Integrand size = 32, antiderivative size = 39 \[ \int \left (b x+c x^2\right ) \left (1+\left (d+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {b x^2}{2}+\frac {c x^3}{3}+\frac {\left (6 d+3 b x^2+2 c x^3\right )^6}{279936} \] Output:
1/2*b*x^2+1/3*c*x^3+1/279936*(2*c*x^3+3*b*x^2+6*d)^6
Leaf count is larger than twice the leaf count of optimal. \(146\) vs. \(2(39)=78\).
Time = 0.07 (sec) , antiderivative size = 146, normalized size of antiderivative = 3.74 \[ \int \left (b x+c x^2\right ) \left (1+\left (d+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {x^2 (3 b+2 c x) \left (46656+46656 d^5+243 b^5 x^{10}+810 b^4 c x^{11}+1080 b^3 c^2 x^{12}+720 b^2 c^3 x^{13}+240 b c^4 x^{14}+32 c^5 x^{15}+19440 d^4 x^2 (3 b+2 c x)+4320 d^3 x^4 (3 b+2 c x)^2+540 d^2 x^6 (3 b+2 c x)^3+36 d x^8 (3 b+2 c x)^4\right )}{279936} \] Input:
Integrate[(b*x + c*x^2)*(1 + (d + (b*x^2)/2 + (c*x^3)/3)^5),x]
Output:
(x^2*(3*b + 2*c*x)*(46656 + 46656*d^5 + 243*b^5*x^10 + 810*b^4*c*x^11 + 10 80*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
Time = 0.36 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2024, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \left (b x+c x^2\right ) \left (\left (\frac {b x^2}{2}+\frac {c x^3}{3}+d\right )^5+1\right ) \, dx\) |
| \(\Big \downarrow \) 2024 |
| \(\displaystyle \int \left (\left (\frac {b x^2}{2}+\frac {c x^3}{3}+d\right )^5+1\right )d\left (\frac {b x^2}{2}+\frac {c x^3}{3}+d\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} \left (\frac {b x^2}{2}+\frac {c x^3}{3}+d\right )^6+\frac {b x^2}{2}+\frac {c x^3}{3}+d\) |
Input:
Int[(b*x + c*x^2)*(1 + (d + (b*x^2)/2 + (c*x^3)/3)^5),x]
Output:
d + (b*x^2)/2 + (c*x^3)/3 + (d + (b*x^2)/2 + (c*x^3)/3)^6/6
Int[((a_.) + (b_.)*(Pq_)^(n_.))^(p_.)*(Qr_), x_Symbol] :> With[{q = Expon[P q, x], r = Expon[Qr, x]}, Simp[Coeff[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]
Time = 0.15 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.85
| method | result | size |
| default | \(\frac {\left (d +\frac {1}{2} b \,x^{2}+\frac {1}{3} c \,x^{3}\right )^{6}}{6}+d +\frac {b \,x^{2}}{2}+\frac {c \,x^{3}}{3}\) | \(33\) |
| norman | \(\left (\frac {1}{2} b \,d^{5}+\frac {1}{2} b \right ) x^{2}+\left (\frac {5}{324} b^{3} c^{3}+\frac {1}{243} c^{5} d \right ) x^{15}+\left (\frac {5}{12} b^{3} d^{3}+\frac {5}{18} c^{2} d^{4}\right ) x^{6}+\left (\frac {5}{288} b^{4} c^{2}+\frac {5}{162} b \,c^{4} d \right ) x^{14}+\left (\frac {5}{32} b^{4} d^{2}+\frac {5}{9} b \,c^{2} d^{3}\right ) x^{8}+\left (\frac {1}{96} b^{5} c +\frac {5}{54} d \,b^{2} c^{3}\right ) x^{13}+\left (\frac {1}{32} b^{5} d +\frac {5}{12} b^{2} c^{2} d^{2}\right ) x^{10}+\left (\frac {1}{3} c \,d^{5}+\frac {1}{3} c \right ) x^{3}+\left (\frac {5}{12} b^{3} c \,d^{2}+\frac {10}{81} c^{3} d^{3}\right ) x^{9}+\left (\frac {5}{48} d c \,b^{4}+\frac {5}{27} b \,c^{3} d^{2}\right ) x^{11}+\left (\frac {1}{384} b^{6}+\frac {5}{36} b^{3} c^{2} d +\frac {5}{162} c^{4} d^{2}\right ) x^{12}+\frac {c^{6} x^{18}}{4374}+\frac {b \,c^{5} x^{17}}{486}+\frac {5 b^{2} c^{4} x^{16}}{648}+\frac {5 b^{2} d^{4} x^{4}}{8}+\frac {5 b \,d^{4} c \,x^{5}}{6}+\frac {5 b^{2} c \,d^{3} x^{7}}{6}\) | \(285\) |
| gosper | \(\frac {x^{2} \left (64 c^{6} x^{16}+576 b \,c^{5} x^{15}+2160 b^{2} c^{4} x^{14}+4320 b^{3} c^{3} x^{13}+1152 c^{5} x^{13} d +4860 b^{4} c^{2} x^{12}+8640 b \,c^{4} d \,x^{12}+2916 b^{5} c \,x^{11}+25920 b^{2} c^{3} d \,x^{11}+729 b^{6} x^{10}+38880 b^{3} c^{2} d \,x^{10}+8640 c^{4} x^{10} d^{2}+29160 b^{4} c d \,x^{9}+51840 b \,c^{3} d^{2} x^{9}+8748 b^{5} d \,x^{8}+116640 b^{2} c^{2} d^{2} x^{8}+116640 b^{3} c \,d^{2} x^{7}+34560 c^{3} x^{7} d^{3}+43740 b^{4} d^{2} x^{6}+155520 b \,c^{2} d^{3} x^{6}+233280 b^{2} c \,d^{3} x^{5}+116640 b^{3} d^{3} x^{4}+77760 c^{2} x^{4} d^{4}+233280 b c \,d^{4} x^{3}+174960 b^{2} d^{4} x^{2}+93312 c x \,d^{5}+139968 b \,d^{5}+93312 c x +139968 b \right )}{279936}\) | \(294\) |
| risch | \(\frac {5}{18} x^{6} c^{2} d^{4}+\frac {1}{3} x^{3} c \,d^{5}+\frac {1}{32} x^{10} b^{5} d +\frac {1}{3} c \,x^{3}+\frac {5}{162} x^{14} b \,c^{4} d +\frac {5}{54} x^{13} d \,b^{2} c^{3}+\frac {1}{2} b \,x^{2}+\frac {5}{36} x^{12} b^{3} c^{2} d +\frac {5}{48} x^{11} d c \,b^{4}+\frac {5}{27} x^{11} b \,c^{3} d^{2}+\frac {5}{12} x^{10} b^{2} c^{2} d^{2}+\frac {5}{12} x^{9} b^{3} c \,d^{2}+\frac {5}{9} x^{8} b \,c^{2} d^{3}+\frac {5}{6} b^{2} c \,d^{3} x^{7}+\frac {5}{6} b \,d^{4} c \,x^{5}+\frac {5}{32} x^{8} b^{4} d^{2}+\frac {1}{4374} c^{6} x^{18}+\frac {5}{162} x^{12} c^{4} d^{2}+\frac {1}{384} b^{6} x^{12}+\frac {5}{648} b^{2} c^{4} x^{16}+\frac {1}{243} c^{5} d \,x^{15}+\frac {10}{81} x^{9} c^{3} d^{3}+\frac {1}{2} b \,d^{5} x^{2}+\frac {1}{486} b \,c^{5} x^{17}+\frac {5}{12} x^{6} b^{3} d^{3}+\frac {5}{8} b^{2} d^{4} x^{4}+\frac {5}{288} b^{4} c^{2} x^{14}+\frac {5}{324} b^{3} c^{3} x^{15}+\frac {1}{96} b^{5} c \,x^{13}\) | \(299\) |
