3.993 \(\int \frac{(c x^2)^p (a+b x)^{1-2 p}}{x^3} \, dx\)

Optimal. Leaf size=35 \[ -\frac{\left (c x^2\right )^p (a+b x)^{2-2 p}}{2 a (1-p) x^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01086, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {15, 37} \[ -\frac{\left (c x^2\right )^p (a+b x)^{2-2 p}}{2 a (1-p) x^2} \]

Antiderivative was successfully verified.

[In]

Int[((c*x^2)^p*(a + b*x)^(1 - 2*p))/x^3,x]

[Out]

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

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)^{1-2 p}}{x^3} \, dx &=\left (x^{-2 p} \left (c x^2\right )^p\right ) \int x^{-3+2 p} (a+b x)^{1-2 p} \, dx\\ &=-\frac{\left (c x^2\right )^p (a+b x)^{2-2 p}}{2 a (1-p) x^2}\\ \end{align*}

Mathematica [A]  time = 0.0106003, size = 32, normalized size = 0.91 \[ \frac{\left (c x^2\right )^p (a+b x)^{2-2 p}}{a (2 p-2) x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[((c*x^2)^p*(a + b*x)^(1 - 2*p))/x^3,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 32, normalized size = 0.9 \begin{align*}{\frac{ \left ( bx+a \right ) ^{2-2\,p} \left ( c{x}^{2} \right ) ^{p}}{2\,{x}^{2}a \left ( p-1 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^p*(b*x+a)^(1-2*p)/x^3,x)

[Out]

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

________________________________________________________________________________________

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 + 1}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^p*(b*x+a)^(1-2*p)/x^3,x, algorithm="maxima")

[Out]

integrate((c*x^2)^p*(b*x + a)^(-2*p + 1)/x^3, x)

________________________________________________________________________________________

Fricas [A]  time = 1.78616, size = 84, normalized size = 2.4 \begin{align*} \frac{{\left (b x + a\right )} \left (c x^{2}\right )^{p}{\left (b x + a\right )}^{-2 \, p + 1}}{2 \,{\left (a p - a\right )} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^p*(b*x+a)^(1-2*p)/x^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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)**(1-2*p)/x**3,x)

[Out]

Exception raised: TypeError

________________________________________________________________________________________

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 + 1}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^p*(b*x+a)^(1-2*p)/x^3,x, algorithm="giac")

[Out]

integrate((c*x^2)^p*(b*x + a)^(-2*p + 1)/x^3, x)