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

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 = 0.0104559, 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, Rules used = {624} \[ -\frac{\left (-\frac{c x}{b}\right )^{-p-1} \left (b x+c x^2\right )^{p+1} \, _2F_1\left (-p,p+1;p+2;\frac{b+c x}{b}\right )}{b (p+1)} \]

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)))

Rule 624

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, -Simp[((a + b*x + c*
x^2)^(p + 1)*Hypergeometric2F1[-p, p + 1, p + 2, (b + q + 2*c*x)/(2*q)])/(q*(p + 1)*((q - b - 2*c*x)/(2*q))^(p
+ 1)), x]] /; FreeQ[{a, b, c, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !IntegerQ[4*p]

Rubi steps

\begin{align*} \int \left (b x+c x^2\right )^p \, dx &=-\frac{\left (-\frac{c x}{b}\right )^{-1-p} \left (b x+c x^2\right )^{1+p} \, _2F_1\left (-p,1+p;2+p;\frac{b+c x}{b}\right )}{b (1+p)}\\ \end{align*}

Mathematica [A]  time = 0.0099882, 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 = 0.441, size = 0, normalized size = 0. \begin{align*} \int \left ( c{x}^{2}+bx \right ) ^{p}\, dx \end{align*}

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. \begin{align*} \int{\left (c x^{2} + b x\right )}^{p}\,{d x} \end{align*}

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. \begin{align*}{\rm integral}\left ({\left (c x^{2} + b x\right )}^{p}, x\right ) \end{align*}

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. \begin{align*} \int \left (b x + c x^{2}\right )^{p}\, dx \end{align*}

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

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)