Optimal. Leaf size=47 \[ a x+\frac {b x^2}{2}+\frac {c x^3}{3}+\frac {1}{6} \left (d+a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^6 \]
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Rubi [A]
time = 0.06, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {1605}
\begin {gather*} \frac {1}{6} \left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}+d\right )^6+a x+\frac {b x^2}{2}+\frac {c x^3}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 1605
Rubi steps
\begin {align*} \int \left (a+b x+c x^2\right ) \left (1+\left (d+a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx &=\text {Subst}\left (\int \left (1+x^5\right ) \, dx,x,d+a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )\\ &=a x+\frac {b x^2}{2}+\frac {c x^3}{3}+\frac {1}{6} \left (d+a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^6\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(248\) vs. \(2(47)=94\).
time = 0.09, size = 248, normalized size = 5.28 \begin {gather*} \frac {x (6 a+x (3 b+2 c x)) \left (46656+46656 d^5+7776 a^5 x^5+243 b^5 x^{10}+810 b^4 c x^{11}+1080 b^3 c^2 x^{12}+720 b^2 c^3 x^{13}+240 b c^4 x^{14}+32 c^5 x^{15}+6480 a^4 x^6 (3 b+2 c x)+2160 a^3 x^7 (3 b+2 c x)^2+360 a^2 x^8 (3 b+2 c x)^3+30 a x^9 (3 b+2 c x)^4+19440 d^4 x (6 a+x (3 b+2 c x))+4320 d^3 x^2 (6 a+x (3 b+2 c x))^2+540 d^2 x^3 (6 a+x (3 b+2 c x))^3+36 d x^4 (6 a+x (3 b+2 c x))^4\right )}{279936} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.29, size = 39, normalized size = 0.83 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. 773 vs.
\(2 (40) = 80\).
time = 0.29, size = 773, normalized size = 16.45 \begin {gather*} \frac {1}{4374} \, c^{6} x^{18} + \frac {1}{486} \, b c^{5} x^{17} + \frac {1}{1944} \, {\left (15 \, b^{2} c^{4} + 8 \, a c^{5}\right )} x^{16} + \frac {1}{972} \, {\left (15 \, b^{3} c^{3} + 30 \, a b c^{4} + 4 \, c^{5} d\right )} x^{15} + \frac {5}{2592} \, {\left (9 \, b^{4} c^{2} + 48 \, a b^{2} c^{3} + 16 \, a^{2} c^{4} + 16 \, b c^{4} d\right )} x^{14} + \frac {1}{2592} \, {\left (27 \, b^{5} c + 360 \, a b^{3} c^{2} + 480 \, a^{2} b c^{3} + 80 \, {\left (3 \, b^{2} c^{3} + 2 \, a c^{4}\right )} d\right )} x^{13} + \frac {1}{10368} \, {\left (27 \, b^{6} + 1080 \, a b^{4} c + 4320 \, a^{2} b^{2} c^{2} + 1280 \, a^{3} c^{3} + 320 \, c^{4} d^{2} + 480 \, {\left (3 \, b^{3} c^{2} + 8 \, a b c^{3}\right )} d\right )} x^{12} + \frac {1}{864} \, {\left (27 \, a b^{5} + 360 \, a^{2} b^{3} c + 480 \, a^{3} b c^{2} + 160 \, b c^{3} d^{2} + 10 \, {\left (9 \, b^{4} c + 72 \, a b^{2} c^{2} + 32 \, a^{2} c^{3}\right )} d\right )} x^{11} + \frac {1}{864} \, {\left (135 \, a^{2} b^{4} + 720 \, a^{3} b^{2} c + 240 \, a^{4} c^{2} + 40 \, {\left (9 \, b^{2} c^{2} + 8 \, a c^{3}\right )} d^{2} + 9 \, {\left (3 \, b^{5} + 80 \, a b^{3} c + 160 \, a^{2} b c^{2}\right )} d\right )} x^{10} + \frac {5}{1296} \, {\left (108 \, a^{3} b^{3} + 216 \, a^{4} b c + 32 \, c^{3} d^{3} + 108 \, {\left (b^{3} c + 4 \, a b c^{2}\right )} d^{2} + 9 \, {\left (9 \, a b^{4} + 72 \, a^{2} b^{2} c + 32 \, a^{3} c^{2}\right )} d\right )} x^{9} + \frac {1}{288} \, {\left (180 \, a^{4} b^{2} + 96 \, a^{5} c + 160 \, b c^{2} d^{3} + 15 \, {\left (3 \, b^{4} + 48 \, a b^{2} c + 32 \, a^{2} c^{2}\right )} d^{2} + 120 \, {\left (3 \, a^{2} b^{3} + 8 \, a^{3} b c\right )} d\right )} x^{8} + \frac {1}{36} \, {\left (18 \, a^{5} b + 10 \, {\left (3 \, b^{2} c + 4 \, a c^{2}\right )} d^{3} + 45 \, {\left (a b^{3} + 4 \, a^{2} b c\right )} d^{2} + 30 \, {\left (3 \, a^{3} b^{2} + 2 \, a^{4} c\right )} d\right )} x^{7} + \frac {1}{36} \, {\left (6 \, a^{6} + 90 \, a^{4} b d + 10 \, c^{2} d^{4} + 15 \, {\left (b^{3} + 8 \, a b c\right )} d^{3} + 15 \, {\left (9 \, a^{2} b^{2} + 8 \, a^{3} c\right )} d^{2}\right )} x^{6} + \frac {1}{6} \, {\left (6 \, a^{5} d + 30 \, a^{3} b d^{2} + 5 \, b c d^{4} + 5 \, {\left (3 \, a b^{2} + 4 \, a^{2} c\right )} d^{3}\right )} x^{5} + \frac {5}{24} \, {\left (12 \, a^{4} d^{2} + 24 \, a^{2} b d^{3} + {\left (3 \, b^{2} + 8 \, a c\right )} d^{4}\right )} x^{4} + \frac {1}{6} \, {\left (20 \, a^{3} d^{3} + 15 \, a b d^{4} + 2 \, c d^{5} + 2 \, c\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{2} d^{4} + b d^{5} + b\right )} x^{2} + {\left (a d^{5} + a\right )} x \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. 773 vs.
\(2 (40) = 80\).
time = 0.38, size = 773, normalized size = 16.45 \begin {gather*} \frac {1}{4374} \, c^{6} x^{18} + \frac {1}{486} \, b c^{5} x^{17} + \frac {1}{1944} \, {\left (15 \, b^{2} c^{4} + 8 \, a c^{5}\right )} x^{16} + \frac {1}{972} \, {\left (15 \, b^{3} c^{3} + 30 \, a b c^{4} + 4 \, c^{5} d\right )} x^{15} + \frac {5}{2592} \, {\left (9 \, b^{4} c^{2} + 48 \, a b^{2} c^{3} + 16 \, a^{2} c^{4} + 16 \, b c^{4} d\right )} x^{14} + \frac {1}{2592} \, {\left (27 \, b^{5} c + 360 \, a b^{3} c^{2} + 480 \, a^{2} b c^{3} + 80 \, {\left (3 \, b^{2} c^{3} + 2 \, a c^{4}\right )} d\right )} x^{13} + \frac {1}{10368} \, {\left (27 \, b^{6} + 1080 \, a b^{4} c + 4320 \, a^{2} b^{2} c^{2} + 1280 \, a^{3} c^{3} + 320 \, c^{4} d^{2} + 480 \, {\left (3 \, b^{3} c^{2} + 8 \, a b c^{3}\right )} d\right )} x^{12} + \frac {1}{864} \, {\left (27 \, a b^{5} + 360 \, a^{2} b^{3} c + 480 \, a^{3} b c^{2} + 160 \, b c^{3} d^{2} + 10 \, {\left (9 \, b^{4} c + 72 \, a b^{2} c^{2} + 32 \, a^{2} c^{3}\right )} d\right )} x^{11} + \frac {1}{864} \, {\left (135 \, a^{2} b^{4} + 720 \, a^{3} b^{2} c + 240 \, a^{4} c^{2} + 40 \, {\left (9 \, b^{2} c^{2} + 8 \, a c^{3}\right )} d^{2} + 9 \, {\left (3 \, b^{5} + 80 \, a b^{3} c + 160 \, a^{2} b c^{2}\right )} d\right )} x^{10} + \frac {5}{1296} \, {\left (108 \, a^{3} b^{3} + 216 \, a^{4} b c + 32 \, c^{3} d^{3} + 108 \, {\left (b^{3} c + 4 \, a b c^{2}\right )} d^{2} + 9 \, {\left (9 \, a b^{4} + 72 \, a^{2} b^{2} c + 32 \, a^{3} c^{2}\right )} d\right )} x^{9} + \frac {1}{288} \, {\left (180 \, a^{4} b^{2} + 96 \, a^{5} c + 160 \, b c^{2} d^{3} + 15 \, {\left (3 \, b^{4} + 48 \, a b^{2} c + 32 \, a^{2} c^{2}\right )} d^{2} + 120 \, {\left (3 \, a^{2} b^{3} + 8 \, a^{3} b c\right )} d\right )} x^{8} + \frac {1}{36} \, {\left (18 \, a^{5} b + 10 \, {\left (3 \, b^{2} c + 4 \, a c^{2}\right )} d^{3} + 45 \, {\left (a b^{3} + 4 \, a^{2} b c\right )} d^{2} + 30 \, {\left (3 \, a^{3} b^{2} + 2 \, a^{4} c\right )} d\right )} x^{7} + \frac {1}{36} \, {\left (6 \, a^{6} + 90 \, a^{4} b d + 10 \, c^{2} d^{4} + 15 \, {\left (b^{3} + 8 \, a b c\right )} d^{3} + 15 \, {\left (9 \, a^{2} b^{2} + 8 \, a^{3} c\right )} d^{2}\right )} x^{6} + \frac {1}{6} \, {\left (6 \, a^{5} d + 30 \, a^{3} b d^{2} + 5 \, b c d^{4} + 5 \, {\left (3 \, a b^{2} + 4 \, a^{2} c\right )} d^{3}\right )} x^{5} + \frac {5}{24} \, {\left (12 \, a^{4} d^{2} + 24 \, a^{2} b d^{3} + {\left (3 \, b^{2} + 8 \, a c\right )} d^{4}\right )} x^{4} + \frac {1}{6} \, {\left (20 \, a^{3} d^{3} + 15 \, a b d^{4} + 2 \, c d^{5} + 2 \, c\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{2} d^{4} + b d^{5} + b\right )} x^{2} + {\left (a d^{5} + a\right )} x \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. 930 vs.
\(2 (37) = 74\).
