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]

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

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Rubi [A]  time = 0.05, antiderivative size = 41, normalized size of antiderivative = 1.00, 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*}

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Mathematica [B]  time = 0.05, size = 146, normalized size = 3.56 \[ \frac {x^2 (3 b+2 c x) \left (243 b^5 x^{10}+810 b^4 c x^{11}+1080 b^3 c^2 x^{12}+720 b^2 c^3 x^{13}+240 b c^4 x^{14}+19440 d^4 x^2 (3 b+2 c x)+4320 d^3 x^4 (3 b+2 c x)^2+540 d^2 x^6 (3 b+2 c x)^3+36 d x^8 (3 b+2 c x)^4+32 c^5 x^{15}+46656 d^5+46656\right )}{279936} \]

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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fricas [B]  time = 0.93, size = 298, normalized size = 7.27 \[ \frac {1}{4374} x^{18} c^{6} + \frac {1}{486} x^{17} c^{5} b + \frac {5}{648} x^{16} c^{4} b^{2} + \frac {1}{243} x^{15} d c^{5} + \frac {5}{324} x^{15} c^{3} b^{3} + \frac {5}{162} x^{14} d c^{4} b + \frac {5}{288} x^{14} c^{2} b^{4} + \frac {5}{54} x^{13} d c^{3} b^{2} + \frac {1}{96} x^{13} c b^{5} + \frac {5}{162} x^{12} d^{2} c^{4} + \frac {5}{36} x^{12} d c^{2} b^{3} + \frac {1}{384} x^{12} b^{6} + \frac {5}{27} x^{11} d^{2} c^{3} b + \frac {5}{48} x^{11} d c b^{4} + \frac {5}{12} x^{10} d^{2} c^{2} b^{2} + \frac {1}{32} x^{10} d b^{5} + \frac {10}{81} x^{9} d^{3} c^{3} + \frac {5}{12} x^{9} d^{2} c b^{3} + \frac {5}{9} x^{8} d^{3} c^{2} b + \frac {5}{32} x^{8} d^{2} b^{4} + \frac {5}{6} x^{7} d^{3} c b^{2} + \frac {5}{18} x^{6} d^{4} c^{2} + \frac {5}{12} x^{6} d^{3} b^{3} + \frac {5}{6} x^{5} d^{4} c b + \frac {5}{8} x^{4} d^{4} b^{2} + \frac {1}{3} x^{3} d^{5} c + \frac {1}{2} x^{2} d^{5} b + \frac {1}{3} x^{3} c + \frac {1}{2} x^{2} b \]

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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giac [B]  time = 0.31, size = 126, normalized size = 3.07 \[ \frac {1}{279936} \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )}^{6} + \frac {1}{7776} \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )}^{5} d + \frac {5}{2592} \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )}^{4} d^{2} + \frac {5}{324} \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )}^{3} d^{3} + \frac {5}{72} \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )}^{2} d^{4} + \frac {1}{6} \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )} d^{5} + \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} \]

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]

1/279936*(2*c*x^3 + 3*b*x^2)^6 + 1/7776*(2*c*x^3 + 3*b*x^2)^5*d + 5/2592*(2*c*x^3 + 3*b*x^2)^4*d^2 + 5/324*(2*
c*x^3 + 3*b*x^2)^3*d^3 + 5/72*(2*c*x^3 + 3*b*x^2)^2*d^4 + 1/6*(2*c*x^3 + 3*b*x^2)*d^5 + 1/3*c*x^3 + 1/2*b*x^2

