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

Optimal. Leaf size=49 \[ \frac{x^3 \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,p+3;p+4;-\frac{c x}{b}\right )}{p+3} \]

[Out]

(x^3*(b*x + c*x^2)^p*Hypergeometric2F1[-p, 3 + p, 4 + p, -((c*x)/b)])/((3 + p)*(
1 + (c*x)/b)^p)

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Rubi [A]  time = 0.065322, antiderivative size = 49, 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 \[ \frac{x^3 \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,p+3;p+4;-\frac{c x}{b}\right )}{p+3} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(b*x + c*x^2)^p,x]

[Out]

(x^3*(b*x + c*x^2)^p*Hypergeometric2F1[-p, 3 + p, 4 + p, -((c*x)/b)])/((3 + p)*(
1 + (c*x)/b)^p)

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Rubi in Sympy [A]  time = 11.6586, size = 49, normalized size = 1. \[ \frac{x^{2} x^{- p - 2} x^{p + 3} \left (1 + \frac{c x}{b}\right )^{- p} \left (b x + c x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, p + 3 \\ p + 4 \end{matrix}\middle |{- \frac{c x}{b}} \right )}}{p + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*x**(-p - 2)*x**(p + 3)*(1 + c*x/b)**(-p)*(b*x + c*x**2)**p*hyper((-p, p + 3
), (p + 4,), -c*x/b)/(p + 3)

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Mathematica [A]  time = 0.0420205, size = 47, normalized size = 0.96 \[ \frac{x^3 (x (b+c x))^p \left (\frac{c x}{b}+1\right )^{-p} \, _2F_1\left (-p,p+3;p+4;-\frac{c x}{b}\right )}{p+3} \]

Antiderivative was successfully verified.

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

[Out]

(x^3*(x*(b + c*x))^p*Hypergeometric2F1[-p, 3 + p, 4 + p, -((c*x)/b)])/((3 + p)*(
1 + (c*x)/b)^p)

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int{x}^{2} \left ( c{x}^{2}+bx \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(c*x^2+b*x)^p,x)

[Out]

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

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2*(x*(b + c*x))**p, x)

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

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)