\(\int (a+b x+c x^2) (1+(a x+\frac {b x^2}{2}+\frac {c x^3}{3})^n) \, dx\) [218]

Optimal result

Integrand size = 35, antiderivative size = 50 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^n\right ) \, dx=a x+\frac {b x^2}{2}+\frac {c x^3}{3}+\frac {\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^{1+n}}{1+n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {1605} \[ \int \left (a+b x+c x^2\right ) \left (1+\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^n\right ) \, dx=\frac {\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^{n+1}}{n+1}+a x+\frac {b x^2}{2}+\frac {c x^3}{3} \]

[In]

Int[(a + b*x + c*x^2)*(1 + (a*x + (b*x^2)/2 + (c*x^3)/3)^n),x]

[Out]

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

Rule 1605

Int[((a_.) + (b_.)*(Pq_)^(n_.))^(p_.)*(Qr_), x_Symbol] :> With[{q = Expon[Pq, x], r = Expon[Qr, x]}, Dist[Coef
f[Qr, x, r]/(q*Coeff[Pq, x, q]), Subst[Int[(a + b*x^n)^p, x], x, Pq], x] /; EqQ[r, q - 1] && EqQ[Coeff[Qr, x,
r]*D[Pq, x], q*Coeff[Pq, x, q]*Qr]] /; FreeQ[{a, b, n, p}, x] && PolyQ[Pq, x] && PolyQ[Qr, x]

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

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.98 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^n\right ) \, dx=\frac {x (6 a+x (3 b+2 c x)) \left (1+n+\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^n\right )}{6 (1+n)} \]

[In]

Integrate[(a + b*x + c*x^2)*(1 + (a*x + (b*x^2)/2 + (c*x^3)/3)^n),x]

[Out]

(x*(6*a + x*(3*b + 2*c*x))*(1 + n + (a*x + (b*x^2)/2 + (c*x^3)/3)^n))/(6*(1 + n))

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.86

method	result	size
derivativedivides	\(a x +\frac {b \,x^{2}}{2}+\frac {c \,x^{3}}{3}+\frac {\left (a x +\frac {1}{2} b \,x^{2}+\frac {1}{3} c \,x^{3}\right )^{1+n}}{1+n}\)	\(43\)
default	\(a x +\frac {b \,x^{2}}{2}+\frac {c \,x^{3}}{3}+\frac {\left (a x +\frac {1}{2} b \,x^{2}+\frac {1}{3} c \,x^{3}\right )^{1+n}}{1+n}\)	\(43\)
risch	\(a x +\frac {b \,x^{2}}{2}+\frac {c \,x^{3}}{3}+\frac {x \left (2 c \,x^{2}+3 b x +6 a \right ) \left (\frac {1}{3}\right )^{n} \left (\frac {1}{2}\right )^{n} {\left (x \left (2 c \,x^{2}+3 b x +6 a \right )\right )}^{n}}{6 n +6}\)	\(63\)
norman	\(a x +\frac {a x \,{\mathrm e}^{n \ln \left (a x +\frac {1}{2} b \,x^{2}+\frac {1}{3} c \,x^{3}\right )}}{1+n}+\frac {b \,x^{2}}{2}+\frac {c \,x^{3}}{3}+\frac {b \,x^{2} {\mathrm e}^{n \ln \left (a x +\frac {1}{2} b \,x^{2}+\frac {1}{3} c \,x^{3}\right )}}{2+2 n}+\frac {c \,x^{3} {\mathrm e}^{n \ln \left (a x +\frac {1}{2} b \,x^{2}+\frac {1}{3} c \,x^{3}\right )}}{3+3 n}\)	\(107\)
parallelrisch	\(\frac {2 x^{3} {\left (\frac {x \left (2 c \,x^{2}+3 b x +6 a \right )}{6}\right )}^{n} c^{2}+2 x^{3} c^{2} n +2 c^{2} x^{3}+3 x^{2} {\left (\frac {x \left (2 c \,x^{2}+3 b x +6 a \right )}{6}\right )}^{n} b c +3 b c n \,x^{2}+3 b c \,x^{2}+6 x {\left (\frac {x \left (2 c \,x^{2}+3 b x +6 a \right )}{6}\right )}^{n} a c +6 x a c n +6 a c x -18 a b n -18 a b}{6 c \left (1+n \right )}\)	\(141\)

[In]

int((c*x^2+b*x+a)*(1+(a*x+1/2*b*x^2+1/3*c*x^3)^n),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.44 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^n\right ) \, dx=\frac {2 \, {\left (c n + c\right )} x^{3} + 3 \, {\left (b n + b\right )} x^{2} + {\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )} {\left (\frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x\right )}^{n} + 6 \, {\left (a n + a\right )} x}{6 \, {\left (n + 1\right )}} \]

[In]

integrate((c*x^2+b*x+a)*(1+(a*x+1/2*b*x^2+1/3*c*x^3)^n),x, algorithm="fricas")

[Out]

1/6*(2*(c*n + c)*x^3 + 3*(b*n + b)*x^2 + (2*c*x^3 + 3*b*x^2 + 6*a*x)*(1/3*c*x^3 + 1/2*b*x^2 + a*x)^n + 6*(a*n
+ a)*x)/(n + 1)

Sympy [F(-1)]

Timed out. \[ \int \left (a+b x+c x^2\right ) \left (1+\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^n\right ) \, dx=\text {Timed out} \]

[In]

integrate((c*x**2+b*x+a)*(1+(a*x+1/2*b*x**2+1/3*c*x**3)**n),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.66 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^n\right ) \, dx=\frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x + \frac {{\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )} e^{\left (n \log \left (2 \, c x^{2} + 3 \, b x + 6 \, a\right ) + n \log \left (x\right )\right )}}{3^{n + 1} 2^{n + 1} n + 3^{n + 1} 2^{n + 1}} \]

[In]

integrate((c*x^2+b*x+a)*(1+(a*x+1/2*b*x^2+1/3*c*x^3)^n),x, algorithm="maxima")

[Out]

1/3*c*x^3 + 1/2*b*x^2 + a*x + (2*c*x^3 + 3*b*x^2 + 6*a*x)*e^(n*log(2*c*x^2 + 3*b*x + 6*a) + n*log(x))/(3^(n +
1)*2^(n + 1)*n + 3^(n + 1)*2^(n + 1))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.84 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^n\right ) \, dx=\frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x + \frac {{\left (\frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x\right )}^{n + 1}}{n + 1} \]

[In]

integrate((c*x^2+b*x+a)*(1+(a*x+1/2*b*x^2+1/3*c*x^3)^n),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.46 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^n\right ) \, dx=a\,x+\left (\frac {3\,b\,x^2}{6\,n+6}+\frac {2\,c\,x^3}{6\,n+6}+\frac {6\,a\,x}{6\,n+6}\right )\,{\left (\frac {c\,x^3}{3}+\frac {b\,x^2}{2}+a\,x\right )}^n+\frac {b\,x^2}{2}+\frac {c\,x^3}{3} \]

[In]

int(((a*x + (b*x^2)/2 + (c*x^3)/3)^n + 1)*(a + b*x + c*x^2),x)

[Out]

a*x + ((3*b*x^2)/(6*n + 6) + (2*c*x^3)/(6*n + 6) + (6*a*x)/(6*n + 6))*(a*x + (b*x^2)/2 + (c*x^3)/3)^n + (b*x^2
)/2 + (c*x^3)/3