3.76 \(\int (\frac{-9+b^2}{4 c}+b x+c x^2)^5 \, dx\)

Optimal. Leaf size=109 \[ -\frac{(-b-2 c x+3)^{11}}{22528 c^6}+\frac{3 (-b-2 c x+3)^{10}}{2048 c^6}-\frac{5 (-b-2 c x+3)^9}{256 c^6}+\frac{135 (-b-2 c x+3)^8}{1024 c^6}-\frac{405 (-b-2 c x+3)^7}{896 c^6}+\frac{81 (-b-2 c x+3)^6}{128 c^6} \]

[Out]

(81*(3 - b - 2*c*x)^6)/(128*c^6) - (405*(3 - b - 2*c*x)^7)/(896*c^6) + (135*(3 - b - 2*c*x)^8)/(1024*c^6) - (5
*(3 - b - 2*c*x)^9)/(256*c^6) + (3*(3 - b - 2*c*x)^10)/(2048*c^6) - (3 - b - 2*c*x)^11/(22528*c^6)

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Rubi [A]  time = 0.142473, 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} \[ -\frac{(-b-2 c x+3)^{11}}{22528 c^6}+\frac{3 (-b-2 c x+3)^{10}}{2048 c^6}-\frac{5 (-b-2 c x+3)^9}{256 c^6}+\frac{135 (-b-2 c x+3)^8}{1024 c^6}-\frac{405 (-b-2 c x+3)^7}{896 c^6}+\frac{81 (-b-2 c x+3)^6}{128 c^6} \]

Antiderivative was successfully verified.

[In]

Int[((-9 + b^2)/(4*c) + b*x + c*x^2)^5,x]

[Out]

(81*(3 - b - 2*c*x)^6)/(128*c^6) - (405*(3 - b - 2*c*x)^7)/(896*c^6) + (135*(3 - b - 2*c*x)^8)/(1024*c^6) - (5
*(3 - b - 2*c*x)^9)/(256*c^6) + (3*(3 - b - 2*c*x)^10)/(2048*c^6) - (3 - 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{-9+b^2}{4 c}+b x+c x^2\right )^5 \, dx &=\frac{\int \left (\frac{1}{2} (-3+b)+c x\right )^5 \left (\frac{3+b}{2}+c x\right )^5 \, dx}{c^5}\\ &=\frac{\int \left (243 \left (\frac{1}{2} (-3+b)+c x\right )^5+405 \left (\frac{1}{2} (-3+b)+c x\right )^6+270 \left (\frac{1}{2} (-3+b)+c x\right )^7+90 \left (\frac{1}{2} (-3+b)+c x\right )^8+15 \left (\frac{1}{2} (-3+b)+c x\right )^9+\left (\frac{1}{2} (-3+b)+c x\right )^{10}\right ) \, dx}{c^5}\\ &=\frac{81 (3-b-2 c x)^6}{128 c^6}-\frac{405 (3-b-2 c x)^7}{896 c^6}+\frac{135 (3-b-2 c x)^8}{1024 c^6}-\frac{5 (3-b-2 c x)^9}{256 c^6}+\frac{3 (3-b-2 c x)^{10}}{2048 c^6}-\frac{(3-b-2 c x)^{11}}{22528 c^6}\\ \end{align*}

Mathematica [A]  time = 0.0312584, size = 199, normalized size = 1.83 \[ \frac{5}{4} \left (b^2-1\right ) c^3 x^9+\frac{15}{8} \left (b^3-3 b\right ) c^2 x^8+\frac{15 b \left (b^2-9\right )^2 \left (b^2-3\right ) x^4}{64 c^2}+\frac{15 \left (b^2-9\right )^3 \left (b^2-1\right ) x^3}{256 c^3}+\frac{5 b \left (b^2-9\right )^4 x^2}{512 c^4}+\frac{\left (b^2-9\right )^5 x}{1024 c^5}+\frac{15}{56} \left (7 b^4-42 b^2+27\right ) c x^7+\frac{3 \left (b^2-9\right ) \left (7 b^4-42 b^2+27\right ) x^5}{32 c}+\frac{3}{16} b \left (7 b^4-70 b^2+135\right ) x^6+\frac{1}{2} b c^4 x^{10}+\frac{c^5 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Integrate[((-9 + b^2)/(4*c) + b*x + c*x^2)^5,x]

