### 3.124 $$\int (d x)^m (b x+c x^2)^p \, dx$$

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0232367, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.176, Rules used = {674, 66, 64} $\frac{x (d x)^m \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,m+p+1;m+p+2;-\frac{c x}{b}\right )}{m+p+1}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d*x)^m*(b*x + c*x^2)^p,x]

[Out]

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

Rule 674

Int[((e_.)*(x_))^(m_)*((b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[((e*x)^m*(b*x + c*x^2)^p)/(x^(m + p)
*(b + c*x)^p), Int[x^(m + p)*(b + c*x)^p, x], x] /; FreeQ[{b, c, e, m}, x] &&  !IntegerQ[p]

Rule 66

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c^IntPart[n]*(c + d*x)^FracPart[n])/(1 + (d
*x)/c)^FracPart[n], Int[(b*x)^m*(1 + (d*x)/c)^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Int
egerQ[n] &&  !GtQ[c, 0] &&  !GtQ[-(d/(b*c)), 0] && ((RationalQ[m] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0]))
||  !RationalQ[n])

Rule 64

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c^n*(b*x)^(m + 1)*Hypergeometric2F1[-n, m +
1, m + 2, -((d*x)/c)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-(d/(b*c)), 0])))

Rubi steps

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

Mathematica [A]  time = 0.0133406, size = 53, normalized size = 0.96 $\frac{x (d x)^m (x (b+c x))^p \left (\frac{c x}{b}+1\right )^{-p} \, _2F_1\left (-p,m+p+1;m+p+2;-\frac{c x}{b}\right )}{m+p+1}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d*x)^m*(b*x + c*x^2)^p,x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.479, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m} \left ( c{x}^{2}+bx \right ) ^{p}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(c*x^2+b*x)^p,x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x)^p*(d*x)^m, x)

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(c*x^2+b*x)^p,x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x)^p*(d*x)^m, x)

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*(c*x**2+b*x)**p,x)

[Out]

Integral((d*x)**m*(x*(b + c*x))**p, x)

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*x^2 + b*x)^p*(d*x)^m, x)