Optimal. Leaf size=123 \[ -\frac{2}{9} \left (-3 a^2+6 a+37\right ) (x-1)^9+\frac{4}{13} (3-a) (x-1)^{13}-\frac{8}{11} (3 a+5) (x-1)^{11}+\frac{8}{7} (a+3) (3 a+5) (x-1)^7+\frac{4}{5} (3-a) (a+3)^2 (x-1)^5-\frac{8}{3} (a+3)^3 (x-1)^3+(a+3)^4 x+\frac{1}{17} (x-1)^{17}+\frac{8}{15} (x-1)^{15} \]
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Rubi [A] time = 0.239745, antiderivative size = 123, normalized size of antiderivative = 1., 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.
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Rule 1106
Rule 1090
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*}
Mathematica [A] time = 0.0280276, size = 195, normalized size = 1.59 \[ \frac{2}{9} \left (3 a^2-1536 a+20480\right ) x^9-6 \left (a^2-128 a+896\right ) x^8+\frac{64}{7} \left (3 a^2-140 a+512\right ) x^7-\frac{16}{3} \left (15 a^2-288 a+512\right ) x^6-\frac{4}{5} \left (a^3-192 a^2+1536 a-1024\right ) x^5+4 a \left (a^2-48 a+128\right ) x^4-\frac{32}{3} (a-12) a^2 x^3+16 a^3 x^2+a^4 x-\frac{4}{13} (a-640) x^{13}+\frac{4}{3} (3 a-464) x^{12}-\frac{32}{11} (9 a-524) x^{11}+\frac{16}{5} (35 a-928) x^{10}+\frac{x^{17}}{17}-x^{16}+\frac{128 x^{15}}{15}-48 x^{14} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 264, normalized size = 2.2 \begin{align*}{\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\,a \left ( -2\,a+128 \right ) +2\, \left ( -16\,a+64 \right ) \left ( 8\,a-128 \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( 2\,{a}^{2} \left ( -2\,a+128 \right ) +32\,a \left ( 8\,a-128 \right ) + \left ( -16\,a+64 \right ) ^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{a}^{2} \left ( 8\,a-128 \right ) +32\,a \left ( -16\,a+64 \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( 2\,{a}^{2} \left ( -16\,a+64 \right ) +256\,{a}^{2} \right ){x}^{3}}{3}}+16\,{a}^{3}{x}^{2}+{a}^{4}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1306, size = 259, normalized size = 2.11 \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.26237, size = 624, normalized size = 5.07 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.112054, size = 199, normalized size = 1.62 \begin{align*} 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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25165, size = 296, normalized size = 2.41 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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