3.116 \(\int (a+8 x-8 x^2+4 x^3-x^4)^4 \, dx\)

Optimal. Leaf size=123 \[ -\frac {2}{9} \left (-3 a^2+6 a+37\right ) (x-1)^9+\frac {4}{13} (3-a) (x-1)^{13}-\frac {8}{11} (3 a+5) (x-1)^{11}+\frac {8}{7} (a+3) (3 a+5) (x-1)^7+\frac {4}{5} (3-a) (a+3)^2 (x-1)^5-\frac {8}{3} (a+3)^3 (x-1)^3+(a+3)^4 x+\frac {1}{17} (x-1)^{17}+\frac {8}{15} (x-1)^{15} \]

[Out]

-8/3*(3+a)^3*(-1+x)^3+4/5*(3-a)*(3+a)^2*(-1+x)^5+8/7*(3+a)*(5+3*a)*(-1+x)^7-2/9*(-3*a^2+6*a+37)*(-1+x)^9-8/11*
(5+3*a)*(-1+x)^11+4/13*(3-a)*(-1+x)^13+8/15*(-1+x)^15+1/17*(-1+x)^17+(3+a)^4*x

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Rubi [A]  time = 0.24, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1106, 1090} \[ -\frac {2}{9} \left (-3 a^2+6 a+37\right ) (x-1)^9+\frac {4}{13} (3-a) (x-1)^{13}-\frac {8}{11} (3 a+5) (x-1)^{11}+\frac {8}{7} (a+3) (3 a+5) (x-1)^7+\frac {4}{5} (3-a) (a+3)^2 (x-1)^5-\frac {8}{3} (a+3)^3 (x-1)^3+(a+3)^4 x+\frac {1}{17} (x-1)^{17}+\frac {8}{15} (x-1)^{15} \]

Antiderivative was successfully verified.

[In]

Int[(a + 8*x - 8*x^2 + 4*x^3 - x^4)^4,x]

[Out]

(-8*(3 + a)^3*(-1 + x)^3)/3 + (4*(3 - a)*(3 + a)^2*(-1 + x)^5)/5 + (8*(3 + a)*(5 + 3*a)*(-1 + x)^7)/7 - (2*(37
+ 6*a - 3*a^2)*(-1 + x)^9)/9 - (8*(5 + 3*a)*(-1 + x)^11)/11 + (4*(3 - a)*(-1 + x)^13)/13 + (8*(-1 + x)^15)/15
+ (-1 + x)^17/17 + (3 + a)^4*x

Rule 1090

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 + c*x^4)^p, x], x]
/; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0]

Rule 1106

Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4
, x, 3], e = Coeff[P4, x, 4]}, Subst[Int[SimplifyIntegrand[(a + d^4/(256*e^3) - (b*d)/(8*e) + (c - (3*d^2)/(8*
e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2, 0] && NeQ[d, 0]] /; FreeQ[p, x] &&
PolyQ[P4, x, 4] && NeQ[p, 2] && NeQ[p, 3]

