Optimal. Leaf size=159 \[ \frac {a^3 f^3 (d+e x)^4}{4 e}+\frac {a^2 b f^3 (d+e x)^6}{2 e}+\frac {3 a \left (b^2+a c\right ) f^3 (d+e x)^8}{8 e}+\frac {b \left (b^2+6 a c\right ) f^3 (d+e x)^{10}}{10 e}+\frac {c \left (b^2+a c\right ) f^3 (d+e x)^{12}}{4 e}+\frac {3 b c^2 f^3 (d+e x)^{14}}{14 e}+\frac {c^3 f^3 (d+e x)^{16}}{16 e} \]
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Rubi [A]
time = 0.21, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1156, 1128,
645} \begin {gather*} \frac {a^3 f^3 (d+e x)^4}{4 e}+\frac {a^2 b f^3 (d+e x)^6}{2 e}+\frac {c f^3 \left (a c+b^2\right ) (d+e x)^{12}}{4 e}+\frac {b f^3 \left (6 a c+b^2\right ) (d+e x)^{10}}{10 e}+\frac {3 a f^3 \left (a c+b^2\right ) (d+e x)^8}{8 e}+\frac {3 b c^2 f^3 (d+e x)^{14}}{14 e}+\frac {c^3 f^3 (d+e x)^{16}}{16 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 645
Rule 1128
Rule 1156
Rubi steps
\begin {align*} \int (d f+e f x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )^3 \, dx &=\frac {f^3 \text {Subst}\left (\int x^3 \left (a+b x^2+c x^4\right )^3 \, dx,x,d+e x\right )}{e}\\ &=\frac {f^3 \text {Subst}\left (\int x \left (a+b x+c x^2\right )^3 \, dx,x,(d+e x)^2\right )}{2 e}\\ &=\frac {f^3 \text {Subst}\left (\int \left (a^3 x+3 a^2 b x^2+3 a \left (b^2+a c\right ) x^3+b \left (b^2+6 a c\right ) x^4+3 c \left (b^2+a c\right ) x^5+3 b c^2 x^6+c^3 x^7\right ) \, dx,x,(d+e x)^2\right )}{2 e}\\ &=\frac {a^3 f^3 (d+e x)^4}{4 e}+\frac {a^2 b f^3 (d+e x)^6}{2 e}+\frac {3 a \left (b^2+a c\right ) f^3 (d+e x)^8}{8 e}+\frac {b \left (b^2+6 a c\right ) f^3 (d+e x)^{10}}{10 e}+\frac {c \left (b^2+a c\right ) f^3 (d+e x)^{12}}{4 e}+\frac {3 b c^2 f^3 (d+e x)^{14}}{14 e}+\frac {c^3 f^3 (d+e x)^{16}}{16 e}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(801\) vs. \(2(159)=318\).
time = 0.03, size = 801, normalized size = 5.04 \begin {gather*} f^3 \left (d^3 \left (a+b d^2+c d^4\right )^3 x+\frac {3}{2} d^2 \left (a+b d^2+c d^4\right )^2 \left (a+3 b d^2+5 c d^4\right ) e x^2+d \left (a^3+10 a^2 b d^2+21 a b^2 d^4+21 a^2 c d^4+12 b^3 d^6+72 a b c d^6+55 b^2 c d^8+55 a c^2 d^8+78 b c^2 d^{10}+35 c^3 d^{12}\right ) e^2 x^3+\frac {1}{4} \left (a^3+30 a^2 b d^2+105 a b^2 d^4+105 a^2 c d^4+84 b^3 d^6+504 a b c d^6+495 b^2 c d^8+495 a c^2 d^8+858 b c^2 d^{10}+455 c^3 d^{12}\right ) e^3 x^4+\frac {3}{5} d \left (5 a^2 b+35 a b^2 d^2+35 a^2 c d^2+42 b^3 d^4+252 a b c d^4+330 b^2 c d^6+330 a c^2 d^6+715 b c^2 d^8+455 c^3 d^{10}\right ) e^4 x^5+\frac {1}{2} \left (a^2 b+21 a b^2 d^2+21 a^2 c d^2+42 b^3 d^4+252 a b c d^4+462 b^2 c d^6+462 a c^2 d^6+1287 b c^2 d^8+1001 c^3 d^{10}\right ) e^5 x^6+\frac {1}{7} d \left (21 a b^2+21 a^2 c+84 b^3 d^2+504 a b c d^2+1386 b^2 c d^4+1386 a c^2 d^4+5148 b c^2 d^6+5005 c^3 d^8\right ) e^6 x^7+\frac {3}{8} \left (a b^2+a^2 c+12 b^3 d^2+72 a b c d^2+330 b^2 c d^4+330 a c^2 d^4+1716 b c^2 d^6+2145 c^3 d^8\right ) e^7 x^8+d \left (b^3+6 a b c+55 b^2 c d^2+55 a c^2 d^2+429 b c^2 d^4+715 c^3 d^6\right ) e^8 x^9+\frac {1}{10} \left (b^3+6 a b c+165 b^2 c d^2+165 a c^2 d^2+2145 b c^2 d^4+5005 c^3 d^6\right ) e^9 x^{10}+3 c d \left (b^2+a c+26 b c d^2+91 c^2 d^4\right ) e^{10} x^{11}+\frac {1}{4} c \left (b^2+a c+78 b c d^2+455 c^2 d^4\right ) e^{11} x^{12}+c^2 d \left (3 b+35 c d^2\right ) e^{12} x^{13}+\frac {3}{14} c^2 \left (b+35 c d^2\right ) e^{13} x^{14}+c^3 d e^{14} x^{15}+\frac {1}{16} c^3 e^{15} x^{16}\right ) \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. \(7696\) vs.
