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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {1588} \[ \frac {\left (c x^2+d x^3\right )^{n+1}}{n+1} \]

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)

Rule 1588

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

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 19, normalized size = 0.90 \[ \frac {\left (x^2 (c+d x)\right )^{n+1}}{n+1} \]

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)

________________________________________________________________________________________

fricas [A]  time = 0.76, size = 30, normalized size = 1.43 \[ \frac {{\left (d x^{3} + c x^{2}\right )} {\left (d x^{3} + c x^{2}\right )}^{n}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.40, size = 21, normalized size = 1.00 \[ \frac {{\left (d x^{3} + c x^{2}\right )}^{n + 1}}{n + 1} \]

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, algorithm="giac")

[Out]

(d*x^3 + c*x^2)^(n + 1)/(n + 1)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 28, normalized size = 1.33 \[ \frac {\left (d x +c \right ) x^{2} \left (d \,x^{3}+c \,x^{2}\right )^{n}}{n +1} \]

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

________________________________________________________________________________________

maxima [A]  time = 0.93, size = 32, normalized size = 1.52 \[ \frac {{\left (d x^{3} + c x^{2}\right )} e^{\left (n \log \left (d x + c\right ) + 2 \, n \log \relax (x)\right )}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(3*d*x+2*c)*(d*x^3+c*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)

________________________________________________________________________________________

mupad [B]  time = 2.15, size = 27, normalized size = 1.29 \[ \frac {x^2\,{\left (d\,x^3+c\,x^2\right )}^n\,\left (c+d\,x\right )}{n+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 1.14, size = 53, normalized size = 2.52 \[ \begin {cases} \frac {c x^{2} \left (c x^{2} + d x^{3}\right )^{n}}{n + 1} + \frac {d x^{3} \left (c x^{2} + d x^{3}\right )^{n}}{n + 1} & \text {for}\: n \neq -1 \\2 \log {\relax (x )} + \log {\left (\frac {c}{d} + x \right )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(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))

________________________________________________________________________________________