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

Optimal. Leaf size=34 \[ \frac{x^{12} (3 b+2 c x)^6}{279936}+\frac{b x^2}{2}+\frac{c x^3}{3} \]

[Out]

(b*x^2)/2 + (c*x^3)/3 + (x^12*(3*b + 2*c*x)^6)/279936

________________________________________________________________________________________

Rubi [A]  time = 0.0329058, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.032, Rules used = {1591} \[ \frac{x^{12} (3 b+2 c x)^6}{279936}+\frac{b x^2}{2}+\frac{c x^3}{3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x^2)/2 + (c*x^3)/3 + (x^12*(3*b + 2*c*x)^6)/279936

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 (b x+c x^2\right ) \left (1+\left (\frac{b x^2}{2}+\frac{c x^3}{3}\right )^5\right ) \, dx &=\operatorname{Subst}\left (\int \left (1+x^5\right ) \, dx,x,\frac{b x^2}{2}+\frac{c x^3}{3}\right )\\ &=\frac{b x^2}{2}+\frac{c x^3}{3}+\frac{x^{12} (3 b+2 c x)^6}{279936}\\ \end{align*}

Mathematica [B]  time = 0.0077667, size = 98, normalized size = 2.88 \[ \frac{5}{648} b^2 c^4 x^{16}+\frac{5}{324} b^3 c^3 x^{15}+\frac{5}{288} b^4 c^2 x^{14}+\frac{1}{96} b^5 c x^{13}+\frac{b^6 x^{12}}{384}+\frac{1}{486} b c^5 x^{17}+\frac{b x^2}{2}+\frac{c^6 x^{18}}{4374}+\frac{c x^3}{3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x^2)/2 + (c*x^3)/3 + (b^6*x^12)/384 + (b^5*c*x^13)/96 + (5*b^4*c^2*x^14)/288 + (5*b^3*c^3*x^15)/324 + (5*b^
2*c^4*x^16)/648 + (b*c^5*x^17)/486 + (c^6*x^18)/4374

________________________________________________________________________________________

Maple [B]  time = 0.003, size = 81, normalized size = 2.4 \begin{align*}{\frac{{c}^{6}{x}^{18}}{4374}}+{\frac{b{c}^{5}{x}^{17}}{486}}+{\frac{5\,{b}^{2}{c}^{4}{x}^{16}}{648}}+{\frac{5\,{b}^{3}{c}^{3}{x}^{15}}{324}}+{\frac{5\,{b}^{4}{c}^{2}{x}^{14}}{288}}+{\frac{{b}^{5}c{x}^{13}}{96}}+{\frac{{b}^{6}{x}^{12}}{384}}+{\frac{c{x}^{3}}{3}}+{\frac{b{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4374*c^6*x^18+1/486*b*c^5*x^17+5/648*b^2*c^4*x^16+5/324*b^3*c^3*x^15+5/288*b^4*c^2*x^14+1/96*b^5*c*x^13+1/38
4*b^6*x^12+1/3*c*x^3+1/2*b*x^2

________________________________________________________________________________________

Maxima [B]  time = 0.975224, size = 108, normalized size = 3.18 \begin{align*} \frac{1}{4374} \, c^{6} x^{18} + \frac{1}{486} \, b c^{5} x^{17} + \frac{5}{648} \, b^{2} c^{4} x^{16} + \frac{5}{324} \, b^{3} c^{3} x^{15} + \frac{5}{288} \, b^{4} c^{2} x^{14} + \frac{1}{96} \, b^{5} c x^{13} + \frac{1}{384} \, b^{6} x^{12} + \frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4374*c^6*x^18 + 1/486*b*c^5*x^17 + 5/648*b^2*c^4*x^16 + 5/324*b^3*c^3*x^15 + 5/288*b^4*c^2*x^14 + 1/96*b^5*c
*x^13 + 1/384*b^6*x^12 + 1/3*c*x^3 + 1/2*b*x^2

________________________________________________________________________________________

Fricas [B]  time = 1.22939, size = 213, normalized size = 6.26 \begin{align*} \frac{1}{4374} x^{18} c^{6} + \frac{1}{486} x^{17} c^{5} b + \frac{5}{648} x^{16} c^{4} b^{2} + \frac{5}{324} x^{15} c^{3} b^{3} + \frac{5}{288} x^{14} c^{2} b^{4} + \frac{1}{96} x^{13} c b^{5} + \frac{1}{384} x^{12} b^{6} + \frac{1}{3} x^{3} c + \frac{1}{2} x^{2} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4374*x^18*c^6 + 1/486*x^17*c^5*b + 5/648*x^16*c^4*b^2 + 5/324*x^15*c^3*b^3 + 5/288*x^14*c^2*b^4 + 1/96*x^13*
c*b^5 + 1/384*x^12*b^6 + 1/3*x^3*c + 1/2*x^2*b

________________________________________________________________________________________

Sympy [B]  time = 0.094561, size = 90, normalized size = 2.65 \begin{align*} \frac{b^{6} x^{12}}{384} + \frac{b^{5} c x^{13}}{96} + \frac{5 b^{4} c^{2} x^{14}}{288} + \frac{5 b^{3} c^{3} x^{15}}{324} + \frac{5 b^{2} c^{4} x^{16}}{648} + \frac{b c^{5} x^{17}}{486} + \frac{b x^{2}}{2} + \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+b*x)*(1+(1/2*b*x**2+1/3*c*x**3)**5),x)

[Out]

b**6*x**12/384 + b**5*c*x**13/96 + 5*b**4*c**2*x**14/288 + 5*b**3*c**3*x**15/324 + 5*b**2*c**4*x**16/648 + b*c
**5*x**17/486 + b*x**2/2 + c**6*x**18/4374 + c*x**3/3

________________________________________________________________________________________

Giac [B]  time = 1.17155, size = 108, normalized size = 3.18 \begin{align*} \frac{1}{4374} \, c^{6} x^{18} + \frac{1}{486} \, b c^{5} x^{17} + \frac{5}{648} \, b^{2} c^{4} x^{16} + \frac{5}{324} \, b^{3} c^{3} x^{15} + \frac{5}{288} \, b^{4} c^{2} x^{14} + \frac{1}{96} \, b^{5} c x^{13} + \frac{1}{384} \, b^{6} x^{12} + \frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4374*c^6*x^18 + 1/486*b*c^5*x^17 + 5/648*b^2*c^4*x^16 + 5/324*b^3*c^3*x^15 + 5/288*b^4*c^2*x^14 + 1/96*b^5*c
*x^13 + 1/384*b^6*x^12 + 1/3*c*x^3 + 1/2*b*x^2