Rubi steps

\begin {align*} \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^4 \, dx &=\operatorname {Subst}\left (\int \left (3+a-2 x^2-x^4\right )^4 \, dx,x,-1+x\right )\\ &=\operatorname {Subst}\left (\int \left (81 \left (1+\frac {1}{81} a \left (108+54 a+12 a^2+a^3\right )\right )-216 \left (1+a \left (1+\frac {1}{27} a (9+a)\right )\right ) x^2+108 \left (1-\frac {1}{27} a \left (-9+3 a+a^2\right )\right ) x^4+120 \left (1+\frac {1}{15} a (14+3 a)\right ) x^6-74 \left (1-\frac {3}{37} (-2+a) a\right ) x^8-40 \left (1+\frac {3 a}{5}\right ) x^{10}+12 \left (1-\frac {a}{3}\right ) x^{12}+8 x^{14}+x^{16}\right ) \, dx,x,-1+x\right )\\ &=-\frac {8}{3} (3+a)^3 (-1+x)^3+\frac {4}{5} (3-a) (3+a)^2 (-1+x)^5+\frac {8}{7} (3+a) (5+3 a) (-1+x)^7-\frac {2}{9} \left (37+6 a-3 a^2\right ) (-1+x)^9-\frac {8}{11} (5+3 a) (-1+x)^{11}+\frac {4}{13} (3-a) (-1+x)^{13}+\frac {8}{15} (-1+x)^{15}+\frac {1}{17} (-1+x)^{17}+(3+a)^4 x\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 195, normalized size = 1.59 \[ a^4 x+16 a^3 x^2+\frac {2}{9} \left (3 a^2-1536 a+20480\right ) x^9-6 \left (a^2-128 a+896\right ) x^8+\frac {64}{7} \left (3 a^2-140 a+512\right ) x^7-\frac {16}{3} \left (15 a^2-288 a+512\right ) x^6+4 a \left (a^2-48 a+128\right ) x^4-\frac {32}{3} (a-12) a^2 x^3-\frac {4}{5} \left (a^3-192 a^2+1536 a-1024\right ) x^5-\frac {4}{13} (a-640) x^{13}+\frac {4}{3} (3 a-464) x^{12}-\frac {32}{11} (9 a-524) x^{11}+\frac {16}{5} (35 a-928) x^{10}+\frac {x^{17}}{17}-x^{16}+\frac {128 x^{15}}{15}-48 x^{14} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + 8*x - 8*x^2 + 4*x^3 - x^4)^4,x]

[Out]

a^4*x + 16*a^3*x^2 - (32*(-12 + a)*a^2*x^3)/3 + 4*a*(128 - 48*a + a^2)*x^4 - (4*(-1024 + 1536*a - 192*a^2 + a^
3)*x^5)/5 - (16*(512 - 288*a + 15*a^2)*x^6)/3 + (64*(512 - 140*a + 3*a^2)*x^7)/7 - 6*(896 - 128*a + a^2)*x^8 +
(2*(20480 - 1536*a + 3*a^2)*x^9)/9 + (16*(-928 + 35*a)*x^10)/5 - (32*(-524 + 9*a)*x^11)/11 + (4*(-464 + 3*a)*
x^12)/3 - (4*(-640 + a)*x^13)/13 - 48*x^14 + (128*x^15)/15 - x^16 + x^17/17

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fricas [B]  time = 0.37, size = 219, normalized size = 1.78 \[ \frac {1}{17} x^{17} - x^{16} + \frac {128}{15} x^{15} - 48 x^{14} - \frac {4}{13} x^{13} a + \frac {2560}{13} x^{13} + 4 x^{12} a - \frac {1856}{3} x^{12} - \frac {288}{11} x^{11} a + \frac {16768}{11} x^{11} + 112 x^{10} a + \frac {2}{3} x^{9} a^{2} - \frac {14848}{5} x^{10} - \frac {1024}{3} x^{9} a - 6 x^{8} a^{2} + \frac {40960}{9} x^{9} + 768 x^{8} a + \frac {192}{7} x^{7} a^{2} - 5376 x^{8} - 1280 x^{7} a - 80 x^{6} a^{2} - \frac {4}{5} x^{5} a^{3} + \frac {32768}{7} x^{7} + 1536 x^{6} a + \frac {768}{5} x^{5} a^{2} + 4 x^{4} a^{3} - \frac {8192}{3} x^{6} - \frac {6144}{5} x^{5} a - 192 x^{4} a^{2} - \frac {32}{3} x^{3} a^{3} + \frac {4096}{5} x^{5} + 512 x^{4} a + 128 x^{3} a^{2} + 16 x^{2} a^{3} + x a^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4+4*x^3-8*x^2+a+8*x)^4,x, algorithm="fricas")

[Out]

1/17*x^17 - x^16 + 128/15*x^15 - 48*x^14 - 4/13*x^13*a + 2560/13*x^13 + 4*x^12*a - 1856/3*x^12 - 288/11*x^11*a
+ 16768/11*x^11 + 112*x^10*a + 2/3*x^9*a^2 - 14848/5*x^10 - 1024/3*x^9*a - 6*x^8*a^2 + 40960/9*x^9 + 768*x^8*
a + 192/7*x^7*a^2 - 5376*x^8 - 1280*x^7*a - 80*x^6*a^2 - 4/5*x^5*a^3 + 32768/7*x^7 + 1536*x^6*a + 768/5*x^5*a^
2 + 4*x^4*a^3 - 8192/3*x^6 - 6144/5*x^5*a - 192*x^4*a^2 - 32/3*x^3*a^3 + 4096/5*x^5 + 512*x^4*a + 128*x^3*a^2
+ 16*x^2*a^3 + x*a^4

