Optimal. Leaf size=88 \[ -\frac {c^2 (b c-a d) (c+d x)^{17}}{17 d^4}+\frac {c (3 b c-2 a d) (c+d x)^{18}}{18 d^4}-\frac {(3 b c-a d) (c+d x)^{19}}{19 d^4}+\frac {b (c+d x)^{20}}{20 d^4} \]
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Rubi [A]
time = 0.13, antiderivative size = 88, normalized size of antiderivative = 1.00, 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 = {77}
\begin {gather*} -\frac {c^2 (c+d x)^{17} (b c-a d)}{17 d^4}-\frac {(c+d x)^{19} (3 b c-a d)}{19 d^4}+\frac {c (c+d x)^{18} (3 b c-2 a d)}{18 d^4}+\frac {b (c+d x)^{20}}{20 d^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int x^2 (a+b x) (c+d x)^{16} \, dx &=\int \left (-\frac {c^2 (b c-a d) (c+d x)^{16}}{d^3}+\frac {c (3 b c-2 a d) (c+d x)^{17}}{d^3}+\frac {(-3 b c+a d) (c+d x)^{18}}{d^3}+\frac {b (c+d x)^{19}}{d^3}\right ) \, dx\\ &=-\frac {c^2 (b c-a d) (c+d x)^{17}}{17 d^4}+\frac {c (3 b c-2 a d) (c+d x)^{18}}{18 d^4}-\frac {(3 b c-a d) (c+d x)^{19}}{19 d^4}+\frac {b (c+d x)^{20}}{20 d^4}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(355\) vs. \(2(88)=176\).
time = 0.05, size = 355, normalized size = 4.03 \begin {gather*} \frac {1}{3} a c^{16} x^3+\frac {1}{4} c^{15} (b c+16 a d) x^4+\frac {8}{5} c^{14} d (2 b c+15 a d) x^5+\frac {20}{3} c^{13} d^2 (3 b c+14 a d) x^6+20 c^{12} d^3 (4 b c+13 a d) x^7+\frac {91}{2} c^{11} d^4 (5 b c+12 a d) x^8+\frac {728}{9} c^{10} d^5 (6 b c+11 a d) x^9+\frac {572}{5} c^9 d^6 (7 b c+10 a d) x^{10}+130 c^8 d^7 (8 b c+9 a d) x^{11}+\frac {715}{6} c^7 d^8 (9 b c+8 a d) x^{12}+88 c^6 d^9 (10 b c+7 a d) x^{13}+52 c^5 d^{10} (11 b c+6 a d) x^{14}+\frac {364}{15} c^4 d^{11} (12 b c+5 a d) x^{15}+\frac {35}{4} c^3 d^{12} (13 b c+4 a d) x^{16}+\frac {40}{17} c^2 d^{13} (14 b c+3 a d) x^{17}+\frac {4}{9} c d^{14} (15 b c+2 a d) x^{18}+\frac {1}{19} d^{15} (16 b c+a d) x^{19}+\frac {1}{20} b d^{16} x^{20} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(387\) vs.
\(2(80)=160\).
time = 0.06, size = 388, normalized size = 4.41
method | result | size |
norman | \(\frac {a \,c^{16} x^{3}}{3}+\left (4 a \,c^{15} d +\frac {1}{4} b \,c^{16}\right ) x^{4}+\left (24 a \,c^{14} d^{2}+\frac {16}{5} b \,c^{15} d \right ) x^{5}+\left (\frac {280}{3} a \,c^{13} d^{3}+20 b \,c^{14} d^{2}\right ) x^{6}+\left (260 a \,c^{12} d^{4}+80 b \,c^{13} d^{3}\right ) x^{7}+\left (546 a \,c^{11} d^{5}+\frac {455}{2} b \,c^{12} d^{4}\right ) x^{8}+\left (\frac {8008}{9} a \,c^{10} d^{6}+\frac {1456}{3} b \,c^{11} d^{5}\right ) x^{9}+\left (1144 a \,c^{9} d^{7}+\frac {4004}{5} b \,c^{10} d^{6}\right ) x^{10}+\left (1170 a \,c^{8} d^{8}+1040 b \,c^{9} d^{7}\right ) x^{11}+\left (\frac {2860}{3} a \,c^{7} d^{9}+\frac {2145}{2} b \,c^{8} d^{8}\right ) x^{12}+\left (616 a \,c^{6} d^{10}+880 b \,c^{7} d^{9}\right ) x^{13}+\left (312 a \,c^{5} d^{11}+572 b \,c^{6} d^{10}\right ) x^{14}+\left (\frac {364}{3} a \,c^{4} d^{12}+\frac {1456}{5} b \,c^{5} d^{11}\right ) x^{15}+\left (35 a \,c^{3} d^{13}+\frac {455}{4} b \,c^{4} d^{12}\right ) x^{16}+\left (\frac {120}{17} a \,c^{2} d^{14}+\frac {560}{17} b \,c^{3} d^{13}\right ) x^{17}+\left (\frac {8}{9} a c \,d^{15}+\frac {20}{3} b \,c^{2} d^{14}\right ) x^{18}+\left (\frac {1}{19} a \,d^{16}+\frac {16}{19} b c \,d^{15}\right ) x^{19}+\frac {b \,d^{16} x^{20}}{20}\) | \(374\) |
default | \(\frac {b \,d^{16} x^{20}}{20}+\frac {\left (a \,d^{16}+16 b c \,d^{15}\right ) x^{19}}{19}+\frac {\left (16 a c \,d^{15}+120 b \,c^{2} d^{14}\right ) x^{18}}{18}+\frac {\left (120 a \,c^{2} d^{14}+560 b \,c^{3} d^{13}\right ) x^{17}}{17}+\frac {\left (560 a \,c^{3} d^{13}+1820 b \,c^{4} d^{12}\right ) x^{16}}{16}+\frac {\left (1820 a \,c^{4} d^{12}+4368 b \,c^{5} d^{11}\right ) x^{15}}{15}+\frac {\left (4368 a \,c^{5} d^{11}+8008 b \,c^{6} d^{10}\right ) x^{14}}{14}+\frac {\left (8008 a \,c^{6} d^{10}+11440 b \,c^{7} d^{9}\right ) x^{13}}{13}+\frac {\left (11440 a \,c^{7} d^{9}+12870 b \,c^{8} d^{8}\right ) x^{12}}{12}+\frac {\left (12870 a \,c^{8} d^{8}+11440 b \,c^{9} d^{7}\right ) x^{11}}{11}+\frac {\left (11440 a \,c^{9} d^{7}+8008 b \,c^{10} d^{6}\right ) x^{10}}{10}+\frac {\left (8008 a \,c^{10} d^{6}+4368 b \,c^{11} d^{5}\right ) x^{9}}{9}+\frac {\left (4368 a \,c^{11} d^{5}+1820 b \,c^{12} d^{4}\right ) x^{8}}{8}+\frac {\left (1820 a \,c^{12} d^{4}+560 b \,c^{13} d^{3}\right ) x^{7}}{7}+\frac {\left (560 a \,c^{13} d^{3}+120 b \,c^{14} d^{2}\right ) x^{6}}{6}+\frac {\left (120 a \,c^{14} d^{2}+16 b \,c^{15} d \right ) x^{5}}{5}+\frac {\left (16 a \,c^{15} d +b \,c^{16}\right ) x^{4}}{4}+\frac {a \,c^{16} x^{3}}{3}\) | \(388\) |
gosper | \(1040 b \,c^{9} d^{7} x^{11}+616 a \,c^{6} d^{10} x^{13}+880 b \,c^{7} d^{9} x^{13}+312 a \,c^{5} d^{11} x^{14}+572 b \,c^{6} d^{10} x^{14}+\frac {1456}{5} x^{15} b \,c^{5} d^{11}+35 x^{16} a \,c^{3} d^{13}+\frac {455}{4} x^{16} b \,c^{4} d^{12}+\frac {120}{17} x^{17} a \,c^{2} d^{14}+\frac {560}{17} x^{17} b \,c^{3} d^{13}+\frac {8}{9} x^{18} a c \,d^{15}+\frac {20}{3} x^{18} b \,c^{2} d^{14}+\frac {16}{19} x^{19} b c \,d^{15}+260 a \,c^{12} d^{4} x^{7}+80 b \,c^{13} d^{3} x^{7}+1170 a \,c^{8} d^{8} x^{11}+\frac {8008}{9} x^{9} a \,c^{10} d^{6}+\frac {1456}{3} x^{9} b \,c^{11} d^{5}+1144 x^{10} a \,c^{9} d^{7}+\frac {4004}{5} x^{10} b \,c^{10} d^{6}+\frac {2860}{3} x^{12} a \,c^{7} d^{9}+\frac {2145}{2} x^{12} b \,c^{8} d^{8}+\frac {364}{3} x^{15} a \,c^{4} d^{12}+20 x^{6} b \,c^{14} d^{2}+546 x^{8} a \,c^{11} d^{5}+\frac {455}{2} x^{8} b \,c^{12} d^{4}+\frac {16}{5} x^{5} b \,c^{15} d +\frac {280}{3} x^{6} a \,c^{13} d^{3}+4 x^{4} a \,c^{15} d +24 x^{5} a \,c^{14} d^{2}+\frac {1}{19} x^{19} a \,d^{16}+\frac {1}{3} a \,c^{16} x^{3}+\frac {1}{20} b \,d^{16} x^{20}+\frac {1}{4} x^{4} b \,c^{16}\) | \(390\) |
