Optimal. Leaf size=62 \[ -\frac{(c+d x)^{18} (2 b c-a d)}{18 d^3}+\frac{c (c+d x)^{17} (b c-a d)}{17 d^3}+\frac{b (c+d x)^{19}}{19 d^3} \]
[Out]
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Rubi [A] time = 0.66482, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{(c+d x)^{18} (2 b c-a d)}{18 d^3}+\frac{c (c+d x)^{17} (b c-a d)}{17 d^3}+\frac{b (c+d x)^{19}}{19 d^3} \]
Antiderivative was successfully verified.
[In] Int[x*(a + b*x)*(c + d*x)^16,x]
[Out]
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Rubi in Sympy [A] time = 65.075, size = 53, normalized size = 0.85 \[ \frac{b \left (c + d x\right )^{19}}{19 d^{3}} - \frac{c \left (c + d x\right )^{17} \left (a d - b c\right )}{17 d^{3}} + \frac{\left (c + d x\right )^{18} \left (a d - 2 b c\right )}{18 d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(b*x+a)*(d*x+c)**16,x)
[Out]
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Mathematica [B] time = 0.119815, size = 347, normalized size = 5.6 \[ \frac{1}{3} c^{15} x^3 (16 a d+b c)+2 c^{14} d x^4 (15 a d+2 b c)+8 c^{13} d^2 x^5 (14 a d+3 b c)+\frac{70}{3} c^{12} d^3 x^6 (13 a d+4 b c)+52 c^{11} d^4 x^7 (12 a d+5 b c)+91 c^{10} d^5 x^8 (11 a d+6 b c)+\frac{1144}{9} c^9 d^6 x^9 (10 a d+7 b c)+143 c^8 d^7 x^{10} (9 a d+8 b c)+130 c^7 d^8 x^{11} (8 a d+9 b c)+\frac{286}{3} c^6 d^9 x^{12} (7 a d+10 b c)+56 c^5 d^{10} x^{13} (6 a d+11 b c)+26 c^4 d^{11} x^{14} (5 a d+12 b c)+\frac{28}{3} c^3 d^{12} x^{15} (4 a d+13 b c)+\frac{5}{2} c^2 d^{13} x^{16} (3 a d+14 b c)+\frac{1}{18} d^{15} x^{18} (a d+16 b c)+\frac{8}{17} c d^{14} x^{17} (2 a d+15 b c)+\frac{1}{2} a c^{16} x^2+\frac{1}{19} b d^{16} x^{19} \]
Antiderivative was successfully verified.
[In] Integrate[x*(a + b*x)*(c + d*x)^16,x]
[Out]
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Maple [B] time = 0.003, size = 388, normalized size = 6.3 \[{\frac{b{d}^{16}{x}^{19}}{19}}+{\frac{ \left ( a{d}^{16}+16\,bc{d}^{15} \right ){x}^{18}}{18}}+{\frac{ \left ( 16\,ac{d}^{15}+120\,b{c}^{2}{d}^{14} \right ){x}^{17}}{17}}+{\frac{ \left ( 120\,a{c}^{2}{d}^{14}+560\,b{c}^{3}{d}^{13} \right ){x}^{16}}{16}}+{\frac{ \left ( 560\,a{c}^{3}{d}^{13}+1820\,b{c}^{4}{d}^{12} \right ){x}^{15}}{15}}+{\frac{ \left ( 1820\,a{c}^{4}{d}^{12}+4368\,b{c}^{5}{d}^{11} \right ){x}^{14}}{14}}+{\frac{ \left ( 4368\,a{c}^{5}{d}^{11}+8008\,b{c}^{6}{d}^{10} \right ){x}^{13}}{13}}+{\frac{ \left ( 8008\,a{c}^{6}{d}^{10}+11440\,b{c}^{7}{d}^{9} \right ){x}^{12}}{12}}+{\frac{ \left ( 11440\,a{c}^{7}{d}^{9}+12870\,b{c}^{8}{d}^{8} \right ){x}^{11}}{11}}+{\frac{ \left ( 12870\,a{c}^{8}{d}^{8}+11440\,b{c}^{9}{d}^{7} \right ){x}^{10}}{10}}+{\frac{ \left ( 11440\,a{c}^{9}{d}^{7}+8008\,b{c}^{10}{d}^{6} \right ){x}^{9}}{9}}+{\frac{ \left ( 8008\,a{c}^{10}{d}^{6}+4368\,b{c}^{11}{d}^{5} \right ){x}^{8}}{8}}+{\frac{ \left ( 4368\,a{c}^{11}{d}^{5}+1820\,b{c}^{12}{d}^{4} \right ){x}^{7}}{7}}+{\frac{ \left ( 1820\,a{c}^{12}{d}^{4}+560\,b{c}^{13}{d}^{3} \right ){x}^{6}}{6}}+{\frac{ \left ( 560\,a{c}^{13}{d}^{3}+120\,b{c}^{14}{d}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 120\,a{c}^{14}{d}^{2}+16\,b{c}^{15}d \right ){x}^{4}}{4}}+{\frac{ \left ( 16\,a{c}^{15}d+b{c}^{16} \right ){x}^{3}}{3}}+{\frac{a{c}^{16}{x}^{2}}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(b*x+a)*(d*x+c)^16,x)
