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

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

[Out]

(c*x^2)^p/(2*a*p*(a + b*x)^(2*p))

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Rubi [A]  time = 0.0217156, antiderivative size = 26, 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 \[ \frac{\left (c x^2\right )^p (a+b x)^{-2 p}}{2 a p} \]

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))

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Rubi in Sympy [A]  time = 14.5611, size = 19, normalized size = 0.73 \[ \frac{\left (c x^{2}\right )^{p} \left (a + b x\right )^{- 2 p}}{2 a p} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*x**2)**p*(a + b*x)**(-2*p)/(2*a*p)

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Mathematica [A]  time = 0.0166298, size = 26, normalized size = 1. \[ \frac{\left (c x^2\right )^p (a+b x)^{-2 p}}{2 a p} \]

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 = 0.004, size = 25, normalized size = 1. \[{\frac{ \left ( bx+a \right ) ^{-2\,p} \left ( c{x}^{2} \right ) ^{p}}{2\,ap}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.35831, size = 36, normalized size = 1.38 \[ \frac{c^{p} e^{\left (-2 \, p \log \left (b x + a\right ) + 2 \, p \log \left (x\right )\right )}}{2 \, a p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^p*(b*x + a)^(-2*p - 1)/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 = 0.220978, size = 42, normalized size = 1.62 \[ \frac{{\left (b x + a\right )} \left (c x^{2}\right )^{p}{\left (b x + a\right )}^{-2 \, p - 1}}{2 \, a p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^p*(b*x + a)^(-2*p - 1)/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 [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x^{2}\right )^{p}{\left (b x + a\right )}^{-2 \, p - 1}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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