| parallelrisch | \(\frac {5}{18} x^{6} c^{2} d^{4}+\frac {1}{3} x^{3} c \,d^{5}+\frac {1}{32} x^{10} b^{5} d +\frac {1}{3} c \,x^{3}+\frac {5}{162} x^{14} b \,c^{4} d +\frac {5}{54} x^{13} d \,b^{2} c^{3}+\frac {1}{2} b \,x^{2}+\frac {5}{36} x^{12} b^{3} c^{2} d +\frac {5}{48} x^{11} d c \,b^{4}+\frac {5}{27} x^{11} b \,c^{3} d^{2}+\frac {5}{12} x^{10} b^{2} c^{2} d^{2}+\frac {5}{12} x^{9} b^{3} c \,d^{2}+\frac {5}{9} x^{8} b \,c^{2} d^{3}+\frac {5}{6} b^{2} c \,d^{3} x^{7}+\frac {5}{6} b \,d^{4} c \,x^{5}+\frac {5}{32} x^{8} b^{4} d^{2}+\frac {1}{4374} c^{6} x^{18}+\frac {5}{162} x^{12} c^{4} d^{2}+\frac {1}{384} b^{6} x^{12}+\frac {5}{648} b^{2} c^{4} x^{16}+\frac {1}{243} c^{5} d \,x^{15}+\frac {10}{81} x^{9} c^{3} d^{3}+\frac {1}{2} b \,d^{5} x^{2}+\frac {1}{486} b \,c^{5} x^{17}+\frac {5}{12} x^{6} b^{3} d^{3}+\frac {5}{8} b^{2} d^{4} x^{4}+\frac {5}{288} b^{4} c^{2} x^{14}+\frac {5}{324} b^{3} c^{3} x^{15}+\frac {1}{96} b^{5} c \,x^{13}\) | \(299\) |
| orering | \(\frac {x \left (64 c^{6} x^{16}+576 b \,c^{5} x^{15}+2160 b^{2} c^{4} x^{14}+4320 b^{3} c^{3} x^{13}+1152 c^{5} x^{13} d +4860 b^{4} c^{2} x^{12}+8640 b \,c^{4} d \,x^{12}+2916 b^{5} c \,x^{11}+25920 b^{2} c^{3} d \,x^{11}+729 b^{6} x^{10}+38880 b^{3} c^{2} d \,x^{10}+8640 c^{4} x^{10} d^{2}+29160 b^{4} c d \,x^{9}+51840 b \,c^{3} d^{2} x^{9}+8748 b^{5} d \,x^{8}+116640 b^{2} c^{2} d^{2} x^{8}+116640 b^{3} c \,d^{2} x^{7}+34560 c^{3} x^{7} d^{3}+43740 b^{4} d^{2} x^{6}+155520 b \,c^{2} d^{3} x^{6}+233280 b^{2} c \,d^{3} x^{5}+116640 b^{3} d^{3} x^{4}+77760 c^{2} x^{4} d^{4}+233280 b c \,d^{4} x^{3}+174960 b^{2} d^{4} x^{2}+93312 c x \,d^{5}+139968 b \,d^{5}+93312 c x +139968 b \right ) \left (c \,x^{2}+b x \right ) \left (1+\left (d +\frac {1}{2} b \,x^{2}+\frac {1}{3} c \,x^{3}\right )^{5}\right )}{36 \left (c x +b \right ) \left (2 c \,x^{3}+3 b \,x^{2}+6 d +6\right ) \left (16 c^{4} x^{12}+96 b \,x^{11} c^{3}+216 b^{2} x^{10} c^{2}+216 b^{3} x^{9} c +192 d \,c^{3} x^{9}+81 b^{4} x^{8}+864 d b \,x^{8} c^{2}-48 c^{3} x^{9}+1296 d \,b^{2} x^{7} c -216 b \,x^{8} c^{2}+648 d \,b^{3} x^{6}-324 b^{2} x^{7} c +864 c^{2} d^{2} x^{6}-162 b^{3} x^{6}+2592 d^{2} b c \,x^{5}-432 c^{2} d \,x^{6}+1944 b^{2} d^{2} x^{4}-1296 d b c \,x^{5}+144 c^{2} x^{6}-972 b^{2} d \,x^{4}+432 b c \,x^{5}+1728 c \,d^{3} x^{3}+324 b^{2} x^{4}+2592 b \,d^{3} x^{2}-1296 c \,d^{2} x^{3}-1944 b \,d^{2} x^{2}+864 c d \,x^{3}+1296 b d \,x^{2}-432 c \,x^{3}+1296 d^{4}-648 b \,x^{2}-1296 d^{3}+1296 d^{2}-1296 d +1296\right )}\) | \(627\) |
Input:
int((c*x^2+b*x)*(1+(d+1/2*b*x^2+1/3*c*x^3)^5),x,method=_RETURNVERBOSE)
Output:
1/6*(d+1/2*b*x^2+1/3*c*x^3)^6+d+1/2*b*x^2+1/3*c*x^3
Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (33) = 66\).
