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

Optimal. Leaf size=26 \[ \frac {\left (c x^2\right )^p (a+b x)^{-2 p}}{2 a p} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 26, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {\left (c x^2\right )^p (a+b x)^{-2 p}}{2 a p} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 0.03, size = 26, normalized size = 1.00 \begin {gather*} \frac {\left (c x^2\right )^p (a+b x)^{-2 p}}{2 a p} \end {gather*}

Antiderivative was successfully verified.

[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))

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Maple [A]
time = 0.12, size = 25, normalized size = 0.96

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 27, normalized size = 1.04 \begin {gather*} \frac {c^{p} e^{\left (-2 \, p \log \left (b x + a\right ) + 2 \, p \log \left (x\right )\right )}}{2 \, a p} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]
time = 1.06, size = 31, normalized size = 1.19 \begin {gather*} \frac {{\left (b x + a\right )} \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{-2 \, p - 1}}{2 \, a p} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (19) = 38\).
time = 17.62, size = 250, normalized size = 9.62 \begin {gather*} \begin {cases} - \frac {\left (b x\right )^{- 2 p} \left (c x^{2}\right )^{p}}{b x} & \text {for}\: a = 0 \\\frac {0^{- 2 p - 1} \left (c x^{2}\right )^{p}}{2 p} & \text {for}\: a = - b x \\\frac {\left (c x^{2}\right )^{p} \left (0^{\frac {1}{p}}\right )^{- 2 p - 1}}{2 p} & \text {for}\: a = 0^{\frac {1}{p}} - b x \\\frac {\log {\left (x \right )}}{a} - \frac {\log {\left (\frac {a}{b} + x \right )}}{a} & \text {for}\: p = 0 \\\frac {a^{2} \left (c x^{2}\right )^{p}}{2 a^{3} p \left (a + b x\right )^{2 p} + 4 a^{2} b p x \left (a + b x\right )^{2 p} + 2 a b^{2} p x^{2} \left (a + b x\right )^{2 p}} + \frac {a b x \left (c x^{2}\right )^{p}}{2 a^{3} p \left (a + b x\right )^{2 p} + 4 a^{2} b p x \left (a + b x\right )^{2 p} + 2 a b^{2} p x^{2} \left (a + b x\right )^{2 p}} + \frac {b x \left (c x^{2}\right )^{p}}{2 a^{2} p \left (a + b x\right )^{2 p} + 2 a b p x \left (a + b x\right )^{2 p}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-(c*x**2)**p/(b*x*(b*x)**(2*p)), Eq(a, 0)), (0**(-2*p - 1)*(c*x**2)**p/(2*p), Eq(a, -b*x)), ((c*x**
2)**p*(0**(1/p))**(-2*p - 1)/(2*p), Eq(a, 0**(1/p) - b*x)), (log(x)/a - log(a/b + x)/a, Eq(p, 0)), (a**2*(c*x*
*2)**p/(2*a**3*p*(a + b*x)**(2*p) + 4*a**2*b*p*x*(a + b*x)**(2*p) + 2*a*b**2*p*x**2*(a + b*x)**(2*p)) + a*b*x*
(c*x**2)**p/(2*a**3*p*(a + b*x)**(2*p) + 4*a**2*b*p*x*(a + b*x)**(2*p) + 2*a*b**2*p*x**2*(a + b*x)**(2*p)) + b
*x*(c*x**2)**p/(2*a**2*p*(a + b*x)**(2*p) + 2*a*b*p*x*(a + b*x)**(2*p)), True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [B]
time = 0.26, size = 26, normalized size = 1.00 \begin {gather*} \frac {{\left (c\,x^2\right )}^p}{2\,a\,p\,{\left (a+b\,x\right )}^{2\,p}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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