time = 0.10, size = 930, normalized size = 19.79 \begin {gather*} \frac {b c^{5} x^{17}}{486} + \frac {c^{6} x^{18}}{4374} + x^{16} \left (\frac {a c^{5}}{243} + \frac {5 b^{2} c^{4}}{648}\right ) + x^{15} \cdot \left (\frac {5 a b c^{4}}{162} + \frac {5 b^{3} c^{3}}{324} + \frac {c^{5} d}{243}\right ) + x^{14} \cdot \left (\frac {5 a^{2} c^{4}}{162} + \frac {5 a b^{2} c^{3}}{54} + \frac {5 b^{4} c^{2}}{288} + \frac {5 b c^{4} d}{162}\right ) + x^{13} \cdot \left (\frac {5 a^{2} b c^{3}}{27} + \frac {5 a b^{3} c^{2}}{36} + \frac {5 a c^{4} d}{81} + \frac {b^{5} c}{96} + \frac {5 b^{2} c^{3} d}{54}\right ) + x^{12} \cdot \left (\frac {10 a^{3} c^{3}}{81} + \frac {5 a^{2} b^{2} c^{2}}{12} + \frac {5 a b^{4} c}{48} + \frac {10 a b c^{3} d}{27} + \frac {b^{6}}{384} + \frac {5 b^{3} c^{2} d}{36} + \frac {5 c^{4} d^{2}}{162}\right ) + x^{11} \cdot \left (\frac {5 a^{3} b c^{2}}{9} + \frac {5 a^{2} b^{3} c}{12} + \frac {10 a^{2} c^{3} d}{27} + \frac {a b^{5}}{32} + \frac {5 a b^{2} c^{2} d}{6} + \frac {5 b^{4} c d}{48} + \frac {5 b c^{3} d^{2}}{27}\right ) + x^{10} \cdot \left (\frac {5 a^{4} c^{2}}{18} + \frac {5 a^{3} b^{2} c}{6} + \frac {5 a^{2} b^{4}}{32} + \frac {5 a^{2} b c^{2} d}{3} + \frac {5 a b^{3} c d}{6} + \frac {10 a c^{3} d^{2}}{27} + \frac {b^{5} d}{32} + \frac {5 b^{2} c^{2} d^{2}}{12}\right ) + x^{9} \cdot \left (\frac {5 a^{4} b c}{6} + \frac {5 a^{3} b^{3}}{12} + \frac {10 a^{3} c^{2} d}{9} + \frac {5 a^{2} b^{2} c d}{2} + \frac {5 a b^{4} d}{16} + \frac {5 a b c^{2} d^{2}}{3} + \frac {5 b^{3} c d^{2}}{12} + \frac {10 c^{3} d^{3}}{81}\right ) + x^{8} \left (\frac {a^{5} c}{3} + \frac {5 a^{4} b^{2}}{8} + \frac {10 a^{3} b c d}{3} + \frac {5 a^{2} b^{3} d}{4} + \frac {5 a^{2} c^{2} d^{2}}{3} + \frac {5 a b^{2} c d^{2}}{2} + \frac {5 b^{4} d^{2}}{32} + \frac {5 b c^{2} d^{3}}{9}\right ) + x^{7} \left (\frac {a^{5} b}{2} + \frac {5 a^{4} c d}{3} + \frac {5 a^{3} b^{2} d}{2} + 5 a^{2} b c d^{2} + \frac {5 a b^{3} d^{2}}{4} + \frac {10 a c^{2} d^{3}}{9} + \frac {5 b^{2} c d^{3}}{6}\right ) + x^{6} \left (\frac {a^{6}}{6} + \frac {5 a^{4} b d}{2} + \frac {10 a^{3} c d^{2}}{3} + \frac {15 a^{2} b^{2} d^{2}}{4} + \frac {10 a b c d^{3}}{3} + \frac {5 b^{3} d^{3}}{12} + \frac {5 c^{2} d^{4}}{18}\right ) + x^{5} \left (a^{5} d + 5 a^{3} b d^{2} + \frac {10 a^{2} c d^{3}}{3} + \frac {5 a b^{2} d^{3}}{2} + \frac {5 b c d^{4}}{6}\right ) + x^{4} \cdot \left (\frac {5 a^{4} d^{2}}{2} + 5 a^{2} b d^{3} + \frac {5 a c d^{4}}{3} + \frac {5 b^{2} d^{4}}{8}\right ) + x^{3} \cdot \left (\frac {10 a^{3} d^{3}}{3} + \frac {5 a b d^{4}}{2} + \frac {c d^{5}}{3} + \frac {c}{3}\right ) + x^{2} \cdot \left (\frac {5 a^{2} d^{4}}{2} + \frac {b d^{5}}{2} + \frac {b}{2}\right ) + x \left (a d^{5} + a\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. 153 vs.
\(2 (40) = 80\).
