3.936 \(\int x^{1+2 n} (a+b x)^n (2 a+3 b x) \, dx\)

Optimal. Leaf size=22 \[ \frac{x^{2 (n+1)} (a+b x)^{n+1}}{n+1} \]

[Out]

(x^(2*(1 + n))*(a + b*x)^(1 + n))/(1 + n)

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Rubi [A]  time = 0.0045588, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {74} \[ \frac{x^{2 (n+1)} (a+b x)^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]

Int[x^(1 + 2*n)*(a + b*x)^n*(2*a + 3*b*x),x]

[Out]

(x^(2*(1 + n))*(a + b*x)^(1 + n))/(1 + n)

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^{1+2 n} (a+b x)^n (2 a+3 b x) \, dx &=\frac{x^{2 (1+n)} (a+b x)^{1+n}}{1+n}\\ \end{align*}

Mathematica [A]  time = 0.0108333, size = 22, normalized size = 1. \[ \frac{x^{2 n+2} (a+b x)^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(1 + 2*n)*(a + b*x)^n*(2*a + 3*b*x),x]

[Out]

(x^(2 + 2*n)*(a + b*x)^(1 + n))/(1 + n)

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Maple [A]  time = 0.003, size = 23, normalized size = 1.1 \begin{align*}{\frac{{x}^{2+2\,n} \left ( bx+a \right ) ^{1+n}}{1+n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1+2*n)*(b*x+a)^n*(3*b*x+2*a),x)

[Out]

x^(2+2*n)*(b*x+a)^(1+n)/(1+n)

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Maxima [A]  time = 2.99243, size = 43, normalized size = 1.95 \begin{align*} \frac{{\left (b x^{3} + a x^{2}\right )} e^{\left (n \log \left (b x + a\right ) + 2 \, n \log \left (x\right )\right )}}{n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1+2*n)*(b*x+a)^n*(3*b*x+2*a),x, algorithm="maxima")

[Out]

(b*x^3 + a*x^2)*e^(n*log(b*x + a) + 2*n*log(x))/(n + 1)

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Fricas [A]  time = 2.00643, size = 63, normalized size = 2.86 \begin{align*} \frac{{\left (b x^{2} + a x\right )}{\left (b x + a\right )}^{n} x^{2 \, n + 1}}{n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1+2*n)*(b*x+a)^n*(3*b*x+2*a),x, algorithm="fricas")

[Out]

(b*x^2 + a*x)*(b*x + a)^n*x^(2*n + 1)/(n + 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(1+2*n)*(b*x+a)**n*(3*b*x+2*a),x)

[Out]

Timed out

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Giac [B]  time = 2.13459, size = 63, normalized size = 2.86 \begin{align*} \frac{{\left (b x + a\right )}^{n} b x^{2} e^{\left (2 \, n \log \left (x\right ) + \log \left (x\right )\right )} +{\left (b x + a\right )}^{n} a x e^{\left (2 \, n \log \left (x\right ) + \log \left (x\right )\right )}}{n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1+2*n)*(b*x+a)^n*(3*b*x+2*a),x, algorithm="giac")

[Out]

((b*x + a)^n*b*x^2*e^(2*n*log(x) + log(x)) + (b*x + a)^n*a*x*e^(2*n*log(x) + log(x)))/(n + 1)