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

Optimal. Leaf size=109 \[ \frac{1152 c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^3}-\frac{384 c^3 \sqrt{c+d x^3}}{d^3}-\frac{128 c^2 \left (c+d x^3\right )^{3/2}}{9 d^3}-\frac{14 c \left (c+d x^3\right )^{5/2}}{15 d^3}-\frac{2 \left (c+d x^3\right )^{7/2}}{21 d^3} \]

[Out]

(-384*c^3*Sqrt[c + d*x^3])/d^3 - (128*c^2*(c + d*x^3)^(3/2))/(9*d^3) - (14*c*(c
+ d*x^3)^(5/2))/(15*d^3) - (2*(c + d*x^3)^(7/2))/(21*d^3) + (1152*c^(7/2)*ArcTan
h[Sqrt[c + d*x^3]/(3*Sqrt[c])])/d^3

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Rubi [A]  time = 0.322432, antiderivative size = 109, 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 \[ \frac{1152 c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^3}-\frac{384 c^3 \sqrt{c+d x^3}}{d^3}-\frac{128 c^2 \left (c+d x^3\right )^{3/2}}{9 d^3}-\frac{14 c \left (c+d x^3\right )^{5/2}}{15 d^3}-\frac{2 \left (c+d x^3\right )^{7/2}}{21 d^3} \]

Antiderivative was successfully verified.

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

[Out]

(-384*c^3*Sqrt[c + d*x^3])/d^3 - (128*c^2*(c + d*x^3)^(3/2))/(9*d^3) - (14*c*(c
+ d*x^3)^(5/2))/(15*d^3) - (2*(c + d*x^3)^(7/2))/(21*d^3) + (1152*c^(7/2)*ArcTan
h[Sqrt[c + d*x^3]/(3*Sqrt[c])])/d^3

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Rubi in Sympy [A]  time = 33.2412, size = 102, normalized size = 0.94 \[ \frac{1152 c^{\frac{7}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{d^{3}} - \frac{384 c^{3} \sqrt{c + d x^{3}}}{d^{3}} - \frac{128 c^{2} \left (c + d x^{3}\right )^{\frac{3}{2}}}{9 d^{3}} - \frac{14 c \left (c + d x^{3}\right )^{\frac{5}{2}}}{15 d^{3}} - \frac{2 \left (c + d x^{3}\right )^{\frac{7}{2}}}{21 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1152*c**(7/2)*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/d**3 - 384*c**3*sqrt(c + d*x**
3)/d**3 - 128*c**2*(c + d*x**3)**(3/2)/(9*d**3) - 14*c*(c + d*x**3)**(5/2)/(15*d
**3) - 2*(c + d*x**3)**(7/2)/(21*d**3)

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Mathematica [A]  time = 0.125944, size = 81, normalized size = 0.74 \[ \frac{362880 c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )-2 \sqrt{c+d x^3} \left (62882 c^3+2579 c^2 d x^3+192 c d^2 x^6+15 d^3 x^9\right )}{315 d^3} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[c + d*x^3]*(62882*c^3 + 2579*c^2*d*x^3 + 192*c*d^2*x^6 + 15*d^3*x^9) +
362880*c^(7/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(315*d^3)

