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

Optimal. Leaf size=41 \[ \frac{1}{6} \left (\frac{b x^2}{2}+\frac{c x^3}{3}+d\right )^6+\frac{b x^2}{2}+\frac{c x^3}{3} \]

[Out]

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

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Rubi [A]  time = 0.0452143, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.031, Rules used = {1591} \[ \frac{1}{6} \left (\frac{b x^2}{2}+\frac{c x^3}{3}+d\right )^6+\frac{b x^2}{2}+\frac{c x^3}{3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x^2)/2 + (c*x^3)/3 + (d + (b*x^2)/2 + (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 (b x+c x^2\right ) \left (1+\left (d+\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,d+\frac{b x^2}{2}+\frac{c x^3}{3}\right )\\ &=\frac{b x^2}{2}+\frac{c x^3}{3}+\frac{1}{6} \left (d+\frac{b x^2}{2}+\frac{c x^3}{3}\right )^6\\ \end{align*}

Mathematica [B]  time = 0.0499823, size = 146, normalized size = 3.56 \[ \frac{x^2 (3 b+2 c x) \left (720 b^2 c^3 x^{13}+1080 b^3 c^2 x^{12}+810 b^4 c x^{11}+243 b^5 x^{10}+240 b c^4 x^{14}+540 d^2 x^6 (3 b+2 c x)^3+4320 d^3 x^4 (3 b+2 c x)^2+19440 d^4 x^2 (3 b+2 c x)+36 d x^8 (3 b+2 c x)^4+32 c^5 x^{15}+46656 d^5+46656\right )}{279936} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(3*b + 2*c*x)*(46656 + 46656*d^5 + 243*b^5*x^10 + 810*b^4*c*x^11 + 1080*b^3*c^2*x^12 + 720*b^2*c^3*x^13 +
240*b*c^4*x^14 + 32*c^5*x^15 + 19440*d^4*x^2*(3*b + 2*c*x) + 4320*d^3*x^4*(3*b + 2*c*x)^2 + 540*d^2*x^6*(3*b
+ 2*c*x)^3 + 36*d*x^8*(3*b + 2*c*x)^4))/279936

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Maple [B]  time = 0.002, size = 646, normalized size = 15.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)*(1+(d+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+1/15*(5/54*b^3*c^3+c*(1/81*d*c^4+1/12*b^3*c^2+1/3*c*(4/27*
c^3*d+1/6*b^3*c)))*x^15+1/14*(b*(1/81*d*c^4+1/12*b^3*c^2+1/3*c*(4/27*c^3*d+1/6*b^3*c))+c*(2/27*d*b*c^3+1/2*b*(
4/27*c^3*d+1/6*b^3*c)+1/3*c*(2/3*b*d*c^2+1/16*b^4)))*x^14+1/13*(b*(2/27*d*b*c^3+1/2*b*(4/27*c^3*d+1/6*b^3*c)+1
/3*c*(2/3*b*d*c^2+1/16*b^4))+c*(1/2*d*c^2*b^2+1/2*b*(2/3*b*d*c^2+1/16*b^4)))*x^13+1/12*(b*(1/2*d*c^2*b^2+1/2*b
*(2/3*b*d*c^2+1/16*b^4))+c*(d*(4/27*c^3*d+1/6*b^3*c)+1/2*b^3*c*d+1/3*c*(2/3*d^2*c^2+1/2*b^3*d)))*x^12+1/11*(b*
(d*(4/27*c^3*d+1/6*b^3*c)+1/2*b^3*c*d+1/3*c*(2/3*d^2*c^2+1/2*b^3*d))+c*(d*(2/3*b*d*c^2+1/16*b^4)+1/2*b*(2/3*d^
2*c^2+1/2*b^3*d)+2/3*c^2*d^2*b))*x^11+1/10*(b*(d*(2/3*b*d*c^2+1/16*b^4)+1/2*b*(2/3*d^2*c^2+1/2*b^3*d)+2/3*c^2*
d^2*b)+5/2*b^2*c^2*d^2)*x^10+1/9*(5/2*b^3*c*d^2+c*(d*(2/3*d^2*c^2+1/2*b^3*d)+3/4*b^3*d^2+4/9*c^2*d^3))*x^9+1/8
*(b*(d*(2/3*d^2*c^2+1/2*b^3*d)+3/4*b^3*d^2+4/9*c^2*d^3)+10/3*c^2*b*d^3)*x^8+5/6*b^2*c*d^3*x^7+1/6*(5/2*b^3*d^3
+5/3*c^2*d^4)*x^6+5/6*b*c*d^4*x^5+5/8*b^2*d^4*x^4+1/3*c*(d^5+1)*x^3+1/2*b*(d^5+1)*x^2

