\(\int \frac {(c x^2)^p (a+b x)^{-1-2 p}}{x} \, dx\) [991]

Optimal result

Integrand size = 22, antiderivative size = 26 \[ \int \frac {\left (c x^2\right )^p (a+b x)^{-1-2 p}}{x} \, dx=\frac {\left (c x^2\right )^p (a+b x)^{-2 p}}{2 a p} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {15, 37} \[ \int \frac {\left (c x^2\right )^p (a+b x)^{-1-2 p}}{x} \, dx=\frac {\left (c x^2\right )^p (a+b x)^{-2 p}}{2 a p} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c x^2\right )^p (a+b x)^{-1-2 p}}{x} \, dx=\frac {\left (c x^2\right )^p (a+b x)^{-2 p}}{2 a p} \]

[In]

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

[Out]

(c*x^2)^p/(2*a*p*(a + b*x)^(2*p))

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96

method	result	size
gosper	\(\frac {\left (c \,x^{2}\right )^{p} \left (b x +a \right )^{-2 p}}{2 a p}\)	\(25\)
parallelrisch	\(\frac {x \left (c \,x^{2}\right )^{p} \left (b x +a \right )^{-1-2 p} b +\left (c \,x^{2}\right )^{p} \left (b x +a \right )^{-1-2 p} a}{2 a p}\)	\(51\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {\left (c x^2\right )^p (a+b x)^{-1-2 p}}{x} \, dx=\frac {{\left (b x + a\right )} \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{-2 \, p - 1}}{2 \, a p} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\left (c x^2\right )^p (a+b x)^{-1-2 p}}{x} \, dx=\int \frac {\left (c x^{2}\right )^{p} \left (a + b x\right )^{- 2 p - 1}}{x}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {\left (c x^2\right )^p (a+b x)^{-1-2 p}}{x} \, dx=\frac {c^{p} e^{\left (-2 \, p \log \left (b x + a\right ) + 2 \, p \log \left (x\right )\right )}}{2 \, a p} \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\left (c x^2\right )^p (a+b x)^{-1-2 p}}{x} \, dx=\int { \frac {\left (c x^{2}\right )^{p} {\left (b x + a\right )}^{-2 \, p - 1}}{x} \,d x } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c x^2\right )^p (a+b x)^{-1-2 p}}{x} \, dx=\frac {{\left (c\,x^2\right )}^p}{2\,a\,p\,{\left (a+b\,x\right )}^{2\,p}} \]

[In]

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

[Out]

(c*x^2)^p/(2*a*p*(a + b*x)^(2*p))