3.270 \(\int \frac{x^8}{\sqrt{c+d x^3} \left (4 c+d x^3\right )} \, dx\)

Optimal. Leaf size=78 \[ \frac{32 c^{3/2} \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{3 \sqrt{3} d^3}-\frac{10 c \sqrt{c+d x^3}}{3 d^3}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^3} \]

[Out]

(-10*c*Sqrt[c + d*x^3])/(3*d^3) + (2*(c + d*x^3)^(3/2))/(9*d^3) + (32*c^(3/2)*Ar
cTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])])/(3*Sqrt[3]*d^3)

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Rubi [A]  time = 0.241893, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{32 c^{3/2} \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{3 \sqrt{3} d^3}-\frac{10 c \sqrt{c+d x^3}}{3 d^3}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^3} \]

Antiderivative was successfully verified.

[In]  Int[x^8/(Sqrt[c + d*x^3]*(4*c + d*x^3)),x]

[Out]

(-10*c*Sqrt[c + d*x^3])/(3*d^3) + (2*(c + d*x^3)^(3/2))/(9*d^3) + (32*c^(3/2)*Ar
cTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])])/(3*Sqrt[3]*d^3)

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Rubi in Sympy [A]  time = 21.8923, size = 75, normalized size = 0.96 \[ \frac{32 \sqrt{3} c^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{3} \sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{9 d^{3}} - \frac{10 c \sqrt{c + d x^{3}}}{3 d^{3}} + \frac{2 \left (c + d x^{3}\right )^{\frac{3}{2}}}{9 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(d*x**3+4*c)/(d*x**3+c)**(1/2),x)

[Out]

32*sqrt(3)*c**(3/2)*atan(sqrt(3)*sqrt(c + d*x**3)/(3*sqrt(c)))/(9*d**3) - 10*c*s
qrt(c + d*x**3)/(3*d**3) + 2*(c + d*x**3)**(3/2)/(9*d**3)

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Mathematica [A]  time = 0.0894404, size = 65, normalized size = 0.83 \[ \frac{32 \sqrt{3} c^{3/2} \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )+2 \left (d x^3-14 c\right ) \sqrt{c+d x^3}}{9 d^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/(Sqrt[c + d*x^3]*(4*c + d*x^3)),x]

[Out]

(2*(-14*c + d*x^3)*Sqrt[c + d*x^3] + 32*Sqrt[3]*c^(3/2)*ArcTan[Sqrt[c + d*x^3]/(
Sqrt[3]*Sqrt[c])])/(9*d^3)

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Maple [C]  time = 0.055, size = 467, normalized size = 6. \[{\frac{1}{{d}^{2}} \left ( d \left ({\frac{2\,{x}^{3}}{9\,d}\sqrt{d{x}^{3}+c}}-{\frac{4\,c}{9\,{d}^{2}}\sqrt{d{x}^{3}+c}} \right ) -{\frac{8\,c}{3\,d}\sqrt{d{x}^{3}+c}} \right ) }-{\frac{{\frac{16\,i}{9}}c\sqrt{2}}{{d}^{5}}\sum _{{\it \_alpha}={\it RootOf} \left ( d{{\it \_Z}}^{3}+4\,c \right ) }{1\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{id\sqrt{3} \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},{\frac{1}{6\,cd} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(d*x^3+4*c)/(d*x^3+c)^(1/2),x)

[Out]

1/d^2*(d*(2/9/d*x^3*(d*x^3+c)^(1/2)-4/9*c*(d*x^3+c)^(1/2)/d^2)-8/3*c*(d*x^3+c)^(
1/2)/d)-16/9*I*c/d^5*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),1/6/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+4*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^8/((d*x^3 + 4*c)*sqrt(d*x^3 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246175, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{3}{\left (24 \, \sqrt{-c} c \log \left (\frac{\sqrt{3}{\left (d x^{3} - 2 \, c\right )} + 6 \, \sqrt{d x^{3} + c} \sqrt{-c}}{d x^{3} + 4 \, c}\right ) + \sqrt{3} \sqrt{d x^{3} + c}{\left (d x^{3} - 14 \, c\right )}\right )}}{27 \, d^{3}}, \frac{2 \, \sqrt{3}{\left (48 \, c^{\frac{3}{2}} \arctan \left (\frac{\sqrt{3} \sqrt{d x^{3} + c}}{3 \, \sqrt{c}}\right ) + \sqrt{3} \sqrt{d x^{3} + c}{\left (d x^{3} - 14 \, c\right )}\right )}}{27 \, d^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/((d*x^3 + 4*c)*sqrt(d*x^3 + c)),x, algorithm="fricas")

[Out]

[2/27*sqrt(3)*(24*sqrt(-c)*c*log((sqrt(3)*(d*x^3 - 2*c) + 6*sqrt(d*x^3 + c)*sqrt
(-c))/(d*x^3 + 4*c)) + sqrt(3)*sqrt(d*x^3 + c)*(d*x^3 - 14*c))/d^3, 2/27*sqrt(3)
*(48*c^(3/2)*arctan(1/3*sqrt(3)*sqrt(d*x^3 + c)/sqrt(c)) + sqrt(3)*sqrt(d*x^3 +
c)*(d*x^3 - 14*c))/d^3]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{\sqrt{c + d x^{3}} \left (4 c + d x^{3}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(d*x**3+4*c)/(d*x**3+c)**(1/2),x)

[Out]

Integral(x**8/(sqrt(c + d*x**3)*(4*c + d*x**3)), x)

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GIAC/XCAS [A]  time = 0.214226, size = 86, normalized size = 1.1 \[ \frac{32 \, \sqrt{3} c^{\frac{3}{2}} \arctan \left (\frac{\sqrt{3} \sqrt{d x^{3} + c}}{3 \, \sqrt{c}}\right )}{9 \, d^{3}} + \frac{2 \,{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} d^{6} - 15 \, \sqrt{d x^{3} + c} c d^{6}\right )}}{9 \, d^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/((d*x^3 + 4*c)*sqrt(d*x^3 + c)),x, algorithm="giac")

[Out]

32/9*sqrt(3)*c^(3/2)*arctan(1/3*sqrt(3)*sqrt(d*x^3 + c)/sqrt(c))/d^3 + 2/9*((d*x
^3 + c)^(3/2)*d^6 - 15*sqrt(d*x^3 + c)*c*d^6)/d^9