3.2.58 \(\int x^m (b x^n)^{3/2} \, dx\) [158]

Optimal. Leaf size=28 \[ \frac {2 b x^{1+m+n} \sqrt {b x^n}}{2+2 m+3 n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {15, 30} \begin {gather*} \frac {2 b \sqrt {b x^n} x^{m+n+1}}{2 m+3 n+2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 25, normalized size = 0.89 \begin {gather*} \frac {x^{1+m} \left (b x^n\right )^{3/2}}{1+m+\frac {3 n}{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.03, size = 25, normalized size = 0.89

method	result	size
gosper	\(\frac {2 x^{1+m} \left (b \,x^{n}\right )^{\frac {3}{2}}}{2+2 m +3 n}\)	\(25\)
risch	\(\frac {2 b^{2} x \,x^{m} x^{2 n}}{\left (2+2 m +3 n \right ) \sqrt {b \,x^{n}}}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.31, size = 24, normalized size = 0.86 \begin {gather*} \frac {2 \, b^{\frac {3}{2}} x x^{m} {\left (x^{n}\right )}^{\frac {3}{2}}}{2 \, m + 3 \, n + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

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^m*(b*x^n)^(3/2),x, algorithm="giac")

[Out]

integrate((b*x^n)^(3/2)*x^m, x)

________________________________________________________________________________________

Mupad [B]
time = 1.06, size = 26, normalized size = 0.93 \begin {gather*} \frac {2\,b\,x^{m+n+1}\,\sqrt {b\,x^n}}{2\,m+3\,n+2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________