3.405 \(\int \frac{x^8}{\sqrt{a+b x^3}} \, dx\)

Optimal. Leaf size=59 \[ \frac{2 a^2 \sqrt{a+b x^3}}{3 b^3}+\frac{2 \left (a+b x^3\right )^{5/2}}{15 b^3}-\frac{4 a \left (a+b x^3\right )^{3/2}}{9 b^3} \]

[Out]

(2*a^2*Sqrt[a + b*x^3])/(3*b^3) - (4*a*(a + b*x^3)^(3/2))/(9*b^3) + (2*(a + b*x^
3)^(5/2))/(15*b^3)

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Rubi [A]  time = 0.0878251, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 a^2 \sqrt{a+b x^3}}{3 b^3}+\frac{2 \left (a+b x^3\right )^{5/2}}{15 b^3}-\frac{4 a \left (a+b x^3\right )^{3/2}}{9 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^8/Sqrt[a + b*x^3],x]

[Out]

(2*a^2*Sqrt[a + b*x^3])/(3*b^3) - (4*a*(a + b*x^3)^(3/2))/(9*b^3) + (2*(a + b*x^
3)^(5/2))/(15*b^3)

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Rubi in Sympy [A]  time = 10.8279, size = 54, normalized size = 0.92 \[ \frac{2 a^{2} \sqrt{a + b x^{3}}}{3 b^{3}} - \frac{4 a \left (a + b x^{3}\right )^{\frac{3}{2}}}{9 b^{3}} + \frac{2 \left (a + b x^{3}\right )^{\frac{5}{2}}}{15 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(b*x**3+a)**(1/2),x)

[Out]

2*a**2*sqrt(a + b*x**3)/(3*b**3) - 4*a*(a + b*x**3)**(3/2)/(9*b**3) + 2*(a + b*x
**3)**(5/2)/(15*b**3)

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Mathematica [A]  time = 0.0264136, size = 39, normalized size = 0.66 \[ \frac{2 \sqrt{a+b x^3} \left (8 a^2-4 a b x^3+3 b^2 x^6\right )}{45 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/Sqrt[a + b*x^3],x]

[Out]

(2*Sqrt[a + b*x^3]*(8*a^2 - 4*a*b*x^3 + 3*b^2*x^6))/(45*b^3)

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Maple [A]  time = 0.008, size = 36, normalized size = 0.6 \[{\frac{6\,{b}^{2}{x}^{6}-8\,ab{x}^{3}+16\,{a}^{2}}{45\,{b}^{3}}\sqrt{b{x}^{3}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(b*x^3+a)^(1/2),x)

[Out]

2/45*(b*x^3+a)^(1/2)*(3*b^2*x^6-4*a*b*x^3+8*a^2)/b^3

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Maxima [A]  time = 1.43605, size = 63, normalized size = 1.07 \[ \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}}}{15 \, b^{3}} - \frac{4 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a}{9 \, b^{3}} + \frac{2 \, \sqrt{b x^{3} + a} a^{2}}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/sqrt(b*x^3 + a),x, algorithm="maxima")

[Out]

2/15*(b*x^3 + a)^(5/2)/b^3 - 4/9*(b*x^3 + a)^(3/2)*a/b^3 + 2/3*sqrt(b*x^3 + a)*a
^2/b^3

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Fricas [A]  time = 0.223603, size = 47, normalized size = 0.8 \[ \frac{2 \,{\left (3 \, b^{2} x^{6} - 4 \, a b x^{3} + 8 \, a^{2}\right )} \sqrt{b x^{3} + a}}{45 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/sqrt(b*x^3 + a),x, algorithm="fricas")

[Out]

2/45*(3*b^2*x^6 - 4*a*b*x^3 + 8*a^2)*sqrt(b*x^3 + a)/b^3

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Sympy [A]  time = 4.84583, size = 70, normalized size = 1.19 \[ \begin{cases} \frac{16 a^{2} \sqrt{a + b x^{3}}}{45 b^{3}} - \frac{8 a x^{3} \sqrt{a + b x^{3}}}{45 b^{2}} + \frac{2 x^{6} \sqrt{a + b x^{3}}}{15 b} & \text{for}\: b \neq 0 \\\frac{x^{9}}{9 \sqrt{a}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(b*x**3+a)**(1/2),x)

[Out]

Piecewise((16*a**2*sqrt(a + b*x**3)/(45*b**3) - 8*a*x**3*sqrt(a + b*x**3)/(45*b*
*2) + 2*x**6*sqrt(a + b*x**3)/(15*b), Ne(b, 0)), (x**9/(9*sqrt(a)), True))

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GIAC/XCAS [A]  time = 0.225241, size = 58, normalized size = 0.98 \[ \frac{2 \,{\left (3 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} - 10 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{b x^{3} + a} a^{2}\right )}}{45 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/sqrt(b*x^3 + a),x, algorithm="giac")

[Out]

2/45*(3*(b*x^3 + a)^(5/2) - 10*(b*x^3 + a)^(3/2)*a + 15*sqrt(b*x^3 + a)*a^2)/b^3