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maple [B]  time = 0.00, size = 646, normalized size = 15.76 \[ \frac {c^{6} x^{18}}{4374}+\frac {b \,c^{5} x^{17}}{486}+\frac {5 b^{2} c^{4} x^{16}}{648}+\frac {\left (\frac {5 b^{3} c^{3}}{54}+\left (\frac {b^{3} c^{2}}{12}+\frac {c^{4} d}{81}+\frac {\left (\frac {1}{6} b^{3} c +\frac {4}{27} c^{3} d \right ) c}{3}\right ) c \right ) x^{15}}{15}+\frac {\left (\left (\frac {b^{3} c^{2}}{12}+\frac {c^{4} d}{81}+\frac {\left (\frac {1}{6} b^{3} c +\frac {4}{27} c^{3} d \right ) c}{3}\right ) b +\left (\frac {2 b \,c^{3} d}{27}+\frac {\left (\frac {1}{6} b^{3} c +\frac {4}{27} c^{3} d \right ) b}{2}+\frac {\left (\frac {1}{16} b^{4}+\frac {2}{3} b \,c^{2} d \right ) c}{3}\right ) c \right ) x^{14}}{14}+\frac {5 b^{2} c \,d^{3} x^{7}}{6}+\frac {\left (\left (\frac {2 b \,c^{3} d}{27}+\frac {\left (\frac {1}{6} b^{3} c +\frac {4}{27} c^{3} d \right ) b}{2}+\frac {\left (\frac {1}{16} b^{4}+\frac {2}{3} b \,c^{2} d \right ) c}{3}\right ) b +\left (\frac {b^{2} c^{2} d}{2}+\frac {\left (\frac {1}{16} b^{4}+\frac {2}{3} b \,c^{2} d \right ) b}{2}\right ) c \right ) x^{13}}{13}+\frac {\left (\left (\frac {b^{2} c^{2} d}{2}+\frac {\left (\frac {1}{16} b^{4}+\frac {2}{3} b \,c^{2} d \right ) b}{2}\right ) b +\left (\frac {b^{3} c d}{2}+\frac {\left (\frac {1}{2} b^{3} d +\frac {2}{3} c^{2} d^{2}\right ) c}{3}+\left (\frac {1}{6} b^{3} c +\frac {4}{27} c^{3} d \right ) d \right ) c \right ) x^{12}}{12}+\frac {5 b c \,d^{4} x^{5}}{6}+\frac {\left (\left (\frac {b^{3} c d}{2}+\frac {\left (\frac {1}{2} b^{3} d +\frac {2}{3} c^{2} d^{2}\right ) c}{3}+\left (\frac {1}{6} b^{3} c +\frac {4}{27} c^{3} d \right ) d \right ) b +\left (\frac {2 b \,c^{2} d^{2}}{3}+\frac {\left (\frac {1}{2} b^{3} d +\frac {2}{3} c^{2} d^{2}\right ) b}{2}+\left (\frac {1}{16} b^{4}+\frac {2}{3} b \,c^{2} d \right ) d \right ) c \right ) x^{11}}{11}+\frac {5 b^{2} d^{4} x^{4}}{8}+\frac {\left (\frac {5 b^{2} c^{2} d^{2}}{2}+\left (\frac {2 b \,c^{2} d^{2}}{3}+\frac {\left (\frac {1}{2} b^{3} d +\frac {2}{3} c^{2} d^{2}\right ) b}{2}+\left (\frac {1}{16} b^{4}+\frac {2}{3} b \,c^{2} d \right ) d \right ) b \right ) x^{10}}{10}+\frac {\left (\frac {5 b^{3} c \,d^{2}}{2}+\left (\frac {3 b^{3} d^{2}}{4}+\frac {4 c^{2} d^{3}}{9}+\left (\frac {1}{2} b^{3} d +\frac {2}{3} c^{2} d^{2}\right ) d \right ) c \right ) x^{9}}{9}+\frac {\left (\frac {10 b \,c^{2} d^{3}}{3}+\left (\frac {3 b^{3} d^{2}}{4}+\frac {4 c^{2} d^{3}}{9}+\left (\frac {1}{2} b^{3} d +\frac {2}{3} c^{2} d^{2}\right ) d \right ) b \right ) x^{8}}{8}+\frac {\left (\frac {5}{2} b^{3} d^{3}+\frac {5}{3} c^{2} d^{4}\right ) x^{6}}{6}+\frac {\left (d^{5}+1\right ) c \,x^{3}}{3}+\frac {\left (d^{5}+1\right ) b \,x^{2}}{2} \]

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

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maxima [B]  time = 0.45, size = 289, normalized size = 7.05 \[ \frac {1}{4374} \, c^{6} x^{18} + \frac {1}{486} \, b c^{5} x^{17} + \frac {5}{648} \, b^{2} c^{4} x^{16} + \frac {1}{972} \, {\left (15 \, b^{3} c^{3} + 4 \, c^{5} d\right )} x^{15} + \frac {5}{2592} \, {\left (9 \, b^{4} c^{2} + 16 \, b c^{4} d\right )} x^{14} + \frac {1}{864} \, {\left (9 \, b^{5} c + 80 \, b^{2} c^{3} d\right )} x^{13} + \frac {5}{6} \, b^{2} c d^{3} x^{7} + \frac {1}{10368} \, {\left (27 \, b^{6} + 1440 \, b^{3} c^{2} d + 320 \, c^{4} d^{2}\right )} x^{12} + \frac {5}{432} \, {\left (9 \, b^{4} c d + 16 \, b c^{3} d^{2}\right )} x^{11} + \frac {5}{6} \, b c d^{4} x^{5} + \frac {1}{96} \, {\left (3 \, b^{5} d + 40 \, b^{2} c^{2} d^{2}\right )} x^{10} + \frac {5}{8} \, b^{2} d^{4} x^{4} + \frac {5}{324} \, {\left (27 \, b^{3} c d^{2} + 8 \, c^{3} d^{3}\right )} x^{9} + \frac {5}{288} \, {\left (9 \, b^{4} d^{2} + 32 \, b c^{2} d^{3}\right )} x^{8} + \frac {5}{36} \, {\left (3 \, b^{3} d^{3} + 2 \, c^{2} d^{4}\right )} x^{6} + \frac {1}{3} \, {\left (c d^{5} + c\right )} x^{3} + \frac {1}{2} \, {\left (b d^{5} + b\right )} x^{2} \]

