3.11.78 \(\int x^p (b+c x)^p (b+2 c x) \, dx\) [1078]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, 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 = {75} \begin {gather*} \frac {x^{p+1} (b+c x)^{p+1}}{p+1} \end {gather*}

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 75

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*}

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Mathematica [A]
time = 0.03, size = 20, normalized size = 1.00 \begin {gather*} \frac {x^{1+p} (b+c x)^{1+p}}{1+p} \end {gather*}

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)

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Maple [A]
time = 0.32, size = 21, normalized size = 1.05

method	result	size
gosper	\(\frac {x^{1+p} \left (c x +b \right )^{1+p}}{1+p}\)	\(21\)
risch	\(\frac {x \left (c x +b \right ) x^{p} \left (c x +b \right )^{p}}{1+p}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.33, size = 29, normalized size = 1.45 \begin {gather*} \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 {gather*}

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)

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Fricas [A]
time = 2.10, size = 25, normalized size = 1.25 \begin {gather*} \frac {{\left (c x^{2} + b x\right )} {\left (c x + b\right )}^{p} x^{p}}{p + 1} \end {gather*}

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)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (15) = 30\).
time = 1.01, size = 46, normalized size = 2.30 \begin {gather*} \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 {gather*}

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))

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Giac [A]
time = 1.12, size = 35, normalized size = 1.75 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 4.80, size = 22, normalized size = 1.10 \begin {gather*} \frac {x\,x^p\,{\left (b+c\,x\right )}^p\,\left (b+c\,x\right )}{p+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^p*(b + c*x)^p*(b + 2*c*x),x)

[Out]

(x*x^p*(b + c*x)^p*(b + c*x))/(p + 1)

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