time = 3.60, size = 153, normalized size = 3.26 \begin {gather*} \frac {1}{279936} \, {\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )}^{6} + \frac {1}{7776} \, {\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )}^{5} d + \frac {5}{2592} \, {\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )}^{4} d^{2} + \frac {5}{324} \, {\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )}^{3} d^{3} + \frac {5}{72} \, {\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )}^{2} d^{4} + \frac {1}{6} \, {\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )} d^{5} + \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.45, size = 753, normalized size = 16.02 \begin {gather*} x^{10}\,\left (\frac {5\,a^4\,c^2}{18}+\frac {5\,a^3\,b^2\,c}{6}+\frac {5\,a^2\,b^4}{32}+\frac {5\,a^2\,b\,c^2\,d}{3}+\frac {5\,a\,b^3\,c\,d}{6}+\frac {10\,a\,c^3\,d^2}{27}+\frac {b^5\,d}{32}+\frac {5\,b^2\,c^2\,d^2}{12}\right )+x^8\,\left (\frac {a^5\,c}{3}+\frac {5\,a^4\,b^2}{8}+\frac {10\,a^3\,b\,c\,d}{3}+\frac {5\,a^2\,b^3\,d}{4}+\frac {5\,a^2\,c^2\,d^2}{3}+\frac {5\,a\,b^2\,c\,d^2}{2}+\frac {5\,b^4\,d^2}{32}+\frac {5\,b\,c^2\,d^3}{9}\right )+x^9\,\left (\frac {5\,a^4\,b\,c}{6}+\frac {5\,a^3\,b^3}{12}+\frac {10\,a^3\,c^2\,d}{9}+\frac {5\,a^2\,b^2\,c\,d}{2}+\frac {5\,a\,b^4\,d}{16}+\frac {5\,a\,b\,c^2\,d^2}{3}+\frac {5\,b^3\,c\,d^2}{12}+\frac {10\,c^3\,d^3}{81}\right )+x^{14}\,\left (\frac {5\,a^2\,c^4}{162}+\frac {5\,a\,b^2\,c^3}{54}+\frac {5\,b^4\,c^2}{288}+\frac {5\,d\,b\,c^4}{162}\right )+x^{12}\,\left (\frac {10\,a^3\,c^3}{81}+\frac {5\,a^2\,b^2\,c^2}{12}+\frac {5\,a\,b^4\,c}{48}+\frac {10\,a\,b\,c^3\,d}{27}+\frac {b^6}{384}+\frac {5\,b^3\,c^2\,d}{36}+\frac {5\,c^4\,d^2}{162}\right )+x^6\,\left (\frac {a^6}{6}+\frac {5\,a^4\,b\,d}{2}+\frac {10\,a^3\,c\,d^2}{3}+\frac {15\,a^2\,b^2\,d^2}{4}+\frac {10\,a\,b\,c\,d^3}{3}+\frac {5\,b^3\,d^3}{12}+\frac {5\,c^2\,d^4}{18}\right )+x^3\,\left (\frac {10\,a^3\,d^3}{3}+\frac {5\,b\,a\,d^4}{2}+\frac {c\,d^5}{3}+\frac {c}{3}\right )+x^{11}\,\left (\frac {5\,a^3\,b\,c^2}{9}+\frac {5\,a^2\,b^3\,c}{12}+\frac {10\,a^2\,c^3\,d}{27}+\frac {a\,b^5}{32}+\frac {5\,a\,b^2\,c^2\,d}{6}+\frac {5\,b^4\,c\,d}{48}+\frac {5\,b\,c^3\,d^2}{27}\right )+x^7\,\left (\frac {a^5\,b}{2}+\frac {5\,a^4\,c\,d}{3}+\frac {5\,a^3\,b^2\,d}{2}+5\,a^2\,b\,c\,d^2+\frac {5\,a\,b^3\,d^2}{4}+\frac {10\,a\,c^2\,d^3}{9}+\frac {5\,b^2\,c\,d^3}{6}\right )+x^2\,\left (\frac {5\,a^2\,d^4}{2}+\frac {b\,d^5}{2}+\frac {b}{2}\right )+x^{13}\,\left (\frac {5\,a^2\,b\,c^3}{27}+\frac {5\,a\,b^3\,c^2}{36}+\frac {5\,d\,a\,c^4}{81}+\frac {b^5\,c}{96}+\frac {5\,d\,b^2\,c^3}{54}\right )+x^5\,\left (a^5\,d+5\,a^3\,b\,d^2+\frac {10\,c\,a^2\,d^3}{3}+\frac {5\,a\,b^2\,d^3}{2}+\frac {5\,c\,b\,d^4}{6}\right )+\frac {c^6\,x^{18}}{4374}+\frac {5\,d^2\,x^4\,\left (12\,a^4+24\,a^2\,b\,d+8\,c\,a\,d^2+3\,b^2\,d^2\right )}{24}+a\,x\,\left (d^5+1\right )+\frac {b\,c^5\,x^{17}}{486}+\frac {c^3\,x^{15}\,\left (15\,b^3+30\,a\,b\,c+4\,d\,c^2\right )}{972}+\frac {c^4\,x^{16}\,\left (15\,b^2+8\,a\,c\right )}{1944} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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