3.2.63 \(\int (c x)^m (b x^n)^{3/2} \, dx\) [163]

Optimal. Leaf size=32 \[ \frac {2 b x^{1+n} (c x)^m \sqrt {b x^n}}{2+2 m+3 n} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {15, 20, 30} \begin {gather*} \frac {2 b x^{n+1} \sqrt {b x^n} (c x)^m}{2 m+3 n+2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*x^(1 + n)*(c*x)^m*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 20

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

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

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Mathematica [A]
time = 0.00, size = 26, normalized size = 0.81 \begin {gather*} \frac {x (c x)^m \left (b x^n\right )^{3/2}}{1+m+\frac {3 n}{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.06, size = 26, normalized size = 0.81

method	result	size
gosper	\(\frac {2 x \left (c x \right )^{m} \left (b \,x^{n}\right )^{\frac {3}{2}}}{2+2 m +3 n}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.32, size = 27, normalized size = 0.84 \begin {gather*} \frac {2 \, b^{\frac {3}{2}} c^{m} 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((c*x)^m*(b*x^n)^(3/2),x, algorithm="maxima")

[Out]

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

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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((c*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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \frac {2 x \left (b x^{n}\right )^{\frac {3}{2}} \left (c x\right )^{m}}{2 m + 3 n + 2} & \text {for}\: m \neq - \frac {3 n}{2} - 1 \\\int \left (b x^{n}\right )^{\frac {3}{2}} \left (c x\right )^{- \frac {3 n}{2} - 1}\, dx & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*x*(b*x**n)**(3/2)*(c*x)**m/(2*m + 3*n + 2), Ne(m, -3*n/2 - 1)), (Integral((b*x**n)**(3/2)*(c*x)**
(-3*n/2 - 1), 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((c*x)^m*(b*x^n)^(3/2),x, algorithm="giac")

[Out]

integrate((b*x^n)^(3/2)*(c*x)^m, x)

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Mupad [B]
time = 1.00, size = 30, normalized size = 0.94 \begin {gather*} \frac {2\,b\,x^{n+1}\,\sqrt {b\,x^n}\,{\left (c\,x\right )}^m}{2\,m+3\,n+2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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