3.2.37 \(\int x (b x^n)^{3/2} \, dx\) [137]

Optimal. Leaf size=24 \[ \frac {2 b x^{2+n} \sqrt {b x^n}}{4+3 n} \]

[Out]

2*b*x^(2+n)*(b*x^n)^(1/2)/(4+3*n)

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Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {15, 30} \begin {gather*} \frac {2 b x^{n+2} \sqrt {b x^n}}{3 n+4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(b*x^n)^(3/2),x]

[Out]

(2*b*x^(2 + n)*Sqrt[b*x^n])/(4 + 3*n)

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int x \left (b x^n\right )^{3/2} \, dx &=\left (b x^{-n/2} \sqrt {b x^n}\right ) \int x^{1+\frac {3 n}{2}} \, dx\\ &=\frac {2 b x^{2+n} \sqrt {b x^n}}{4+3 n}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 22, normalized size = 0.92 \begin {gather*} \frac {x^2 \left (b x^n\right )^{3/2}}{2+\frac {3 n}{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(b*x^n)^(3/2),x]

[Out]

(x^2*(b*x^n)^(3/2))/(2 + (3*n)/2)

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Maple [A]
time = 0.01, size = 20, normalized size = 0.83

method	result	size
gosper	\(\frac {2 x^{2} \left (b \,x^{n}\right )^{\frac {3}{2}}}{4+3 n}\)	\(20\)
risch	\(\frac {2 b^{2} x^{2} x^{2 n}}{\left (4+3 n \right ) \sqrt {b \,x^{n}}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x^2/(4+3*n)*(b*x^n)^(3/2)

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Maxima [A]
time = 0.30, size = 19, normalized size = 0.79 \begin {gather*} \frac {2 \, \left (b x^{n}\right )^{\frac {3}{2}} x^{2}}{3 \, n + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(b*x^n)^(3/2)*x^2/(3*n + 4)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \frac {2 x^{2} \left (b x^{n}\right )^{\frac {3}{2}}}{3 n + 4} & \text {for}\: n \neq - \frac {4}{3} \\\int x \left (\frac {b}{x^{\frac {4}{3}}}\right )^{\frac {3}{2}}\, dx & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x**n)**(3/2),x)

[Out]

Piecewise((2*x**2*(b*x**n)**(3/2)/(3*n + 4), Ne(n, -4/3)), (Integral(x*(b/x**(4/3))**(3/2), x), True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^n)^(3/2)*x, x)

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Mupad [B]
time = 0.98, size = 22, normalized size = 0.92 \begin {gather*} \frac {2\,b\,x^{n+2}\,\sqrt {b\,x^n}}{3\,n+4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x^n)^(3/2),x)

[Out]

(2*b*x^(n + 2)*(b*x^n)^(1/2))/(3*n + 4)

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