3.388 \(\int x^8 \left (a+b x^3\right )^{3/2} \, dx\)

Optimal. Leaf size=59 \[ \frac{2 a^2 \left (a+b x^3\right )^{5/2}}{15 b^3}+\frac{2 \left (a+b x^3\right )^{9/2}}{27 b^3}-\frac{4 a \left (a+b x^3\right )^{7/2}}{21 b^3} \]

[Out]

(2*a^2*(a + b*x^3)^(5/2))/(15*b^3) - (4*a*(a + b*x^3)^(7/2))/(21*b^3) + (2*(a +
b*x^3)^(9/2))/(27*b^3)

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Rubi [A]  time = 0.0869064, 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 \[ \frac{2 a^2 \left (a+b x^3\right )^{5/2}}{15 b^3}+\frac{2 \left (a+b x^3\right )^{9/2}}{27 b^3}-\frac{4 a \left (a+b x^3\right )^{7/2}}{21 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^8*(a + b*x^3)^(3/2),x]

[Out]

(2*a^2*(a + b*x^3)^(5/2))/(15*b^3) - (4*a*(a + b*x^3)^(7/2))/(21*b^3) + (2*(a +
b*x^3)^(9/2))/(27*b^3)

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Rubi in Sympy [A]  time = 10.9935, size = 54, normalized size = 0.92 \[ \frac{2 a^{2} \left (a + b x^{3}\right )^{\frac{5}{2}}}{15 b^{3}} - \frac{4 a \left (a + b x^{3}\right )^{\frac{7}{2}}}{21 b^{3}} + \frac{2 \left (a + b x^{3}\right )^{\frac{9}{2}}}{27 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*(a + b*x**3)**(5/2)/(15*b**3) - 4*a*(a + b*x**3)**(7/2)/(21*b**3) + 2*(a
+ b*x**3)**(9/2)/(27*b**3)

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Mathematica [A]  time = 0.038141, size = 39, normalized size = 0.66 \[ \frac{2 \left (a+b x^3\right )^{5/2} \left (8 a^2-20 a b x^3+35 b^2 x^6\right )}{945 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8*(a + b*x^3)^(3/2),x]

[Out]

(2*(a + b*x^3)^(5/2)*(8*a^2 - 20*a*b*x^3 + 35*b^2*x^6))/(945*b^3)

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Maple [A]  time = 0.007, size = 36, normalized size = 0.6 \[{\frac{70\,{b}^{2}{x}^{6}-40\,ab{x}^{3}+16\,{a}^{2}}{945\,{b}^{3}} \left ( b{x}^{3}+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*(b*x^3+a)^(3/2),x)

[Out]

2/945*(b*x^3+a)^(5/2)*(35*b^2*x^6-20*a*b*x^3+8*a^2)/b^3

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Maxima [A]  time = 1.43586, size = 63, normalized size = 1.07 \[ \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{9}{2}}}{27 \, b^{3}} - \frac{4 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} a}{21 \, b^{3}} + \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a^{2}}{15 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(3/2)*x^8,x, algorithm="maxima")

[Out]

2/27*(b*x^3 + a)^(9/2)/b^3 - 4/21*(b*x^3 + a)^(7/2)*a/b^3 + 2/15*(b*x^3 + a)^(5/
2)*a^2/b^3

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Fricas [A]  time = 0.211186, size = 77, normalized size = 1.31 \[ \frac{2 \,{\left (35 \, b^{4} x^{12} + 50 \, a b^{3} x^{9} + 3 \, a^{2} b^{2} x^{6} - 4 \, a^{3} b x^{3} + 8 \, a^{4}\right )} \sqrt{b x^{3} + a}}{945 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(3/2)*x^8,x, algorithm="fricas")

[Out]

2/945*(35*b^4*x^12 + 50*a*b^3*x^9 + 3*a^2*b^2*x^6 - 4*a^3*b*x^3 + 8*a^4)*sqrt(b*
x^3 + a)/b^3

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Sympy [A]  time = 12.7318, size = 112, normalized size = 1.9 \[ \begin{cases} \frac{16 a^{4} \sqrt{a + b x^{3}}}{945 b^{3}} - \frac{8 a^{3} x^{3} \sqrt{a + b x^{3}}}{945 b^{2}} + \frac{2 a^{2} x^{6} \sqrt{a + b x^{3}}}{315 b} + \frac{20 a x^{9} \sqrt{a + b x^{3}}}{189} + \frac{2 b x^{12} \sqrt{a + b x^{3}}}{27} & \text{for}\: b \neq 0 \\\frac{a^{\frac{3}{2}} x^{9}}{9} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((16*a**4*sqrt(a + b*x**3)/(945*b**3) - 8*a**3*x**3*sqrt(a + b*x**3)/(9
45*b**2) + 2*a**2*x**6*sqrt(a + b*x**3)/(315*b) + 20*a*x**9*sqrt(a + b*x**3)/189
+ 2*b*x**12*sqrt(a + b*x**3)/27, Ne(b, 0)), (a**(3/2)*x**9/9, True))

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GIAC/XCAS [A]  time = 0.224002, size = 143, normalized size = 2.42 \[ \frac{2 \,{\left (\frac{3 \,{\left (15 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{2}\right )} a}{b^{2}} + \frac{35 \,{\left (b x^{3} + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{3}}{b^{2}}\right )}}{945 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(3/2)*x^8,x, algorithm="giac")

[Out]

2/945*(3*(15*(b*x^3 + a)^(7/2) - 42*(b*x^3 + a)^(5/2)*a + 35*(b*x^3 + a)^(3/2)*a
^2)*a/b^2 + (35*(b*x^3 + a)^(9/2) - 135*(b*x^3 + a)^(7/2)*a + 189*(b*x^3 + a)^(5
/2)*a^2 - 105*(b*x^3 + a)^(3/2)*a^3)/b^2)/b