3.77 \(\int \left (\frac{-16+b^2}{4 c}+b x+c x^2\right )^5 \, dx\)

Optimal. Leaf size=109 \[ -\frac{(-b-2 c x+4)^{11}}{22528 c^6}+\frac{(-b-2 c x+4)^{10}}{512 c^6}-\frac{5 (-b-2 c x+4)^9}{144 c^6}+\frac{5 (-b-2 c x+4)^8}{16 c^6}-\frac{10 (-b-2 c x+4)^7}{7 c^6}+\frac{8 (-b-2 c x+4)^6}{3 c^6} \]

[Out]

(8*(4 - b - 2*c*x)^6)/(3*c^6) - (10*(4 - b - 2*c*x)^7)/(7*c^6) + (5*(4 - b - 2*c
*x)^8)/(16*c^6) - (5*(4 - b - 2*c*x)^9)/(144*c^6) + (4 - b - 2*c*x)^10/(512*c^6)
- (4 - b - 2*c*x)^11/(22528*c^6)

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Rubi [A]  time = 0.29194, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ -\frac{(-b-2 c x+4)^{11}}{22528 c^6}+\frac{(-b-2 c x+4)^{10}}{512 c^6}-\frac{5 (-b-2 c x+4)^9}{144 c^6}+\frac{5 (-b-2 c x+4)^8}{16 c^6}-\frac{10 (-b-2 c x+4)^7}{7 c^6}+\frac{8 (-b-2 c x+4)^6}{3 c^6} \]

Antiderivative was successfully verified.

[In]  Int[((-16 + b^2)/(4*c) + b*x + c*x^2)^5,x]

[Out]

(8*(4 - b - 2*c*x)^6)/(3*c^6) - (10*(4 - b - 2*c*x)^7)/(7*c^6) + (5*(4 - b - 2*c
*x)^8)/(16*c^6) - (5*(4 - b - 2*c*x)^9)/(144*c^6) + (4 - b - 2*c*x)^10/(512*c^6)
- (4 - b - 2*c*x)^11/(22528*c^6)

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Rubi in Sympy [A]  time = 44.2048, size = 97, normalized size = 0.89 \[ - \frac{\left (- b - 2 c x + 4\right )^{11}}{22528 c^{6}} + \frac{\left (- b - 2 c x + 4\right )^{10}}{512 c^{6}} - \frac{5 \left (- b - 2 c x + 4\right )^{9}}{144 c^{6}} + \frac{5 \left (- b - 2 c x + 4\right )^{8}}{16 c^{6}} - \frac{10 \left (- b - 2 c x + 4\right )^{7}}{7 c^{6}} + \frac{8 \left (- b - 2 c x + 4\right )^{6}}{3 c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1/4*(b**2-16)/c+b*x+c*x**2)**5,x)

[Out]

-(-b - 2*c*x + 4)**11/(22528*c**6) + (-b - 2*c*x + 4)**10/(512*c**6) - 5*(-b - 2
*c*x + 4)**9/(144*c**6) + 5*(-b - 2*c*x + 4)**8/(16*c**6) - 10*(-b - 2*c*x + 4)*
*7/(7*c**6) + 8*(-b - 2*c*x + 4)**6/(3*c**6)

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Mathematica [A]  time = 0.0776292, size = 207, normalized size = 1.9 \[ \frac{5}{8} \left (3 b^3-16 b\right ) c^2 x^8+\frac{\left (b^2-16\right )^5 x}{1024 c^5}+\frac{5 b \left (b^2-16\right )^4 x^2}{512 c^4}+\frac{5}{36} \left (9 b^2-16\right ) c^3 x^9+\frac{5 \left (b^2-16\right )^3 \left (9 b^2-16\right ) x^3}{768 c^3}+\frac{5 b \left (b^2-16\right )^2 \left (3 b^2-16\right ) x^4}{64 c^2}+\frac{5}{56} \left (21 b^4-224 b^2+256\right ) c x^7+\frac{\left (b^2-16\right ) \left (21 b^4-224 b^2+256\right ) x^5}{32 c}+\frac{1}{48} b \left (63 b^4-1120 b^2+3840\right ) x^6+\frac{1}{2} b c^4 x^{10}+\frac{c^5 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]  Integrate[((-16 + b^2)/(4*c) + b*x + c*x^2)^5,x]

[Out]

((-16 + b^2)^5*x)/(1024*c^5) + (5*b*(-16 + b^2)^4*x^2)/(512*c^4) + (5*(-16 + b^2
)^3*(-16 + 9*b^2)*x^3)/(768*c^3) + (5*b*(-16 + b^2)^2*(-16 + 3*b^2)*x^4)/(64*c^2
) + ((-16 + b^2)*(256 - 224*b^2 + 21*b^4)*x^5)/(32*c) + (b*(3840 - 1120*b^2 + 63
*b^4)*x^6)/48 + (5*(256 - 224*b^2 + 21*b^4)*c*x^7)/56 + (5*(-16*b + 3*b^3)*c^2*x
^8)/8 + (5*(-16 + 9*b^2)*c^3*x^9)/36 + (b*c^4*x^10)/2 + (c^5*x^11)/11