[Out]

((-9 + b^2)^5*x)/(1024*c^5) + (5*b*(-9 + b^2)^4*x^2)/(512*c^4) + (15*(-9 + b^2)^3*(-1 + b^2)*x^3)/(256*c^3) +
(15*b*(-9 + b^2)^2*(-3 + b^2)*x^4)/(64*c^2) + (3*(-9 + b^2)*(27 - 42*b^2 + 7*b^4)*x^5)/(32*c) + (3*b*(135 - 70
*b^2 + 7*b^4)*x^6)/16 + (15*(27 - 42*b^2 + 7*b^4)*c*x^7)/56 + (15*(-3*b + b^3)*c^2*x^8)/8 + (5*(-1 + b^2)*c^3*
x^9)/4 + (b*c^4*x^10)/2 + (c^5*x^11)/11

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Maple [B]  time = 0.072, 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^2-9)/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^2-9)*c^3+4*b^2*c^3+c*(2*(3/2*b^2-9/2)*c^2+4*b^2*c^2))*x^9+1/8*((b^2-9
)*c^2*b+b*(2*(3/2*b^2-9/2)*c^2+4*b^2*c^2)+c*((b^2-9)*c*b+4*(3/2*b^2-9/2)*b*c))*x^8+1/7*(1/4*(b^2-9)/c*(2*(3/2*
b^2-9/2)*c^2+4*b^2*c^2)+b*((b^2-9)*c*b+4*(3/2*b^2-9/2)*b*c)+c*(1/8*(b^2-9)^2+2*(b^2-9)*b^2+(3/2*b^2-9/2)^2))*x
^7+1/6*(1/4*(b^2-9)/c*((b^2-9)*c*b+4*(3/2*b^2-9/2)*b*c)+b*(1/8*(b^2-9)^2+2*(b^2-9)*b^2+(3/2*b^2-9/2)^2)+c*(1/4
*(b^2-9)^2/c*b+(b^2-9)/c*b*(3/2*b^2-9/2)))*x^6+1/5*(1/4*(b^2-9)/c*(1/8*(b^2-9)^2+2*(b^2-9)*b^2+(3/2*b^2-9/2)^2
)+b*(1/4*(b^2-9)^2/c*b+(b^2-9)/c*b*(3/2*b^2-9/2))+c*(1/8*(b^2-9)^2/c^2*(3/2*b^2-9/2)+1/4*(b^2-9)^2/c^2*b^2))*x
^5+1/4*(1/4*(b^2-9)/c*(1/4*(b^2-9)^2/c*b+(b^2-9)/c*b*(3/2*b^2-9/2))+b*(1/8*(b^2-9)^2/c^2*(3/2*b^2-9/2)+1/4*(b^
2-9)^2/c^2*b^2)+1/16/c^2*(b^2-9)^3*b)*x^4+1/3*(1/4*(b^2-9)/c*(1/8*(b^2-9)^2/c^2*(3/2*b^2-9/2)+1/4*(b^2-9)^2/c^
2*b^2)+1/16*b^2*(b^2-9)^3/c^3+1/256/c^3*(b^2-9)^4)*x^3+5/512*(b^2-9)^4/c^4*b*x^2+1/1024*(b^2-9)^5/c^5*x

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Maxima [B]  time = 1.14294, 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} - 9\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} - 9\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} - 9\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} - 9\right )}}{504 \, c} + \frac{{\left (b^{2} - 9\right )}^{5} x}{1024 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/4*(b^2-9)/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 - 9)^4/c^4 + 1/192*(6*c^2*x^5 + 15*b*c*x^4 + 10*b^2*x^3)*(b^2 - 9)^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 - 9)^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 - 9)/c + 1/1024*(b^2 - 9)^5*x/c^5