[Out]
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Maxima [A] time = 1.35692, size = 522, normalized size = 8.42 \[ \frac{1}{19} \, b d^{16} x^{19} + \frac{1}{2} \, a c^{16} x^{2} + \frac{1}{18} \,{\left (16 \, b c d^{15} + a d^{16}\right )} x^{18} + \frac{8}{17} \,{\left (15 \, b c^{2} d^{14} + 2 \, a c d^{15}\right )} x^{17} + \frac{5}{2} \,{\left (14 \, b c^{3} d^{13} + 3 \, a c^{2} d^{14}\right )} x^{16} + \frac{28}{3} \,{\left (13 \, b c^{4} d^{12} + 4 \, a c^{3} d^{13}\right )} x^{15} + 26 \,{\left (12 \, b c^{5} d^{11} + 5 \, a c^{4} d^{12}\right )} x^{14} + 56 \,{\left (11 \, b c^{6} d^{10} + 6 \, a c^{5} d^{11}\right )} x^{13} + \frac{286}{3} \,{\left (10 \, b c^{7} d^{9} + 7 \, a c^{6} d^{10}\right )} x^{12} + 130 \,{\left (9 \, b c^{8} d^{8} + 8 \, a c^{7} d^{9}\right )} x^{11} + 143 \,{\left (8 \, b c^{9} d^{7} + 9 \, a c^{8} d^{8}\right )} x^{10} + \frac{1144}{9} \,{\left (7 \, b c^{10} d^{6} + 10 \, a c^{9} d^{7}\right )} x^{9} + 91 \,{\left (6 \, b c^{11} d^{5} + 11 \, a c^{10} d^{6}\right )} x^{8} + 52 \,{\left (5 \, b c^{12} d^{4} + 12 \, a c^{11} d^{5}\right )} x^{7} + \frac{70}{3} \,{\left (4 \, b c^{13} d^{3} + 13 \, a c^{12} d^{4}\right )} x^{6} + 8 \,{\left (3 \, b c^{14} d^{2} + 14 \, a c^{13} d^{3}\right )} x^{5} + 2 \,{\left (2 \, b c^{15} d + 15 \, a c^{14} d^{2}\right )} x^{4} + \frac{1}{3} \,{\left (b c^{16} + 16 \, a c^{15} d\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(d*x + c)^16*x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.177916, size = 1, normalized size = 0.02 \[ \frac{1}{19} x^{19} d^{16} b + \frac{8}{9} x^{18} d^{15} c b + \frac{1}{18} x^{18} d^{16} a + \frac{120}{17} x^{17} d^{14} c^{2} b + \frac{16}{17} x^{17} d^{15} c a + 35 x^{16} d^{13} c^{3} b + \frac{15}{2} x^{16} d^{14} c^{2} a + \frac{364}{3} x^{15} d^{12} c^{4} b + \frac{112}{3} x^{15} d^{13} c^{3} a + 312 x^{14} d^{11} c^{5} b + 130 x^{14} d^{12} c^{4} a + 616 x^{13} d^{10} c^{6} b + 336 x^{13} d^{11} c^{5} a + \frac{2860}{3} x^{12} d^{9} c^{7} b + \frac{2002}{3} x^{12} d^{10} c^{6} a + 1170 x^{11} d^{8} c^{8} b + 1040 x^{11} d^{9} c^{7} a + 1144 x^{10} d^{7} c^{9} b + 1287 x^{10} d^{8} c^{8} a + \frac{8008}{9} x^{9} d^{6} c^{10} b + \frac{11440}{9} x^{9} d^{7} c^{9} a + 546 x^{8} d^{5} c^{11} b + 1001 x^{8} d^{6} c^{10} a + 260 x^{7} d^{4} c^{12} b + 624 x^{7} d^{5} c^{11} a + \frac{280}{3} x^{6} d^{3} c^{13} b + \frac{910}{3} x^{6} d^{4} c^{12} a + 24 x^{5} d^{2} c^{14} b + 112 x^{5} d^{3} c^{13} a + 4 x^{4} d c^{15} b + 30 