3.127 \(\int x \left (b x+c x^2\right )^p \, dx\)

Optimal. Leaf size=49 \[ \frac{x^2 \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,p+2;p+3;-\frac{c x}{b}\right )}{p+2} \]

[Out]

(x^2*(b*x + c*x^2)^p*Hypergeometric2F1[-p, 2 + p, 3 + p, -((c*x)/b)])/((2 + p)*(
1 + (c*x)/b)^p)

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Rubi [A]  time = 0.0702721, antiderivative size = 83, normalized size of antiderivative = 1.69, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\left (b x+c x^2\right )^{p+1} \left (-\frac{c x}{b}\right )^{-p-1} \, _2F_1\left (-p,p+1;p+2;\frac{b+c x}{b}\right )}{2 c (p+1)}+\frac{\left (b x+c x^2\right )^{p+1}}{2 c (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[x*(b*x + c*x^2)^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 10.7041, size = 48, normalized size = 0.98 \[ \frac{x x^{- p - 1} x^{p + 2} \left (1 + \frac{c x}{b}\right )^{- p} \left (b x + c x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, p + 2 \\ p + 3 \end{matrix}\middle |{- \frac{c x}{b}} \right )}}{p + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*x**(-p - 1)*x**(p + 2)*(1 + c*x/b)**(-p)*(b*x + c*x**2)**p*hyper((-p, p + 2),
(p + 3,), -c*x/b)/(p + 2)

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Mathematica [A]  time = 0.0351655, size = 47, normalized size = 0.96 \[ \frac{x^2 (x (b+c x))^p \left (\frac{c x}{b}+1\right )^{-p} \, _2F_1\left (-p,p+2;p+3;-\frac{c x}{b}\right )}{p+2} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(b*x + c*x^2)^p,x]

[Out]

(x^2*(x*(b + c*x))^p*Hypergeometric2F1[-p, 2 + p, 3 + p, -((c*x)/b)])/((2 + p)*(
1 + (c*x)/b)^p)

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \[ \int x \left ( c{x}^{2}+bx \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(c*x^2+b*x)^p,x)

[Out]

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

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

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. \[{\rm integral}\left ({\left (c x^{2} + b x\right )}^{p} x, x\right ) \]

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. \[ \int x \left (x \left (b + c x\right )\right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(c*x**2+b*x)**p,x)

[Out]

Integral(x*(x*(b + c*x))**p, x)

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

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)