3.1078 \(\int x^p (b+c x)^p (b+2 c x) \, dx\)

Optimal. Leaf size=20 \[ \frac{x^{p+1} (b+c x)^{p+1}}{p+1} \]

[Out]

(x^(1 + p)*(b + c*x)^(1 + p))/(1 + p)

________________________________________________________________________________________

Rubi [A]  time = 0.003549, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {74} \[ \frac{x^{p+1} (b+c x)^{p+1}}{p+1} \]

Antiderivative was successfully verified.

[In]

Int[x^p*(b + c*x)^p*(b + 2*c*x),x]

[Out]

(x^(1 + p)*(b + c*x)^(1 + p))/(1 + p)

Rule 74

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

Rubi steps

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

Mathematica [A]  time = 0.0109254, size = 20, normalized size = 1. \[ \frac{x^{p+1} (b+c x)^{p+1}}{p+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^(1 + p)*(b + c*x)^(1 + p))/(1 + p)

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 21, normalized size = 1.1 \begin{align*}{\frac{{x}^{1+p} \left ( cx+b \right ) ^{1+p}}{1+p}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.08433, size = 39, normalized size = 1.95 \begin{align*} \frac{{\left (c x^{2} + b x\right )} e^{\left (p \log \left (c x + b\right ) + p \log \left (x\right )\right )}}{p + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.01109, size = 53, normalized size = 2.65 \begin{align*} \frac{{\left (c x^{2} + b x\right )}{\left (c x + b\right )}^{p} x^{p}}{p + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*x^2 + b*x)*(c*x + b)^p*x^p/(p + 1)

________________________________________________________________________________________

Sympy [A]  time = 2.89365, size = 46, normalized size = 2.3 \begin{align*} \begin{cases} \frac{b x x^{p} \left (b + c x\right )^{p}}{p + 1} + \frac{c x^{2} x^{p} \left (b + c x\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(x**p*(c*x+b)**p*(2*c*x+b),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.57675, size = 47, normalized size = 2.35 \begin{align*} \frac{{\left (c x + b\right )}^{p} c x^{2} x^{p} +{\left (c x + b\right )}^{p} b x x^{p}}{p + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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