \[ \frac{1024 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{9 d^4}-\frac{38 c^2 \sqrt{c+d x^3}}{d^4}-\frac{4 c \left (c+d x^3\right )^{3/2}}{3 d^4}-\frac{2 \left (c+d x^3\right )^{5/2}}{15 d^4} \]

[Out]

(-38*c^2*Sqrt[c + d*x^3])/d^4 - (4*c*(c + d*x^3)^(3/2))/(3*d^4) - (2*(c + d*x^3)
^(5/2))/(15*d^4) + (1024*c^(5/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(9*d^4)

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Rubi [A]  time = 0.228862, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{1024 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{9 d^4}-\frac{38 c^2 \sqrt{c+d x^3}}{d^4}-\frac{4 c \left (c+d x^3\right )^{3/2}}{3 d^4}-\frac{2 \left (c+d x^3\right )^{5/2}}{15 d^4} \]

Antiderivative was successfully verified.

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

[Out]

(-38*c^2*Sqrt[c + d*x^3])/d^4 - (4*c*(c + d*x^3)^(3/2))/(3*d^4) - (2*(c + d*x^3)
^(5/2))/(15*d^4) + (1024*c^(5/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(9*d^4)

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Rubi in Sympy [A]  time = 26.2305, size = 83, normalized size = 0.92 \[ \frac{1024 c^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{9 d^{4}} - \frac{38 c^{2} \sqrt{c + d x^{3}}}{d^{4}} - \frac{4 c \left (c + d x^{3}\right )^{\frac{3}{2}}}{3 d^{4}} - \frac{2 \left (c + d x^{3}\right )^{\frac{5}{2}}}{15 d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(-d*x**3+8*c)/(d*x**3+c)**(1/2),x)

[Out]

1024*c**(5/2)*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/(9*d**4) - 38*c**2*sqrt(c + d*
x**3)/d**4 - 4*c*(c + d*x**3)**(3/2)/(3*d**4) - 2*(c + d*x**3)**(5/2)/(15*d**4)

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Mathematica [A]  time = 0.109898, size = 69, normalized size = 0.77 \[ \frac{5120 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )-6 \sqrt{c+d x^3} \left (296 c^2+12 c d x^3+d^2 x^6\right )}{45 d^4} \]

Antiderivative was successfully verified.

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

[Out]

(-6*Sqrt[c + d*x^3]*(296*c^2 + 12*c*d*x^3 + d^2*x^6) + 5120*c^(5/2)*ArcTanh[Sqrt
[c + d*x^3]/(3*Sqrt[c])])/(45*d^4)

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Maple [C]  time = 0.064, size = 528, normalized size = 5.9 \[ -{\frac{1}{d} \left ({\frac{2\,{x}^{6}}{15\,d}\sqrt{d{x}^{3}+c}}-{\frac{8\,c{x}^{3}}{45\,{d}^{2}}\sqrt{d{x}^{3}+c}}+{\frac{16\,{c}^{2}}{45\,{d}^{3}}\sqrt{d{x}^{3}+c}} \right ) }-8\,{\frac{c}{{d}^{2}} \left ( 2/9\,{\frac{{x}^{3}\sqrt{d{x}^{3}+c}}{d}}-4/9\,{\frac{c\sqrt{d{x}^{3}+c}}{{d}^{2}}} \right ) }-{\frac{128\,{c}^{2}}{3\,{d}^{4}}\sqrt{d{x}^{3}+c}}-{\frac{{\frac{512\,i}{27}}{c}^{2}\sqrt{2}}{{d}^{6}}\sum _{{\it \_alpha}={\it RootOf} \left ( d{{\it \_Z}}^{3}-8\,c \right ) }{1\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{id\sqrt{3} \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},-{\frac{1}{18\,cd} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(-d*x^3+8*c)/(d*x^3+c)^(1/2),x)

[Out]

-1/d*(2/15/d*x^6*(d*x^3+c)^(1/2)-8/45*c/d^2*x^3*(d*x^3+c)^(1/2)+16/45*c^2*(d*x^3
+c)^(1/2)/d^3)-8*c/d^2*(2/9/d*x^3*(d*x^3+c)^(1/2)-4/9*c*(d*x^3+c)^(1/2)/d^2)-128
/3*c^2*(d*x^3+c)^(1/2)/d^4-512/27*I*c^2/d^6*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/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241403, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (1280 \, c^{\frac{5}{2}} \log \left (\frac{d x^{3} + 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) - 3 \,{\left (d^{2} x^{6} + 12 \, c d x^{3} + 296 \, c^{2}\right )} \sqrt{d x^{3} + c}\right )}}{45 \, d^{4}}, \frac{2 \,{\left (2560 \, \sqrt{-c} c^{2} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right ) - 3 \,{\left (d^{2} x^{6} + 12 \, c d x^{3} + 296 \, c^{2}\right )} \sqrt{d x^{3} + c}\right )}}{45 \, d^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[2/45*(1280*c^(5/2)*log((d*x^3 + 6*sqrt(d*x^3 + c)*sqrt(c) + 10*c)/(d*x^3 - 8*c)
) - 3*(d^2*x^6 + 12*c*d*x^3 + 296*c^2)*sqrt(d*x^3 + c))/d^4, 2/45*(2560*sqrt(-c)
*c^2*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c)) - 3*(d^2*x^6 + 12*c*d*x^3 + 296*c^2)*s
qrt(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)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218956, size = 111, normalized size = 1.23 \[ -\frac{1024 \, c^{3} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{9 \, \sqrt{-c} d^{4}} - \frac{2 \,{\left ({\left (d x^{3} + c\right )}^{\frac{5}{2}} d^{16} + 10 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} c d^{16} + 285 \, \sqrt{d x^{3} + c} c^{2} d^{16}\right )}}{15 \, d^{20}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1024/9*c^3*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*d^4) - 2/15*((d*x^3 +
c)^(5/2)*d^16 + 10*(d*x^3 + c)^(3/2)*c*d^16 + 285*sqrt(d*x^3 + c)*c^2*d^16)/d^2
0