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

Optimal. Leaf size=55 \[ -\frac{\left (-\frac{c x}{b}\right )^{-p-1} \left (b x+c x^2\right )^{p+1} \, _2F_1\left (-p,p+1;p+2;\frac{b+c x}{b}\right )}{b (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0321692, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{\left (-\frac{c x}{b}\right )^{-p-1} \left (b x+c x^2\right )^{p+1} \, _2F_1\left (-p,p+1;p+2;\frac{b+c x}{b}\right )}{b (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 2.75342, size = 42, normalized size = 0.76 \[ - \frac{\left (- \frac{c x}{b}\right )^{- p - 1} \left (b x + c x^{2}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} - p, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{b + c x}{b}} \right )}}{b \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0407748, size = 45, normalized size = 0.82 \[ \frac{x (x (b+c x))^p \left (\frac{c x}{b}+1\right )^{-p} \, _2F_1\left (-p,p+1;p+2;-\frac{c x}{b}\right )}{p+1} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^p,x]

[Out]

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

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \[ \int \left ( c{x}^{2}+bx \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^p,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*x^2 + b*x)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**p,x)

[Out]

Integral((b*x + c*x**2)**p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x)^p, x)