3.10.95 \(\int x^m (c x^2)^p (a+b x)^{-2-m-2 p} \, dx\) [995]

Optimal. Leaf size=38 \[ \frac {x^{1+m} \left (c x^2\right )^p (a+b x)^{-1-m-2 p}}{a (1+m+2 p)} \]

[Out]

x^(1+m)*(c*x^2)^p*(b*x+a)^(-1-m-2*p)/a/(1+m+2*p)

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Rubi [A]
time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {15, 37} \begin {gather*} \frac {x^{m+1} \left (c x^2\right )^p (a+b x)^{-m-2 p-1}}{a (m+2 p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^m*(c*x^2)^p*(a + b*x)^(-2 - m - 2*p),x]

[Out]

(x^(1 + m)*(c*x^2)^p*(a + b*x)^(-1 - m - 2*p))/(a*(1 + m + 2*p))

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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin {align*} \int x^m \left (c x^2\right )^p (a+b x)^{-2-m-2 p} \, dx &=\left (x^{-2 p} \left (c x^2\right )^p\right ) \int x^{m+2 p} (a+b x)^{-2-m-2 p} \, dx\\ &=\frac {x^{1+m} \left (c x^2\right )^p (a+b x)^{-1-m-2 p}}{a (1+m+2 p)}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 38, normalized size = 1.00 \begin {gather*} \frac {x^{1+m} \left (c x^2\right )^p (a+b x)^{-1-m-2 p}}{a (1+m+2 p)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^m*(c*x^2)^p*(a + b*x)^(-2 - m - 2*p),x]

[Out]

(x^(1 + m)*(c*x^2)^p*(a + b*x)^(-1 - m - 2*p))/(a*(1 + m + 2*p))

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Maple [A]
time = 0.16, size = 39, normalized size = 1.03

method	result	size
gosper	\(\frac {x^{1+m} \left (c \,x^{2}\right )^{p} \left (b x +a \right )^{-1-m -2 p}}{a \left (1+m +2 p \right )}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(c*x^2)^p*(b*x+a)^(-2-m-2*p),x,method=_RETURNVERBOSE)

[Out]

x^(1+m)*(c*x^2)^p*(b*x+a)^(-1-m-2*p)/a/(1+m+2*p)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(c*x^2)^p*(b*x+a)^(-2-m-2*p),x, algorithm="maxima")

[Out]

integrate((c*x^2)^p*(b*x + a)^(-m - 2*p - 2)*x^m, x)

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Fricas [A]
time = 0.65, size = 49, normalized size = 1.29 \begin {gather*} \frac {{\left (b x^{2} + a x\right )} {\left (b x + a\right )}^{-m - 2 \, p - 2} x^{m} e^{\left (p \log \left (c\right ) + 2 \, p \log \left (x\right )\right )}}{a m + 2 \, a p + a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(c*x^2)^p*(b*x+a)^(-2-m-2*p),x, algorithm="fricas")

[Out]

(b*x^2 + a*x)*(b*x + a)^(-m - 2*p - 2)*x^m*e^(p*log(c) + 2*p*log(x))/(a*m + 2*a*p + a)

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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(x**m*(c*x**2)**p*(b*x+a)**(-2-m-2*p),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 8012 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(x^m*(c*x^2)^p*(b*x+a)^(-2-m-2*p),x, algorithm="giac")

[Out]

integrate((c*x^2)^p*(b*x + a)^(-m - 2*p - 2)*x^m, x)

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Mupad [B]
time = 0.34, size = 50, normalized size = 1.32 \begin {gather*} \frac {x\,x^m\,{\left (c\,x^2\right )}^p}{a\,{\left (a+b\,x\right )}^m\,{\left (a+b\,x\right )}^{2\,p}\,\left (a+b\,x\right )\,\left (m+2\,p+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^m*(c*x^2)^p)/(a + b*x)^(m + 2*p + 2),x)

[Out]

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

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