risch | \(1040 b \,c^{9} d^{7} x^{11}+616 a \,c^{6} d^{10} x^{13}+880 b \,c^{7} d^{9} x^{13}+312 a \,c^{5} d^{11} x^{14}+572 b \,c^{6} d^{10} x^{14}+\frac {1456}{5} x^{15} b \,c^{5} d^{11}+35 x^{16} a \,c^{3} d^{13}+\frac {455}{4} x^{16} b \,c^{4} d^{12}+\frac {120}{17} x^{17} a \,c^{2} d^{14}+\frac {560}{17} x^{17} b \,c^{3} d^{13}+\frac {8}{9} x^{18} a c \,d^{15}+\frac {20}{3} x^{18} b \,c^{2} d^{14}+\frac {16}{19} x^{19} b c \,d^{15}+260 a \,c^{12} d^{4} x^{7}+80 b \,c^{13} d^{3} x^{7}+1170 a \,c^{8} d^{8} x^{11}+\frac {8008}{9} x^{9} a \,c^{10} d^{6}+\frac {1456}{3} x^{9} b \,c^{11} d^{5}+1144 x^{10} a \,c^{9} d^{7}+\frac {4004}{5} x^{10} b \,c^{10} d^{6}+\frac {2860}{3} x^{12} a \,c^{7} d^{9}+\frac {2145}{2} x^{12} b \,c^{8} d^{8}+\frac {364}{3} x^{15} a \,c^{4} d^{12}+20 x^{6} b \,c^{14} d^{2}+546 x^{8} a \,c^{11} d^{5}+\frac {455}{2} x^{8} b \,c^{12} d^{4}+\frac {16}{5} x^{5} b \,c^{15} d +\frac {280}{3} x^{6} a \,c^{13} d^{3}+4 x^{4} a \,c^{15} d +24 x^{5} a \,c^{14} d^{2}+\frac {1}{19} x^{19} a \,d^{16}+\frac {1}{3} a \,c^{16} x^{3}+\frac {1}{20} b \,d^{16} x^{20}+\frac {1}{4} x^{4} b \,c^{16}\) | \(390\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 387 vs.
\(2 (80) = 160\).
time = 0.28, size = 387, normalized size = 4.40 \begin {gather*} \frac {1}{20} \, b d^{16} x^{20} + \frac {1}{3} \, a c^{16} x^{3} + \frac {1}{19} \, {\left (16 \, b c d^{15} + a d^{16}\right )} x^{19} + \frac {4}{9} \, {\left (15 \, b c^{2} d^{14} + 2 \, a c d^{15}\right )} x^{18} + \frac {40}{17} \, {\left (14 \, b c^{3} d^{13} + 3 \, a c^{2} d^{14}\right )} x^{17} + \frac {35}{4} \, {\left (13 \, b c^{4} d^{12} + 4 \, a c^{3} d^{13}\right )} x^{16} + \frac {364}{15} \, {\left (12 \, b c^{5} d^{11} + 5 \, a c^{4} d^{12}\right )} x^{15} + 52 \, {\left (11 \, b c^{6} d^{10} + 6 \, a c^{5} d^{11}\right )} x^{14} + 88 \, {\left (10 \, b c^{7} d^{9} + 7 \, a c^{6} d^{10}\right )} x^{13} + \frac {715}{6} \, {\left (9 \, b c^{8} d^{8} + 8 \, a c^{7} d^{9}\right )} x^{12} + 130 \, {\left (8 \, b c^{9} d^{7} + 9 \, a c^{8} d^{8}\right )} x^{11} + \frac {572}{5} \, {\left (7 \, b c^{10} d^{6} + 10 \, a c^{9} d^{7}\right )} x^{10} + \frac {728}{9} \, {\left (6 \, b c^{11} d^{5} + 11 \, a c^{10} d^{6}\right )} x^{9} + \frac {91}{2} \, {\left (5 \, b c^{12} d^{4} + 12 \, a c^{11} d^{5}\right )} x^{8} + 20 \, {\left (4 \, b c^{13} d^{3} + 13 \, a c^{12} d^{4}\right )} x^{7} + \frac {20}{3} \, {\left (3 \, b c^{14} d^{2} + 14 \, a c^{13} d^{3}\right )} x^{6} + \frac {8}{5} \, {\left (2 \, b c^{15} d + 15 \, a c^{14} d^{2}\right )} x^{5} + \frac {1}{4} \, {\left (b c^{16} + 16 \, a c^{15} d\right )} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 387 vs.
