3.996 \(\int (d x)^m (c x^2)^p (a+b x)^{-2-m-2 p} \, dx\)

Optimal. Leaf size=39 \[ \frac{x \left (c x^2\right )^p (d x)^m (a+b x)^{-m-2 p-1}}{a (m+2 p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0103912, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {15, 20, 37} \[ \frac{x \left (c x^2\right )^p (d x)^m (a+b x)^{-m-2 p-1}}{a (m+2 p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.015029, size = 39, normalized size = 1. \[ \frac{x \left (c x^2\right )^p (d x)^m (a+b x)^{-m-2 p-1}}{a (m+2 p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 40, normalized size = 1. \begin{align*}{\frac{x \left ( dx \right ) ^{m} \left ( c{x}^{2} \right ) ^{p} \left ( bx+a \right ) ^{-1-m-2\,p}}{a \left ( 1+m+2\,p \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(d*x)^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., size = 0, normalized size = 0. \begin{align*} \int \left (c x^{2}\right )^{p}{\left (b x + a\right )}^{-m - 2 \, p - 2} \left (d x\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*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)*(d*x)^m, x)

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Fricas [A]  time = 1.59401, size = 132, normalized size = 3.38 \begin{align*} \frac{{\left (b x^{2} + a x\right )}{\left (b x + a\right )}^{-m - 2 \, p - 2} \left (d x\right )^{m} e^{\left (2 \, p \log \left (d x\right ) + p \log \left (\frac{c}{d^{2}}\right )\right )}}{a m + 2 \, a p + a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*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)*(d*x)^m*e^(2*p*log(d*x) + p*log(c/d^2))/(a*m + 2*a*p + a)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c x^{2}\right )^{p}{\left (b x + a\right )}^{-m - 2 \, p - 2} \left (d x\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*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)*(d*x)^m, x)