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giac [B]  time = 0.33, size = 219, normalized size = 1.78 \[ \frac {1}{17} \, x^{17} - x^{16} + \frac {128}{15} \, x^{15} - \frac {4}{13} \, a x^{13} - 48 \, x^{14} + 4 \, a x^{12} + \frac {2560}{13} \, x^{13} - \frac {288}{11} \, a x^{11} - \frac {1856}{3} \, x^{12} + \frac {2}{3} \, a^{2} x^{9} + 112 \, a x^{10} + \frac {16768}{11} \, x^{11} - 6 \, a^{2} x^{8} - \frac {1024}{3} \, a x^{9} - \frac {14848}{5} \, x^{10} + \frac {192}{7} \, a^{2} x^{7} + 768 \, a x^{8} + \frac {40960}{9} \, x^{9} - \frac {4}{5} \, a^{3} x^{5} - 80 \, a^{2} x^{6} - 1280 \, a x^{7} - 5376 \, x^{8} + 4 \, a^{3} x^{4} + \frac {768}{5} \, a^{2} x^{5} + 1536 \, a x^{6} + \frac {32768}{7} \, x^{7} - \frac {32}{3} \, a^{3} x^{3} - 192 \, a^{2} x^{4} - \frac {6144}{5} \, a x^{5} - \frac {8192}{3} \, x^{6} + a^{4} x + 16 \, a^{3} x^{2} + 128 \, a^{2} x^{3} + 512 \, a x^{4} + \frac {4096}{5} \, x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4+4*x^3-8*x^2+a+8*x)^4,x, algorithm="giac")

[Out]

1/17*x^17 - x^16 + 128/15*x^15 - 4/13*a*x^13 - 48*x^14 + 4*a*x^12 + 2560/13*x^13 - 288/11*a*x^11 - 1856/3*x^12
+ 2/3*a^2*x^9 + 112*a*x^10 + 16768/11*x^11 - 6*a^2*x^8 - 1024/3*a*x^9 - 14848/5*x^10 + 192/7*a^2*x^7 + 768*a*
x^8 + 40960/9*x^9 - 4/5*a^3*x^5 - 80*a^2*x^6 - 1280*a*x^7 - 5376*x^8 + 4*a^3*x^4 + 768/5*a^2*x^5 + 1536*a*x^6
+ 32768/7*x^7 - 32/3*a^3*x^3 - 192*a^2*x^4 - 6144/5*a*x^5 - 8192/3*x^6 + a^4*x + 16*a^3*x^2 + 128*a^2*x^3 + 51
2*a*x^4 + 4096/5*x^5

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maple [B]  time = 0.00, size = 264, normalized size = 2.15 \[ \frac {x^{17}}{17}-x^{16}+\frac {128 x^{15}}{15}-48 x^{14}+\frac {\left (-4 a +2560\right ) x^{13}}{13}+\frac {\left (48 a -7424\right ) x^{12}}{12}+\frac {\left (-288 a +16768\right ) x^{11}}{11}+\frac {\left (1120 a -29696\right ) x^{10}}{10}+\frac {\left (2 a^{2}-2560 a +24576+\left (-2 a +128\right )^{2}\right ) x^{9}}{9}+\frac {\left (-16 a^{2}+3584 a -10240+2 \left (8 a -128\right ) \left (-2 a +128\right )\right ) x^{8}}{8}+\frac {\left (64 a^{2}-2560 a +2 \left (-16 a +64\right ) \left (-2 a +128\right )+\left (8 a -128\right )^{2}\right ) x^{7}}{7}+\frac {\left (-160 a^{2}+32 \left (-2 a +128\right ) a +2 \left (-16 a +64\right ) \left (8 a -128\right )\right ) x^{6}}{6}+a^{4} x +16 a^{3} x^{2}+\frac {\left (2 \left (-2 a +128\right ) a^{2}+32 \left (8 a -128\right ) a +\left (-16 a +64\right )^{2}\right ) x^{5}}{5}+\frac {\left (2 \left (8 a -128\right ) a^{2}+32 \left (-16 a +64\right ) a \right ) x^{4}}{4}+\frac {\left (2 \left (-16 a +64\right ) a^{2}+256 a^{2}\right ) x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^4+4*x^3-8*x^2+a+8*x)^4,x)