\(2 (80) = 160\).
time = 1.50, size = 387, normalized size = 4.40 \begin {gather*} \frac {1}{20} \, b d^{16} x^{20} + \frac {1}{3} \, a c^{16} x^{3} + \frac {1}{19} \, {\left (16 \, b c d^{15} + a d^{16}\right )} x^{19} + \frac {4}{9} \, {\left (15 \, b c^{2} d^{14} + 2 \, a c d^{15}\right )} x^{18} + \frac {40}{17} \, {\left (14 \, b c^{3} d^{13} + 3 \, a c^{2} d^{14}\right )} x^{17} + \frac {35}{4} \, {\left (13 \, b c^{4} d^{12} + 4 \, a c^{3} d^{13}\right )} x^{16} + \frac {364}{15} \, {\left (12 \, b c^{5} d^{11} + 5 \, a c^{4} d^{12}\right )} x^{15} + 52 \, {\left (11 \, b c^{6} d^{10} + 6 \, a c^{5} d^{11}\right )} x^{14} + 88 \, {\left (10 \, b c^{7} d^{9} + 7 \, a c^{6} d^{10}\right )} x^{13} + \frac {715}{6} \, {\left (9 \, b c^{8} d^{8} + 8 \, a c^{7} d^{9}\right )} x^{12} + 130 \, {\left (8 \, b c^{9} d^{7} + 9 \, a c^{8} d^{8}\right )} x^{11} + \frac {572}{5} \, {\left (7 \, b c^{10} d^{6} + 10 \, a c^{9} d^{7}\right )} x^{10} + \frac {728}{9} \, {\left (6 \, b c^{11} d^{5} + 11 \, a c^{10} d^{6}\right )} x^{9} + \frac {91}{2} \, {\left (5 \, b c^{12} d^{4} + 12 \, a c^{11} d^{5}\right )} x^{8} + 20 \, {\left (4 \, b c^{13} d^{3} + 13 \, a c^{12} d^{4}\right )} x^{7} + \frac {20}{3} \, {\left (3 \, b c^{14} d^{2} + 14 \, a c^{13} d^{3}\right )} x^{6} + \frac {8}{5} \, {\left (2 \, b c^{15} d + 15 \, a c^{14} d^{2}\right )} x^{5} + \frac {1}{4} \, {\left (b c^{16} + 16 \, a c^{15} d\right )} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 413 vs.
\(2 (80) = 160\).
time = 0.05, size = 413, normalized size = 4.69 \begin {gather*} \frac {a c^{16} x^{3}}{3} + \frac {b d^{16} x^{20}}{20} + x^{19} \left (\frac {a d^{16}}{19} + \frac {16 b c d^{15}}{19}\right ) + x^{18} \cdot \left (\frac {8 a c d^{15}}{9} + \frac {20 b c^{2} d^{14}}{3}\right ) + x^{17} \cdot \left (\frac {120 a c^{2} d^{14}}{17} + \frac {560 b c^{3} d^{13}}{17}\right ) + x^{16} \cdot \left (35 a c^{3} d^{13} + \frac {455 b c^{4} d^{12}}{4}\right ) + x^{15} \cdot \left (\frac {364 a c^{4} d^{12}}{3} + \frac {1456 b c^{5} d^{11}}{5}\right ) + x^{14} \cdot \left (312 a c^{5} d^{11} + 572 b c^{6} d^{10}\right ) + x^{13} \cdot \left (616 a c^{6} d^{10} + 880 b c^{7} d^{9}\right ) + x^{12} \cdot \left (\frac {2860 a c^{7} d^{9}}{3} + \frac {2145 b c^{8} d^{8}}{2}\right ) + x^{11} \cdot \left (1170 a c^{8} d^{8} + 1040 b c^{9} d^{7}\right ) + x^{10} \cdot \left (1144 a c^{9} d^{7} + \frac {4004 b c^{10} d^{6}}{5}\right ) + x^{9} \cdot \left (\frac {8008 a c^{10} d^{6}}{9} + \frac {1456 b c^{11} d^{5}}{3}\right ) + x^{8} \cdot \left (546 a c^{11} d^{5} + \frac {455 b c^{12} d^{4}}{2}\right ) + x^{7} \cdot \left (260 a c^{12} d^{4} + 80 b c^{13} d^{3}\right ) + x^{6} \cdot \left (\frac {280 a c^{13} d^{3}}{3} + 20 b c^{14} d^{2}\right ) + x^{5} \cdot \left (24 a c^{14} d^{2} + \frac {16 b c^{15} d}{5}\right ) + x^{4} \cdot \left (4 a c^{15} d + \frac {b c^{16}}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 389 vs.
