3.74 \(\int (\frac{-1+b^2}{4 c}+b x+c x^2)^5 \, dx\)

Optimal. Leaf size=109 \[ -\frac{(-b-2 c x+1)^{11}}{22528 c^6}+\frac{(-b-2 c x+1)^{10}}{2048 c^6}-\frac{5 (-b-2 c x+1)^9}{2304 c^6}+\frac{5 (-b-2 c x+1)^8}{1024 c^6}-\frac{5 (-b-2 c x+1)^7}{896 c^6}+\frac{(-b-2 c x+1)^6}{384 c^6} \]

[Out]

(1 - b - 2*c*x)^6/(384*c^6) - (5*(1 - b - 2*c*x)^7)/(896*c^6) + (5*(1 - b - 2*c*x)^8)/(1024*c^6) - (5*(1 - b -
2*c*x)^9)/(2304*c^6) + (1 - b - 2*c*x)^10/(2048*c^6) - (1 - b - 2*c*x)^11/(22528*c^6)

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Rubi [A]  time = 0.139885, 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, Rules used = {610, 43} \[ -\frac{(-b-2 c x+1)^{11}}{22528 c^6}+\frac{(-b-2 c x+1)^{10}}{2048 c^6}-\frac{5 (-b-2 c x+1)^9}{2304 c^6}+\frac{5 (-b-2 c x+1)^8}{1024 c^6}-\frac{5 (-b-2 c x+1)^7}{896 c^6}+\frac{(-b-2 c x+1)^6}{384 c^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 - b - 2*c*x)^6/(384*c^6) - (5*(1 - b - 2*c*x)^7)/(896*c^6) + (5*(1 - b - 2*c*x)^8)/(1024*c^6) - (5*(1 - b -
2*c*x)^9)/(2304*c^6) + (1 - b - 2*c*x)^10/(2048*c^6) - (1 - b - 2*c*x)^11/(22528*c^6)

Rule 610

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[1/c^p, Int[Simp[
b/2 - q/2 + c*x, x]^p*Simp[b/2 + q/2 + c*x, x]^p, x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && IGt
Q[p, 0] && PerfectSquareQ[b^2 - 4*a*c]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \left (\frac{-1+b^2}{4 c}+b x+c x^2\right )^5 \, dx &=\frac{\int \left (\frac{1}{2} (-1+b)+c x\right )^5 \left (\frac{1+b}{2}+c x\right )^5 \, dx}{c^5}\\ &=\frac{\int \left (\left (\frac{1}{2} (-1+b)+c x\right )^5+5 \left (\frac{1}{2} (-1+b)+c x\right )^6+10 \left (\frac{1}{2} (-1+b)+c x\right )^7+10 \left (\frac{1}{2} (-1+b)+c x\right )^8+5 \left (\frac{1}{2} (-1+b)+c x\right )^9+\left (\frac{1}{2} (-1+b)+c x\right )^{10}\right ) \, dx}{c^5}\\ &=\frac{(1-b-2 c x)^6}{384 c^6}-\frac{5 (1-b-2 c x)^7}{896 c^6}+\frac{5 (1-b-2 c x)^8}{1024 c^6}-\frac{5 (1-b-2 c x)^9}{2304 c^6}+\frac{(1-b-2 c x)^{10}}{2048 c^6}-\frac{(1-b-2 c x)^{11}}{22528 c^6}\\ \end{align*}

