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

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 = 0.0238234, 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, Rules used = {266, 43} \[ \frac{2 \left (a+b x^3\right )^{7/2}}{21 b^2}-\frac{2 a \left (a+b x^3\right )^{5/2}}{15 b^2} \]

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)

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^5 \left (a+b x^3\right )^{3/2} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x (a+b x)^{3/2} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^{3/2}}{b}+\frac{(a+b x)^{5/2}}{b}\right ) \, dx,x,x^3\right )\\ &=-\frac{2 a \left (a+b x^3\right )^{5/2}}{15 b^2}+\frac{2 \left (a+b x^3\right )^{7/2}}{21 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0143225, 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 = 0.004, size = 25, normalized size = 0.7 \begin{align*} -{\frac{-10\,b{x}^{3}+4\,a}{105\,{b}^{2}} \left ( b{x}^{3}+a \right ) ^{{\frac{5}{2}}}} \end{align*}

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 = 1.26559, size = 41, normalized size = 1.08 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^3+a)^(3/2),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 = 1.41838, size = 99, normalized size = 2.61 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^3+a)^(3/2),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 = 2.3229, size = 88, normalized size = 2.32 \begin{align*} \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} \end{align*}

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)/(105*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 [B]  time = 1.13056, size = 105, normalized size = 2.76 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^3+a)^(3/2),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