\(2(145)=290\).
time = 0.24, size = 7697, normalized size = 48.41
method | result | size |
gosper | \(\frac {f^{3} x \left (35 e^{15} c^{3} x^{15}+560 d \,e^{14} c^{3} x^{14}+4200 x^{13} d^{2} e^{13} c^{3}+19600 c^{3} d^{3} e^{12} x^{12}+120 x^{13} b \,c^{2} e^{13}+63700 x^{11} d^{4} e^{11} c^{3}+1680 b \,c^{2} d \,e^{12} x^{12}+152880 c^{3} d^{5} e^{10} x^{10}+10920 x^{11} b \,c^{2} d^{2} e^{11}+280280 x^{9} c^{3} d^{6} e^{9}+43680 b \,c^{2} d^{3} e^{10} x^{10}+400400 c^{3} d^{7} e^{8} x^{8}+140 x^{11} a \,c^{2} e^{11}+140 x^{11} b^{2} c \,e^{11}+120120 x^{9} b \,c^{2} d^{4} e^{9}+450450 x^{7} c^{3} d^{8} e^{7}+1680 a \,c^{2} d \,e^{10} x^{10}+1680 b^{2} c d \,e^{10} x^{10}+240240 b \,c^{2} d^{5} e^{8} x^{8}+400400 x^{6} c^{3} d^{9} e^{6}+9240 x^{9} a \,c^{2} d^{2} e^{9}+9240 x^{9} b^{2} c \,d^{2} e^{9}+360360 x^{7} b \,c^{2} d^{6} e^{7}+280280 x^{5} c^{3} d^{10} e^{5}+30800 a \,c^{2} d^{3} e^{8} x^{8}+30800 b^{2} c \,d^{3} e^{8} x^{8}+411840 x^{6} b \,c^{2} d^{7} e^{6}+152880 x^{4} c^{3} d^{11} e^{4}+336 x^{9} a b c \,e^{9}+69300 x^{7} a \,c^{2} d^{4} e^{7}+56 x^{9} b^{3} e^{9}+69300 x^{7} b^{2} c \,d^{4} e^{7}+360360 x^{5} b \,c^{2} d^{8} e^{5}+63700 x^{3} c^{3} d^{12} e^{3}+3360 a b c d \,e^{8} x^{8}+110880 x^{6} a \,c^{2} d^{5} e^{6}+560 b^{3} d \,e^{8} x^{8}+110880 x^{6} b^{2} c \,d^{5} e^{6}+240240 x^{4} b \,c^{2} d^{9} e^{4}+19600 c^{3} d^{13} e^{2} x^{2}+15120 x^{7} a b c \,d^{2} e^{7}+129360 x^{5} a \,c^{2} d^{6} e^{5}+2520 x^{7} b^{3} d^{2} e^{7}+129360 x^{5} b^{2} c \,d^{6} e^{5}+120120 x^{3} b \,c^{2} d^{10} e^{3}+4200 x \,c^{3} d^{14} e +40320 x^{6} a b c \,d^{3} e^{6}+110880 x^{4} a \,c^{2} d^{7} e^{4}+6720 x^{6} b^{3} d^{3} e^{6}+110880 x^{4} b^{2} c \,d^{7} e^{4}+43680 b \,c^{2} d^{11} e^{2} x^{2}+560 c^{3} d^{15}+210 x^{7} a^{2} c \,e^{7}+210 x^{7} a \,b^{2} e^{7}+70560 x^{5} a b c \,d^{4} e^{5}+69300 x^{3} a \,c^{2} d^{8} e^{3}+11760 x^{5} b^{3} d^{4} e^{5}+69300 x^{3} b^{2} c \,d^{8} e^{3}+10920 x b \,c^{2} d^{12} e +1680 x^{6} a^{2} c d \,e^{6}+1680 x^{6} a \,b^{2} d \,e^{6}+84672 x^{4} a b c \,d^{5} e^{4}+30800 a \,c^{2} d^{9} e^{2} x^{2}+14112 x^{4} b^{3} d^{5} e^{4}+30800 b^{2} c \,d^{9} e^{2} x^{2}+1680 b \,c^{2} d^{13}+5880 x^{5} a^{2} c \,d^{2} e^{5}+5880 x^{5} a \,b^{2} d^{2} e^{5}+70560 x^{3} a b c \,d^{6} e^{3}+9240 x a \,c^{2} d^{10} e +11760 x^{3} b^{3} d^{6} e^{3}+9240 x \,b^{2} c \,d^{10} e +11760 x^{4} a^{2} c \,d^{3} e^{4}+11760 x^{4} a \,b^{2} d^{3} e^{4}+40320 a b c \,d^{7} e^{2} x^{2}+1680 a \,c^{2} d^{11}+6720 b^{3} d^{7} e^{2} x^{2}+1680 b^{2} c \,d^{11}+280 x^{5} a^{2} b \,e^{5}+14700 x^{3} a^{2} c \,d^{4} e^{3}+14700 x^{3} a \,b^{2} d^{4} e^{3}+15120 x a b c \,d^{8} e +2520 x \,b^{3} d^{8} e +1680 x^{4} a^{2} b d \,e^{4}+11760 a^{2} c \,d^{5} e^{2} x^{2}+11760 a \,b^{2} d^{5} e^{2} x^{2}+3360 a b c \,d^{9}+560 b^{3} d^{9}+4200 x^{3} a^{2} b \,d^{2} e^{3}+5880 x \,a^{2} c \,d^{6} e +5880 x a \,b^{2} d^{6} e +5600 a^{2} b \,d^{3} e^{2} x^{2}+1680 a^{2} c \,d^{7}+1680 a \,b^{2} d^{7}+140 x^{3} a^{3} e^{3}+4200 x \,a^{2} b \,d^{4} e +560 a^{3} d \,e^{2} x^{2}+1680 a^{2} b \,d^{5}+840 x \,a^{3} d^{2} e +560 a^{3} d^{3}\right )}{560}\) | \(1318\) |
