3.2.91 \(\int x (2 c+3 d x) (c x^2+d x^3)^n \, dx\) [191]

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

[Out]

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

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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 = {1602} \begin {gather*} \frac {\left (c x^2+d x^3\right )^{n+1}}{n+1} \end {gather*}

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 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*} \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*}

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Mathematica [A]
time = 0.01, size = 19, normalized size = 0.90 \begin {gather*} \frac {\left (x^2 (c+d x)\right )^{1+n}}{1+n} \end {gather*}

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)

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Maple [A]
time = 0.17, size = 26, normalized size = 1.24

method	result	size
risch	\(\frac {x^{2} \left (d x +c \right ) \left (x^{2} \left (d x +c \right )\right )^{n}}{1+n}\)	\(26\)
gosper	\(\frac {\left (d \,x^{3}+c \,x^{2}\right )^{n} x^{2} \left (d x +c \right )}{1+n}\)	\(28\)
norman	\(\frac {c \,x^{2} {\mathrm e}^{n \ln \left (d \,x^{3}+c \,x^{2}\right )}}{1+n}+\frac {d \,x^{3} {\mathrm e}^{n \ln \left (d \,x^{3}+c \,x^{2}\right )}}{1+n}\)	\(52\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.31, size = 32, normalized size = 1.52 \begin {gather*} \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} \end {gather*}

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)

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Fricas [A]
time = 0.39, size = 30, normalized size = 1.43 \begin {gather*} \frac {{\left (d x^{3} + c x^{2}\right )} {\left (d x^{3} + c x^{2}\right )}^{n}}{n + 1} \end {gather*}

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)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (15) = 30\).
time = 0.40, size = 53, normalized size = 2.52 \begin {gather*} \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} \end {gather*}

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))

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Giac [A]
time = 4.81, size = 21, normalized size = 1.00 \begin {gather*} \frac {{\left (d x^{3} + c x^{2}\right )}^{n + 1}}{n + 1} \end {gather*}

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)

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

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)

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