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

Optimal. Leaf size=111 \[ \frac{1024 c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^4}-\frac{1024 c^3 \sqrt{c+d x^3}}{3 d^4}-\frac{38 c^2 \left (c+d x^3\right )^{3/2}}{3 d^4}-\frac{4 c \left (c+d x^3\right )^{5/2}}{5 d^4}-\frac{2 \left (c+d x^3\right )^{7/2}}{21 d^4} \]

[Out]

(-1024*c^3*Sqrt[c + d*x^3])/(3*d^4) - (38*c^2*(c + d*x^3)^(3/2))/(3*d^4) - (4*c*
(c + d*x^3)^(5/2))/(5*d^4) - (2*(c + d*x^3)^(7/2))/(21*d^4) + (1024*c^(7/2)*ArcT
anh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/d^4

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Rubi [A]  time = 0.291983, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{1024 c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^4}-\frac{1024 c^3 \sqrt{c+d x^3}}{3 d^4}-\frac{38 c^2 \left (c+d x^3\right )^{3/2}}{3 d^4}-\frac{4 c \left (c+d x^3\right )^{5/2}}{5 d^4}-\frac{2 \left (c+d x^3\right )^{7/2}}{21 d^4} \]

Antiderivative was successfully verified.

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

[Out]

(-1024*c^3*Sqrt[c + d*x^3])/(3*d^4) - (38*c^2*(c + d*x^3)^(3/2))/(3*d^4) - (4*c*
(c + d*x^3)^(5/2))/(5*d^4) - (2*(c + d*x^3)^(7/2))/(21*d^4) + (1024*c^(7/2)*ArcT
anh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/d^4

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Rubi in Sympy [A]  time = 31.3084, size = 104, normalized size = 0.94 \[ \frac{1024 c^{\frac{7}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{d^{4}} - \frac{1024 c^{3} \sqrt{c + d x^{3}}}{3 d^{4}} - \frac{38 c^{2} \left (c + d x^{3}\right )^{\frac{3}{2}}}{3 d^{4}} - \frac{4 c \left (c + d x^{3}\right )^{\frac{5}{2}}}{5 d^{4}} - \frac{2 \left (c + d x^{3}\right )^{\frac{7}{2}}}{21 d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1024*c**(7/2)*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/d**4 - 1024*c**3*sqrt(c + d*x*
*3)/(3*d**4) - 38*c**2*(c + d*x**3)**(3/2)/(3*d**4) - 4*c*(c + d*x**3)**(5/2)/(5
*d**4) - 2*(c + d*x**3)**(7/2)/(21*d**4)

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Mathematica [A]  time = 0.118387, size = 81, normalized size = 0.73 \[ \frac{107520 c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )-2 \sqrt{c+d x^3} \left (18632 c^3+764 c^2 d x^3+57 c d^2 x^6+5 d^3 x^9\right )}{105 d^4} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[c + d*x^3]*(18632*c^3 + 764*c^2*d*x^3 + 57*c*d^2*x^6 + 5*d^3*x^9) + 107
520*c^(7/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(105*d^4)