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Fricas [B]  time = 2.20788, size = 618, normalized size = 5.67 \begin{align*} \frac{7168 \, c^{10} x^{11} + 39424 \, b c^{9} x^{10} + 98560 \,{\left (b^{2} - 1\right )} c^{8} x^{9} + 147840 \,{\left (b^{3} - 3 \, b\right )} c^{7} x^{8} + 21120 \,{\left (7 \, b^{4} - 42 \, b^{2} + 27\right )} c^{6} x^{7} + 14784 \,{\left (7 \, b^{5} - 70 \, b^{3} + 135 \, b\right )} c^{5} x^{6} + 7392 \,{\left (7 \, b^{6} - 105 \, b^{4} + 405 \, b^{2} - 243\right )} c^{4} x^{5} + 18480 \,{\left (b^{7} - 21 \, b^{5} + 135 \, b^{3} - 243 \, b\right )} c^{3} x^{4} + 4620 \,{\left (b^{8} - 28 \, b^{6} + 270 \, b^{4} - 972 \, b^{2} + 729\right )} c^{2} x^{3} + 770 \,{\left (b^{9} - 36 \, b^{7} + 486 \, b^{5} - 2916 \, b^{3} + 6561 \, b\right )} c x^{2} + 77 \,{\left (b^{10} - 45 \, b^{8} + 810 \, b^{6} - 7290 \, b^{4} + 32805 \, b^{2} - 59049\right )} x}{78848 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/4*(b^2-9)/c+b*x+c*x^2)^5,x, algorithm="fricas")

[Out]

1/78848*(7168*c^10*x^11 + 39424*b*c^9*x^10 + 98560*(b^2 - 1)*c^8*x^9 + 147840*(b^3 - 3*b)*c^7*x^8 + 21120*(7*b
^4 - 42*b^2 + 27)*c^6*x^7 + 14784*(7*b^5 - 70*b^3 + 135*b)*c^5*x^6 + 7392*(7*b^6 - 105*b^4 + 405*b^2 - 243)*c^
4*x^5 + 18480*(b^7 - 21*b^5 + 135*b^3 - 243*b)*c^3*x^4 + 4620*(b^8 - 28*b^6 + 270*b^4 - 972*b^2 + 729)*c^2*x^3
+ 770*(b^9 - 36*b^7 + 486*b^5 - 2916*b^3 + 6561*b)*c*x^2 + 77*(b^10 - 45*b^8 + 810*b^6 - 7290*b^4 + 32805*b^2
- 59049)*x)/c^5

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Sympy [B]  time = 0.292203, 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}}{4}\right ) + x^{8} \left (\frac{15 b^{3} c^{2}}{8} - \frac{45 b c^{2}}{8}\right ) + x^{7} \left (\frac{15 b^{4} c}{8} - \frac{45 b^{2} c}{4} + \frac{405 c}{56}\right ) + x^{6} \left (\frac{21 b^{5}}{16} - \frac{105 b^{3}}{8} + \frac{405 b}{16}\right ) + \frac{x^{5} \left (21 b^{6} - 315 b^{4} + 1215 b^{2} - 729\right )}{32 c} + \frac{x^{4} \left (15 b^{7} - 315 b^{5} + 2025 b^{3} - 3645 b\right )}{64 c^{2}} + \frac{x^{3} \left (15 b^{8} - 420 b^{6} + 4050 b^{4} - 14580 b^{2} + 10935\right )}{256 c^{3}} + \frac{x^{2} \left (5 b^{9} - 180 b^{7} + 2430 b^{5} - 14580 b^{3} + 32805 b\right )}{512 c^{4}} + \frac{x \left (b^{10} - 45 b^{8} + 810 b^{6} - 7290 b^{4} + 32805 b^{2} - 59049\right )}{1024 c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/4*(b**2-9)/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**3/4) + x**8*(15*b**3*c**2/8 - 45*b*c**2/8) + x**7*
(15*b**4*c/8 - 45*b**2*c/4 + 405*c/56) + x**6*(21*b**5/16 - 105*b**3/8 + 405*b/16) + x**5*(21*b**6 - 315*b**4
+ 1215*b**2 - 729)/(32*c) + x**4*(15*b**7 - 315*b**5 + 2025*b**3 - 3645*b)/(64*c**2) + x**3*(15*b**8 - 420*b**
6 + 4050*b**4 - 14580*b**2 + 10935)/(256*c**3) + x**2*(5*b**9 - 180*b**7 + 2430*b**5 - 14580*b**3 + 32805*b)/(
512*c**4) + x*(b**10 - 45*b**8 + 810*b**6 - 7290*b**4 + 32805*b**2 - 59049)/(1024*c**5)