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Maple [B]  time = 0.006, size = 636, normalized size = 5.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1/4*(b^2-16)/c+b*x+c*x^2)^5,x)

[Out]

1/11*c^5*x^11+1/2*b*c^4*x^10+1/9*(1/4*(b^2-16)*c^3+4*b^2*c^3+c*(2*(3/2*b^2-8)*c^
2+4*b^2*c^2))*x^9+1/8*((b^2-16)*c^2*b+b*(2*(3/2*b^2-8)*c^2+4*b^2*c^2)+c*((b^2-16
)*c*b+4*(3/2*b^2-8)*b*c))*x^8+1/7*(1/4*(b^2-16)/c*(2*(3/2*b^2-8)*c^2+4*b^2*c^2)+
b*((b^2-16)*c*b+4*(3/2*b^2-8)*b*c)+c*(1/8*(b^2-16)^2+2*(b^2-16)*b^2+(3/2*b^2-8)^
2))*x^7+1/6*(1/4*(b^2-16)/c*((b^2-16)*c*b+4*(3/2*b^2-8)*b*c)+b*(1/8*(b^2-16)^2+2
*(b^2-16)*b^2+(3/2*b^2-8)^2)+c*(1/4*(b^2-16)^2/c*b+(b^2-16)/c*b*(3/2*b^2-8)))*x^
6+1/5*(1/4*(b^2-16)/c*(1/8*(b^2-16)^2+2*(b^2-16)*b^2+(3/2*b^2-8)^2)+b*(1/4*(b^2-
16)^2/c*b+(b^2-16)/c*b*(3/2*b^2-8))+c*(1/8*(b^2-16)^2/c^2*(3/2*b^2-8)+1/4*(b^2-1
6)^2/c^2*b^2))*x^5+1/4*(1/4*(b^2-16)/c*(1/4*(b^2-16)^2/c*b+(b^2-16)/c*b*(3/2*b^2
-8))+b*(1/8*(b^2-16)^2/c^2*(3/2*b^2-8)+1/4*(b^2-16)^2/c^2*b^2)+1/16/c^2*(b^2-16)
^3*b)*x^4+1/3*(1/4*(b^2-16)/c*(1/8*(b^2-16)^2/c^2*(3/2*b^2-8)+1/4*(b^2-16)^2/c^2
*b^2)+1/16*b^2*(b^2-16)^3/c^3+1/256/c^3*(b^2-16)^4)*x^3+5/512*(b^2-16)^4/c^4*b*x
^2+1/1024*(b^2-16)^5/c^5*x

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Maxima [A]  time = 0.726255, size = 316, normalized size = 2.9 \[ \frac{1}{11} \, c^{5} x^{11} + \frac{1}{2} \, b c^{4} x^{10} + \frac{10}{9} \, b^{2} c^{3} x^{9} + \frac{5}{4} \, b^{3} c^{2} x^{8} + \frac{5}{7} \, b^{4} c x^{7} + \frac{1}{6} \, b^{5} x^{6} + \frac{5 \,{\left (2 \, c x^{3} + 3 \, b x^{2}\right )}{\left (b^{2} - 16\right )}^{4}}{1536 \, c^{4}} + \frac{{\left (6 \, c^{2} x^{5} + 15 \, b c x^{4} + 10 \, b^{2} x^{3}\right )}{\left (b^{2} - 16\right )}^{3}}{192 \, c^{3}} + \frac{{\left (20 \, c^{3} x^{7} + 70 \, b c^{2} x^{6} + 84 \, b^{2} c x^{5} + 35 \, b^{3} x^{4}\right )}{\left (b^{2} - 16\right )}^{2}}{224 \, c^{2}} + \frac{{\left (70 \, c^{4} x^{9} + 315 \, b c^{3} x^{8} + 540 \, b^{2} c^{2} x^{7} + 420 \, b^{3} c x^{6} + 126 \, b^{4} x^{5}\right )}{\left (b^{2} - 16\right )}}{504 \, c} + \frac{{\left (b^{2} - 16\right )}^{5} x}{1024 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/1024*(4*c*x^2 + 4*b*x + (b^2 - 16)/c)^5,x, algorithm="maxima")

[Out]