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Maxima [B]  time = 0.987637, size = 390, normalized size = 9.51 \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{1}{972} \,{\left (15 \, b^{3} c^{3} + 4 \, c^{5} d\right )} x^{15} + \frac{5}{2592} \,{\left (9 \, b^{4} c^{2} + 16 \, b c^{4} d\right )} x^{14} + \frac{1}{864} \,{\left (9 \, b^{5} c + 80 \, b^{2} c^{3} d\right )} x^{13} + \frac{5}{6} \, b^{2} c d^{3} x^{7} + \frac{1}{10368} \,{\left (27 \, b^{6} + 1440 \, b^{3} c^{2} d + 320 \, c^{4} d^{2}\right )} x^{12} + \frac{5}{432} \,{\left (9 \, b^{4} c d + 16 \, b c^{3} d^{2}\right )} x^{11} + \frac{5}{6} \, b c d^{4} x^{5} + \frac{1}{96} \,{\left (3 \, b^{5} d + 40 \, b^{2} c^{2} d^{2}\right )} x^{10} + \frac{5}{8} \, b^{2} d^{4} x^{4} + \frac{5}{324} \,{\left (27 \, b^{3} c d^{2} + 8 \, c^{3} d^{3}\right )} x^{9} + \frac{5}{288} \,{\left (9 \, b^{4} d^{2} + 32 \, b c^{2} d^{3}\right )} x^{8} + \frac{5}{36} \,{\left (3 \, b^{3} d^{3} + 2 \, c^{2} d^{4}\right )} x^{6} + \frac{1}{3} \,{\left (c d^{5} + c\right )} x^{3} + \frac{1}{2} \,{\left (b d^{5} + b\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)*(1+(d+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 + 1/972*(15*b^3*c^3 + 4*c^5*d)*x^15 + 5/2592*(9*b^4*c^
2 + 16*b*c^4*d)*x^14 + 1/864*(9*b^5*c + 80*b^2*c^3*d)*x^13 + 5/6*b^2*c*d^3*x^7 + 1/10368*(27*b^6 + 1440*b^3*c^
2*d + 320*c^4*d^2)*x^12 + 5/432*(9*b^4*c*d + 16*b*c^3*d^2)*x^11 + 5/6*b*c*d^4*x^5 + 1/96*(3*b^5*d + 40*b^2*c^2
*d^2)*x^10 + 5/8*b^2*d^4*x^4 + 5/324*(27*b^3*c*d^2 + 8*c^3*d^3)*x^9 + 5/288*(9*b^4*d^2 + 32*b*c^2*d^3)*x^8 + 5
/36*(3*b^3*d^3 + 2*c^2*d^4)*x^6 + 1/3*(c*d^5 + c)*x^3 + 1/2*(b*d^5 + b)*x^2

