3.2.86 \(\int (2 c x+3 d x^2) (a+c x^2+d x^3)^n \, dx\) [186]

Optimal. Leaf size=22 \[ \frac {\left (a+c x^2+d x^3\right )^{1+n}}{1+n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {1602} \begin {gather*} \frac {\left (a+c x^2+d x^3\right )^{n+1}}{n+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*c*x + 3*d*x^2)*(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*} \int \left (2 c x+3 d x^2\right ) \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}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.06, size = 21, normalized size = 0.95 \begin {gather*} \frac {\left (a+x^2 (c+d x)\right )^{1+n}}{1+n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a + x^2*(c + d*x))^(1 + n)/(1 + n)

________________________________________________________________________________________

Maple [A]
time = 0.02, size = 23, normalized size = 1.05

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 22, normalized size = 1.00 \begin {gather*} \frac {{\left (d x^{3} + c x^{2} + a\right )}^{n + 1}}{n + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*x^3 + c*x^2 + a)^(n + 1)/(n + 1)

________________________________________________________________________________________

Fricas [A]
time = 0.38, size = 32, normalized size = 1.45 \begin {gather*} \frac {{\left (d x^{3} + c x^{2} + a\right )} {\left (d x^{3} + c x^{2} + a\right )}^{n}}{n + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*d*x^2+2*c*x)*(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
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 4.16, size = 22, normalized size = 1.00 \begin {gather*} \frac {{\left (d x^{3} + c x^{2} + a\right )}^{n + 1}}{n + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*d*x^2+2*c*x)*(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]
time = 2.14, size = 43, normalized size = 1.95 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x + 3*d*x^2)*(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

________________________________________________________________________________________