Mathematica [A]  time = 0.0338732, size = 207, normalized size = 1.9 \[ \frac{5}{36} \left (9 b^2-1\right ) c^3 x^9+\frac{5}{8} \left (3 b^3-b\right ) c^2 x^8+\frac{5 b \left (b^2-1\right )^2 \left (3 b^2-1\right ) x^4}{64 c^2}+\frac{5 \left (b^2-1\right )^3 \left (9 b^2-1\right ) x^3}{768 c^3}+\frac{5 b \left (b^2-1\right )^4 x^2}{512 c^4}+\frac{\left (b^2-1\right )^5 x}{1024 c^5}+\frac{5}{56} \left (21 b^4-14 b^2+1\right ) c x^7+\frac{\left (b^2-1\right ) \left (21 b^4-14 b^2+1\right ) x^5}{32 c}+\frac{1}{48} b \left (63 b^4-70 b^2+15\right ) x^6+\frac{1}{2} b c^4 x^{10}+\frac{c^5 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-1 + b^2)^5*x)/(1024*c^5) + (5*b*(-1 + b^2)^4*x^2)/(512*c^4) + (5*(-1 + b^2)^3*(-1 + 9*b^2)*x^3)/(768*c^3) +
(5*b*(-1 + b^2)^2*(-1 + 3*b^2)*x^4)/(64*c^2) + ((-1 + b^2)*(1 - 14*b^2 + 21*b^4)*x^5)/(32*c) + (b*(15 - 70*b^
2 + 63*b^4)*x^6)/48 + (5*(1 - 14*b^2 + 21*b^4)*c*x^7)/56 + (5*(-b + 3*b^3)*c^2*x^8)/8 + (5*(-1 + 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 = 0.079, size = 636, normalized size = 5.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [B]  time = 1.12539, size = 316, normalized size = 2.9 \begin{align*} \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} - 1\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} - 1\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} - 1\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} - 1\right )}}{504 \, c} + \frac{{\left (b^{2} - 1\right )}^{5} x}{1024 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/4*(b^2-1)/c+b*x+c*x^2)^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 - 1)^4/c^4 + 1/192*(6*c^2*x^5 + 15*b*c*x^4 + 10*b^2*x^3)*(b^2 - 1)^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 - 1)^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 - 1)/c + 1/1024*(b^2 - 1)^5*x/c^5

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Fricas [B]  time = 2.10253, size = 585, normalized size = 5.37 \begin{align*} \frac{64512 \, c^{10} x^{11} + 354816 \, b c^{9} x^{10} + 98560 \,{\left (9 \, b^{2} - 1\right )} c^{8} x^{9} + 443520 \,{\left (3 \, b^{3} - b\right )} c^{7} x^{8} + 63360 \,{\left (21 \, b^{4} - 14 \, b^{2} + 1\right )} c^{6} x^{7} + 14784 \,{\left (63 \, b^{5} - 70 \, b^{3} + 15 \, b\right )} c^{5} x^{6} + 22176 \,{\left (21 \, b^{6} - 35 \, b^{4} + 15 \, b^{2} - 1\right )} c^{4} x^{5} + 55440 \,{\left (3 \, b^{7} - 7 \, b^{5} + 5 \, b^{3} - b\right )} c^{3} x^{4} + 4620 \,{\left (9 \, b^{8} - 28 \, b^{6} + 30 \, b^{4} - 12 \, b^{2} + 1\right )} c^{2} x^{3} + 6930 \,{\left (b^{9} - 4 \, b^{7} + 6 \, b^{5} - 4 \, b^{3} + b\right )} c x^{2} + 693 \,{\left (b^{10} - 5 \, b^{8} + 10 \, b^{6} - 10 \, b^{4} + 5 \, b^{2} - 1\right )} x}{709632 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/709632*(64512*c^10*x^11 + 354816*b*c^9*x^10 + 98560*(9*b^2 - 1)*c^8*x^9 + 443520*(3*b^3 - b)*c^7*x^8 + 63360
*(21*b^4 - 14*b^2 + 1)*c^6*x^7 + 14784*(63*b^5 - 70*b^3 + 15*b)*c^5*x^6 + 22176*(21*b^6 - 35*b^4 + 15*b^2 - 1)
*c^4*x^5 + 55440*(3*b^7 - 7*b^5 + 5*b^3 - b)*c^3*x^4 + 4620*(9*b^8 - 28*b^6 + 30*b^4 - 12*b^2 + 1)*c^2*x^3 + 6
930*(b^9 - 4*b^7 + 6*b^5 - 4*b^3 + b)*c*x^2 + 693*(b^10 - 5*b^8 + 10*b^6 - 10*b^4 + 5*b^2 - 1)*x)/c^5

