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

Optimal. Leaf size=21 \[ \frac{\left (c x^2+d x^3\right )^{n+1}}{n+1} \]

[Out]

(c*x^2 + d*x^3)^(1 + n)/(1 + n)

_______________________________________________________________________________________

Rubi [A]  time = 0.0146223, antiderivative size = 21, 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 \[ \frac{\left (c x^2+d x^3\right )^{n+1}}{n+1} \]

Antiderivative was successfully verified.

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

[Out]

(c*x^2 + d*x^3)^(1 + n)/(1 + n)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.42921, size = 15, normalized size = 0.71 \[ \frac{\left (c x^{2} + d x^{3}\right )^{n + 1}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*x**2 + d*x**3)**(n + 1)/(n + 1)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0303184, size = 19, normalized size = 0.9 \[ \frac{\left (x^2 (c+d x)\right )^{n+1}}{n+1} \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(c + d*x))^(1 + n)/(1 + n)

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 28, normalized size = 1.3 \[{\frac{ \left ( d{x}^{3}+c{x}^{2} \right ) ^{n}{x}^{2} \left ( dx+c \right ) }{1+n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.906612, size = 43, normalized size = 2.05 \[ \frac{{\left (d x^{3} + c x^{2}\right )} e^{\left (n \log \left (d x + c\right ) + 2 \, n \log \left (x\right )\right )}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.275777, size = 41, normalized size = 1.95 \[ \frac{{\left (d x^{3} + c x^{2}\right )}{\left (d x^{3} + c x^{2}\right )}^{n}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 2.40536, size = 53, normalized size = 2.52 \[ \begin{cases} \frac{c x^{2} \left (c x^{2} + d x^{3}\right )^{n}}{n + 1} + \frac{d x^{3} \left (c x^{2} + d x^{3}\right )^{n}}{n + 1} & \text{for}\: n \neq -1 \\2 \log{\left (x \right )} + \log{\left (\frac{c}{d} + x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.266766, size = 63, normalized size = 3. \[ \frac{d x^{3} e^{\left (n{\rm ln}\left (d x^{3} + c x^{2}\right )\right )} + c x^{2} e^{\left (n{\rm ln}\left (d x^{3} + c x^{2}\right )\right )}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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