3.2.62 \(\int (c x)^m (b x^n)^{5/2} \, dx\) [162]

Optimal. Leaf size=31 \[ \frac {2 (c x)^{1+m} \left (b x^n\right )^{5/2}}{c (2+2 m+5 n)} \]

[Out]

2*(c*x)^(1+m)*(b*x^n)^(5/2)/c/(2+2*m+5*n)

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Rubi [A]
time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.16, 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^2 x^{2 n+1} \sqrt {b x^n} (c x)^m}{2 m+5 n+2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*x)^m*(b*x^n)^(5/2),x]

[Out]

(2*b^2*x^(1 + 2*n)*(c*x)^m*Sqrt[b*x^n])/(2 + 2*m + 5*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 )^{5/2} \, dx &=\left (b^2 x^{-n/2} \sqrt {b x^n}\right ) \int x^{5 n/2} (c x)^m \, dx\\ &=\left (b^2 x^{-m-\frac {n}{2}} (c x)^m \sqrt {b x^n}\right ) \int x^{m+\frac {5 n}{2}} \, dx\\ &=\frac {2 b^2 x^{1+2 n} (c x)^m \sqrt {b x^n}}{2+2 m+5 n}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 26, normalized size = 0.84 \begin {gather*} \frac {x (c x)^m \left (b x^n\right )^{5/2}}{1+m+\frac {5 n}{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^m*(b*x^n)^(5/2),x]

[Out]

(x*(c*x)^m*(b*x^n)^(5/2))/(1 + m + (5*n)/2)

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Maple [A]
time = 0.06, size = 26, normalized size = 0.84

method	result	size
gosper	\(\frac {2 x \left (c x \right )^{m} \left (b \,x^{n}\right )^{\frac {5}{2}}}{2+2 m +5 n}\)	\(26\)
risch	\(\frac {2 b^{3} x \,{\mathrm e}^{\frac {m \left (-i \mathrm {csgn}\left (i c x \right )^{3} \pi +i \mathrm {csgn}\left (i c x \right )^{2} \mathrm {csgn}\left (i c \right ) \pi +i \mathrm {csgn}\left (i c x \right )^{2} \mathrm {csgn}\left (i x \right ) \pi -i \mathrm {csgn}\left (i c x \right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x \right ) \pi +2 \ln \left (x \right )+2 \ln \left (c \right )\right )}{2}} x^{3 n}}{\left (2+2 m +5 n \right ) \sqrt {b \,x^{n}}}\)	\(108\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^m*(b*x^n)^(5/2),x,method=_RETURNVERBOSE)

[Out]

2*x/(2+2*m+5*n)*(c*x)^m*(b*x^n)^(5/2)

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Maxima [A]
time = 0.31, size = 27, normalized size = 0.87 \begin {gather*} \frac {2 \, b^{\frac {5}{2}} c^{m} x x^{m} {\left (x^{n}\right )}^{\frac {5}{2}}}{2 \, m + 5 \, n + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m*(b*x^n)^(5/2),x, algorithm="maxima")

[Out]

2*b^(5/2)*c^m*x*x^m*(x^n)^(5/2)/(2*m + 5*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)^(5/2),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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)**m*(b*x**n)**(5/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 8569 deep

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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)^(5/2),x, algorithm="giac")

[Out]

integrate((b*x^n)^(5/2)*(c*x)^m, x)

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Mupad [B]
time = 1.05, size = 34, normalized size = 1.10 \begin {gather*} \frac {2\,b^2\,x^{2\,n+1}\,\sqrt {b\,x^n}\,{\left (c\,x\right )}^m}{2\,m+5\,n+2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^n)^(5/2)*(c*x)^m,x)

[Out]

(2*b^2*x^(2*n + 1)*(b*x^n)^(1/2)*(c*x)^m)/(2*m + 5*n + 2)

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