### 3.137 $$\int (b+2 c x) (b x+c x^2)^p \, dx$$

Optimal. Leaf size=19 $\frac{\left (b x+c x^2\right )^{p+1}}{p+1}$

[Out]

(b*x + c*x^2)^(1 + p)/(1 + p)

________________________________________________________________________________________

Rubi [A]  time = 0.004136, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.056, Rules used = {629} $\frac{\left (b x+c x^2\right )^{p+1}}{p+1}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b + 2*c*x)*(b*x + c*x^2)^p,x]

[Out]

(b*x + c*x^2)^(1 + p)/(1 + p)

Rule 629

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

Rubi steps

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

Mathematica [A]  time = 0.0098645, size = 17, normalized size = 0.89 $\frac{(x (b+c x))^{p+1}}{p+1}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(x*(b + c*x))^(1 + p)/(1 + p)

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 24, normalized size = 1.3 \begin{align*}{\frac{x \left ( cx+b \right ) \left ( c{x}^{2}+bx \right ) ^{p}}{1+p}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(c*x+b)/(1+p)*(c*x^2+b*x)^p

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.0047, size = 53, normalized size = 2.79 \begin{align*} \frac{{\left (c x^{2} + b x\right )}{\left (c x^{2} + b x\right )}^{p}}{p + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*x^2 + b*x)*(c*x^2 + b*x)^p/(p + 1)

________________________________________________________________________________________

Sympy [A]  time = 0.576551, size = 46, normalized size = 2.42 \begin{align*} \begin{cases} \frac{b x \left (b x + c x^{2}\right )^{p}}{p + 1} + \frac{c x^{2} \left (b x + c x^{2}\right )^{p}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (x \right )} + \log{\left (\frac{b}{c} + x \right )} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((b*x*(b*x + c*x**2)**p/(p + 1) + c*x**2*(b*x + c*x**2)**p/(p + 1), Ne(p, -1)), (log(x) + log(b/c + x
), True))

________________________________________________________________________________________

Giac [A]  time = 1.11184, size = 50, normalized size = 2.63 \begin{align*} \frac{{\left (c x^{2} + b x\right )}^{p} c x^{2} +{\left (c x^{2} + b x\right )}^{p} b x}{p + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((c*x^2 + b*x)^p*c*x^2 + (c*x^2 + b*x)^p*b*x)/(p + 1)