3.2.64 \(\int (c x)^m \sqrt {b x^n} \, dx\) [164]

Optimal. Leaf size=29 \[ \frac {2 (c x)^{1+m} \sqrt {b x^n}}{c (2+2 m+n)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 25, normalized size of antiderivative = 0.86, 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 x \sqrt {b x^n} (c x)^m}{2 m+n+2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*x)^m*Sqrt[b*x^n],x]

[Out]

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

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 26, normalized size = 0.90 \begin {gather*} \frac {x (c x)^m \sqrt {b x^n}}{1+m+\frac {n}{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^m*Sqrt[b*x^n],x]

[Out]

(x*(c*x)^m*Sqrt[b*x^n])/(1 + m + n/2)

________________________________________________________________________________________

Maple [A]
time = 0.03, size = 24, normalized size = 0.83

method	result	size
gosper	\(\frac {2 x \left (c x \right )^{m} \sqrt {b \,x^{n}}}{2+2 m +n}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^m*(b*x^n)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.31, size = 25, normalized size = 0.86 \begin {gather*} \frac {2 \, \sqrt {b} c^{m} x x^{m} \sqrt {x^{n}}}{2 \, m + n + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m*(b*x^n)^(1/2),x, algorithm="maxima")

[Out]

2*sqrt(b)*c^m*x*x^m*sqrt(x^n)/(2*m + 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((c*x)^m*(b*x^n)^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

[Out]

Piecewise((2*x*sqrt(b*x**n)*(c*x)**m/(2*m + n + 2), Ne(m, -n/2 - 1)), (Integral(sqrt(b*x**n)*(c*x)**(-n/2 - 1)
, x), True))

________________________________________________________________________________________

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)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(b*x^n)*(c*x)^m, x)

________________________________________________________________________________________

Mupad [B]
time = 1.01, size = 23, normalized size = 0.79 \begin {gather*} \frac {2\,x\,\sqrt {b\,x^n}\,{\left (c\,x\right )}^m}{2\,m+n+2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^n)^(1/2)*(c*x)^m,x)

[Out]

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

________________________________________________________________________________________