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Fricas [B]  time = 1.12347, size = 747, normalized size = 18.22 \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{1}{243} x^{15} d c^{5} + \frac{5}{324} x^{15} c^{3} b^{3} + \frac{5}{162} x^{14} d c^{4} b + \frac{5}{288} x^{14} c^{2} b^{4} + \frac{5}{54} x^{13} d c^{3} b^{2} + \frac{1}{96} x^{13} c b^{5} + \frac{5}{162} x^{12} d^{2} c^{4} + \frac{5}{36} x^{12} d c^{2} b^{3} + \frac{1}{384} x^{12} b^{6} + \frac{5}{27} x^{11} d^{2} c^{3} b + \frac{5}{48} x^{11} d c b^{4} + \frac{5}{12} x^{10} d^{2} c^{2} b^{2} + \frac{1}{32} x^{10} d b^{5} + \frac{10}{81} x^{9} d^{3} c^{3} + \frac{5}{12} x^{9} d^{2} c b^{3} + \frac{5}{9} x^{8} d^{3} c^{2} b + \frac{5}{32} x^{8} d^{2} b^{4} + \frac{5}{6} x^{7} d^{3} c b^{2} + \frac{5}{18} x^{6} d^{4} c^{2} + \frac{5}{12} x^{6} d^{3} b^{3} + \frac{5}{6} x^{5} d^{4} c b + \frac{5}{8} x^{4} d^{4} b^{2} + \frac{1}{3} x^{3} d^{5} c + \frac{1}{2} x^{2} d^{5} b + \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+(d+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 + 1/243*x^15*d*c^5 + 5/324*x^15*c^3*b^3 + 5/162*x^14*d
*c^4*b + 5/288*x^14*c^2*b^4 + 5/54*x^13*d*c^3*b^2 + 1/96*x^13*c*b^5 + 5/162*x^12*d^2*c^4 + 5/36*x^12*d*c^2*b^3
+ 1/384*x^12*b^6 + 5/27*x^11*d^2*c^3*b + 5/48*x^11*d*c*b^4 + 5/12*x^10*d^2*c^2*b^2 + 1/32*x^10*d*b^5 + 10/81*
x^9*d^3*c^3 + 5/12*x^9*d^2*c*b^3 + 5/9*x^8*d^3*c^2*b + 5/32*x^8*d^2*b^4 + 5/6*x^7*d^3*c*b^2 + 5/18*x^6*d^4*c^2
+ 5/12*x^6*d^3*b^3 + 5/6*x^5*d^4*c*b + 5/8*x^4*d^4*b^2 + 1/3*x^3*d^5*c + 1/2*x^2*d^5*b + 1/3*x^3*c + 1/2*x^2*
b

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Sympy [B]  time = 0.146376, size = 321, normalized size = 7.83 \begin{align*} \frac{5 b^{2} c^{4} x^{16}}{648} + \frac{5 b^{2} c d^{3} x^{7}}{6} + \frac{5 b^{2} d^{4} x^{4}}{8} + \frac{b c^{5} x^{17}}{486} + \frac{5 b c d^{4} x^{5}}{6} + \frac{c^{6} x^{18}}{4374} + x^{15} \left (\frac{5 b^{3} c^{3}}{324} + \frac{c^{5} d}{243}\right ) + x^{14} \left (\frac{5 b^{4} c^{2}}{288} + \frac{5 b c^{4} d}{162}\right ) + x^{13} \left (\frac{b^{5} c}{96} + \frac{5 b^{2} c^{3} d}{54}\right ) + x^{12} \left (\frac{b^{6}}{384} + \frac{5 b^{3} c^{2} d}{36} + \frac{5 c^{4} d^{2}}{162}\right ) + x^{11} \left (\frac{5 b^{4} c d}{48} + \frac{5 b c^{3} d^{2}}{27}\right ) + x^{10} \left (\frac{b^{5} d}{32} + \frac{5 b^{2} c^{2} d^{2}}{12}\right ) + x^{9} \left (\frac{5 b^{3} c d^{2}}{12} + \frac{10 c^{3} d^{3}}{81}\right ) + x^{8} \left (\frac{5 b^{4} d^{2}}{32} + \frac{5 b c^{2} d^{3}}{9}\right ) + x^{6} \left (\frac{5 b^{3} d^{3}}{12} + \frac{5 c^{2} d^{4}}{18}\right ) + x^{3} \left (\frac{c d^{5}}{3} + \frac{c}{3}\right ) + x^{2} \left (\frac{b d^{5}}{2} + \frac{b}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*b**2*c**4*x**16/648 + 5*b**2*c*d**3*x**7/6 + 5*b**2*d**4*x**4/8 + b*c**5*x**17/486 + 5*b*c*d**4*x**5/6 + c**
6*x**18/4374 + x**15*(5*b**3*c**3/324 + c**5*d/243) + x**14*(5*b**4*c**2/288 + 5*b*c**4*d/162) + x**13*(b**5*c
/96 + 5*b**2*c**3*d/54) + x**12*(b**6/384 + 5*b**3*c**2*d/36 + 5*c**4*d**2/162) + x**11*(5*b**4*c*d/48 + 5*b*c
**3*d**2/27) + x**10*(b**5*d/32 + 5*b**2*c**2*d**2/12) + x**9*(5*b**3*c*d**2/12 + 10*c**3*d**3/81) + x**8*(5*b
**4*d**2/32 + 5*b*c**2*d**3/9) + x**6*(5*b**3*d**3/12 + 5*c**2*d**4/18) + x**3*(c*d**5/3 + c/3) + x**2*(b*d**5
/2 + b/2)

