3.4.90 \(\int x^2 (a+b x^3)^{3/2} \, dx\) [390]

Optimal. Leaf size=18 \[ \frac {2 \left (a+b x^3\right )^{5/2}}{15 b} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {267} \begin {gather*} \frac {2 \left (a+b x^3\right )^{5/2}}{15 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^3)^(3/2),x]

[Out]

(2*(a + b*x^3)^(5/2))/(15*b)

Rule 267

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

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} \frac {2 \left (a+b x^3\right )^{5/2}}{15 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x^3)^(3/2),x]

[Out]

(2*(a + b*x^3)^(5/2))/(15*b)

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Maple [A]
time = 0.12, size = 15, normalized size = 0.83

method	result	size
gosper	\(\frac {2 \left (b \,x^{3}+a \right )^{\frac {5}{2}}}{15 b}\)	\(15\)
derivativedivides	\(\frac {2 \left (b \,x^{3}+a \right )^{\frac {5}{2}}}{15 b}\)	\(15\)
default	\(\frac {2 \left (b \,x^{3}+a \right )^{\frac {5}{2}}}{15 b}\)	\(15\)
trager	\(\frac {2 \left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right ) \sqrt {b \,x^{3}+a}}{15 b}\)	\(33\)
risch	\(\frac {2 \left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right ) \sqrt {b \,x^{3}+a}}{15 b}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x^3+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 14, normalized size = 0.78 \begin {gather*} \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}}}{15 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(b*x^3 + a)^(5/2)/b

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (14) = 28\).
time = 0.34, size = 32, normalized size = 1.78 \begin {gather*} \frac {2 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \sqrt {b x^{3} + a}}{15 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(b^2*x^6 + 2*a*b*x^3 + a^2)*sqrt(b*x^3 + a)/b

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (14) = 28\).
time = 0.14, size = 65, normalized size = 3.61 \begin {gather*} \begin {cases} \frac {2 a^{2} \sqrt {a + b x^{3}}}{15 b} + \frac {4 a x^{3} \sqrt {a + b x^{3}}}{15} + \frac {2 b x^{6} \sqrt {a + b x^{3}}}{15} & \text {for}\: b \neq 0 \\\frac {a^{\frac {3}{2}} x^{3}}{3} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*a**2*sqrt(a + b*x**3)/(15*b) + 4*a*x**3*sqrt(a + b*x**3)/15 + 2*b*x**6*sqrt(a + b*x**3)/15, Ne(b,
0)), (a**(3/2)*x**3/3, True))

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Giac [A]
time = 1.97, size = 14, normalized size = 0.78 \begin {gather*} \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}}}{15 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(b*x^3 + a)^(5/2)/b

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Mupad [B]
time = 1.09, size = 14, normalized size = 0.78 \begin {gather*} \frac {2\,{\left (b\,x^3+a\right )}^{5/2}}{15\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*x^3)^(3/2),x)

[Out]

(2*(a + b*x^3)^(5/2))/(15*b)

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