[Out]

1/17*x^17-x^16+128/15*x^15-48*x^14+1/13*(-4*a+2560)*x^13+1/12*(48*a-7424)*x^12+1/11*(-288*a+16768)*x^11+1/10*(
1120*a-29696)*x^10+1/9*(2*a^2-2560*a+24576+(-2*a+128)^2)*x^9+1/8*(-16*a^2+3584*a-10240+2*(8*a-128)*(-2*a+128))
*x^8+1/7*(64*a^2-2560*a+2*(-16*a+64)*(-2*a+128)+(8*a-128)^2)*x^7+1/6*(-160*a^2+32*a*(-2*a+128)+2*(-16*a+64)*(8
*a-128))*x^6+1/5*(2*a^2*(-2*a+128)+32*a*(8*a-128)+(-16*a+64)^2)*x^5+1/4*(2*a^2*(8*a-128)+32*a*(-16*a+64))*x^4+
1/3*(2*a^2*(-16*a+64)+256*a^2)*x^3+16*a^3*x^2+a^4*x

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maxima [A]  time = 0.69, size = 192, normalized size = 1.56 \[ \frac {1}{17} \, x^{17} - x^{16} + \frac {128}{15} \, x^{15} - 48 \, x^{14} + \frac {2560}{13} \, x^{13} - \frac {1856}{3} \, x^{12} + \frac {16768}{11} \, x^{11} - \frac {14848}{5} \, x^{10} + \frac {40960}{9} \, x^{9} - 5376 \, x^{8} + \frac {32768}{7} \, x^{7} - \frac {8192}{3} \, x^{6} + a^{4} x + \frac {4096}{5} \, x^{5} - \frac {4}{15} \, {\left (3 \, x^{5} - 15 \, x^{4} + 40 \, x^{3} - 60 \, x^{2}\right )} a^{3} + \frac {2}{105} \, {\left (35 \, x^{9} - 315 \, x^{8} + 1440 \, x^{7} - 4200 \, x^{6} + 8064 \, x^{5} - 10080 \, x^{4} + 6720 \, x^{3}\right )} a^{2} - \frac {4}{2145} \, {\left (165 \, x^{13} - 2145 \, x^{12} + 14040 \, x^{11} - 60060 \, x^{10} + 183040 \, x^{9} - 411840 \, x^{8} + 686400 \, x^{7} - 823680 \, x^{6} + 658944 \, x^{5} - 274560 \, x^{4}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4+4*x^3-8*x^2+a+8*x)^4,x, algorithm="maxima")

[Out]

1/17*x^17 - x^16 + 128/15*x^15 - 48*x^14 + 2560/13*x^13 - 1856/3*x^12 + 16768/11*x^11 - 14848/5*x^10 + 40960/9
*x^9 - 5376*x^8 + 32768/7*x^7 - 8192/3*x^6 + a^4*x + 4096/5*x^5 - 4/15*(3*x^5 - 15*x^4 + 40*x^3 - 60*x^2)*a^3
+ 2/105*(35*x^9 - 315*x^8 + 1440*x^7 - 4200*x^6 + 8064*x^5 - 10080*x^4 + 6720*x^3)*a^2 - 4/2145*(165*x^13 - 21
45*x^12 + 14040*x^11 - 60060*x^10 + 183040*x^9 - 411840*x^8 + 686400*x^7 - 823680*x^6 + 658944*x^5 - 274560*x^
4)*a

