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

Optimal. Leaf size=24 \[ \frac{x^{n+1} \left (c x+d x^2\right )^{n+1}}{n+1} \]

[Out]

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

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Rubi [A]  time = 0.0175274, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{x^{n+1} \left (c x+d x^2\right )^{n+1}}{n+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 40.9879, size = 124, normalized size = 5.17 \[ \frac{c x^{- 2 n - 1} x^{n + 1} x^{2 n + 2} \left (1 + \frac{d x}{c}\right )^{- n} \left (c x + d x^{2}\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} - n, 2 n + 2 \\ 2 n + 3 \end{matrix}\middle |{- \frac{d x}{c}} \right )}}{n + 1} + \frac{3 d x^{- 2 n - 2} x^{n + 2} x^{2 n + 3} \left (1 + \frac{d x}{c}\right )^{- n} \left (c x + d x^{2}\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} - n, 2 n + 3 \\ 2 n + 4 \end{matrix}\middle |{- \frac{d x}{c}} \right )}}{2 n + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*x**(-2*n - 1)*x**(n + 1)*x**(2*n + 2)*(1 + d*x/c)**(-n)*(c*x + d*x**2)**n*hype
r((-n, 2*n + 2), (2*n + 3,), -d*x/c)/(n + 1) + 3*d*x**(-2*n - 2)*x**(n + 2)*x**(
2*n + 3)*(1 + d*x/c)**(-n)*(c*x + d*x**2)**n*hyper((-n, 2*n + 3), (2*n + 4,), -d
*x/c)/(2*n + 3)

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Mathematica [A]  time = 0.0413796, size = 22, normalized size = 0.92 \[ \frac{x^{n+1} (x (c+d x))^{n+1}}{n+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 28, normalized size = 1.2 \[{\frac{ \left ( d{x}^{2}+cx \right ) ^{n}{x}^{2+n} \left ( dx+c \right ) }{1+n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.911642, size = 43, normalized size = 1.79 \[ \frac{{\left (d x^{3} + c x^{2}\right )} e^{\left (n \log \left (d x + c\right ) + 2 \, n \log \left (x\right )\right )}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*d*x^2 + 2*c*x)*(d*x^2 + c*x)^n*x^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 = 0.271137, size = 42, normalized size = 1.75 \[ \frac{{\left (d x^{3} + c x^{2}\right )}{\left (d x^{2} + c x\right )}^{n} x^{n}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 18.3019, size = 56, normalized size = 2.33 \[ \begin{cases} \frac{c x^{2} x^{n} \left (c x + d x^{2}\right )^{n}}{n + 1} + \frac{d x^{3} x^{n} \left (c x + d x^{2}\right )^{n}}{n + 1} & \text{for}\: n \neq -1 \\2 \log{\left (x \right )} + \log{\left (\frac{c}{d} + x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.274939, size = 63, normalized size = 2.62 \[ \frac{d x^{3} e^{\left (n{\rm ln}\left (d x + c\right ) + 2 \, n{\rm ln}\left (x\right )\right )} + c x^{2} e^{\left (n{\rm ln}\left (d x + c\right ) + 2 \, n{\rm ln}\left (x\right )\right )}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*d*x^2 + 2*c*x)*(d*x^2 + c*x)^n*x^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)