1/11*c^5*x^11 + 1/2*b*c^4*x^10 + 10/9*b^2*c^3*x^9 + 5/4*b^3*c^2*x^8 + 5/7*b^4*c*
x^7 + 1/6*b^5*x^6 + 5/1536*(2*c*x^3 + 3*b*x^2)*(b^2 - 16)^4/c^4 + 1/192*(6*c^2*x
^5 + 15*b*c*x^4 + 10*b^2*x^3)*(b^2 - 16)^3/c^3 + 1/224*(20*c^3*x^7 + 70*b*c^2*x^
6 + 84*b^2*c*x^5 + 35*b^3*x^4)*(b^2 - 16)^2/c^2 + 1/504*(70*c^4*x^9 + 315*b*c^3*
x^8 + 540*b^2*c^2*x^7 + 420*b^3*c*x^6 + 126*b^4*x^5)*(b^2 - 16)/c + 1/1024*(b^2
- 16)^5*x/c^5

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Fricas [A]  time = 0.217172, size = 317, normalized size = 2.91 \[ \frac{64512 \, c^{10} x^{11} + 354816 \, b c^{9} x^{10} + 98560 \,{\left (9 \, b^{2} - 16\right )} c^{8} x^{9} + 443520 \,{\left (3 \, b^{3} - 16 \, b\right )} c^{7} x^{8} + 63360 \,{\left (21 \, b^{4} - 224 \, b^{2} + 256\right )} c^{6} x^{7} + 14784 \,{\left (63 \, b^{5} - 1120 \, b^{3} + 3840 \, b\right )} c^{5} x^{6} + 22176 \,{\left (21 \, b^{6} - 560 \, b^{4} + 3840 \, b^{2} - 4096\right )} c^{4} x^{5} + 55440 \,{\left (3 \, b^{7} - 112 \, b^{5} + 1280 \, b^{3} - 4096 \, b\right )} c^{3} x^{4} + 4620 \,{\left (9 \, b^{8} - 448 \, b^{6} + 7680 \, b^{4} - 49152 \, b^{2} + 65536\right )} c^{2} x^{3} + 6930 \,{\left (b^{9} - 64 \, b^{7} + 1536 \, b^{5} - 16384 \, b^{3} + 65536 \, b\right )} c x^{2} + 693 \,{\left (b^{10} - 80 \, b^{8} + 2560 \, b^{6} - 40960 \, b^{4} + 327680 \, b^{2} - 1048576\right )} x}{709632 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/1024*(4*c*x^2 + 4*b*x + (b^2 - 16)/c)^5,x, algorithm="fricas")

[Out]

1/709632*(64512*c^10*x^11 + 354816*b*c^9*x^10 + 98560*(9*b^2 - 16)*c^8*x^9 + 443
520*(3*b^3 - 16*b)*c^7*x^8 + 63360*(21*b^4 - 224*b^2 + 256)*c^6*x^7 + 14784*(63*
b^5 - 1120*b^3 + 3840*b)*c^5*x^6 + 22176*(21*b^6 - 560*b^4 + 3840*b^2 - 4096)*c^
4*x^5 + 55440*(3*b^7 - 112*b^5 + 1280*b^3 - 4096*b)*c^3*x^4 + 4620*(9*b^8 - 448*
b^6 + 7680*b^4 - 49152*b^2 + 65536)*c^2*x^3 + 6930*(b^9 - 64*b^7 + 1536*b^5 - 16
384*b^3 + 65536*b)*c*x^2 + 693*(b^10 - 80*b^8 + 2560*b^6 - 40960*b^4 + 327680*b^
2 - 1048576)*x)/c^5

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Sympy [A]  time = 0.411293, size = 248, normalized size = 2.28 \[ \frac{b c^{4} x^{10}}{2} + \frac{c^{5} x^{11}}{11} + x^{9} \left (\frac{5 b^{2} c^{3}}{4} - \frac{20 c^{3}}{9}\right ) + x^{8} \left (\frac{15 b^{3} c^{2}}{8} - 10 b c^{2}\right ) + x^{7} \left (\frac{15 b^{4} c}{8} - 20 b^{2} c + \frac{160 c}{7}\right ) + x^{6} \left (\frac{21 b^{5}}{16} - \frac{70 b^{3}}{3} + 80 b\right ) + \frac{x^{5} \left (21 b^{6} - 560 b^{4} + 3840 b^{2} - 4096\right )}{32 c} + \frac{x^{4} \left (15 b^{7} - 560 b^{5} + 6400 b^{3} - 20480 b\right )}{64 c^{2}} + \frac{x^{3} \left (45 b^{8} - 2240 b^{6} + 38400 b^{4} - 245760 b^{2} + 327680\right )}{768 c^{3}} + \frac{x^{2} \left (5 b^{9} - 320 b^{7} + 7680 b^{5} - 81920 b^{3} + 327680 b\right )}{512 c^{4}} + \frac{x \left (b^{10} - 80 b^{8} + 2560 b^{6} - 40960 b^{4} + 327680 b^{2} - 1048576\right )}{1024 c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1/4*(b**2-16)/c+b*x+c*x**2)**5,x)