norman | \(\text {Expression too large to display}\) | \(1430\) |
risch | \(\text {Expression too large to display}\) | \(1636\) |
default | \(\text {Expression too large to display}\) | \(7697\) |
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. 1067 vs.
\(2 (145) = 290\).
time = 0.28, size = 1067, normalized size = 6.71 \begin {gather*} \frac {1}{16} \, c^{3} f^{3} x^{16} e^{15} + c^{3} d f^{3} x^{15} e^{14} + \frac {3}{14} \, {\left (35 \, c^{3} d^{2} e^{13} + b c^{2} e^{13}\right )} f^{3} x^{14} + {\left (35 \, c^{3} d^{3} e^{12} + 3 \, b c^{2} d e^{12}\right )} f^{3} x^{13} + \frac {1}{4} \, {\left (455 \, c^{3} d^{4} e^{11} + 78 \, b c^{2} d^{2} e^{11} + b^{2} c e^{11} + a c^{2} e^{11}\right )} f^{3} x^{12} + 3 \, {\left (91 \, c^{3} d^{5} e^{10} + 26 \, b c^{2} d^{3} e^{10} + {\left (b^{2} c e^{10} + a c^{2} e^{10}\right )} d\right )} f^{3} x^{11} + \frac {1}{10} \, {\left (5005 \, c^{3} d^{6} e^{9} + 2145 \, b c^{2} d^{4} e^{9} + b^{3} e^{9} + 6 \, a b c e^{9} + 165 \, {\left (b^{2} c e^{9} + a c^{2} e^{9}\right )} d^{2}\right )} f^{3} x^{10} + {\left (715 \, c^{3} d^{7} e^{8} + 429 \, b c^{2} d^{5} e^{8} + 55 \, {\left (b^{2} c e^{8} + a c^{2} e^{8}\right )} d^{3} + {\left (b^{3} e^{8} + 6 \, a b c e^{8}\right )} d\right )} f^{3} x^{9} + \frac {3}{8} \, {\left (2145 \, c^{3} d^{8} e^{7} + 1716 \, b c^{2} d^{6} e^{7} + 330 \, {\left (b^{2} c e^{7} + a c^{2} e^{7}\right )} d^{4} + a b^{2} e^{7} + a^{2} c e^{7} + 12 \, {\left (b^{3} e^{7} + 6 \, a b c e^{7}\right )} d^{2}\right )} f^{3} x^{8} + \frac {1}{7} \, {\left (5005 \, c^{3} d^{9} e^{6} + 5148 \, b c^{2} d^{7} e^{6} + 1386 \, {\left (b^{2} c e^{6} + a c^{2} e^{6}\right )} d^{5} + 84 \, {\left (b^{3} e^{6} + 6 \, a b c e^{6}\right )} d^{3} + 21 \, {\left (a b^{2} e^{6} + a^{2} c e^{6}\right )} d\right )} f^{3} x^{7} + \frac {1}{2} \, {\left (1001 \, c^{3} d^{10} e^{5} + 1287 \, b c^{2} d^{8} e^{5} + 462 \, {\left (b^{2} c e^{5} + a c^{2} e^{5}\right )} d^{6} + 42 \, {\left (b^{3} e^{5} + 6 \, a b c e^{5}\right )} d^{4} + a^{2} b e^{5} + 21 \, {\left (a b^{2} e^{5} + a^{2} c e^{5}\right )} d^{2}\right )} f^{3} x^{6} + \frac {3}{5} \, {\left (455 \, c^{3} d^{11} e^{4} + 715 \, b c^{2} d^{9} e^{4} + 330 \, {\left (b^{2} c e^{4} + a c^{2} e^{4}\right )} d^{7} + 42 \, {\left (b^{3} e^{4} + 6 \, a b c e^{4}\right )} d^{5} + 