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mupad [B]  time = 0.21, size = 175, normalized size = 1.42 \[ x^{12}\,\left (4\,a-\frac {1856}{3}\right )-x^{13}\,\left (\frac {4\,a}{13}-\frac {2560}{13}\right )+x^{10}\,\left (112\,a-\frac {14848}{5}\right )-x^{11}\,\left (\frac {288\,a}{11}-\frac {16768}{11}\right )-x^8\,\left (6\,a^2-768\,a+5376\right )-x^6\,\left (80\,a^2-1536\,a+\frac {8192}{3}\right )+x^7\,\left (\frac {192\,a^2}{7}-1280\,a+\frac {32768}{7}\right )+x^9\,\left (\frac {2\,a^2}{3}-\frac {1024\,a}{3}+\frac {40960}{9}\right )-x^5\,\left (\frac {4\,a^3}{5}-\frac {768\,a^2}{5}+\frac {6144\,a}{5}-\frac {4096}{5}\right )+a^4\,x-48\,x^{14}+\frac {128\,x^{15}}{15}-x^{16}+\frac {x^{17}}{17}+16\,a^3\,x^2+4\,a\,x^4\,\left (a^2-48\,a+128\right )-\frac {32\,a^2\,x^3\,\left (a-12\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + 8*x - 8*x^2 + 4*x^3 - x^4)^4,x)

[Out]

x^12*(4*a - 1856/3) - x^13*((4*a)/13 - 2560/13) + x^10*(112*a - 14848/5) - x^11*((288*a)/11 - 16768/11) - x^8*
(6*a^2 - 768*a + 5376) - x^6*(80*a^2 - 1536*a + 8192/3) + x^7*((192*a^2)/7 - 1280*a + 32768/7) + x^9*((2*a^2)/
3 - (1024*a)/3 + 40960/9) - x^5*((6144*a)/5 - (768*a^2)/5 + (4*a^3)/5 - 4096/5) + a^4*x - 48*x^14 + (128*x^15)
/15 - x^16 + x^17/17 + 16*a^3*x^2 + 4*a*x^4*(a^2 - 48*a + 128) - (32*a^2*x^3*(a - 12))/3

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sympy [A]  time = 0.12, size = 199, normalized size = 1.62 \[ a^{4} x + 16 a^{3} x^{2} + \frac {x^{17}}{17} - x^{16} + \frac {128 x^{15}}{15} - 48 x^{14} + x^{13} \left (\frac {2560}{13} - \frac {4 a}{13}\right ) + x^{12} \left (4 a - \frac {1856}{3}\right ) + x^{11} \left (\frac {16768}{11} - \frac {288 a}{11}\right ) + x^{10} \left (112 a - \frac {14848}{5}\right ) + x^{9} \left (\frac {2 a^{2}}{3} - \frac {1024 a}{3} + \frac {40960}{9}\right ) + x^{8} \left (- 6 a^{2} + 768 a - 5376\right ) + x^{7} \left (\frac {192 a^{2}}{7} - 1280 a + \frac {32768}{7}\right ) + x^{6} \left (- 80 a^{2} + 1536 a - \frac {8192}{3}\right ) + x^{5} \left (- \frac {4 a^{3}}{5} + \frac {768 a^{2}}{5} - \frac {6144 a}{5} + \frac {4096}{5}\right ) + x^{4} \left (4 a^{3} - 192 a^{2} + 512 a\right ) + x^{3} \left (- \frac {32 a^{3}}{3} + 128 a^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**4+4*x**3-8*x**2+a+8*x)**4,x)

[Out]

a**4*x + 16*a**3*x**2 + x**17/17 - x**16 + 128*x**15/15 - 48*x**14 + x**13*(2560/13 - 4*a/13) + x**12*(4*a - 1
856/3) + x**11*(16768/11 - 288*a/11) + x**10*(112*a - 14848/5) + x**9*(2*a**2/3 - 1024*a/3 + 40960/9) + x**8*(
-6*a**2 + 768*a - 5376) + x**7*(192*a**2/7 - 1280*a + 32768/7) + x**6*(-80*a**2 + 1536*a - 8192/3) + x**5*(-4*
a**3/5 + 768*a**2/5 - 6144*a/5 + 4096/5) + x**4*(4*a**3 - 192*a**2 + 512*a) + x**3*(-32*a**3/3 + 128*a**2)

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