\(\int x (2 c+3 d x) (a+c x^2+d x^3)^n \, dx\) [190]

Optimal result

Integrand size = 24, antiderivative size = 22 \[ \int x (2 c+3 d x) \left (a+c x^2+d x^3\right )^n \, dx=\frac {\left (a+c x^2+d x^3\right )^{1+n}}{1+n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {1602} \[ \int x (2 c+3 d x) \left (a+c x^2+d x^3\right )^n \, dx=\frac {\left (a+c x^2+d x^3\right )^{n+1}}{n+1} \]

[In]

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

[Out]

(a + c*x^2 + d*x^3)^(1 + n)/(1 + n)

Rule 1602

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[Coeff[Pp, x, p]*x^(p - q +
1)*(Qq^(m + 1)/((p + m*q + 1)*Coeff[Qq, x, q])), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (a+c x^2+d x^3\right )^{1+n}}{1+n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int x (2 c+3 d x) \left (a+c x^2+d x^3\right )^n \, dx=\frac {\left (a+x^2 (c+d x)\right )^{1+n}}{1+n} \]

[In]

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

[Out]

(a + x^2*(c + d*x))^(1 + n)/(1 + n)

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05

method	result	size
gosper	\(\frac {\left (x^{3} d +c \,x^{2}+a \right )^{1+n}}{1+n}\)	\(23\)
risch	\(\frac {\left (x^{3} d +c \,x^{2}+a \right )^{n} \left (x^{3} d +c \,x^{2}+a \right )}{1+n}\)	\(33\)
parallelrisch	\(\frac {x^{3} \left (x^{3} d +c \,x^{2}+a \right )^{n} d^{2}+x^{2} \left (x^{3} d +c \,x^{2}+a \right )^{n} c d +\left (x^{3} d +c \,x^{2}+a \right )^{n} a d}{d \left (1+n \right )}\)	\(69\)
norman	\(\frac {a \,{\mathrm e}^{n \ln \left (x^{3} d +c \,x^{2}+a \right )}}{1+n}+\frac {c \,x^{2} {\mathrm e}^{n \ln \left (x^{3} d +c \,x^{2}+a \right )}}{1+n}+\frac {d \,x^{3} {\mathrm e}^{n \ln \left (x^{3} d +c \,x^{2}+a \right )}}{1+n}\)	\(77\)

[In]

int(x*(3*d*x+2*c)*(d*x^3+c*x^2+a)^n,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int x (2 c+3 d x) \left (a+c x^2+d x^3\right )^n \, dx=\frac {{\left (d x^{3} + c x^{2} + a\right )} {\left (d x^{3} + c x^{2} + a\right )}^{n}}{n + 1} \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int x (2 c+3 d x) \left (a+c x^2+d x^3\right )^n \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int x (2 c+3 d x) \left (a+c x^2+d x^3\right )^n \, dx=\frac {{\left (d x^{3} + c x^{2} + a\right )} {\left (d x^{3} + c x^{2} + a\right )}^{n}}{n + 1} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int x (2 c+3 d x) \left (a+c x^2+d x^3\right )^n \, dx=\frac {{\left (d x^{3} + c x^{2} + a\right )}^{n + 1}}{n + 1} \]

[In]

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

[Out]

(d*x^3 + c*x^2 + a)^(n + 1)/(n + 1)

Mupad [B] (verification not implemented)

Time = 9.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.95 \[ \int x (2 c+3 d x) \left (a+c x^2+d x^3\right )^n \, dx=\left (\frac {a}{n+1}+\frac {c\,x^2}{n+1}+\frac {d\,x^3}{n+1}\right )\,{\left (d\,x^3+c\,x^2+a\right )}^n \]

[In]

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

[Out]

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