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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mupad [B]  time = 2.28, size = 273, normalized size = 6.66 \[ x^{13}\,\left (\frac {b^5\,c}{96}+\frac {5\,d\,b^2\,c^3}{54}\right )+x^{14}\,\left (\frac {5\,b^4\,c^2}{288}+\frac {5\,d\,b\,c^4}{162}\right )+x^{12}\,\left (\frac {b^6}{384}+\frac {5\,b^3\,c^2\,d}{36}+\frac {5\,c^4\,d^2}{162}\right )+\frac {c^6\,x^{18}}{4374}+x^{15}\,\left (\frac {5\,b^3\,c^3}{324}+\frac {d\,c^5}{243}\right )+\frac {5\,d^3\,x^6\,\left (3\,b^3+2\,d\,c^2\right )}{36}+\frac {b\,c^5\,x^{17}}{486}+\frac {5\,b^2\,c^4\,x^{16}}{648}+\frac {b\,x^2\,\left (d^5+1\right )}{2}+\frac {5\,b^2\,d^4\,x^4}{8}+\frac {c\,x^3\,\left (d^5+1\right )}{3}+\frac {5\,b^2\,c\,d^3\,x^7}{6}+\frac {5\,b\,d^2\,x^8\,\left (9\,b^3+32\,d\,c^2\right )}{288}+\frac {b^2\,d\,x^{10}\,\left (3\,b^3+40\,d\,c^2\right )}{96}+\frac {5\,c\,d^2\,x^9\,\left (27\,b^3+8\,d\,c^2\right )}{324}+\frac {5\,b\,c\,d^4\,x^5}{6}+\frac {5\,b\,c\,d\,x^{11}\,\left (9\,b^3+16\,d\,c^2\right )}{432} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^13*((b^5*c)/96 + (5*b^2*c^3*d)/54) + x^14*((5*b^4*c^2)/288 + (5*b*c^4*d)/162) + x^12*(b^6/384 + (5*c^4*d^2)/
162 + (5*b^3*c^2*d)/36) + (c^6*x^18)/4374 + x^15*((c^5*d)/243 + (5*b^3*c^3)/324) + (5*d^3*x^6*(2*c^2*d + 3*b^3
))/36 + (b*c^5*x^17)/486 + (5*b^2*c^4*x^16)/648 + (b*x^2*(d^5 + 1))/2 + (5*b^2*d^4*x^4)/8 + (c*x^3*(d^5 + 1))/
3 + (5*b^2*c*d^3*x^7)/6 + (5*b*d^2*x^8*(32*c^2*d + 9*b^3))/288 + (b^2*d*x^10*(40*c^2*d + 3*b^3))/96 + (5*c*d^2
*x^9*(8*c^2*d + 27*b^3))/324 + (5*b*c*d^4*x^5)/6 + (5*b*c*d*x^11*(16*c^2*d + 9*b^3))/432

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sympy [B]  time = 0.15, size = 321, normalized size = 7.83 \[ \frac {5 b^{2} c^{4} x^{16}}{648} + \frac {5 b^{2} c d^{3} x^{7}}{6} + \frac {5 b^{2} d^{4} x^{4}}{8} + \frac {b c^{5} x^{17}}{486} + \frac {5 b c d^{4} x^{5}}{6} + \frac {c^{6} x^{18}}{4374} + x^{15} \left (\frac {5 b^{3} c^{3}}{324} + \frac {c^{5} d}{243}\right ) + x^{14} \left (\frac {5 b^{4} c^{2}}{288} + \frac {5 b c^{4} d}{162}\right ) + x^{13} \left (\frac {b^{5} c}{96} + \frac {5 b^{2} c^{3} d}{54}\right ) + x^{12} \left (\frac {b^{6}}{384} + \frac {5 b^{3} c^{2} d}{36} + \frac {5 c^{4} d^{2}}{162}\right ) + x^{11} \left (\frac {5 b^{4} c d}{48} + \frac {5 b c^{3} d^{2}}{27}\right ) + x^{10} \left (\frac {b^{5} d}{32} + \frac {5 b^{2} c^{2} d^{2}}{12}\right ) + x^{9} \left (\frac {5 b^{3} c d^{2}}{12} + \frac {10 c^{3} d^{3}}{81}\right ) + x^{8} \left (\frac {5 b^{4} d^{2}}{32} + \frac {5 b c^{2} d^{3}}{9}\right ) + x^{6} \left (\frac {5 b^{3} d^{3}}{12} + \frac {5 c^{2} d^{4}}{18}\right ) + x^{3} \left (\frac {c d^{5}}{3} + \frac {c}{3}\right ) + x^{2} \left (\frac {b d^{5}}{2} + \frac {b}{2}\right ) \]

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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