3.210 \(\int (a+c x^2) (1+(a x+\frac{c x^3}{3})^5) \, dx\)

Optimal. Leaf size=30 \[ \frac{1}{6} \left (a x+\frac{c x^3}{3}\right )^6+a x+\frac{c x^3}{3} \]

[Out]

a*x + (c*x^3)/3 + (a*x + (c*x^3)/3)^6/6

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Rubi [A]  time = 0.0198891, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {1591} \[ \frac{1}{6} \left (a x+\frac{c x^3}{3}\right )^6+a x+\frac{c x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)*(1 + (a*x + (c*x^3)/3)^5),x]

[Out]

a*x + (c*x^3)/3 + (a*x + (c*x^3)/3)^6/6

Rule 1591

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*} \int \left (a+c x^2\right ) \left (1+\left (a x+\frac{c x^3}{3}\right )^5\right ) \, dx &=\operatorname{Subst}\left (\int \left (1+x^5\right ) \, dx,x,a x+\frac{c x^3}{3}\right )\\ &=a x+\frac{c x^3}{3}+\frac{1}{6} \left (a x+\frac{c x^3}{3}\right )^6\\ \end{align*}

Mathematica [B]  time = 0.0057631, size = 93, normalized size = 3.1 \[ \frac{5}{162} a^2 c^4 x^{14}+\frac{10}{81} a^3 c^3 x^{12}+\frac{5}{18} a^4 c^2 x^{10}+\frac{1}{3} a^5 c x^8+\frac{a^6 x^6}{6}+\frac{1}{243} a c^5 x^{16}+a x+\frac{c^6 x^{18}}{4374}+\frac{c x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)*(1 + (a*x + (c*x^3)/3)^5),x]

[Out]

a*x + (c*x^3)/3 + (a^6*x^6)/6 + (a^5*c*x^8)/3 + (5*a^4*c^2*x^10)/18 + (10*a^3*c^3*x^12)/81 + (5*a^2*c^4*x^14)/
162 + (a*c^5*x^16)/243 + (c^6*x^18)/4374

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Maple [B]  time = 0.002, size = 78, normalized size = 2.6 \begin{align*}{\frac{{c}^{6}{x}^{18}}{4374}}+{\frac{{c}^{5}a{x}^{16}}{243}}+{\frac{5\,{c}^{4}{a}^{2}{x}^{14}}{162}}+{\frac{10\,{a}^{3}{c}^{3}{x}^{12}}{81}}+{\frac{5\,{a}^{4}{c}^{2}{x}^{10}}{18}}+{\frac{{a}^{5}c{x}^{8}}{3}}+{\frac{{a}^{6}{x}^{6}}{6}}+{\frac{c{x}^{3}}{3}}+ax \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)*(1+(a*x+1/3*c*x^3)^5),x)

[Out]

1/4374*c^6*x^18+1/243*c^5*a*x^16+5/162*c^4*a^2*x^14+10/81*a^3*c^3*x^12+5/18*a^4*c^2*x^10+1/3*a^5*c*x^8+1/6*a^6
*x^6+1/3*c*x^3+a*x

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Maxima [B]  time = 1.00994, size = 104, normalized size = 3.47 \begin{align*} \frac{1}{4374} \, c^{6} x^{18} + \frac{1}{243} \, a c^{5} x^{16} + \frac{5}{162} \, a^{2} c^{4} x^{14} + \frac{10}{81} \, a^{3} c^{3} x^{12} + \frac{5}{18} \, a^{4} c^{2} x^{10} + \frac{1}{3} \, a^{5} c x^{8} + \frac{1}{6} \, a^{6} x^{6} + \frac{1}{3} \, c x^{3} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)*(1+(a*x+1/3*c*x^3)^5),x, algorithm="maxima")

[Out]

1/4374*c^6*x^18 + 1/243*a*c^5*x^16 + 5/162*a^2*c^4*x^14 + 10/81*a^3*c^3*x^12 + 5/18*a^4*c^2*x^10 + 1/3*a^5*c*x
^8 + 1/6*a^6*x^6 + 1/3*c*x^3 + a*x

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Fricas [B]  time = 1.11426, size = 197, normalized size = 6.57 \begin{align*} \frac{1}{4374} x^{18} c^{6} + \frac{1}{243} x^{16} c^{5} a + \frac{5}{162} x^{14} c^{4} a^{2} + \frac{10}{81} x^{12} c^{3} a^{3} + \frac{5}{18} x^{10} c^{2} a^{4} + \frac{1}{3} x^{8} c a^{5} + \frac{1}{6} x^{6} a^{6} + \frac{1}{3} x^{3} c + x a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)*(1+(a*x+1/3*c*x^3)^5),x, algorithm="fricas")

[Out]

1/4374*x^18*c^6 + 1/243*x^16*c^5*a + 5/162*x^14*c^4*a^2 + 10/81*x^12*c^3*a^3 + 5/18*x^10*c^2*a^4 + 1/3*x^8*c*a
^5 + 1/6*x^6*a^6 + 1/3*x^3*c + x*a

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Sympy [B]  time = 0.090498, size = 87, normalized size = 2.9 \begin{align*} \frac{a^{6} x^{6}}{6} + \frac{a^{5} c x^{8}}{3} + \frac{5 a^{4} c^{2} x^{10}}{18} + \frac{10 a^{3} c^{3} x^{12}}{81} + \frac{5 a^{2} c^{4} x^{14}}{162} + \frac{a c^{5} x^{16}}{243} + a x + \frac{c^{6} x^{18}}{4374} + \frac{c x^{3}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)*(1+(a*x+1/3*c*x**3)**5),x)

[Out]

a**6*x**6/6 + a**5*c*x**8/3 + 5*a**4*c**2*x**10/18 + 10*a**3*c**3*x**12/81 + 5*a**2*c**4*x**14/162 + a*c**5*x*
*16/243 + a*x + c**6*x**18/4374 + c*x**3/3

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Giac [B]  time = 1.22165, size = 104, normalized size = 3.47 \begin{align*} \frac{1}{4374} \, c^{6} x^{18} + \frac{1}{243} \, a c^{5} x^{16} + \frac{5}{162} \, a^{2} c^{4} x^{14} + \frac{10}{81} \, a^{3} c^{3} x^{12} + \frac{5}{18} \, a^{4} c^{2} x^{10} + \frac{1}{3} \, a^{5} c x^{8} + \frac{1}{6} \, a^{6} x^{6} + \frac{1}{3} \, c x^{3} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)*(1+(a*x+1/3*c*x^3)^5),x, algorithm="giac")

[Out]

1/4374*c^6*x^18 + 1/243*a*c^5*x^16 + 5/162*a^2*c^4*x^14 + 10/81*a^3*c^3*x^12 + 5/18*a^4*c^2*x^10 + 1/3*a^5*c*x
^8 + 1/6*a^6*x^6 + 1/3*c*x^3 + a*x