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Giac [B]  time = 1.37238, size = 451, normalized size = 4.14 \begin{align*} \frac{7168 \, c^{10} x^{11} + 39424 \, b c^{9} x^{10} + 98560 \, b^{2} c^{8} x^{9} + 147840 \, b^{3} c^{7} x^{8} + 147840 \, b^{4} c^{6} x^{7} - 98560 \, c^{8} x^{9} + 103488 \, b^{5} c^{5} x^{6} - 443520 \, b c^{7} x^{8} + 51744 \, b^{6} c^{4} x^{5} - 887040 \, b^{2} c^{6} x^{7} + 18480 \, b^{7} c^{3} x^{4} - 1034880 \, b^{3} c^{5} x^{6} + 4620 \, b^{8} c^{2} x^{3} - 776160 \, b^{4} c^{4} x^{5} + 570240 \, c^{6} x^{7} + 770 \, b^{9} c x^{2} - 388080 \, b^{5} c^{3} x^{4} + 1995840 \, b c^{5} x^{6} + 77 \, b^{10} x - 129360 \, b^{6} c^{2} x^{3} + 2993760 \, b^{2} c^{4} x^{5} - 27720 \, b^{7} c x^{2} + 2494800 \, b^{3} c^{3} x^{4} - 3465 \, b^{8} x + 1247400 \, b^{4} c^{2} x^{3} - 1796256 \, c^{4} x^{5} + 374220 \, b^{5} c x^{2} - 4490640 \, b c^{3} x^{4} + 62370 \, b^{6} x - 4490640 \, b^{2} c^{2} x^{3} - 2245320 \, b^{3} c x^{2} - 561330 \, b^{4} x + 3367980 \, c^{2} x^{3} + 5051970 \, b c x^{2} + 2525985 \, b^{2} x - 4546773 \, x}{78848 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/4*(b^2-9)/c+b*x+c*x^2)^5,x, algorithm="giac")

[Out]

1/78848*(7168*c^10*x^11 + 39424*b*c^9*x^10 + 98560*b^2*c^8*x^9 + 147840*b^3*c^7*x^8 + 147840*b^4*c^6*x^7 - 985
60*c^8*x^9 + 103488*b^5*c^5*x^6 - 443520*b*c^7*x^8 + 51744*b^6*c^4*x^5 - 887040*b^2*c^6*x^7 + 18480*b^7*c^3*x^
4 - 1034880*b^3*c^5*x^6 + 4620*b^8*c^2*x^3 - 776160*b^4*c^4*x^5 + 570240*c^6*x^7 + 770*b^9*c*x^2 - 388080*b^5*
c^3*x^4 + 1995840*b*c^5*x^6 + 77*b^10*x - 129360*b^6*c^2*x^3 + 2993760*b^2*c^4*x^5 - 27720*b^7*c*x^2 + 2494800
*b^3*c^3*x^4 - 3465*b^8*x + 1247400*b^4*c^2*x^3 - 1796256*c^4*x^5 + 374220*b^5*c*x^2 - 4490640*b*c^3*x^4 + 623
70*b^6*x - 4490640*b^2*c^2*x^3 - 2245320*b^3*c*x^2 - 561330*b^4*x + 3367980*c^2*x^3 + 5051970*b*c*x^2 + 252598
5*b^2*x - 4546773*x)/c^5