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

Optimal. Leaf size=42 \[ \frac{\left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,p;p+1;-\frac{c x}{b}\right )}{p} \]

[Out]

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

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Rubi [A]  time = 0.0603126, antiderivative size = 42, 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 \[ \frac{\left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,p;p+1;-\frac{c x}{b}\right )}{p} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^p/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 10.8091, size = 41, normalized size = 0.98 \[ \frac{x^{p} x^{- p + 1} \left (1 + \frac{c x}{b}\right )^{- p} \left (b x + c x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, p \\ p + 1 \end{matrix}\middle |{- \frac{c x}{b}} \right )}}{p x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0305421, size = 40, normalized size = 0.95 \[ \frac{(x (b+c x))^p \left (\frac{c x}{b}+1\right )^{-p} \, _2F_1\left (-p,p;p+1;-\frac{c x}{b}\right )}{p} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^p/x,x]

[Out]

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

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \[ \int{\frac{ \left ( c{x}^{2}+bx \right ) ^{p}}{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^p/x,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x)^p/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*x^2 + b*x)^p/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**p/x,x)

[Out]

Integral((x*(b + c*x))**p/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x)^p/x, x)