5 \, a^{2} b d e^{4} + 35 \, {\left (a b^{2} e^{4} + a^{2} c e^{4}\right )} d^{3}\right )} f^{3} x^{5} + \frac {1}{4} \, {\left (455 \, c^{3} d^{12} e^{3} + 858 \, b c^{2} d^{10} e^{3} + 495 \, {\left (b^{2} c e^{3} + a c^{2} e^{3}\right )} d^{8} + 84 \, {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d^{6} + 30 \, a^{2} b d^{2} e^{3} + 105 \, {\left (a b^{2} e^{3} + a^{2} c e^{3}\right )} d^{4} + a^{3} e^{3}\right )} f^{3} x^{4} + {\left (35 \, c^{3} d^{13} e^{2} + 78 \, b c^{2} d^{11} e^{2} + 55 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{9} + 12 \, {\left (b^{3} e^{2} + 6 \, a b c e^{2}\right )} d^{7} + 10 \, a^{2} b d^{3} e^{2} + 21 \, {\left (a b^{2} e^{2} + a^{2} c e^{2}\right )} d^{5} + a^{3} d e^{2}\right )} f^{3} x^{3} + \frac {3}{2} \, {\left (5 \, c^{3} d^{14} e + 13 \, b c^{2} d^{12} e + 11 \, {\left (b^{2} c e + a c^{2} e\right )} d^{10} + 3 \, {\left (b^{3} e + 6 \, a b c e\right )} d^{8} + 5 \, a^{2} b d^{4} e + 7 \, {\left (a b^{2} e + a^{2} c e\right )} d^{6} + a^{3} d^{2} e\right )} f^{3} x^{2} + {\left (c^{3} d^{15} + 3 \, b c^{2} d^{13} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{11} + {\left (b^{3} + 6 \, a b c\right )} d^{9} + 3 \, a^{2} b d^{5} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{7} + a^{3} d^{3}\right )} f^{3} 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. 907 vs.
\(2 (145) = 290\).
time = 0.36, size = 907, normalized size = 5.70 \begin {gather*} \frac {1}{16} \, c^{3} f^{3} x^{16} e^{15} + c^{3} d f^{3} x^{15} e^{14} + \frac {3}{14} \, {\left (35 \, c^{3} d^{2} + b c^{2}\right )} f^{3} x^{14} e^{13} + {\left (35 \, c^{3} d^{3} + 3 \, b c^{2} d\right )} f^{3} x^{13} e^{12} + \frac {1}{4} \, {\left (455 \, c^{3} d^{4} + 78 \, b c^{2} d^{2} + b^{2} c + a c^{2}\right )} f^{3} x^{12} e^{11} + 3 \, {\left (91 \, c^{3} d^{5} + 26 \, b c^{2} d^{3} + {\left (b^{2} c + a c^{2}\right )} d\right )} f^{3} x^{11} e^{10} + \frac {1}{10} \, {\left (5005 \, c^{3} d^{6} + 2145 \, b c^{2} d^{4} + b^{3} + 6 \, a b c + 165 \, {\left (b^{2} c + a c^{2}\right )} d^{2}\right )} f^{3} x^{10} e^{9} + {\left (715 \, c^{3} d^{7} + 429 \, b c^{2} d^{5} + 55 \, {\left (b^{2} c + a c^{2}\right )} d^{3} + {\left (b^{3} + 6 \, a b c\right )} d\right )} f^{3} x^{9} e^{8} + \frac {3}{8} \, {\left (2145 \, c^{3} d^{8} + 1716 \, b c^{2} d^{6} + 330 \, {\left (b^{2} c + a c^{2}\right )} d^{4} + a b^{2} + a^{2} c + 12 \, {\left (b^{3} + 6 \, a b c\right )} d^{2}\right )} f^{3} x^{8} e^{7} + \frac {1}{7} \, {\left (5005 \, c^{3} d^{9} + 5148 \, b c^{2} d^{7} + 1386 \, {\left (b^{2} c + a c^{2}\right )} d^{5} + 84 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} + 21 \, {\left (a b^{2} + a^{2} c\right )} d\right )} f^{3} x^{7} e^{6} + \frac {1}{2} \, {\left (1001 \, c^{3} d^{10} + 1287 \, b c^{2} d^{8} + 462 \, {\left (b^{2} c + a c^{2}\right )} d^{6} + 42 \, {\left (b^{3} + 6 \, a b c\right )} d^{4} + a^{2} b + 21 \, {\left (a b^{2} + a^{2} c\right )} d^{2}\right )} f^{3} x^{6} e^{5} + \frac {3}{5} \, {\left (455 \, c^{3} d^{11} + 715 \, b c^{2} d^{9} + 330 \, {\left (b^{2} c + a c^{2}\right )} d^{7} + 42 \, {\left (b^{3} + 6 \, a b c\right )} d^{5} + 5 \, a^{2} b d + 35 \, {\left (a b^{2} + a^{2} c\right )} d^{3}\right )} f^{3} x^{5} e^{4} + \frac {1}{4} \, {\left (455 \, c^{3} d^{12} + 858 \, b c^{2} d^{10} + 495 \, {\left (b^{2} c + a c^{2}\right )} d^{8} + 84 \, {\left (b^{3} + 6 \, a b c\right )} d^{6} + 30 \, a^{2} b d^{2} + 105 \, {\left (a b^{2} + a^{2} c\right )} d^{4} + a^{3}\right )} f^{3} x^{4} e^{3} + {\left (35 \, c^{3} d^{13} + 78 \, b c^{2} d^{11} + 55 \, {\left (b^{2} c + a c^{2}\right )} d^{9} + 12 \, {\left (b^{3} + 6 \, a b c\right )} d^{7} + 10 \, a^{2} b d^{3} + 21 \, {\left (a b^{2} + a^{2} c\right )} d^{5} + a^{3} d\right )} f^{3} x^{3} e^{2} + \frac {3}{2} \, {\left (5 \, c^{3} d^{14} + 13 \, b c^{2} d^{12} + 11 \, {\left (b^{2} c + a c^{2}\right )} d^{10} + 3 \, {\left (b^{3} + 6 \, a b c\right )} d^{8} + 5 \, a^{2} b d^{4} + 7 \, {\left (a b^{2} + a^{2} c\right )} d^{6} + a^{3} d^{2}\right )} f^{3} x^{2} e + {\left (c^{3} d^{15} + 3 \, b c^{2} d^{13} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{11} + {\left (b^{3} + 6 \, a b c\right )} d^{9} + 3 \, a^{2} b d^{5} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{7} + a^{3} d^{3}\right )} f^{3} 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. 1654 vs.
\(2 (141) = 282\).
time = 0.15, size = 1654, normalized size = 10.40 \begin {gather*} c^{3} d e^{14} f^{3} x^{15} + \frac {c^{3} e^{15} f^{3} x^{16}}{16} + x^{14} \cdot \left (\frac {3 b c^{2} e^{13} f^{3}}{14} + \frac {15 c^{3} d^{2} e^{13} f^{3}}{2}\right ) + x^{13} \cdot \left (3 b c^{2} d e^{12} f^{3} + 35 c^{3} d^{3} e^{12} f^{3}\right ) + x^{12} \left (\frac {a c^{2} e^{11} f^{3}}{4} + \frac {b^{2} c e^{11} f^{3}}{4} + \frac {39 b c^{2} d^{2} e^{11} f^{3}}{2} + \frac {455 c^{3} d^{4} e^{11} f^{3}}{4}\right ) + x^{11} \cdot \left (3 a c^{2} d e^{10} f^{3} + 3 b^{2} c d e^{10} f^{3} + 78 b c^{2} d^{3} e^{10} f^{3} + 273 c^{3} d^{5} e^{10} f^{3}\right ) + x^{10} \cdot \left (\frac {3 a b c e^{9} f^{3}}{5} + \frac {33 a c^{2} d^{2} e^{9} f^{3}}{2} + \frac {b^{3} e^{9} f^{3}}{10} + \frac {33 b^{2} c d^{2} e^{9} f^{3}}{2} + \frac {429 b c^{2} d^{4} e^{9} f^{3}}{2} + \frac {1001 c^{3} d^{6} e^{9} f^{3}}{2}\right ) + x^{9} \cdot \left (6 a b c d e^{8} f^{3} + 55 a c^{2} d^{3} e^{8} f^{3} + b^{3} d e^{8} f^{3} + 55 b^{2} c d^{3} e^{8} f^{3} + 429 b c^{2} d^{5} e^{8} f^{3} + 715 