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Maple [C]  time = 0.016, size = 541, normalized size = 5. \[ -{\frac{1}{{d}^{2}} \left ( d \left ({\frac{2\,d{x}^{9}}{21}\sqrt{d{x}^{3}+c}}+{\frac{16\,c{x}^{6}}{105}\sqrt{d{x}^{3}+c}}+{\frac{2\,{c}^{2}{x}^{3}}{105\,d}\sqrt{d{x}^{3}+c}}-{\frac{4\,{c}^{3}}{105\,{d}^{2}}\sqrt{d{x}^{3}+c}} \right ) +{\frac{16\,c}{15\,d} \left ( d{x}^{3}+c \right ) ^{{\frac{5}{2}}}} \right ) }-64\,{\frac{{c}^{2}}{{d}^{2}} \left ( 2/9\,{x}^{3}\sqrt{d{x}^{3}+c}+{\frac{56\,c\sqrt{d{x}^{3}+c}}{9\,d}}+{\frac{3\,ic\sqrt{2}}{{d}^{3}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}d-8\,c \right ) }{\frac{\sqrt [3]{-c{d}^{2}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{2/3}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{2/3} \right ) }{\sqrt{d{x}^{3}+c}}\sqrt{{\frac{i/2d}{\sqrt [3]{-c{d}^{2}}} \left ( 2\,x+{\frac{-i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}\sqrt{{\frac{d}{-3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}}} \left ( x-{\frac{\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}\sqrt{{\frac{-i/2d}{\sqrt [3]{-c{d}^{2}}} \left ( 2\,x+{\frac{i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}{\it EllipticPi} \left ( 1/3\,\sqrt{3}\sqrt{{\frac{i\sqrt{3}d}{\sqrt [3]{-c{d}^{2}}} \left ( x+1/2\,{\frac{\sqrt [3]{-c{d}^{2}}}{d}}-{\frac{i/2\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d}} \right ) }},-1/18\,{\frac{2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd}{cd}},\sqrt{{\frac{i\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d} \left ( -3/2\,{\frac{\sqrt [3]{-c{d}^{2}}}{d}}+{\frac{i/2\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d}} \right ) ^{-1}}} \right ) }} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/d^2*(d*(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))+16/15*c/d*(d*x^3+c)^(5/2))-64*
c^2/d^2*(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*_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))*Elliptic
Pi(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*_alph
a*(-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^8/(d*x^3 - 8*c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24195, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (90720 \, c^{\frac{7}{2}} \log \left (\frac{d x^{3} + 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) -{\left (15 \, d^{3} x^{9} + 192 \, c d^{2} x^{6} + 2579 \, c^{2} d x^{3} + 62882 \, c^{3}\right )} \sqrt{d x^{3} + c}\right )}}{315 \, d^{3}}, \frac{2 \,{\left (181440 \, \sqrt{-c} c^{3} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right ) -{\left (15 \, d^{3} x^{9} + 192 \, c d^{2} x^{6} + 2579 \, c^{2} d x^{3} + 62882 \, c^{3}\right )} \sqrt{d x^{3} + c}\right )}}{315 \, d^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[2/315*(90720*c^(7/2)*log((d*x^3 + 6*sqrt(d*x^3 + c)*sqrt(c) + 10*c)/(d*x^3 - 8*
c)) - (15*d^3*x^9 + 192*c*d^2*x^6 + 2579*c^2*d*x^3 + 62882*c^3)*sqrt(d*x^3 + c))
/d^3, 2/315*(181440*sqrt(-c)*c^3*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c)) - (15*d^3*
x^9 + 192*c*d^2*x^6 + 2579*c^2*d*x^3 + 62882*c^3)*sqrt(d*x^3 + c))/d^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(x**8*(d*x**3+c)**(3/2)/(-d*x**3+8*c),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215404, size = 135, normalized size = 1.24 \[ -\frac{1152 \, c^{4} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} d^{3}} - \frac{2 \,{\left (15 \,{\left (d x^{3} + c\right )}^{\frac{7}{2}} d^{18} + 147 \,{\left (d x^{3} + c\right )}^{\frac{5}{2}} c d^{18} + 2240 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} c^{2} d^{18} + 60480 \, \sqrt{d x^{3} + c} c^{3} d^{18}\right )}}{315 \, d^{21}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1152*c^4*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*d^3) - 2/315*(15*(d*x^3
+ c)^(7/2)*d^18 + 147*(d*x^3 + c)^(5/2)*c*d^18 + 2240*(d*x^3 + c)^(3/2)*c^2*d^1
8 + 60480*sqrt(d*x^3 + c)*c^3*d^18)/d^21