Optimal. Leaf size=114 \[ -\frac{c^2 (c+d x)^{18} (4 b c-3 a d)}{18 d^5}+\frac{c^3 (c+d x)^{17} (b c-a d)}{17 d^5}-\frac{(c+d x)^{20} (4 b c-a d)}{20 d^5}+\frac{3 c (c+d x)^{19} (2 b c-a d)}{19 d^5}+\frac{b (c+d x)^{21}}{21 d^5} \]
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Rubi [A] time = 0.364003, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {76} \[ -\frac{c^2 (c+d x)^{18} (4 b c-3 a d)}{18 d^5}+\frac{c^3 (c+d x)^{17} (b c-a d)}{17 d^5}-\frac{(c+d x)^{20} (4 b c-a d)}{20 d^5}+\frac{3 c (c+d x)^{19} (2 b c-a d)}{19 d^5}+\frac{b (c+d x)^{21}}{21 d^5} \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin{align*} \int x^3 (a+b x) (c+d x)^{16} \, dx &=\int \left (\frac{c^3 (b c-a d) (c+d x)^{16}}{d^4}-\frac{c^2 (4 b c-3 a d) (c+d x)^{17}}{d^4}+\frac{3 c (2 b c-a d) (c+d x)^{18}}{d^4}+\frac{(-4 b c+a d) (c+d x)^{19}}{d^4}+\frac{b (c+d x)^{20}}{d^4}\right ) \, dx\\ &=\frac{c^3 (b c-a d) (c+d x)^{17}}{17 d^5}-\frac{c^2 (4 b c-3 a d) (c+d x)^{18}}{18 d^5}+\frac{3 c (2 b c-a d) (c+d x)^{19}}{19 d^5}-\frac{(4 b c-a d) (c+d x)^{20}}{20 d^5}+\frac{b (c+d x)^{21}}{21 d^5}\\ \end{align*}
Mathematica [B] time = 0.0719858, size = 359, normalized size = 3.15 \[ \frac{20}{9} c^2 d^{13} x^{18} (3 a d+14 b c)+\frac{140}{17} c^3 d^{12} x^{17} (4 a d+13 b c)+\frac{91}{4} c^4 d^{11} x^{16} (5 a d+12 b c)+\frac{728}{15} c^5 d^{10} x^{15} (6 a d+11 b c)+\frac{572}{7} c^6 d^9 x^{14} (7 a d+10 b c)+110 c^7 d^8 x^{13} (8 a d+9 b c)+\frac{715}{6} c^8 d^7 x^{12} (9 a d+8 b c)+104 c^9 d^6 x^{11} (10 a d+7 b c)+\frac{364}{5} c^{10} d^5 x^{10} (11 a d+6 b c)+\frac{364}{9} c^{11} d^4 x^9 (12 a d+5 b c)+\frac{35}{2} c^{12} d^3 x^8 (13 a d+4 b c)+\frac{40}{7} c^{13} d^2 x^7 (14 a d+3 b c)+\frac{4}{3} c^{14} d x^6 (15 a d+2 b c)+\frac{1}{5} c^{15} x^5 (16 a d+b c)+\frac{1}{20} d^{15} x^{20} (a d+16 b c)+\frac{8}{19} c d^{14} x^{19} (2 a d+15 b c)+\frac{1}{4} a c^{16} x^4+\frac{1}{21} b d^{16} x^{21} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 388, normalized size = 3.4 \begin{align*}{\frac{b{d}^{16}{x}^{21}}{21}}+{\frac{ \left ( a{d}^{16}+16\,bc{d}^{15} \right ){x}^{20}}{20}}+{\frac{ \left ( 16\,ac{d}^{15}+120\,b{c}^{2}{d}^{14} \right ){x}^{19}}{19}}+{\frac{ \left ( 120\,a{c}^{2}{d}^{14}+560\,b{c}^{3}{d}^{13} \right ){x}^{18}}{18}}+{\frac{ \left ( 560\,a{c}^{3}{d}^{13}+1820\,b{c}^{4}{d}^{12} \right ){x}^{17}}{17}}+{\frac{ \left ( 1820\,a{c}^{4}{d}^{12}+4368\,b{c}^{5}{d}^{11} \right ){x}^{16}}{16}}+{\frac{ \left ( 4368\,a{c}^{5}{d}^{11}+8008\,b{c}^{6}{d}^{10} \right ){x}^{15}}{15}}+{\frac{ \left ( 8008\,a{c}^{6}{d}^{10}+11440\,b{c}^{7}{d}^{9} \right ){x}^{14}}{14}}+{\frac{ \left ( 11440\,a{c}^{7}{d}^{9}+12870\,b{c}^{8}{d}^{8} \right ){x}^{13}}{13}}+{\frac{ \left ( 12870\,a{c}^{8}{d}^{8}+11440\,b{c}^{9}{d}^{7} \right ){x}^{12}}{12}}+{\frac{ \left ( 11440\,a{c}^{9}{d}^{7}+8008\,b{c}^{10}{d}^{6} \right ){x}^{11}}{11}}+{\frac{ \left ( 8008\,a{c}^{10}{d}^{6}+4368\,b{c}^{11}{d}^{5} \right ){x}^{10}}{10}}+{\frac{ \left ( 