3.164 \(\int (c x)^m \sqrt {b x^n} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.01, 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} \[ \frac {2 x \sqrt {b x^n} (c x)^m}{2 m+n+2} \]

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*}

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

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)

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b x^{n}} \left (c x\right )^{m}\,{d x} \]

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)

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maple [A]  time = 0.00, size = 24, normalized size = 0.83 \[ \frac {2 \sqrt {b \,x^{n}}\, x \left (c x \right )^{m}}{2 m +n +2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.47, size = 25, normalized size = 0.86 \[ \frac {2 \, \sqrt {b} c^{m} x x^{m} \sqrt {x^{n}}}{2 \, m + n + 2} \]

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)

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

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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