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

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)

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Rubi [A]  time = 0.0309804, 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 \[ -\frac{\left (c x^2\right )^p (a+b x)^{2-2 p}}{2 a (1-p) x^2} \]

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)

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Rubi in Sympy [A]  time = 14.6016, size = 36, normalized size = 1.03 \[ - \frac{x^{- 2 p} x^{2 p - 2} \left (c x^{2}\right )^{p} \left (a + b x\right )^{- 2 p + 2}}{2 a \left (- p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0442335, size = 33, normalized size = 0.94 \[ \frac{\left (c x^2\right )^p (a+b x)^{2-2 p}}{2 a (p-1) x^2} \]

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))/(2*a*(-1 + p)*x^2)

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Maple [A]  time = 0.004, size = 32, normalized size = 0.9 \[{\frac{ \left ( bx+a \right ) ^{2-2\,p} \left ( c{x}^{2} \right ) ^{p}}{2\,{x}^{2}a \left ( p-1 \right ) }} \]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x^{2}\right )^{p}{\left (b x + a\right )}^{-2 \, p + 1}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.232915, size = 50, normalized size = 1.43 \[ \frac{{\left (b x + a\right )} \left (c x^{2}\right )^{p}{\left (b x + a\right )}^{-2 \, p + 1}}{2 \,{\left (a p - a\right )} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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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**3,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x^{2}\right )^{p}{\left (b x + a\right )}^{-2 \, p + 1}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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