3.389 \(\int x^5 \left (a+b x^3\right )^{3/2} \, dx\)

Optimal. Leaf size=38 \[ \frac{2 \left (a+b x^3\right )^{7/2}}{21 b^2}-\frac{2 a \left (a+b x^3\right )^{5/2}}{15 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0601747, antiderivative size = 38, 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 \[ \frac{2 \left (a+b x^3\right )^{7/2}}{21 b^2}-\frac{2 a \left (a+b x^3\right )^{5/2}}{15 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a + b*x^3)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.3807, size = 34, normalized size = 0.89 \[ - \frac{2 a \left (a + b x^{3}\right )^{\frac{5}{2}}}{15 b^{2}} + \frac{2 \left (a + b x^{3}\right )^{\frac{7}{2}}}{21 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(b*x**3+a)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0330856, size = 28, normalized size = 0.74 \[ \frac{2 \left (a+b x^3\right )^{5/2} \left (5 b x^3-2 a\right )}{105 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*(a + b*x^3)^(3/2),x]

[Out]

(2*(a + b*x^3)^(5/2)*(-2*a + 5*b*x^3))/(105*b^2)

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Maple [A]  time = 0.008, size = 25, normalized size = 0.7 \[ -{\frac{-10\,b{x}^{3}+4\,a}{105\,{b}^{2}} \left ( b{x}^{3}+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b*x^3+a)^(3/2),x)

[Out]

-2/105*(b*x^3+a)^(5/2)*(-5*b*x^3+2*a)/b^2

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Maxima [A]  time = 1.42802, size = 41, normalized size = 1.08 \[ \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}}}{21 \, b^{2}} - \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a}{15 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(3/2)*x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.2125, size = 61, normalized size = 1.61 \[ \frac{2 \,{\left (5 \, b^{3} x^{9} + 8 \, a b^{2} x^{6} + a^{2} b x^{3} - 2 \, a^{3}\right )} \sqrt{b x^{3} + a}}{105 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(3/2)*x^5,x, algorithm="fricas")

[Out]

2/105*(5*b^3*x^9 + 8*a*b^2*x^6 + a^2*b*x^3 - 2*a^3)*sqrt(b*x^3 + a)/b^2

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Sympy [A]  time = 5.66199, size = 88, normalized size = 2.32 \[ \begin{cases} - \frac{4 a^{3} \sqrt{a + b x^{3}}}{105 b^{2}} + \frac{2 a^{2} x^{3} \sqrt{a + b x^{3}}}{105 b} + \frac{16 a x^{6} \sqrt{a + b x^{3}}}{105} + \frac{2 b x^{9} \sqrt{a + b x^{3}}}{21} & \text{for}\: b \neq 0 \\\frac{a^{\frac{3}{2}} x^{6}}{6} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(b*x**3+a)**(3/2),x)

[Out]

Piecewise((-4*a**3*sqrt(a + b*x**3)/(105*b**2) + 2*a**2*x**3*sqrt(a + b*x**3)/(1
05*b) + 16*a*x**6*sqrt(a + b*x**3)/105 + 2*b*x**9*sqrt(a + b*x**3)/21, Ne(b, 0))
, (a**(3/2)*x**6/6, True))

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GIAC/XCAS [A]  time = 0.231698, size = 105, normalized size = 2.76 \[ \frac{2 \,{\left (\frac{7 \,{\left (3 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a\right )} a}{b} + \frac{15 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{2}}{b}\right )}}{315 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(3/2)*x^5,x, algorithm="giac")

[Out]

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