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

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

[Out]

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

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Rubi [A]  time = 0.0166558, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{x^{2 (n+1)} (c+d x)^{n+1}}{n+1} \]

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)

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Rubi in Sympy [A]  time = 17.1062, size = 31, normalized size = 1.41 \[ \frac{x x^{2 n + 1} \left (c + d x\right )^{n + 1} \left (9 n + 6\right )}{3 \left (n + 1\right ) \left (3 n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(2*n)*(d*x+c)**n*(3*d*x**2+2*c*x),x)

[Out]

x*x**(2*n + 1)*(c + d*x)**(n + 1)*(9*n + 6)/(3*(n + 1)*(3*n + 2))

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Mathematica [A]  time = 0.0357936, size = 22, normalized size = 1. \[ \frac{x^{2 n+2} (c+d x)^{n+1}}{n+1} \]

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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Maple [A]  time = 0.004, size = 23, normalized size = 1.1 \[{\frac{{x}^{2+2\,n} \left ( dx+c \right ) ^{1+n}}{1+n}} \]

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)^(1+n)/(1+n)

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Maxima [A]  time = 0.895376, size = 43, normalized size = 1.95 \[ \frac{{\left (d x^{3} + c x^{2}\right )} e^{\left (n \log \left (d x + c\right ) + 2 \, n \log \left (x\right )\right )}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*d*x^2 + 2*c*x)*(d*x + c)^n*x^(2*n),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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Fricas [A]  time = 0.281125, size = 39, normalized size = 1.77 \[ \frac{{\left (d x^{3} + c x^{2}\right )}{\left (d x + c\right )}^{n} x^{2 \, n}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 17.7628, size = 53, normalized size = 2.41 \[ \begin{cases} \frac{c x^{2} x^{2 n} \left (c + d x\right )^{n}}{n + 1} + \frac{d x^{3} x^{2 n} \left (c + d x\right )^{n}}{n + 1} & \text{for}\: n \neq -1 \\2 \log{\left (x \right )} + \log{\left (\frac{c}{d} + x \right )} & \text{otherwise} \end{cases} \]

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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GIAC/XCAS [A]  time = 0.266437, size = 63, normalized size = 2.86 \[ \frac{d x^{3} e^{\left (n{\rm ln}\left (d x + c\right ) + 2 \, n{\rm ln}\left (x\right )\right )} + c x^{2} e^{\left (n{\rm ln}\left (d x + c\right ) + 2 \, n{\rm ln}\left (x\right )\right )}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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