3.129 \(\int \frac{\left (b x+c x^2\right )^p}{x^2} \, dx\)

Optimal. Leaf size=50 \[ -\frac{\left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (p-1,-p;p;-\frac{c x}{b}\right )}{(1-p) x} \]

[Out]

-(((b*x + c*x^2)^p*Hypergeometric2F1[-1 + p, -p, p, -((c*x)/b)])/((1 - p)*x*(1 +
(c*x)/b)^p))

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Rubi [A]  time = 0.0651665, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (p-1,-p;p;-\frac{c x}{b}\right )}{(1-p) x} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^p/x^2,x]

[Out]

-(((b*x + c*x^2)^p*Hypergeometric2F1[-1 + p, -p, p, -((c*x)/b)])/((1 - p)*x*(1 +
(c*x)/b)^p))

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Rubi in Sympy [A]  time = 11.2132, size = 48, normalized size = 0.96 \[ - \frac{x^{- p + 2} x^{p - 1} \left (1 + \frac{c x}{b}\right )^{- p} \left (b x + c x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, p - 1 \\ p \end{matrix}\middle |{- \frac{c x}{b}} \right )}}{x^{2} \left (- p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0390914, size = 45, normalized size = 0.9 \[ \frac{(x (b+c x))^p \left (\frac{c x}{b}+1\right )^{-p} \, _2F_1\left (p-1,-p;p;-\frac{c x}{b}\right )}{(p-1) x} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^p/x^2,x]

[Out]

((x*(b + c*x))^p*Hypergeometric2F1[-1 + p, -p, p, -((c*x)/b)])/((-1 + p)*x*(1 +
(c*x)/b)^p)

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \[ \int{\frac{ \left ( c{x}^{2}+bx \right ) ^{p}}{{x}^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^p/x^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x)^p/x^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c x^{2} + b x\right )}^{p}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^p/x^2,x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x)^p/x^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{p}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**p/x**2,x)

[Out]

Integral((x*(b + c*x))**p/x**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x)^p/x^2, x)