Optimal. Leaf size=177 \[ -\frac {c^3 (b c-a d)^2 (c+d x)^{17}}{17 d^6}+\frac {c^2 (5 b c-3 a d) (b c-a d) (c+d x)^{18}}{18 d^6}-\frac {c \left (10 b^2 c^2-12 a b c d+3 a^2 d^2\right ) (c+d x)^{19}}{19 d^6}+\frac {\left (10 b^2 c^2-8 a b c d+a^2 d^2\right ) (c+d x)^{20}}{20 d^6}-\frac {b (5 b c-2 a d) (c+d x)^{21}}{21 d^6}+\frac {b^2 (c+d x)^{22}}{22 d^6} \]
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Rubi [A]
time = 0.40, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {90}
\begin {gather*} \frac {(c+d x)^{20} \left (a^2 d^2-8 a b c d+10 b^2 c^2\right )}{20 d^6}-\frac {c (c+d x)^{19} \left (3 a^2 d^2-12 a b c d+10 b^2 c^2\right )}{19 d^6}-\frac {c^3 (c+d x)^{17} (b c-a d)^2}{17 d^6}+\frac {c^2 (c+d x)^{18} (5 b c-3 a d) (b c-a d)}{18 d^6}-\frac {b (c+d x)^{21} (5 b c-2 a d)}{21 d^6}+\frac {b^2 (c+d x)^{22}}{22 d^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rubi steps
\begin {align*} \int x^3 (a+b x)^2 (c+d x)^{16} \, dx &=\int \left (-\frac {c^3 (b c-a d)^2 (c+d x)^{16}}{d^5}+\frac {c^2 (5 b c-3 a d) (b c-a d) (c+d x)^{17}}{d^5}-\frac {c \left (10 b^2 c^2-12 a b c d+3 a^2 d^2\right ) (c+d x)^{18}}{d^5}+\frac {\left (10 b^2 c^2-8 a b c d+a^2 d^2\right ) (c+d x)^{19}}{d^5}-\frac {b (5 b c-2 a d) (c+d x)^{20}}{d^5}+\frac {b^2 (c+d x)^{21}}{d^5}\right ) \, dx\\ &=-\frac {c^3 (b c-a d)^2 (c+d x)^{17}}{17 d^6}+\frac {c^2 (5 b c-3 a d) (b c-a d) (c+d x)^{18}}{18 d^6}-\frac {c \left (10 b^2 c^2-12 a b c d+3 a^2 d^2\right ) (c+d x)^{19}}{19 d^6}+\frac {\left (10 b^2 c^2-8 a b c d+a^2 d^2\right ) (c+d x)^{20}}{20 d^6}-\frac {b (5 b c-2 a d) (c+d x)^{21}}{21 d^6}+\frac {b^2 (c+d x)^{22}}{22 d^6}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(589\) vs. \(2(177)=354\).
time = 0.11, size = 589, normalized size = 3.33 \begin {gather*} \frac {1}{4} a^2 c^{16} x^4+\frac {2}{5} a c^{15} (b c+8 a d) x^5+\frac {1}{6} c^{14} \left (b^2 c^2+32 a b c d+120 a^2 d^2\right ) x^6+\frac {16}{7} c^{13} d \left (b^2 c^2+15 a b c d+35 a^2 d^2\right ) x^7+\frac {5}{2} c^{12} d^2 \left (6 b^2 c^2+56 a b c d+91 a^2 d^2\right ) x^8+\frac {56}{9} c^{11} d^3 \left (10 b^2 c^2+65 a b c d+78 a^2 d^2\right ) x^9+\frac {182}{5} c^{10} d^4 \left (5 b^2 c^2+24 a b c d+22 a^2 d^2\right ) x^{10}+\frac {208}{11} c^9 d^5 \left (21 b^2 c^2+77 a b c d+55 a^2 d^2\right ) x^{11}+\frac {143}{6} c^8 d^6 \left (28 b^2 c^2+80 a b c d+45 a^2 d^2\right ) x^{12}+220 c^7 d^7 \left (4 b^2 c^2+9 a b c d+4 a^2 d^2\right ) x^{13}+\frac {143}{7} c^6 d^8 \left (45 b^2 c^2+80 a b c d+28 a^2 d^2\right ) x^{14}+\frac {208}{15} c^5 d^9 \left (55 b^2 c^2+77 a b c d+21 a^2 d^2\right ) x^{15}+\frac {91}{4} c^4 d^{10} \left (22 b^2 c^2+24 a b c d+5 a^2 d^2\right ) x^{16}+\frac {56}{17} c^3 d^{11} \left (78 b^2 c^2+65 a b c d+10 a^2 d^2\right ) x^{17}+\frac {10}{9} c^2 d^{12} \left (91 b^2 c^2+56 a b c d+6 a^2 d^2\right ) x^{18}+\frac {16}{19} c d^{13} \left (35 b^2 c^2+15 a b c d+a^2 d^2\right ) x^{19}+\frac {1}{20} d^{14} \left (120 b^2 c^2+32 a b c d+a^2 d^2\right ) x^{20}+\frac {2}{21} b d^{15} (8 b c+a d) x^{21}+\frac {1}{22} b^2 d^{16} x^{22} \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. \(621\) vs.
