### 3.127 $$\int x (b x+c x^2)^p \, dx$$

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 = 0.02233, 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, Rules used = {640, 624} $\frac{\left (b x+c x^2\right )^{p+1} \left (-\frac{c x}{b}\right )^{-p-1} \, _2F_1\left (-p,p+1;p+2;\frac{b+c x}{b}\right )}{2 c (p+1)}+\frac{\left (b x+c x^2\right )^{p+1}}{2 c (p+1)}$

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))

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

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

Mathematica [A]  time = 0.007747, 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 = 0.413, size = 0, normalized size = 0. \begin{align*} \int x \left ( c{x}^{2}+bx \right ) ^{p}\, dx \end{align*}

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*x^2 + b*x)^p*x, x)