3.189 \(\int x^{2 n} (c+d x)^n (2 c x+3 d x^2) \, dx\)

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

[Out]

x^(2+2*n)*(d*x+c)^(1+n)/(1+n)

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Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {845} \[ \frac {x^{2 (n+1)} (c+d x)^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]

Int[x^(2*n)*(c + d*x)^n*(2*c*x + 3*d*x^2),x]

[Out]

(x^(2*(1 + n))*(c + d*x)^(1 + n))/(1 + n)

Rule 845

Int[(x_)^(m_.)*((f_) + (g_.)*(x_))^(n_.)*((b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(c*x^(m + 2)*(f + g*x)
^(n + 1))/(g*(m + n + 3)), x] /; FreeQ[{b, c, f, g, m, n}, x] && EqQ[c*f*(m + 2) - b*g*(m + n + 3), 0] && NeQ[
m + n + 3, 0]

Rubi steps

\begin {align*} \int x^{2 n} (c+d x)^n \left (2 c x+3 d x^2\right ) \, dx &=\frac {x^{2 (1+n)} (c+d x)^{1+n}}{1+n}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 1.00 \[ \frac {x^{2 n+2} (c+d x)^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(2*n)*(c + d*x)^n*(2*c*x + 3*d*x^2),x]

[Out]

(x^(2 + 2*n)*(c + d*x)^(1 + n))/(1 + n)

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fricas [A]  time = 0.98, size = 29, normalized size = 1.32 \[ \frac {{\left (d x^{3} + c x^{2}\right )} {\left (d x + c\right )}^{n} x^{2 \, n}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(2*n)*(d*x+c)^n*(3*d*x^2+2*c*x),x, algorithm="fricas")

[Out]

(d*x^3 + c*x^2)*(d*x + c)^n*x^(2*n)/(n + 1)

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giac [A]  time = 0.31, size = 41, normalized size = 1.86 \[ \frac {{\left (d x + c\right )}^{n} d x^{3} x^{2 \, n} + {\left (d x + c\right )}^{n} c x^{2} x^{2 \, n}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(2*n)*(d*x+c)^n*(3*d*x^2+2*c*x),x, algorithm="giac")

[Out]

((d*x + c)^n*d*x^3*x^(2*n) + (d*x + c)^n*c*x^2*x^(2*n))/(n + 1)

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maple [A]  time = 0.00, size = 23, normalized size = 1.05 \[ \frac {x^{2 n +2} \left (d x +c \right )^{n +1}}{n +1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(2*n)*(d*x+c)^n*(3*d*x^2+2*c*x),x)

[Out]

x^(2+2*n)*(d*x+c)^(n+1)/(n+1)

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maxima [A]  time = 0.90, size = 32, normalized size = 1.45 \[ \frac {{\left (d x^{3} + c x^{2}\right )} e^{\left (n \log \left (d x + c\right ) + 2 \, n \log \relax (x)\right )}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(2*n)*(d*x+c)^n*(3*d*x^2+2*c*x),x, algorithm="maxima")

[Out]

(d*x^3 + c*x^2)*e^(n*log(d*x + c) + 2*n*log(x))/(n + 1)

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mupad [B]  time = 2.19, size = 26, normalized size = 1.18 \[ \frac {x^{2\,n}\,x^2\,{\left (c+d\,x\right )}^n\,\left (c+d\,x\right )}{n+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(2*n)*(2*c*x + 3*d*x^2)*(c + d*x)^n,x)

[Out]

(x^(2*n)*x^2*(c + d*x)^n*(c + d*x))/(n + 1)

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sympy [A]  time = 6.15, size = 53, normalized size = 2.41 \[ \begin {cases} \frac {c x^{2} x^{2 n} \left (c + d x\right )^{n}}{n + 1} + \frac {d x^{3} x^{2 n} \left (c + d x\right )^{n}}{n + 1} & \text {for}\: n \neq -1 \\2 \log {\relax (x )} + \log {\left (\frac {c}{d} + x \right )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(2*n)*(d*x+c)**n*(3*d*x**2+2*c*x),x)

[Out]

Piecewise((c*x**2*x**(2*n)*(c + d*x)**n/(n + 1) + d*x**3*x**(2*n)*(c + d*x)**n/(n + 1), Ne(n, -1)), (2*log(x)
+ log(c/d + x), True))

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