Time = 0.07 (sec) , antiderivative size = 289, normalized size of antiderivative = 7.41 \[ \int \left (b x+c x^2\right ) \left (1+\left (d+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {1}{4374} \, c^{6} x^{18} + \frac {1}{486} \, b c^{5} x^{17} + \frac {5}{648} \, b^{2} c^{4} x^{16} + \frac {1}{972} \, {\left (15 \, b^{3} c^{3} + 4 \, c^{5} d\right )} x^{15} + \frac {5}{2592} \, {\left (9 \, b^{4} c^{2} + 16 \, b c^{4} d\right )} x^{14} + \frac {1}{864} \, {\left (9 \, b^{5} c + 80 \, b^{2} c^{3} d\right )} x^{13} + \frac {5}{6} \, b^{2} c d^{3} x^{7} + \frac {1}{10368} \, {\left (27 \, b^{6} + 1440 \, b^{3} c^{2} d + 320 \, c^{4} d^{2}\right )} x^{12} + \frac {5}{432} \, {\left (9 \, b^{4} c d + 16 \, b c^{3} d^{2}\right )} x^{11} + \frac {5}{6} \, b c d^{4} x^{5} + \frac {1}{96} \, {\left (3 \, b^{5} d + 40 \, b^{2} c^{2} d^{2}\right )} x^{10} + \frac {5}{8} \, b^{2} d^{4} x^{4} + \frac {5}{324} \, {\left (27 \, b^{3} c d^{2} + 8 \, c^{3} d^{3}\right )} x^{9} + \frac {5}{288} \, {\left (9 \, b^{4} d^{2} + 32 \, b c^{2} d^{3}\right )} x^{8} + \frac {5}{36} \, {\left (3 \, b^{3} d^{3} + 2 \, c^{2} d^{4}\right )} x^{6} + \frac {1}{3} \, {\left (c d^{5} + c\right )} x^{3} + \frac {1}{2} \, {\left (b d^{5} + b\right )} x^{2} \] Input:
integrate((c*x^2+b*x)*(1+(d+1/2*b*x^2+1/3*c*x^3)^5),x, algorithm="fricas")
Output:
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
Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (32) = 64\).
Time = 0.08 (sec) , antiderivative size = 321, normalized size of antiderivative = 8.23 \[ \int \left (b x+c x^2\right ) \left (1+\left (d+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {5 b^{2} c^{4} x^{16}}{648} + \frac {5 b^{2} c d^{3} x^{7}}{6} + \frac {5 b^{2} d^{4} x^{4}}{8} + \frac {b c^{5} x^{17}}{486} + \frac {5 b c d^{4} x^{5}}{6} + \frac {c^{6} x^{18}}{4374} + x^{15} \cdot \left (\frac {5 b^{3} c^{3}}{324} + \frac {c^{5} d}{243}\right ) + x^{14} \cdot \left (\frac {5 b^{4} c^{2}}{288} + \frac {5 b c^{4} d}{162}\right ) + x^{13} \left (\frac {b^{5} c}{96} + \frac {5 b^{2} c^{3} d}{54}\right ) + x^{12} \left (\frac {b^{6}}{384} + \frac {5 b^{3} c^{2} d}{36} + \frac {5 c^{4} d^{2}}{162}\right ) + x^{11} \cdot \left (\frac {5 b^{4} c d}{48} + \frac {5 b c^{3} d^{2}}{27}\right ) + x^{10} \left (\frac {b^{5} d}{32} + \frac {5 b^{2} c^{2} d^{2}}{12}\right ) + x^{9} \cdot \left (\frac {5 b^{3} c d^{2}}{12} + \frac {10 c^{3} d^{3}}{81}\right ) + x^{8} \cdot \left (\frac {5 b^{4} d^{2}}{32} + \frac {5 b c^{2} d^{3}}{9}\right ) + x^{6} \cdot \left (\frac {5 b^{3} d^{3}}{12} + \frac {5 c^{2} d^{4}}{18}\right ) + x^{3} \left (\frac {c d^{5}}{3} + \frac {c}{3}\right ) + x^{2} \left (\frac {b d^{5}}{2} + \frac {b}{2}\right ) \] Input:
integrate((c*x**2+b*x)*(1+(d+1/2*b*x**2+1/3*c*x**3)**5),x)
Output:
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/9 6 + 5*b**2*c**3*d/54) + x**12*(b**6/384 + 5*b**3*c**2*d/36 + 5*c**4*d**2/1 62) + 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)
Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (33) = 66\).
