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

Optimal. Leaf size=59 \[ \frac{2 a^2 \left (a+b x^3\right )^{3/2}}{9 b^3}+\frac{2 \left (a+b x^3\right )^{7/2}}{21 b^3}-\frac{4 a \left (a+b x^3\right )^{5/2}}{15 b^3} \]

[Out]

(2*a^2*(a + b*x^3)^(3/2))/(9*b^3) - (4*a*(a + b*x^3)^(5/2))/(15*b^3) + (2*(a + b
*x^3)^(7/2))/(21*b^3)

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Rubi [A]  time = 0.0842506, 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 \[ \frac{2 a^2 \left (a+b x^3\right )^{3/2}}{9 b^3}+\frac{2 \left (a+b x^3\right )^{7/2}}{21 b^3}-\frac{4 a \left (a+b x^3\right )^{5/2}}{15 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^8*Sqrt[a + b*x^3],x]

[Out]

(2*a^2*(a + b*x^3)^(3/2))/(9*b^3) - (4*a*(a + b*x^3)^(5/2))/(15*b^3) + (2*(a + b
*x^3)^(7/2))/(21*b^3)

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Rubi in Sympy [A]  time = 11.0251, size = 54, normalized size = 0.92 \[ \frac{2 a^{2} \left (a + b x^{3}\right )^{\frac{3}{2}}}{9 b^{3}} - \frac{4 a \left (a + b x^{3}\right )^{\frac{5}{2}}}{15 b^{3}} + \frac{2 \left (a + b x^{3}\right )^{\frac{7}{2}}}{21 b^{3}} \]

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*(a + b*x**3)**(3/2)/(9*b**3) - 4*a*(a + b*x**3)**(5/2)/(15*b**3) + 2*(a +
b*x**3)**(7/2)/(21*b**3)

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Mathematica [A]  time = 0.0239034, size = 50, normalized size = 0.85 \[ \frac{2 \sqrt{a+b x^3} \left (8 a^3-4 a^2 b x^3+3 a b^2 x^6+15 b^3 x^9\right )}{315 b^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a + b*x^3]*(8*a^3 - 4*a^2*b*x^3 + 3*a*b^2*x^6 + 15*b^3*x^9))/(315*b^3)

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Maple [A]  time = 0.009, size = 36, normalized size = 0.6 \[{\frac{30\,{b}^{2}{x}^{6}-24\,ab{x}^{3}+16\,{a}^{2}}{315\,{b}^{3}} \left ( b{x}^{3}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*(b*x^3+a)^(1/2),x)

[Out]

2/315*(b*x^3+a)^(3/2)*(15*b^2*x^6-12*a*b*x^3+8*a^2)/b^3

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Maxima [A]  time = 1.44107, size = 63, normalized size = 1.07 \[ \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}}}{21 \, b^{3}} - \frac{4 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a}{15 \, b^{3}} + \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{2}}{9 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*(b*x^3 + a)^(7/2)/b^3 - 4/15*(b*x^3 + a)^(5/2)*a/b^3 + 2/9*(b*x^3 + a)^(3/2
)*a^2/b^3

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Fricas [A]  time = 0.211916, size = 62, normalized size = 1.05 \[ \frac{2 \,{\left (15 \, b^{3} x^{9} + 3 \, a b^{2} x^{6} - 4 \, a^{2} b x^{3} + 8 \, a^{3}\right )} \sqrt{b x^{3} + a}}{315 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(15*b^3*x^9 + 3*a*b^2*x^6 - 4*a^2*b*x^3 + 8*a^3)*sqrt(b*x^3 + a)/b^3

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Sympy [A]  time = 4.43762, size = 90, normalized size = 1.53 \[ \begin{cases} \frac{16 a^{3} \sqrt{a + b x^{3}}}{315 b^{3}} - \frac{8 a^{2} x^{3} \sqrt{a + b x^{3}}}{315 b^{2}} + \frac{2 a x^{6} \sqrt{a + b x^{3}}}{105 b} + \frac{2 x^{9} \sqrt{a + b x^{3}}}{21} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{9}}{9} & \text{otherwise} \end{cases} \]

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**3*sqrt(a + b*x**3)/(315*b**3) - 8*a**2*x**3*sqrt(a + b*x**3)/(3
15*b**2) + 2*a*x**6*sqrt(a + b*x**3)/(105*b) + 2*x**9*sqrt(a + b*x**3)/21, Ne(b,
0)), (sqrt(a)*x**9/9, True))

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GIAC/XCAS [A]  time = 0.261105, size = 58, normalized size = 0.98 \[ \frac{2 \,{\left (15 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{2}\right )}}{315 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(15*(b*x^3 + a)^(7/2) - 42*(b*x^3 + a)^(5/2)*a + 35*(b*x^3 + a)^(3/2)*a^2)
/b^3