c^{3} d^{7} e^{8} f^{3}\right ) + x^{8} \cdot \left (\frac {3 a^{2} c e^{7} f^{3}}{8} + \frac {3 a b^{2} e^{7} f^{3}}{8} + 27 a b c d^{2} e^{7} f^{3} + \frac {495 a c^{2} d^{4} e^{7} f^{3}}{4} + \frac {9 b^{3} d^{2} e^{7} f^{3}}{2} + \frac {495 b^{2} c d^{4} e^{7} f^{3}}{4} + \frac {1287 b c^{2} d^{6} e^{7} f^{3}}{2} + \frac {6435 c^{3} d^{8} e^{7} f^{3}}{8}\right ) + x^{7} \cdot \left (3 a^{2} c d e^{6} f^{3} + 3 a b^{2} d e^{6} f^{3} + 72 a b c d^{3} e^{6} f^{3} + 198 a c^{2} d^{5} e^{6} f^{3} + 12 b^{3} d^{3} e^{6} f^{3} + 198 b^{2} c d^{5} e^{6} f^{3} + \frac {5148 b c^{2} d^{7} e^{6} f^{3}}{7} + 715 c^{3} d^{9} e^{6} f^{3}\right ) + x^{6} \left (\frac {a^{2} b e^{5} f^{3}}{2} + \frac {21 a^{2} c d^{2} e^{5} f^{3}}{2} + \frac {21 a b^{2} d^{2} e^{5} f^{3}}{2} + 126 a b c d^{4} e^{5} f^{3} + 231 a c^{2} d^{6} e^{5} f^{3} + 21 b^{3} d^{4} e^{5} f^{3} + 231 b^{2} c d^{6} e^{5} f^{3} + \frac {1287 b c^{2} d^{8} e^{5} f^{3}}{2} + \frac {1001 c^{3} d^{10} e^{5} f^{3}}{2}\right ) + x^{5} \cdot \left (3 a^{2} b d e^{4} f^{3} + 21 a^{2} c d^{3} e^{4} f^{3} + 21 a b^{2} d^{3} e^{4} f^{3} + \frac {756 a b c d^{5} e^{4} f^{3}}{5} + 198 a c^{2} d^{7} e^{4} f^{3} + \frac {126 b^{3} d^{5} e^{4} f^{3}}{5} + 198 b^{2} c d^{7} e^{4} f^{3} + 429 b c^{2} d^{9} e^{4} f^{3} + 273 c^{3} d^{11} e^{4} f^{3}\right ) + x^{4} \left (\frac {a^{3} e^{3} f^{3}}{4} + \frac {15 a^{2} b d^{2} e^{3} f^{3}}{2} + \frac {105 a^{2} c d^{4} e^{3} f^{3}}{4} + \frac {105 a b^{2} d^{4} e^{3} f^{3}}{4} + 126 a b c d^{6} e^{3} f^{3} + \frac {495 a c^{2} d^{8} e^{3} f^{3}}{4} + 21 b^{3} d^{6} e^{3} f^{3} + \frac {495 b^{2} c d^{8} e^{3} f^{3}}{4} + \frac {429 b c^{2} d^{10} e^{3} f^{3}}{2} + \frac {455 c^{3} d^{12} e^{3} f^{3}}{4}\right ) + x^{3} \left (a^{3} d e^{2} f^{3} + 10 a^{2} b d^{3} e^{2} f^{3} + 21 a^{2} c d^{5} e^{2} f^{3} + 21 a b^{2} d^{5} e^{2} f^{3} + 72 a b c d^{7} e^{2} f^{3} + 55 a c^{2} d^{9} e^{2} f^{3} + 12 b^{3} d^{7} e^{2} f^{3} + 55 b^{2} c d^{9} e^{2} f^{3} + 78 b c^{2} d^{11} e^{2} f^{3} + 35 c^{3} d^{13} e^{2} f^{3}\right ) + x^{2} \cdot \left (\frac {3 a^{3} d^{2} e f^{3}}{2} + \frac {15 a^{2} b d^{4} e f^{3}}{2} + \frac {21 a^{2} c d^{6} e f^{3}}{2} + \frac {21 a b^{2} d^{6} e f^{3}}{2} + 27 a b c d^{8} e f^{3} + \frac {33 a c^{2} d^{10} e f^{3}}{2} + \frac {9 b^{3} d^{8} e f^{3}}{2} + \frac {33 b^{2} c d^{10} e f^{3}}{2} + \frac {39 b c^{2} d^{12} e f^{3}}{2} + \frac {15 c^{3} d^{14} e f^{3}}{2}\right ) + x \left (a^{3} d^{3} f^{3} + 3 a^{2} b d^{5} f^{3} + 3 a^{2} c d^{7} f^{3} + 3 a b^{2} d^{7} f^{3} + 6 a b c d^{9} f^{3} + 3 a c^{2} d^{11} f^{3} + b^{3} d^{9} f^{3} + 3 b^{2} c d^{11} f^{3} + 3 b c^{2} d^{13} f^{3} + c^{3} d^{15} f^{3}\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. 1360 vs.