\(2(165)=330\).
time = 0.06, size = 622, normalized size = 3.51 Too large to display
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. 617 vs.
\(2 (165) = 330\).
time = 0.30, size = 617, normalized size = 3.49 \begin {gather*} \frac {1}{22} \, b^{2} d^{16} x^{22} + \frac {1}{4} \, a^{2} c^{16} x^{4} + \frac {2}{21} \, {\left (8 \, b^{2} c d^{15} + a b d^{16}\right )} x^{21} + \frac {1}{20} \, {\left (120 \, b^{2} c^{2} d^{14} + 32 \, a b c d^{15} + a^{2} d^{16}\right )} x^{20} + \frac {16}{19} \, {\left (35 \, b^{2} c^{3} d^{13} + 15 \, a b c^{2} d^{14} + a^{2} c d^{15}\right )} x^{19} + \frac {10}{9} \, {\left (91 \, b^{2} c^{4} d^{12} + 56 \, a b c^{3} d^{13} + 6 \, a^{2} c^{2} d^{14}\right )} x^{18} + \frac {56}{17} \, {\left (78 \, b^{2} c^{5} d^{11} + 65 \, a b c^{4} d^{12} + 10 \, a^{2} c^{3} d^{13}\right )} x^{17} + \frac {91}{4} \, {\left (22 \, b^{2} c^{6} d^{10} + 24 \, a b c^{5} d^{11} + 5 \, a^{2} c^{4} d^{12}\right )} x^{16} + \frac {208}{15} \, {\left (55 \, b^{2} c^{7} d^{9} + 77 \, a b c^{6} d^{10} + 21 \, a^{2} c^{5} d^{11}\right )} x^{15} + \frac {143}{7} \, {\left (45 \, b^{2} c^{8} d^{8} + 80 \, a b c^{7} d^{9} + 28 \, a^{2} c^{6} d^{10}\right )} x^{14} + 220 \, {\left (4 \, b^{2} c^{9} d^{7} + 9 \, a b c^{8} d^{8} + 4 \, a^{2} c^{7} d^{9}\right )} x^{13} + \frac {143}{6} \, {\left (28 \, b^{2} c^{10} d^{6} + 80 \, a b c^{9} d^{7} + 45 \, a^{2} c^{8} d^{8}\right )} x^{12} + \frac {208}{11} \, {\left (21 \, b^{2} c^{11} d^{5} + 77 \, a b c^{10} d^{6} + 55 \, a^{2} c^{9} d^{7}\right )} x^{11} + \frac {182}{5} \, {\left (5 \, b^{2} c^{12} d^{4} + 24 \, a b c^{11} d^{5} + 22 \, a^{2} c^{10} d^{6}\right )} x^{10} + \frac {56}{9} \, {\left (10 \, b^{2} c^{13} d^{3} + 65 \, a b c^{12} d^{4} + 78 \, a^{2} c^{11} d^{5}\right )} x^{9} + \frac {5}{2} \, {\left (6 \, b^{2} c^{14} d^{2} + 56 \, a b c^{13} d^{3} + 91 \, a^{2} c^{12} d^{4}\right )} x^{8} + \frac {16}{7} \, {\left (b^{2} c^{15} d + 15 \, a b c^{14} d^{2} + 35 \, a^{2} c^{13} d^{3}\right )} x^{7} + \frac {1}{6} \, {\left (b^{2} c^{16} + 32 \, a b c^{15} d + 120 \, a^{2} c^{14} d^{2}\right )} x^{6} + \frac {2}{5} \, {\left (a b c^{16} + 8 \, a^{2} c^{15} d\right )} x^{5} \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. 617 vs.
