3.125 \(\int x^3 \left (b x+c x^2\right )^p \, dx\)

Optimal. Leaf size=49 \[ \frac{x^4 \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,p+4;p+5;-\frac{c x}{b}\right )}{p+4} \]

[Out]

(x^4*(b*x + c*x^2)^p*Hypergeometric2F1[-p, 4 + p, 5 + p, -((c*x)/b)])/((4 + p)*(
1 + (c*x)/b)^p)

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Rubi [A]  time = 0.0664326, 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 \[ \frac{x^4 \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,p+4;p+5;-\frac{c x}{b}\right )}{p+4} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(b*x + c*x^2)^p,x]

[Out]

(x^4*(b*x + c*x^2)^p*Hypergeometric2F1[-p, 4 + p, 5 + p, -((c*x)/b)])/((4 + p)*(
1 + (c*x)/b)^p)

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Rubi in Sympy [A]  time = 11.5467, size = 49, normalized size = 1. \[ \frac{x^{3} x^{- p - 3} x^{p + 4} \left (1 + \frac{c x}{b}\right )^{- p} \left (b x + c x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, p + 4 \\ p + 5 \end{matrix}\middle |{- \frac{c x}{b}} \right )}}{p + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(c*x**2+b*x)**p,x)

[Out]

x**3*x**(-p - 3)*x**(p + 4)*(1 + c*x/b)**(-p)*(b*x + c*x**2)**p*hyper((-p, p + 4
), (p + 5,), -c*x/b)/(p + 4)

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Mathematica [A]  time = 0.0457032, size = 47, normalized size = 0.96 \[ \frac{x^4 (x (b+c x))^p \left (\frac{c x}{b}+1\right )^{-p} \, _2F_1\left (-p,p+4;p+5;-\frac{c x}{b}\right )}{p+4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(b*x + c*x^2)^p,x]

[Out]

(x^4*(x*(b + c*x))^p*Hypergeometric2F1[-p, 4 + p, 5 + p, -((c*x)/b)])/((4 + p)*(
1 + (c*x)/b)^p)

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{x}^{3} \left ( c{x}^{2}+bx \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(c*x^2+b*x)^p,x)

[Out]

int(x^3*(c*x^2+b*x)^p,x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x)^p*x^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{2} + b x\right )}^{p} x^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^p*x^3,x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x)^p*x^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{3} \left (x \left (b + c x\right )\right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(c*x**2+b*x)**p,x)

[Out]

Integral(x**3*(x*(b + c*x))**p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x)^p*x^3, x)