\(2 (145) = 290\).
time = 5.03, size = 1360, normalized size = 8.55 \begin {gather*} \frac {1}{2} \, {\left (f x^{2} e + 2 \, d f x\right )} c^{3} d^{14} f^{2} + \frac {3}{2} \, {\left (f x^{2} e + 2 \, d f x\right )} b c^{2} d^{12} f^{2} + \frac {3}{2} \, {\left (f x^{2} e + 2 \, d f x\right )} b^{2} c d^{10} f^{2} + \frac {3}{2} \, {\left (f x^{2} e + 2 \, d f x\right )} a c^{2} d^{10} f^{2} + \frac {1}{2} \, {\left (f x^{2} e + 2 \, d f x\right )} b^{3} d^{8} f^{2} + 3 \, {\left (f x^{2} e + 2 \, d f x\right )} a b c d^{8} f^{2} + \frac {3}{2} \, {\left (f x^{2} e + 2 \, d f x\right )} a b^{2} d^{6} f^{2} + \frac {3}{2} \, {\left (f x^{2} e + 2 \, d f x\right )} a^{2} c d^{6} f^{2} + \frac {3}{2} \, {\left (f x^{2} e + 2 \, d f x\right )} a^{2} b d^{4} f^{2} + \frac {1}{2} \, {\left (f x^{2} e + 2 \, d f x\right )} a^{3} d^{2} f^{2} + \frac {980 \, {\left (f x^{2} e + 2 \, d f x\right )}^{2} c^{3} d^{12} f^{6} e + 1960 \, {\left (f x^{2} e + 2 \, d f x\right )}^{3} c^{3} d^{10} f^{5} e^{2} + 2520 \, {\left (f x^{2} e + 2 \, d f x\right )}^{2} b c^{2} d^{10} f^{6} e + 2450 \, {\left (f x^{2} e + 2 \, d f x\right )}^{4} c^{3} d^{8} f^{4} e^{3} + 4200 \, {\left (f x^{2} e + 2 \, d f x\right )}^{3} b c^{2} d^{8} f^{5} e^{2} + 2100 \, {\left (f x^{2} e + 2 \, d f x\right )}^{2} b^{2} c d^{8} f^{6} e + 2100 \, {\left (f x^{2} e + 2 \, d f x\right )}^{2} a c^{2} d^{8} f^{6} e + 1960 \, {\left (f x^{2} e + 2 \, d f x\right )}^{5} c^{3} d^{6} f^{3} e^{4} + 4200 \, {\left (f x^{2} e + 2 \, d f x\right )}^{4} b c^{2} d^{6} f^{4} e^{3} + 2800 \, {\left (f x^{2} e + 2 \, d f x\right )}^{3} b^{2} c d^{6} f^{5} e^{2} + 2800 \, {\left (f x^{2} e + 2 \, d f x\right )}^{3} a c^{2} d^{6} f^{5} e^{2} + 560 \, {\left (f x^{2} e + 2 \, d f x\right )}^{2} b^{3} d^{6} f^{6} e + 3360 \, {\left (f x^{2} e + 2 \, d f x\right )}^{2} a b c d^{6} f^{6} e + 980 \, {\left (f x^{2} e + 2 \, d f x\right )}^{6} c^{3} d^{4} f^{2} e^{5} + 2520 \, {\left (f x^{2} e + 2 \, d f x\right )}^{5} b c^{2} d^{4} f^{3} e^{4} + 2100 \, {\left (f x^{2} e + 2 \, d f x\right )}^{4} b^{2} c d^{4} f^{4} e^{3} + 2100 \, {\left (f x^{2} e + 2 \, d f x\right )}^{4} a c^{2} d^{4} f^{4} e^{3} + 560 \, {\left (f x^{2} e + 2 \, d f x\right )}^{3} b^{3} d^{4} f^{5} e^{2} + 3360 \, {\left (f x^{2} e + 2 \, d f x\right )}^{3} a b c d^{4} f^{5} e^{2} + 1260 \, {\left (f x^{2} e + 2 \, d f x\right )}^{2} a b^{2} d^{4} f^{6} e + 1260 \, {\left (f x^{2} e + 2 \, d f x\right )}^{2} a^{2} c d^{4} f^{6} e + 280 \, {\left (f x^{2} e + 2 \, d f x\right )}^{7} c^{3} d^{2} f e^{6} + 840 \, {\left (f x^{2} e + 2 \, d f x\right )}^{6} b c^{2} d^{2} f^{2} e^{5} + 840 \, {\left (f