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Giac [B]  time = 1.20062, size = 402, normalized size = 9.8 \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{1}{243} \, c^{5} d x^{15} + \frac{5}{288} \, b^{4} c^{2} x^{14} + \frac{5}{162} \, b c^{4} d x^{14} + \frac{1}{96} \, b^{5} c x^{13} + \frac{5}{54} \, b^{2} c^{3} d x^{13} + \frac{1}{384} \, b^{6} x^{12} + \frac{5}{36} \, b^{3} c^{2} d x^{12} + \frac{5}{162} \, c^{4} d^{2} x^{12} + \frac{5}{48} \, b^{4} c d x^{11} + \frac{5}{27} \, b c^{3} d^{2} x^{11} + \frac{1}{32} \, b^{5} d x^{10} + \frac{5}{12} \, b^{2} c^{2} d^{2} x^{10} + \frac{5}{12} \, b^{3} c d^{2} x^{9} + \frac{10}{81} \, c^{3} d^{3} x^{9} + \frac{5}{32} \, b^{4} d^{2} x^{8} + \frac{5}{9} \, b c^{2} d^{3} x^{8} + \frac{5}{6} \, b^{2} c d^{3} x^{7} + \frac{5}{12} \, b^{3} d^{3} x^{6} + \frac{5}{18} \, c^{2} d^{4} x^{6} + \frac{5}{6} \, b c d^{4} x^{5} + \frac{5}{8} \, b^{2} d^{4} x^{4} + \frac{1}{3} \, c d^{5} x^{3} + \frac{1}{2} \, b d^{5} x^{2} + \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+(d+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 + 1/243*c^5*d*x^15 + 5/288*b^4*c^
2*x^14 + 5/162*b*c^4*d*x^14 + 1/96*b^5*c*x^13 + 5/54*b^2*c^3*d*x^13 + 1/384*b^6*x^12 + 5/36*b^3*c^2*d*x^12 + 5
/162*c^4*d^2*x^12 + 5/48*b^4*c*d*x^11 + 5/27*b*c^3*d^2*x^11 + 1/32*b^5*d*x^10 + 5/12*b^2*c^2*d^2*x^10 + 5/12*b
^3*c*d^2*x^9 + 10/81*c^3*d^3*x^9 + 5/32*b^4*d^2*x^8 + 5/9*b*c^2*d^3*x^8 + 5/6*b^2*c*d^3*x^7 + 5/12*b^3*d^3*x^6
+ 5/18*c^2*d^4*x^6 + 5/6*b*c*d^4*x^5 + 5/8*b^2*d^4*x^4 + 1/3*c*d^5*x^3 + 1/2*b*d^5*x^2 + 1/3*c*x^3 + 1/2*b*x^
2