Time = 0.03 (sec) , antiderivative size = 289, normalized size of antiderivative = 7.41 \[ \int \left (b x+c x^2\right ) \left (1+\left (d+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {1}{4374} \, c^{6} x^{18} + \frac {1}{486} \, b c^{5} x^{17} + \frac {5}{648} \, b^{2} c^{4} x^{16} + \frac {1}{972} \, {\left (15 \, b^{3} c^{3} + 4 \, c^{5} d\right )} x^{15} + \frac {5}{2592} \, {\left (9 \, b^{4} c^{2} + 16 \, b c^{4} d\right )} x^{14} + \frac {1}{864} \, {\left (9 \, b^{5} c + 80 \, b^{2} c^{3} d\right )} x^{13} + \frac {5}{6} \, b^{2} c d^{3} x^{7} + \frac {1}{10368} \, {\left (27 \, b^{6} + 1440 \, b^{3} c^{2} d + 320 \, c^{4} d^{2}\right )} x^{12} + \frac {5}{432} \, {\left (9 \, b^{4} c d + 16 \, b c^{3} d^{2}\right )} x^{11} + \frac {5}{6} \, b c d^{4} x^{5} + \frac {1}{96} \, {\left (3 \, b^{5} d + 40 \, b^{2} c^{2} d^{2}\right )} x^{10} + \frac {5}{8} \, b^{2} d^{4} x^{4} + \frac {5}{324} \, {\left (27 \, b^{3} c d^{2} + 8 \, c^{3} d^{3}\right )} x^{9} + \frac {5}{288} \, {\left (9 \, b^{4} d^{2} + 32 \, b c^{2} d^{3}\right )} x^{8} + \frac {5}{36} \, {\left (3 \, b^{3} d^{3} + 2 \, c^{2} d^{4}\right )} x^{6} + \frac {1}{3} \, {\left (c d^{5} + c\right )} x^{3} + \frac {1}{2} \, {\left (b d^{5} + b\right )} x^{2} \] Input:
integrate((c*x^2+b*x)*(1+(d+1/2*b*x^2+1/3*c*x^3)^5),x, algorithm="maxima")
Output:
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
Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (33) = 66\).
Time = 0.13 (sec) , antiderivative size = 126, normalized size of antiderivative = 3.23 \[ \int \left (b x+c x^2\right ) \left (1+\left (d+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {1}{279936} \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )}^{6} + \frac {1}{7776} \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )}^{5} d + \frac {5}{2592} \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )}^{4} d^{2} + \frac {5}{324} \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )}^{3} d^{3} + \frac {5}{72} \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )}^{2} d^{4} + \frac {1}{6} \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )} d^{5} + \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} \] Input:
integrate((c*x^2+b*x)*(1+(d+1/2*b*x^2+1/3*c*x^3)^5),x, algorithm="giac")
Output:
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