[Out]

b*c**4*x**10/2 + c**5*x**11/11 + x**9*(5*b**2*c**3/4 - 20*c**3/9) + x**8*(15*b**
3*c**2/8 - 10*b*c**2) + x**7*(15*b**4*c/8 - 20*b**2*c + 160*c/7) + x**6*(21*b**5
/16 - 70*b**3/3 + 80*b) + x**5*(21*b**6 - 560*b**4 + 3840*b**2 - 4096)/(32*c) +
x**4*(15*b**7 - 560*b**5 + 6400*b**3 - 20480*b)/(64*c**2) + x**3*(45*b**8 - 2240
*b**6 + 38400*b**4 - 245760*b**2 + 327680)/(768*c**3) + x**2*(5*b**9 - 320*b**7
+ 7680*b**5 - 81920*b**3 + 327680*b)/(512*c**4) + x*(b**10 - 80*b**8 + 2560*b**6
- 40960*b**4 + 327680*b**2 - 1048576)/(1024*c**5)

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GIAC/XCAS [A]  time = 0.208897, size = 489, normalized size = 4.49 \[ \frac{64512 \, c^{60} x^{11} + 354816 \, b c^{59} x^{10} + 887040 \, b^{2} c^{58} x^{9} + 1330560 \, b^{3} c^{57} x^{8} + 1330560 \, b^{4} c^{56} x^{7} - 1576960 \, c^{58} x^{9} + 931392 \, b^{5} c^{55} x^{6} - 7096320 \, b c^{57} x^{8} + 465696 \, b^{6} c^{54} x^{5} - 14192640 \, b^{2} c^{56} x^{7} + 166320 \, b^{7} c^{53} x^{4} - 16558080 \, b^{3} c^{55} x^{6} + 41580 \, b^{8} c^{52} x^{3} - 12418560 \, b^{4} c^{54} x^{5} + 16220160 \, c^{56} x^{7} + 6930 \, b^{9} c^{51} x^{2} - 6209280 \, b^{5} c^{53} x^{4} + 56770560 \, b c^{55} x^{6} + 693 \, b^{10} c^{50} x - 2069760 \, b^{6} c^{52} x^{3} + 85155840 \, b^{2} c^{54} x^{5} - 443520 \, b^{7} c^{51} x^{2} + 70963200 \, b^{3} c^{53} x^{4} - 55440 \, b^{8} c^{50} x + 35481600 \, b^{4} c^{52} x^{3} - 90832896 \, c^{54} x^{5} + 10644480 \, b^{5} c^{51} x^{2} - 227082240 \, b c^{53} x^{4} + 1774080 \, b^{6} c^{50} x - 227082240 \, b^{2} c^{52} x^{3} - 113541120 \, b^{3} c^{51} x^{2} - 28385280 \, b^{4} c^{50} x + 302776320 \, c^{52} x^{3} + 454164480 \, b c^{51} x^{2} + 227082240 \, b^{2} c^{50} x - 726663168 \, c^{50} x}{709632 \, c^{55}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/1024*(4*c*x^2 + 4*b*x + (b^2 - 16)/c)^5,x, algorithm="giac")

[Out]

1/709632*(64512*c^60*x^11 + 354816*b*c^59*x^10 + 887040*b^2*c^58*x^9 + 1330560*b
^3*c^57*x^8 + 1330560*b^4*c^56*x^7 - 1576960*c^58*x^9 + 931392*b^5*c^55*x^6 - 70
96320*b*c^57*x^8 + 465696*b^6*c^54*x^5 - 14192640*b^2*c^56*x^7 + 166320*b^7*c^53
*x^4 - 16558080*b^3*c^55*x^6 + 41580*b^8*c^52*x^3 - 12418560*b^4*c^54*x^5 + 1622
0160*c^56*x^7 + 6930*b^9*c^51*x^2 - 6209280*b^5*c^53*x^4 + 56770560*b*c^55*x^6 +
693*b^10*c^50*x - 2069760*b^6*c^52*x^3 + 85155840*b^2*c^54*x^5 - 443520*b^7*c^5
1*x^2 + 70963200*b^3*c^53*x^4 - 55440*b^8*c^50*x + 35481600*b^4*c^52*x^3 - 90832
896*c^54*x^5 + 10644480*b^5*c^51*x^2 - 227082240*b*c^53*x^4 + 1774080*b^6*c^50*x
- 227082240*b^2*c^52*x^3 - 113541120*b^3*c^51*x^2 - 28385280*b^4*c^50*x + 30277
6320*c^52*x^3 + 454164480*b*c^51*x^2 + 227082240*b^2*c^50*x - 726663168*c^50*x)/
c^55