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Maple [C]  time = 0.089, size = 582, normalized size = 5.2 \[ -{\frac{1}{d} \left ({\frac{2\,{x}^{9}}{21}\sqrt{d{x}^{3}+c}}+{\frac{2\,c{x}^{6}}{105\,d}\sqrt{d{x}^{3}+c}}-{\frac{8\,{c}^{2}{x}^{3}}{315\,{d}^{2}}\sqrt{d{x}^{3}+c}}+{\frac{16\,{c}^{3}}{315\,{d}^{3}}\sqrt{d{x}^{3}+c}} \right ) }-8\,{\frac{c}{{d}^{2}} \left ( 2/15\,{x}^{6}\sqrt{d{x}^{3}+c}+{\frac{2\,c{x}^{3}\sqrt{d{x}^{3}+c}}{45\,d}}-{\frac{4\,{c}^{2}\sqrt{d{x}^{3}+c}}{45\,{d}^{2}}} \right ) }-{\frac{128\,{c}^{2}}{9\,{d}^{4}} \left ( d{x}^{3}+c \right ) ^{{\frac{3}{2}}}}-512\,{\frac{{c}^{3}}{{d}^{3}} \left ( 2/3\,{\frac{\sqrt{d{x}^{3}+c}}{d}}+{\frac{i/3\sqrt{2}}{{d}^{3}}\sum _{{\it \_alpha}={\it RootOf} \left ( d{{\it \_Z}}^{3}-8\,c \right ) }{\frac{\sqrt [3]{-c{d}^{2}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{2/3}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{2/3} \right ) }{\sqrt{d{x}^{3}+c}}\sqrt{{\frac{i/2d}{\sqrt [3]{-c{d}^{2}}} \left ( 2\,x+{\frac{-i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}\sqrt{{\frac{d}{-3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}}} \left ( x-{\frac{\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}\sqrt{{\frac{-i/2d}{\sqrt [3]{-c{d}^{2}}} \left ( 2\,x+{\frac{i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}{\it EllipticPi} \left ( 1/3\,\sqrt{3}\sqrt{{\frac{id\sqrt{3}}{\sqrt [3]{-c{d}^{2}}} \left ( x+1/2\,{\frac{\sqrt [3]{-c{d}^{2}}}{d}}-{\frac{i/2\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d}} \right ) }},-1/18\,{\frac{2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd}{cd}},\sqrt{{\frac{i\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d} \left ( -3/2\,{\frac{\sqrt [3]{-c{d}^{2}}}{d}}+{\frac{i/2\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d}} \right ) ^{-1}}} \right ) }} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/d*(2/21*x^9*(d*x^3+c)^(1/2)+2/105*c/d*x^6*(d*x^3+c)^(1/2)-8/315*c^2/d^2*x^3*(
d*x^3+c)^(1/2)+16/315*c^3*(d*x^3+c)^(1/2)/d^3)-8*c/d^2*(2/15*x^6*(d*x^3+c)^(1/2)
+2/45*c/d*x^3*(d*x^3+c)^(1/2)-4/45*c^2*(d*x^3+c)^(1/2)/d^2)-128/9*c^2*(d*x^3+c)^
(3/2)/d^4-512*c^3/d^3*(2/3*(d*x^3+c)^(1/2)/d+1/3*I/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)*_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=Roo
tOf(_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(-sqrt(d*x^3 + c)*x^11/(d*x^3 - 8*c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.243944, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (26880 \, c^{\frac{7}{2}} \log \left (\frac{d x^{3} + 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) -{\left (5 \, d^{3} x^{9} + 57 \, c d^{2} x^{6} + 764 \, c^{2} d x^{3} + 18632 \, c^{3}\right )} \sqrt{d x^{3} + c}\right )}}{105 \, d^{4}}, \frac{2 \,{\left (53760 \, \sqrt{-c} c^{3} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right ) -{\left (5 \, d^{3} x^{9} + 57 \, c d^{2} x^{6} + 764 \, c^{2} d x^{3} + 18632 \, c^{3}\right )} \sqrt{d x^{3} + c}\right )}}{105 \, d^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[2/105*(26880*c^(7/2)*log((d*x^3 + 6*sqrt(d*x^3 + c)*sqrt(c) + 10*c)/(d*x^3 - 8*
c)) - (5*d^3*x^9 + 57*c*d^2*x^6 + 764*c^2*d*x^3 + 18632*c^3)*sqrt(d*x^3 + c))/d^
4, 2/105*(53760*sqrt(-c)*c^3*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c)) - (5*d^3*x^9 +
57*c*d^2*x^6 + 764*c^2*d*x^3 + 18632*c^3)*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+c)**(1/2)/(-d*x**3+8*c),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.214813, size = 135, normalized size = 1.22 \[ -\frac{1024 \, c^{4} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} d^{4}} - \frac{2 \,{\left (5 \,{\left (d x^{3} + c\right )}^{\frac{7}{2}} d^{24} + 42 \,{\left (d x^{3} + c\right )}^{\frac{5}{2}} c d^{24} + 665 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} c^{2} d^{24} + 17920 \, \sqrt{d x^{3} + c} c^{3} d^{24}\right )}}{105 \, d^{28}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1024*c^4*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*d^4) - 2/105*(5*(d*x^3
+ c)^(7/2)*d^24 + 42*(d*x^3 + c)^(5/2)*c*d^24 + 665*(d*x^3 + c)^(3/2)*c^2*d^24 +
17920*sqrt(d*x^3 + c)*c^3*d^24)/d^28