\(2 (165) = 330\).
time = 1.50, size = 617, normalized size = 3.49 \begin {gather*} \frac {1}{22} \, b^{2} d^{16} x^{22} + \frac {1}{4} \, a^{2} c^{16} x^{4} + \frac {2}{21} \, {\left (8 \, b^{2} c d^{15} + a b d^{16}\right )} x^{21} + \frac {1}{20} \, {\left (120 \, b^{2} c^{2} d^{14} + 32 \, a b c d^{15} + a^{2} d^{16}\right )} x^{20} + \frac {16}{19} \, {\left (35 \, b^{2} c^{3} d^{13} + 15 \, a b c^{2} d^{14} + a^{2} c d^{15}\right )} x^{19} + \frac {10}{9} \, {\left (91 \, b^{2} c^{4} d^{12} + 56 \, a b c^{3} d^{13} + 6 \, a^{2} c^{2} d^{14}\right )} x^{18} + \frac {56}{17} \, {\left (78 \, b^{2} c^{5} d^{11} + 65 \, a b c^{4} d^{12} + 10 \, a^{2} c^{3} d^{13}\right )} x^{17} + \frac {91}{4} \, {\left (22 \, b^{2} c^{6} d^{10} + 24 \, a b c^{5} d^{11} + 5 \, a^{2} c^{4} d^{12}\right )} x^{16} + \frac {208}{15} \, {\left (55 \, b^{2} c^{7} d^{9} + 77 \, a b c^{6} d^{10} + 21 \, a^{2} c^{5} d^{11}\right )} x^{15} + \frac {143}{7} \, {\left (45 \, b^{2} c^{8} d^{8} + 80 \, a b c^{7} d^{9} + 28 \, a^{2} c^{6} d^{10}\right )} x^{14} + 220 \, {\left (4 \, b^{2} c^{9} d^{7} + 9 \, a b c^{8} d^{8} + 4 \, a^{2} c^{7} d^{9}\right )} x^{13} + \frac {143}{6} \, {\left (28 \, b^{2} c^{10} d^{6} + 80 \, a b c^{9} d^{7} + 45 \, a^{2} c^{8} d^{8}\right )} x^{12} + \frac {208}{11} \, {\left (21 \, b^{2} c^{11} d^{5} + 77 \, a b c^{10} d^{6} + 55 \, a^{2} c^{9} d^{7}\right )} x^{11} + \frac {182}{5} \, {\left (5 \, b^{2} c^{12} d^{4} + 24 \, a b c^{11} d^{5} + 22 \, a^{2} c^{10} d^{6}\right )} x^{10} + \frac {56}{9} \, {\left (10 \, b^{2} c^{13} d^{3} + 65 \, a b c^{12} d^{4} + 78 \, a^{2} c^{11} d^{5}\right )} x^{9} + \frac {5}{2} \, {\left (6 \, b^{2} c^{14} d^{2} + 56 \, a b c^{13} d^{3} + 91 \, a^{2} c^{12} d^{4}\right )} x^{8} + \frac {16}{7} \, {\left (b^{2} c^{15} d + 15 \, a b c^{14} d^{2} + 35 \, a^{2} c^{13} d^{3}\right )} x^{7} + \frac {1}{6} \, {\left (b^{2} c^{16} + 32 \, a b c^{15} d + 120 \, a^{2} c^{14} d^{2}\right )} x^{6} + \frac {2}{5} \, {\left (a b c^{16} + 8 \, a^{2} c^{15} d\right )} x^{5} \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. 697 vs.
\(2 (170) = 340\).
