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

Optimal. Leaf size=130 \[ \frac{9216 c^{9/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^4}-\frac{3072 c^4 \sqrt{c+d x^3}}{d^4}-\frac{1024 c^3 \left (c+d x^3\right )^{3/2}}{9 d^4}-\frac{38 c^2 \left (c+d x^3\right )^{5/2}}{5 d^4}-\frac{4 c \left (c+d x^3\right )^{7/2}}{7 d^4}-\frac{2 \left (c+d x^3\right )^{9/2}}{27 d^4} \]

[Out]

(-3072*c^4*Sqrt[c + d*x^3])/d^4 - (1024*c^3*(c + d*x^3)^(3/2))/(9*d^4) - (38*c^2
*(c + d*x^3)^(5/2))/(5*d^4) - (4*c*(c + d*x^3)^(7/2))/(7*d^4) - (2*(c + d*x^3)^(
9/2))/(27*d^4) + (9216*c^(9/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/d^4

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Rubi [A]  time = 0.342433, antiderivative size = 130, 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 \[ \frac{9216 c^{9/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^4}-\frac{3072 c^4 \sqrt{c+d x^3}}{d^4}-\frac{1024 c^3 \left (c+d x^3\right )^{3/2}}{9 d^4}-\frac{38 c^2 \left (c+d x^3\right )^{5/2}}{5 d^4}-\frac{4 c \left (c+d x^3\right )^{7/2}}{7 d^4}-\frac{2 \left (c+d x^3\right )^{9/2}}{27 d^4} \]

Antiderivative was successfully verified.

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

[Out]

(-3072*c^4*Sqrt[c + d*x^3])/d^4 - (1024*c^3*(c + d*x^3)^(3/2))/(9*d^4) - (38*c^2
*(c + d*x^3)^(5/2))/(5*d^4) - (4*c*(c + d*x^3)^(7/2))/(7*d^4) - (2*(c + d*x^3)^(
9/2))/(27*d^4) + (9216*c^(9/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/d^4

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Rubi in Sympy [A]  time = 36.6091, size = 122, normalized size = 0.94 \[ \frac{9216 c^{\frac{9}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{d^{4}} - \frac{3072 c^{4} \sqrt{c + d x^{3}}}{d^{4}} - \frac{1024 c^{3} \left (c + d x^{3}\right )^{\frac{3}{2}}}{9 d^{4}} - \frac{38 c^{2} \left (c + d x^{3}\right )^{\frac{5}{2}}}{5 d^{4}} - \frac{4 c \left (c + d x^{3}\right )^{\frac{7}{2}}}{7 d^{4}} - \frac{2 \left (c + d x^{3}\right )^{\frac{9}{2}}}{27 d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9216*c**(9/2)*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/d**4 - 3072*c**4*sqrt(c + d*x*
*3)/d**4 - 1024*c**3*(c + d*x**3)**(3/2)/(9*d**4) - 38*c**2*(c + d*x**3)**(5/2)/
(5*d**4) - 4*c*(c + d*x**3)**(7/2)/(7*d**4) - 2*(c + d*x**3)**(9/2)/(27*d**4)

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Mathematica [A]  time = 0.161684, size = 93, normalized size = 0.72 \[ \frac{9216 c^{9/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^4}-\frac{2 \sqrt{c+d x^3} \left (1509176 c^4+61892 c^3 d x^3+4611 c^2 d^2 x^6+410 c d^3 x^9+35 d^4 x^{12}\right )}{945 d^4} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[c + d*x^3]*(1509176*c^4 + 61892*c^3*d*x^3 + 4611*c^2*d^2*x^6 + 410*c*d^
3*x^9 + 35*d^4*x^12))/(945*d^4) + (9216*c^(9/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[
c])])/d^4

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Maple [C]  time = 0.07, size = 634, normalized size = 4.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/d*(2/27*d*x^12*(d*x^3+c)^(1/2)+20/189*c*x^9*(d*x^3+c)^(1/2)+2/315*c^2/d*x^6*(
d*x^3+c)^(1/2)-8/945*c^3/d^2*x^3*(d*x^3+c)^(1/2)+16/945*c^4/d^3*(d*x^3+c)^(1/2))
-8*c/d^2*(2/21*d*x^9*(d*x^3+c)^(1/2)+16/105*c*x^6*(d*x^3+c)^(1/2)+2/105*c^2/d*x^
3*(d*x^3+c)^(1/2)-4/105*c^3/d^2*(d*x^3+c)^(1/2))-128/15*c^2*(d*x^3+c)^(5/2)/d^4-
512*c^3/d^3*(2/9*x^3*(d*x^3+c)^(1/2)+56/9*c*(d*x^3+c)^(1/2)/d+3*I*c/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*_a
lpha^2*d^2-I*3^(1/2)*(-c*d^2)^(2/3)-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*Elli
pticPi(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(-(d*x^3 + c)^(3/2)*x^11/(d*x^3 - 8*c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.253619, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2177280 \, c^{\frac{9}{2}} \log \left (\frac{d x^{3} + 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) -{\left (35 \, d^{4} x^{12} + 410 \, c d^{3} x^{9} + 4611 \, c^{2} d^{2} x^{6} + 61892 \, c^{3} d x^{3} + 1509176 \, c^{4}\right )} \sqrt{d x^{3} + c}\right )}}{945 \, d^{4}}, \frac{2 \,{\left (4354560 \, \sqrt{-c} c^{4} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right ) -{\left (35 \, d^{4} x^{12} + 410 \, c d^{3} x^{9} + 4611 \, c^{2} d^{2} x^{6} + 61892 \, c^{3} d x^{3} + 1509176 \, c^{4}\right )} \sqrt{d x^{3} + c}\right )}}{945 \, d^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[2/945*(2177280*c^(9/2)*log((d*x^3 + 6*sqrt(d*x^3 + c)*sqrt(c) + 10*c)/(d*x^3 -
8*c)) - (35*d^4*x^12 + 410*c*d^3*x^9 + 4611*c^2*d^2*x^6 + 61892*c^3*d*x^3 + 1509
176*c^4)*sqrt(d*x^3 + c))/d^4, 2/945*(4354560*sqrt(-c)*c^4*arctan(1/3*sqrt(d*x^3
+ c)/sqrt(-c)) - (35*d^4*x^12 + 410*c*d^3*x^9 + 4611*c^2*d^2*x^6 + 61892*c^3*d*
x^3 + 1509176*c^4)*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)**(3/2)/(-d*x**3+8*c),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.217599, size = 158, normalized size = 1.22 \[ -\frac{9216 \, c^{5} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} d^{4}} - \frac{2 \,{\left (35 \,{\left (d x^{3} + c\right )}^{\frac{9}{2}} d^{32} + 270 \,{\left (d x^{3} + c\right )}^{\frac{7}{2}} c d^{32} + 3591 \,{\left (d x^{3} + c\right )}^{\frac{5}{2}} c^{2} d^{32} + 53760 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} c^{3} d^{32} + 1451520 \, \sqrt{d x^{3} + c} c^{4} d^{32}\right )}}{945 \, d^{36}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9216*c^5*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*d^4) - 2/945*(35*(d*x^3
+ c)^(9/2)*d^32 + 270*(d*x^3 + c)^(7/2)*c*d^32 + 3591*(d*x^3 + c)^(5/2)*c^2*d^3
2 + 53760*(d*x^3 + c)^(3/2)*c^3*d^32 + 1451520*sqrt(d*x^3 + c)*c^4*d^32)/d^36