Time = 22.37 (sec) , antiderivative size = 273, normalized size of antiderivative = 7.00 \[ \int \left (b x+c x^2\right ) \left (1+\left (d+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=x^{13}\,\left (\frac {b^5\,c}{96}+\frac {5\,d\,b^2\,c^3}{54}\right )+x^{14}\,\left (\frac {5\,b^4\,c^2}{288}+\frac {5\,d\,b\,c^4}{162}\right )+x^{12}\,\left (\frac {b^6}{384}+\frac {5\,b^3\,c^2\,d}{36}+\frac {5\,c^4\,d^2}{162}\right )+\frac {c^6\,x^{18}}{4374}+x^{15}\,\left (\frac {5\,b^3\,c^3}{324}+\frac {d\,c^5}{243}\right )+\frac {5\,d^3\,x^6\,\left (3\,b^3+2\,d\,c^2\right )}{36}+\frac {b\,c^5\,x^{17}}{486}+\frac {5\,b^2\,c^4\,x^{16}}{648}+\frac {b\,x^2\,\left (d^5+1\right )}{2}+\frac {5\,b^2\,d^4\,x^4}{8}+\frac {c\,x^3\,\left (d^5+1\right )}{3}+\frac {5\,b^2\,c\,d^3\,x^7}{6}+\frac {5\,b\,d^2\,x^8\,\left (9\,b^3+32\,d\,c^2\right )}{288}+\frac {b^2\,d\,x^{10}\,\left (3\,b^3+40\,d\,c^2\right )}{96}+\frac {5\,c\,d^2\,x^9\,\left (27\,b^3+8\,d\,c^2\right )}{324}+\frac {5\,b\,c\,d^4\,x^5}{6}+\frac {5\,b\,c\,d\,x^{11}\,\left (9\,b^3+16\,d\,c^2\right )}{432} \] Input:
int((b*x + c*x^2)*((d + (b*x^2)/2 + (c*x^3)/3)^5 + 1),x)
Output:
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)/4 374 + 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
Time = 0.15 (sec) , antiderivative size = 293, normalized size of antiderivative = 7.51 \[ \int \left (b x+c x^2\right ) \left (1+\left (d+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {x^{2} \left (64 c^{6} x^{16}+576 b \,c^{5} x^{15}+2160 b^{2} c^{4} x^{14}+4320 b^{3} c^{3} x^{13}+1152 c^{5} d \,x^{13}+4860 b^{4} c^{2} x^{12}+8640 b \,c^{4} d \,x^{12}+2916 b^{5} c \,x^{11}+25920 b^{2} c^{3} d \,x^{11}+729 b^{6} x^{10}+38880 b^{3} c^{2} d \,x^{10}+8640 c^{4} d^{2} x^{10}+29160 b^{4} c d \,x^{9}+51840 b \,c^{3} d^{2} x^{9}+8748 b^{5} d \,x^{8}+116640 b^{2} c^{2} d^{2} x^{8}+116640 b^{3} c \,d^{2} x^{7}+34560 c^{3} d^{3} x^{7}+43740 b^{4} d^{2} x^{6}+155520 b \,c^{2} d^{3} x^{6}+233280 b^{2} c \,d^{3} x^{5}+116640 b^{3} d^{3} x^{4}+77760 c^{2} d^{4} x^{4}+233280 b c \,d^{4} x^{3}+174960 b^{2} d^{4} x^{2}+93312 c \,d^{5} x +139968 b \,d^{5}+93312 c x +139968 b \right )}{279936} \] Input:
int((c*x^2+b*x)*(1+(d+1/2*b*x^2+1/3*c*x^3)^5),x)
Output:
(x**2*(729*b**6*x**10 + 2916*b**5*c*x**11 + 8748*b**5*d*x**8 + 4860*b**4*c **2*x**12 + 29160*b**4*c*d*x**9 + 43740*b**4*d**2*x**6 + 4320*b**3*c**3*x* *13 + 38880*b**3*c**2*d*x**10 + 116640*b**3*c*d**2*x**7 + 116640*b**3*d**3 *x**4 + 2160*b**2*c**4*x**14 + 25920*b**2*c**3*d*x**11 + 116640*b**2*c**2* d**2*x**8 + 233280*b**2*c*d**3*x**5 + 174960*b**2*d**4*x**2 + 576*b*c**5*x **15 + 8640*b*c**4*d*x**12 + 51840*b*c**3*d**2*x**9 + 155520*b*c**2*d**3*x **6 + 233280*b*c*d**4*x**3 + 139968*b*d**5 + 139968*b + 64*c**6*x**16 + 11 52*c**5*d*x**13 + 8640*c**4*d**2*x**10 + 34560*c**3*d**3*x**7 + 77760*c**2 *d**4*x**4 + 93312*c*d**5*x + 93312*c*x))/279936