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Sympy [B]  time = 0.209606, size = 253, normalized size = 2.32 \begin{align*} \frac{b c^{4} x^{10}}{2} + \frac{c^{5} x^{11}}{11} + x^{9} \left (\frac{5 b^{2} c^{3}}{4} - \frac{5 c^{3}}{36}\right ) + x^{8} \left (\frac{15 b^{3} c^{2}}{8} - \frac{5 b c^{2}}{8}\right ) + x^{7} \left (\frac{15 b^{4} c}{8} - \frac{5 b^{2} c}{4} + \frac{5 c}{56}\right ) + x^{6} \left (\frac{21 b^{5}}{16} - \frac{35 b^{3}}{24} + \frac{5 b}{16}\right ) + \frac{x^{5} \left (21 b^{6} - 35 b^{4} + 15 b^{2} - 1\right )}{32 c} + \frac{x^{4} \left (15 b^{7} - 35 b^{5} + 25 b^{3} - 5 b\right )}{64 c^{2}} + \frac{x^{3} \left (45 b^{8} - 140 b^{6} + 150 b^{4} - 60 b^{2} + 5\right )}{768 c^{3}} + \frac{x^{2} \left (5 b^{9} - 20 b^{7} + 30 b^{5} - 20 b^{3} + 5 b\right )}{512 c^{4}} + \frac{x \left (b^{10} - 5 b^{8} + 10 b^{6} - 10 b^{4} + 5 b^{2} - 1\right )}{1024 c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/4*(b**2-1)/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 - 5*c**3/36) + x**8*(15*b**3*c**2/8 - 5*b*c**2/8) + x**7*
(15*b**4*c/8 - 5*b**2*c/4 + 5*c/56) + x**6*(21*b**5/16 - 35*b**3/24 + 5*b/16) + x**5*(21*b**6 - 35*b**4 + 15*b
**2 - 1)/(32*c) + x**4*(15*b**7 - 35*b**5 + 25*b**3 - 5*b)/(64*c**2) + x**3*(45*b**8 - 140*b**6 + 150*b**4 - 6
0*b**2 + 5)/(768*c**3) + x**2*(5*b**9 - 20*b**7 + 30*b**5 - 20*b**3 + 5*b)/(512*c**4) + x*(b**10 - 5*b**8 + 10
*b**6 - 10*b**4 + 5*b**2 - 1)/(1024*c**5)

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Giac [B]  time = 1.33613, size = 451, normalized size = 4.14 \begin{align*} \frac{64512 \, c^{10} x^{11} + 354816 \, b c^{9} x^{10} + 887040 \, b^{2} c^{8} x^{9} + 1330560 \, b^{3} c^{7} x^{8} + 1330560 \, b^{4} c^{6} x^{7} - 98560 \, c^{8} x^{9} + 931392 \, b^{5} c^{5} x^{6} - 443520 \, b c^{7} x^{8} + 465696 \, b^{6} c^{4} x^{5} - 887040 \, b^{2} c^{6} x^{7} + 166320 \, b^{7} c^{3} x^{4} - 1034880 \, b^{3} c^{5} x^{6} + 41580 \, b^{8} c^{2} x^{3} - 776160 \, b^{4} c^{4} x^{5} + 63360 \, c^{6} x^{7} + 6930 \, b^{9} c x^{2} - 388080 \, b^{5} c^{3} x^{4} + 221760 \, b c^{5} x^{6} + 693 \, b^{10} x - 129360 \, b^{6} c^{2} x^{3} + 332640 \, b^{2} c^{4} x^{5} - 27720 \, b^{7} c x^{2} + 277200 \, b^{3} c^{3} x^{4} - 3465 \, b^{8} x + 138600 \, b^{4} c^{2} x^{3} - 22176 \, c^{4} x^{5} + 41580 \, b^{5} c x^{2} - 55440 \, b c^{3} x^{4} + 6930 \, b^{6} x - 55440 \, b^{2} c^{2} x^{3} - 27720 \, b^{3} c x^{2} - 6930 \, b^{4} x + 4620 \, c^{2} x^{3} + 6930 \, b c x^{2} + 3465 \, b^{2} x - 693 \, x}{709632 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/709632*(64512*c^10*x^11 + 354816*b*c^9*x^10 + 887040*b^2*c^8*x^9 + 1330560*b^3*c^7*x^8 + 1330560*b^4*c^6*x^7
- 98560*c^8*x^9 + 931392*b^5*c^5*x^6 - 443520*b*c^7*x^8 + 465696*b^6*c^4*x^5 - 887040*b^2*c^6*x^7 + 166320*b^
7*c^3*x^4 - 1034880*b^3*c^5*x^6 + 41580*b^8*c^2*x^3 - 776160*b^4*c^4*x^5 + 63360*c^6*x^7 + 6930*b^9*c*x^2 - 38
8080*b^5*c^3*x^4 + 221760*b*c^5*x^6 + 693*b^10*x - 129360*b^6*c^2*x^3 + 332640*b^2*c^4*x^5 - 27720*b^7*c*x^2 +
277200*b^3*c^3*x^4 - 3465*b^8*x + 138600*b^4*c^2*x^3 - 22176*c^4*x^5 + 41580*b^5*c*x^2 - 55440*b*c^3*x^4 + 69
30*b^6*x - 55440*b^2*c^2*x^3 - 27720*b^3*c*x^2 - 6930*b^4*x + 4620*c^2*x^3 + 6930*b*c*x^2 + 3465*b^2*x - 693*x
)/c^5