time = 0.06, size = 697, normalized size = 3.94 \begin {gather*} \frac {a^{2} c^{16} x^{4}}{4} + \frac {b^{2} d^{16} x^{22}}{22} + x^{21} \cdot \left (\frac {2 a b d^{16}}{21} + \frac {16 b^{2} c d^{15}}{21}\right ) + x^{20} \left (\frac {a^{2} d^{16}}{20} + \frac {8 a b c d^{15}}{5} + 6 b^{2} c^{2} d^{14}\right ) + x^{19} \cdot \left (\frac {16 a^{2} c d^{15}}{19} + \frac {240 a b c^{2} d^{14}}{19} + \frac {560 b^{2} c^{3} d^{13}}{19}\right ) + x^{18} \cdot \left (\frac {20 a^{2} c^{2} d^{14}}{3} + \frac {560 a b c^{3} d^{13}}{9} + \frac {910 b^{2} c^{4} d^{12}}{9}\right ) + x^{17} \cdot \left (\frac {560 a^{2} c^{3} d^{13}}{17} + \frac {3640 a b c^{4} d^{12}}{17} + \frac {4368 b^{2} c^{5} d^{11}}{17}\right ) + x^{16} \cdot \left (\frac {455 a^{2} c^{4} d^{12}}{4} + 546 a b c^{5} d^{11} + \frac {1001 b^{2} c^{6} d^{10}}{2}\right ) + x^{15} \cdot \left (\frac {1456 a^{2} c^{5} d^{11}}{5} + \frac {16016 a b c^{6} d^{10}}{15} + \frac {2288 b^{2} c^{7} d^{9}}{3}\right ) + x^{14} \cdot \left (572 a^{2} c^{6} d^{10} + \frac {11440 a b c^{7} d^{9}}{7} + \frac {6435 b^{2} c^{8} d^{8}}{7}\right ) + x^{13} \cdot \left (880 a^{2} c^{7} d^{9} + 1980 a b c^{8} d^{8} + 880 b^{2} c^{9} d^{7}\right ) + x^{12} \cdot \left (\frac {2145 a^{2} c^{8} d^{8}}{2} + \frac {5720 a b c^{9} d^{7}}{3} + \frac {2002 b^{2} c^{10} d^{6}}{3}\right ) + x^{11} \cdot \left (1040 a^{2} c^{9} d^{7} + 1456 a b c^{10} d^{6} + \frac {4368 b^{2} c^{11} d^{5}}{11}\right ) + x^{10} \cdot \left (\frac {4004 a^{2} c^{10} d^{6}}{5} + \frac {4368 a b c^{11} d^{5}}{5} + 182 b^{2} c^{12} d^{4}\right ) + x^{9} \cdot \left (\frac {1456 a^{2} c^{11} d^{5}}{3} + \frac {3640 a b c^{12} d^{4}}{9} + \frac {560 b^{2} c^{13} d^{3}}{9}\right ) + x^{8} \cdot \left (\frac {455 a^{2} c^{12} d^{4}}{2} + 140 a b c^{13} d^{3} + 15 b^{2} c^{14} d^{2}\right ) + x^{7} \cdot \left (80 a^{2} c^{13} d^{3} + \frac {240 a b c^{14} d^{2}}{7} + \frac {16 b^{2} c^{15} d}{7}\right ) + x^{6} \cdot \left (20 a^{2} c^{14} d^{2} + \frac {16 a b c^{15} d}{3} + \frac {b^{2} c^{16}}{6}\right ) + x^{5} \cdot \left (\frac {16 a^{2} c^{15} d}{5} + \frac {2 a b c^{16}}{5}\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. 668 vs.
\(2 (165) = 330\).
time = 1.60, size = 668, normalized size = 3.77 \begin {gather*} \frac {1}{22} \, b^{2} d^{16} x^{22} + \frac {16}{21} \, b^{2} c d^{15} x^{21} + \frac {2}{21} \, a b d^{16} x^{21} + 6 \, b^{2} c^{2} d^{14} x^{20} + \frac {8}{5} \, a b c d^{15} x^{20} + \frac {1}{20} \, a^{2} d^{16} x^{20} + \frac {560}{19} \, b^{2} c^{3} d^{13} x^{19} + \frac {240}{19} \, a b c^{2} d^{14} x^{19} + \frac {16}{19} \, a^{2} c d^{15} x^{19} + \frac {910}{9} \, b^{2} c^{4} d^{12} x^{18} + \frac {560}{9} \, a b c^{3} d^{13} x^{18} + \frac {20}{3} \, a^{2} c^{2} d^{14} x^{18} + \frac {4368}{17} \, b^{2} c^{5} d^{11} x^{17} + \frac {3640}{17} \, a b c^{4} d^{12} x^{17} + \frac {560}{17} \, a^{2} c^{3} d^{13} x^{17} + \frac {1001}{2} \, b^{2} c^{6} d^{10} x^{16} + 546 \, a b c^{5} d^{11} x^{16} + \frac {455}{4} \, a^{2} c^{4} d^{12} x^{16} + \frac {2288}{3} \, b^{2} c^{7} d^{9} x^{15} + \frac {16016}{15} \, a b c^{6} d^{10} x^{15} + \frac {1456}{5} \, a^{2} c^{5} d^{11} x^{15} + \frac {6435}{7} \, b^{2} c^{8} d^{8} x^{14} + \frac {11440}{7} \, a b c^{7} d^{9} x^{14} + 572 \, a^{2} c^{6} d^{10} x^{14} + 880 \, b^{2} c^{9} d^{7} x^{13} + 1980 \, a b c^{8} d^{8} x^{13} + 880 \, a^{2} c^{7} d^{9} x^{13} + \frac {2002}{3} \, b^{2} c^{10} d^{6} x^{12} + \frac {5720}{3} \, a b c^{9} d^{7} x^{12} + \frac {2145}{2} \, a^{2} c^{8} d^{8} x^{12} + \frac {4368}{11} \, b^{2} c^{11} d^{5} x^{11} + 1456 \, a b c^{10} d^{6} x^{11} + 1040 \, a^{2} c^{9} d^{7} x^{11} + 182 \, b^{2} c^{12} d^{4} x^{10} + \frac {4368}{5} \, a b c^{11} d^{5} x^{10} + \frac {4004}{5} \, a^{2} c^{10} d^{6} x^{10} + \frac {560}{9} \, b^{2} c^{13} d^{3} x^{9} + \frac {3640}{9} \, a b c^{12} d^{4} x^{9} + \frac {1456}{3} \, a^{2} c^{11} d^{5} x^{9} + 15 \, b^{2} c^{14} d^{2} x^{8} + 140 \, a b c^{13} d^{3} x^{8} + \frac {455}{2} \, a^{2} c^{12} d^{4} x^{8} + \frac {16}{7} \, b^{2} c^{15} d x^{7} + \frac {240}{7} \, a b c^{14} d^{2} x^{7} + 80 \, a^{2} c^{13} d^{3} x^{7} + \frac {1}{6} \, b^{2} c^{16} x^{6} + \frac {16}{3} \, a b c^{15} d x^{6} + 20 \, a^{2} c^{14} d^{2} x^{6} + \frac {2}{5} \, a b c^{16} x^{5} + \frac {16}{5} \, a^{2} c^{15} d x^{5} + \frac {1}{4} \, a^{2} c^{16} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.44, size = 557, normalized size = 3.15 \begin {gather*} x^6\,\left (20\,a^2\,c^{14}\,d^2+\frac {16\,a\,b\,c^{15}\,d}{3}+\frac {b^2\,c^{16}}{6}\right )+x^{20}\,\left (\frac {a^2\,d^{16}}{20}+\frac {8\,a\,b\,c\,d^{15}}{5}+6\,b^2\,c^2\,d^{14}\right )+\frac {a^2\,c^{16}\,x^4}{4}+\frac {b^2\,d^{16}\,x^{22}}{22}+\frac {2\,a\,c^{15}\,x^5\,\left (8\,a\,d+b\,c\right )}{5}+\frac {2\,b\,d^{15}\,x^{21}\,\left (a\,d+8\,b\,c\right )}{21}+\frac {16\,c^{13}\,d\,x^7\,\left (35\,a^2\,d^2+15\,a\,b\,c\,d+b^2\,c^2\right )}{7}+\frac {16\,c\,d^{13}\,x^{19}\,\left (a^2\,d^2+15\,a\,b\,c\,d+35\,b^2\,c^2\right )}{19}+220\,c^7\,d^7\,x^{13}\,\left (4\,a^2\,d^2+9\,a\,b\,c\,d+4\,b^2\,c^2\right )+\frac {182\,c^{10}\,d^4\,x^{10}\,\left (22\,a^2\,d^2+24\,a\,b\,c\,d+5\,b^2\,c^2\right )}{5}+\frac {91\,c^4\,d^{10}\,x^{16}\,\left (5\,a^2\,d^2+24\,a\,b\,c\,d+22\,b^2\,c^2\right )}{4}+\frac {5\,c^{12}\,d^2\,x^8\,\left (91\,a^2\,d^2+56\,a\,b\,c\,d+6\,b^2\,c^2\right )}{2}+\frac {56\,c^{11}\,d^3\,x^9\,\left (78\,a^2\,d^2+65\,a\,b\,c\,d+10\,b^2\,c^2\right )}{9}+\frac {208\,c^9\,d^5\,x^{11}\,\left (55\,a^2\,d^2+77\,a\,b\,c\,d+21\,b^2\,c^2\right )}{11}+\frac {143\,c^8\,d^6\,x^{12}\,\left (45\,a^2\,d^2+80\,a\,b\,c\,d+28\,b^2\,c^2\right )}{6}+\frac {143\,c^6\,d^8\,x^{14}\,\left (28\,a^2\,d^2+80\,a\,b\,c\,d+45\,b^2\,c^2\right )}{7}+\frac {208\,c^5\,d^9\,x^{15}\,\left (21\,a^2\,d^2+77\,a\,b\,c\,d+55\,b^2\,c^2\right )}{15}+\frac {56\,c^3\,d^{11}\,x^{17}\,\left (10\,a^2\,d^2+65\,a\,b\,c\,d+78\,b^2\,c^2\right )}{17}+\frac {10\,c^2\,d^{12}\,x^{18}\,\left (6\,a^2\,d^2+56\,a\,b\,c\,d+91\,b^2\,c^2\right )}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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