x^{4} d^{2} c^{14} a + \frac{1}{3} x^{3} c^{16} b + \frac{16}{3} x^{3} d c^{15} a + \frac{1}{2} x^{2} c^{16} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(d*x + c)^16*x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.54633, size = 408, normalized size = 6.58 \[ \frac{a c^{16} x^{2}}{2} + \frac{b d^{16} x^{19}}{19} + x^{18} \left (\frac{a d^{16}}{18} + \frac{8 b c d^{15}}{9}\right ) + x^{17} \left (\frac{16 a c d^{15}}{17} + \frac{120 b c^{2} d^{14}}{17}\right ) + x^{16} \left (\frac{15 a c^{2} d^{14}}{2} + 35 b c^{3} d^{13}\right ) + x^{15} \left (\frac{112 a c^{3} d^{13}}{3} + \frac{364 b c^{4} d^{12}}{3}\right ) + x^{14} \left (130 a c^{4} d^{12} + 312 b c^{5} d^{11}\right ) + x^{13} \left (336 a c^{5} d^{11} + 616 b c^{6} d^{10}\right ) + x^{12} \left (\frac{2002 a c^{6} d^{10}}{3} + \frac{2860 b c^{7} d^{9}}{3}\right ) + x^{11} \left (1040 a c^{7} d^{9} + 1170 b c^{8} d^{8}\right ) + x^{10} \left (1287 a c^{8} d^{8} + 1144 b c^{9} d^{7}\right ) + x^{9} \left (\frac{11440 a c^{9} d^{7}}{9} + \frac{8008 b c^{10} d^{6}}{9}\right ) + x^{8} \left (1001 a c^{10} d^{6} + 546 b c^{11} d^{5}\right ) + x^{7} \left (624 a c^{11} d^{5} + 260 b c^{12} d^{4}\right ) + x^{6} \left (\frac{910 a c^{12} d^{4}}{3} + \frac{280 b c^{13} d^{3}}{3}\right ) + x^{5} \left (112 a c^{13} d^{3} + 24 b c^{14} d^{2}\right ) + x^{4} \left (30 a c^{14} d^{2} + 4 b c^{15} d\right ) + x^{3} \left (\frac{16 a c^{15} d}{3} + \frac{b c^{16}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(b*x+a)*(d*x+c)**16,x)
[Out]
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GIAC/XCAS [A] time = 0.30246, size = 525, normalized size = 8.47 \[ \frac{1}{19} \, b d^{16} x^{19} + \frac{8}{9} \, b c d^{15} x^{18} + \frac{1}{18} \, a d^{16} x^{18} + \frac{120}{17} \, b c^{2} d^{14} x^{17} + \frac{16}{17} \, a c d^{15} x^{17} + 35 \, b c^{3} d^{13} x^{16} + \frac{15}{2} \, a c^{2} d^{14} x^{16} + \frac{364}{3} \, b c^{4} d^{12} x^{15} + \frac{112}{3} \, a c^{3} d^{13} x^{15} + 312 \, b c^{5} d^{11} x^{14} + 130 \, a c^{4} d^{12} x^{14} + 616 \, b c^{6} d^{10} x^{13} + 336 \, a c^{5} d^{11} x^{13} + \frac{2860}{3} \, b c^{7} d^{9} x^{12} + \frac{2002}{3} \, a c^{6} d^{10} x^{12} + 1170 \, b c^{8} d^{8} x^{11} + 1040 \, a c^{7} d^{9} x^{11} + 1144 \, b c^{9} d^{7} x^{10} + 1287 \, a c^{8} d^{8} x^{10} + \frac{8008}{9} \, b c^{10} d^{6} x^{9} + \frac{11440}{9} \, a c^{9} d^{7} x^{9} + 546 \, b c^{11} d^{5} x^{8} + 1001 \, a c^{10} d^{6} x^{8} + 260 \, b c^{12} d^{4} x^{7} + 624 \, a c^{11} d^{5} x^{7} + \frac{280}{3} \, b c^{13} d^{3} x^{6} + \frac{910}{3} \, a c^{12} d^{4} x^{6} + 24 \, b c^{14} d^{2} x^{5} + 112 \, a c^{13} d^{3} x^{5} + 4 \, b c^{15} d x^{4} + 30 \, a c^{14} d^{2} x^{4} + \frac{1}{3} \, b c^{16} x^{3} + \frac{16}{3} \, a c^{15} d x^{3} + \frac{1}{2} \, a c^{16} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(d*x + c)^16*x,x, algorithm="giac")
[Out]