4368\,a{c}^{11}{d}^{5}+1820\,b{c}^{12}{d}^{4} \right ){x}^{9}}{9}}+{\frac{ \left ( 1820\,a{c}^{12}{d}^{4}+560\,b{c}^{13}{d}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( 560\,a{c}^{13}{d}^{3}+120\,b{c}^{14}{d}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 120\,a{c}^{14}{d}^{2}+16\,b{c}^{15}d \right ){x}^{6}}{6}}+{\frac{ \left ( 16\,a{c}^{15}d+b{c}^{16} \right ){x}^{5}}{5}}+{\frac{a{c}^{16}{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.05267, size = 522, normalized size = 4.58 \begin{align*} \frac{1}{21} \, b d^{16} x^{21} + \frac{1}{4} \, a c^{16} x^{4} + \frac{1}{20} \,{\left (16 \, b c d^{15} + a d^{16}\right )} x^{20} + \frac{8}{19} \,{\left (15 \, b c^{2} d^{14} + 2 \, a c d^{15}\right )} x^{19} + \frac{20}{9} \,{\left (14 \, b c^{3} d^{13} + 3 \, a c^{2} d^{14}\right )} x^{18} + \frac{140}{17} \,{\left (13 \, b c^{4} d^{12} + 4 \, a c^{3} d^{13}\right )} x^{17} + \frac{91}{4} \,{\left (12 \, b c^{5} d^{11} + 5 \, a c^{4} d^{12}\right )} x^{16} + \frac{728}{15} \,{\left (11 \, b c^{6} d^{10} + 6 \, a c^{5} d^{11}\right )} x^{15} + \frac{572}{7} \,{\left (10 \, b c^{7} d^{9} + 7 \, a c^{6} d^{10}\right )} x^{14} + 110 \,{\left (9 \, b c^{8} d^{8} + 8 \, a c^{7} d^{9}\right )} x^{13} + \frac{715}{6} \,{\left (8 \, b c^{9} d^{7} + 9 \, a c^{8} d^{8}\right )} x^{12} + 104 \,{\left (7 \, b c^{10} d^{6} + 10 \, a c^{9} d^{7}\right )} x^{11} + \frac{364}{5} \,{\left (6 \, b c^{11} d^{5} + 11 \, a c^{10} d^{6}\right )} x^{10} + \frac{364}{9} \,{\left (5 \, b c^{12} d^{4} + 12 \, a c^{11} d^{5}\right )} x^{9} + \frac{35}{2} \,{\left (4 \, b c^{13} d^{3} + 13 \, a c^{12} d^{4}\right )} x^{8} + \frac{40}{7} \,{\left (3 \, b c^{14} d^{2} + 14 \, a c^{13} d^{3}\right )} x^{7} + \frac{4}{3} \,{\left (2 \, b c^{15} d + 15 \, a c^{14} d^{2}\right )} x^{6} + \frac{1}{5} \,{\left (b c^{16} + 16 \, a c^{15} d\right )} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.20033, size = 1021, normalized size = 8.96 \begin{align*} \frac{1}{21} x^{21} d^{16} b + \frac{4}{5} x^{20} d^{15} c b + \frac{1}{20} x^{20} d^{16} a + \frac{120}{19} x^{19} d^{14} c^{2} b + \frac{16}{19} x^{19} d^{15} c a + \frac{280}{9} x^{18} d^{13} c^{3} b + \frac{20}{3} x^{18} d^{14} c^{2} a + \frac{1820}{17} x^{17} d^{12} c^{4} b + \frac{560}{17} x^{17} d^{13} c^{3} a + 273 x^{16} d^{11} c^{5} b + \frac{455}{4} x^{16} d^{12} c^{4} a + \frac{8008}{15} x^{15} d^{10} c^{6} b + \frac{1456}{5} x^{15} d^{11} c^{5} a + \frac{5720}{7} x^{14} d^{9} c^{7} b + 572 x^{14} d^{10} c^{6} a + 990 x^{13} d^{8} c^{8} b + 880 x^{13} d^{9} c^{7} a + \frac{2860}{3} x^{12} d^{7} c^{9} b + \frac{2145}{2} x^{12} d^{8} c^{8} a + 728 x^{11} d^{6} c^{10} b + 1040 x^{11} d^{7} c^{9} a + \frac{2184}{5} x^{10} d^{5} c^{11} b + \frac{4004}{5} x^{10} d^{6} c^{10} a + \frac{1820}{9} x^{9} d^{4} c^{12} b + \frac{1456}{3} x^{9} d^{5} c^{11} a + 70 x^{8} d^{3} c^{13} b + \frac{455}{2} x^{8} d^{4} c^{12} a + \frac{120}{7} x^{7} d^{2} c^{14} b + 80 x^{7} d^{3} c^{13} a + \frac{8}{3} x^{6} d c^{15} b + 20 x^{6} d^{2} c^{14} a + \frac{1}{5} x^{5} c^{16} b + \frac{16}{5} x^{5} d c^{15} a + \frac{1}{4} x^{4} c^{16} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.172255, size = 422, normalized size = 3.7 \begin{align*} \frac{a c^{16} x^{4}}{4} + \frac{b d^{16} x^{21}}{21} + x^{20} \left (\frac{a d^{16}}{20} + \frac{4 b c d^{15}}{5}\right ) + x^{19} \left (\frac{16 a c d^{15}}{19} + \frac{120 b c^{2} d^{14}}{19}\right ) + x^{18} \left (\frac{20 a c^{2} d^{14}}{3} + \frac{280 b c^{3} d^{13}}{9}\right ) + x^{17} \left (\frac{560 a c^{3} d^{13}}{17} + \frac{1820 b c^{4} d^{12}}{17}\right ) + x^{16} \left (\frac{455 a c^{4} d^{12}}{4} + 273 b c^{5} d^{11}\right ) + x^{15} \left (\frac{1456 a c^{5} d^{11}}{5} + \frac{8008 b c^{6} d^{10}}{15}\right ) + x^{14} \left (572 a c^{6} d^{10} + \frac{5720 b c^{7} d^{9}}{7}\right ) + x^{13} \left (880 a c^{7} d^{9} + 990 b c^{8} d^{8}\right ) + x^{12} \left (\frac{2145 a c^{8} d^{8}}{2} + \frac{2860 b c^{9} d^{7}}{3}\right ) + x^{11} \left (1040 a c^{9} d^{7} + 728 b c^{10} d^{6}\right ) + x^{10} \left (\frac{4004 a c^{10} d^{6}}{5} + \frac{2184 b c^{11} d^{5}}{5}\right ) + x^{9} \left (\frac{1456 a c^{11} d^{5}}{3} + \frac{1820 b c^{12} d^{4}}{9}\right ) + x^{8} \left (\frac{455 a c^{12} d^{4}}{2} + 70 b c^{13} d^{3}\right ) + x^{7} \left (80 a c^{13} d^{3} + \frac{120 b c^{14} d^{2}}{7}\right ) + x^{6} \left (20 a c^{14} d^{2} + \frac{8 b c^{15} d}{3}\right ) + x^{5} \left (\frac{16 a c^{15} d}{5} + \frac{b c^{16}}{5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18905, size = 525, normalized size = 4.61 \begin{align*} \frac{1}{21} \, b d^{16} x^{21} + \frac{4}{5} \, b c d^{15} x^{20} + \frac{1}{20} \, a d^{16} x^{20} + \frac{120}{19} \, b c^{2} d^{14} x^{19} + \frac{16}{19} \, a c d^{15} x^{19} + \frac{280}{9} \, b c^{3} d^{13} x^{18} + \frac{20}{3} \, a c^{2} d^{14} x^{18} + \frac{1820}{17} \, b c^{4} d^{12} x^{17} + \frac{560}{17} \, a c^{3} d^{13} x^{17} + 273 \, b c^{5} d^{11} x^{16} + \frac{455}{4} \, a c^{4} d^{12} x^{16} + \frac{8008}{15} \, b c^{6} d^{10} x^{15} + \frac{1456}{5} \, a c^{5} d^{11} x^{15} + \frac{5720}{7} \, b c^{7} d^{9} x^{14} + 572 \, a c^{6} d^{10} x^{14} + 990 \, b c^{8} d^{8} x^{13} + 880 \, a c^{7} d^{9} x^{13} + \frac{2860}{3} \, b c^{9} d^{7} x^{12} + \frac{2145}{2} \, a c^{8} d^{8} x^{12} + 728 \, b c^{10} d^{6} x^{11} + 1040 \, a c^{9} d^{7} x^{11} + \frac{2184}{5} \, b c^{11} d^{5} x^{10} + \frac{4004}{5} \, a c^{10} d^{6} x^{10} + \frac{1820}{9} \, b c^{12} d^{4} x^{9} + \frac{1456}{3} \, a c^{11} d^{5} x^{9} + 70 \, b c^{13} d^{3} x^{8} + \frac{455}{2} \, a c^{12} d^{4} x^{8} + \frac{120}{7} \, b c^{14} d^{2} x^{7} + 80 \, a c^{13} d^{3} x^{7} + \frac{8}{3} \, b c^{15} d x^{6} + 20 \, a c^{14} d^{2} x^{6} + \frac{1}{5} \, b c^{16} x^{5} + \frac{16}{5} \, a c^{15} d x^{5} + \frac{1}{4} \, a c^{16} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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