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

Optimal. Leaf size=33 \[ -\frac{\left (c x^2\right )^p (a+b x)^{1-2 p}}{a (1-2 p) x} \]

[Out]

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

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Rubi [A]  time = 0.0108223, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 37} \[ -\frac{\left (c x^2\right )^p (a+b x)^{1-2 p}}{a (1-2 p) x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0098735, size = 32, normalized size = 0.97 \[ \frac{\left (c x^2\right )^p (a+b x)^{1-2 p}}{a (2 p-1) x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 38, normalized size = 1.2 \begin{align*}{\frac{ \left ( bx+a \right ) \left ( c{x}^{2} \right ) ^{p}}{ \left ( 2\,p-1 \right ) ax \left ( bx+a \right ) ^{2\,p}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.63047, size = 72, normalized size = 2.18 \begin{align*} \frac{{\left (b x + a\right )} \left (c x^{2}\right )^{p}}{{\left (2 \, a p - a\right )}{\left (b x + a\right )}^{2 \, p} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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/x**2/((b*x+a)**(2*p)),x)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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