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

Optimal. Leaf size=30 \[ \frac{x \left (c x^2\right )^p (a+b x)^{-2 p-1}}{a (2 p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0079995, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {15, 37} \[ \frac{x \left (c x^2\right )^p (a+b x)^{-2 p-1}}{a (2 p+1)} \]

Antiderivative was successfully verified.

[In]

Int[(c*x^2)^p*(a + b*x)^(-2 - 2*p),x]

[Out]

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

Mathematica [A]  time = 0.0161289, size = 28, normalized size = 0.93 \[ \frac{x \left (c x^2\right )^p (a+b x)^{-2 p-1}}{2 a p+a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 31, normalized size = 1. \begin{align*}{\frac{x \left ( c{x}^{2} \right ) ^{p} \left ( bx+a \right ) ^{-1-2\,p}}{a \left ( 1+2\,p \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(c*x^2)^p*(b*x+a)^(-1-2*p)/a/(1+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 )}^{-2 \, p - 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.56111, size = 78, normalized size = 2.6 \begin{align*} \frac{{\left (b x^{2} + a x\right )} \left (c x^{2}\right )^{p}{\left (b x + a\right )}^{-2 \, p - 2}}{2 \, a p + a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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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 )}^{-2 \, p - 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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