\(2 (80) = 160\).
time = 0.89, size = 389, normalized size = 4.42 \begin {gather*} \frac {1}{20} \, b d^{16} x^{20} + \frac {16}{19} \, b c d^{15} x^{19} + \frac {1}{19} \, a d^{16} x^{19} + \frac {20}{3} \, b c^{2} d^{14} x^{18} + \frac {8}{9} \, a c d^{15} x^{18} + \frac {560}{17} \, b c^{3} d^{13} x^{17} + \frac {120}{17} \, a c^{2} d^{14} x^{17} + \frac {455}{4} \, b c^{4} d^{12} x^{16} + 35 \, a c^{3} d^{13} x^{16} + \frac {1456}{5} \, b c^{5} d^{11} x^{15} + \frac {364}{3} \, a c^{4} d^{12} x^{15} + 572 \, b c^{6} d^{10} x^{14} + 312 \, a c^{5} d^{11} x^{14} + 880 \, b c^{7} d^{9} x^{13} + 616 \, a c^{6} d^{10} x^{13} + \frac {2145}{2} \, b c^{8} d^{8} x^{12} + \frac {2860}{3} \, a c^{7} d^{9} x^{12} + 1040 \, b c^{9} d^{7} x^{11} + 1170 \, a c^{8} d^{8} x^{11} + \frac {4004}{5} \, b c^{10} d^{6} x^{10} + 1144 \, a c^{9} d^{7} x^{10} + \frac {1456}{3} \, b c^{11} d^{5} x^{9} + \frac {8008}{9} \, a c^{10} d^{6} x^{9} + \frac {455}{2} \, b c^{12} d^{4} x^{8} + 546 \, a c^{11} d^{5} x^{8} + 80 \, b c^{13} d^{3} x^{7} + 260 \, a c^{12} d^{4} x^{7} + 20 \, b c^{14} d^{2} x^{6} + \frac {280}{3} \, a c^{13} d^{3} x^{6} + \frac {16}{5} \, b c^{15} d x^{5} + 24 \, a c^{14} d^{2} x^{5} + \frac {1}{4} \, b c^{16} x^{4} + 4 \, a c^{15} d x^{4} + \frac {1}{3} \, a c^{16} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.54, size = 331, normalized size = 3.76 \begin {gather*} x^4\,\left (\frac {b\,c^{16}}{4}+4\,a\,d\,c^{15}\right )+x^{19}\,\left (\frac {a\,d^{16}}{19}+\frac {16\,b\,c\,d^{15}}{19}\right )+\frac {a\,c^{16}\,x^3}{3}+\frac {b\,d^{16}\,x^{20}}{20}+\frac {8\,c^{14}\,d\,x^5\,\left (15\,a\,d+2\,b\,c\right )}{5}+\frac {4\,c\,d^{14}\,x^{18}\,\left (2\,a\,d+15\,b\,c\right )}{9}+\frac {20\,c^{13}\,d^2\,x^6\,\left (14\,a\,d+3\,b\,c\right )}{3}+20\,c^{12}\,d^3\,x^7\,\left (13\,a\,d+4\,b\,c\right )+\frac {91\,c^{11}\,d^4\,x^8\,\left (12\,a\,d+5\,b\,c\right )}{2}+\frac {728\,c^{10}\,d^5\,x^9\,\left (11\,a\,d+6\,b\,c\right )}{9}+\frac {572\,c^9\,d^6\,x^{10}\,\left (10\,a\,d+7\,b\,c\right )}{5}+130\,c^8\,d^7\,x^{11}\,\left (9\,a\,d+8\,b\,c\right )+\frac {715\,c^7\,d^8\,x^{12}\,\left (8\,a\,d+9\,b\,c\right )}{6}+88\,c^6\,d^9\,x^{13}\,\left (7\,a\,d+10\,b\,c\right )+52\,c^5\,d^{10}\,x^{14}\,\left (6\,a\,d+11\,b\,c\right )+\frac {364\,c^4\,d^{11}\,x^{15}\,\left (5\,a\,d+12\,b\,c\right )}{15}+\frac {35\,c^3\,d^{12}\,x^{16}\,\left (4\,a\,d+13\,b\,c\right )}{4}+\frac {40\,c^2\,d^{13}\,x^{17}\,\left (3\,a\,d+14\,b\,c\right )}{17} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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