x^{2} e + 2 \, d f x\right )}^{5} b^{2} c d^{2} f^{3} e^{4} + 840 \, {\left (f x^{2} e + 2 \, d f x\right )}^{5} a c^{2} d^{2} f^{3} e^{4} + 280 \, {\left (f x^{2} e + 2 \, d f x\right )}^{4} b^{3} d^{2} f^{4} e^{3} + 1680 \, {\left (f x^{2} e + 2 \, d f x\right )}^{4} a b c d^{2} f^{4} e^{3} + 840 \, {\left (f x^{2} e + 2 \, d f x\right )}^{3} a b^{2} d^{2} f^{5} e^{2} + 840 \, {\left (f x^{2} e + 2 \, d f x\right )}^{3} a^{2} c d^{2} f^{5} e^{2} + 840 \, {\left (f x^{2} e + 2 \, d f x\right )}^{2} a^{2} b d^{2} f^{6} e + 35 \, {\left (f x^{2} e + 2 \, d f x\right )}^{8} c^{3} e^{7} + 120 \, {\left (f x^{2} e + 2 \, d f x\right )}^{7} b c^{2} f e^{6} + 140 \, {\left (f x^{2} e + 2 \, d f x\right )}^{6} b^{2} c f^{2} e^{5} + 140 \, {\left (f x^{2} e + 2 \, d f x\right )}^{6} a c^{2} f^{2} e^{5} + 56 \, {\left (f x^{2} e + 2 \, d f x\right )}^{5} b^{3} f^{3} e^{4} + 336 \, {\left (f x^{2} e + 2 \, d f x\right )}^{5} a b c f^{3} e^{4} + 210 \, {\left (f x^{2} e + 2 \, d f x\right )}^{4} a b^{2} f^{4} e^{3} + 210 \, {\left (f x^{2} e + 2 \, d f x\right )}^{4} a^{2} c f^{4} e^{3} + 280 \, {\left (f x^{2} e + 2 \, d f x\right )}^{3} a^{2} b f^{5} e^{2} + 140 \, {\left (f x^{2} e + 2 \, d f x\right )}^{2} a^{3} f^{6} e}{560 \, f^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.65, size = 825, normalized size = 5.19 \begin {gather*} \frac {3\,e^7\,f^3\,x^8\,\left (a^2\,c+a\,b^2+72\,a\,b\,c\,d^2+330\,a\,c^2\,d^4+12\,b^3\,d^2+330\,b^2\,c\,d^4+1716\,b\,c^2\,d^6+2145\,c^3\,d^8\right )}{8}+\frac {e^5\,f^3\,x^6\,\left (a^2\,b+21\,a^2\,c\,d^2+21\,a\,b^2\,d^2+252\,a\,b\,c\,d^4+462\,a\,c^2\,d^6+42\,b^3\,d^4+462\,b^2\,c\,d^6+1287\,b\,c^2\,d^8+1001\,c^3\,d^{10}\right )}{2}+\frac {e^9\,f^3\,x^{10}\,\left (b^3+165\,b^2\,c\,d^2+2145\,b\,c^2\,d^4+6\,a\,b\,c+5005\,c^3\,d^6+165\,a\,c^2\,d^2\right )}{10}+\frac {c^3\,e^{15}\,f^3\,x^{16}}{16}+d^3\,f^3\,x\,{\left (c\,d^4+b\,d^2+a\right )}^3+\frac {e^3\,f^3\,x^4\,\left (a^3+30\,a^2\,b\,d^2+105\,a^2\,c\,d^4+105\,a\,b^2\,d^4+504\,a\,b\,c\,d^6+495\,a\,c^2\,d^8+84\,b^3\,d^6+495\,b^2\,c\,d^8+858\,b\,c^2\,d^{10}+455\,c^3\,d^{12}\right )}{4}+\frac {c\,e^{11}\,f^3\,x^{12}\,\left (b^2+78\,b\,c\,d^2+455\,c^2\,d^4+a\,c\right )}{4}+\frac {d\,e^6\,f^3\,x^7\,\left (21\,a^2\,c+21\,a\,b^2+504\,a\,b\,c\,d^2+1386\,a\,c^2\,d^4+84\,b^3\,d^2+1386\,b^2\,c\,d^4+5148\,b\,c^2\,d^6+5005\,c^3\,d^8\right )}{7}+\frac {3\,d\,e^4\,f^3\,x^5\,\left (5\,a^2\,b+35\,a^2\,c\,d^2+35\,a\,b^2\,d^2+252\,a\,b\,c\,d^4+330\,a\,c^2\,d^6+42\,b^3\,d^4+330\,b^2\,c\,d^6+715\,b\,c^2\,d^8+455\,c^3\,d^{10}\right )}{5}+d\,e^8\,f^3\,x^9\,\left (b^3+55\,b^2\,c\,d^2+429\,b\,c^2\,d^4+6\,a\,b\,c+715\,c^3\,d^6+55\,a\,c^2\,d^2\right )+\frac {3\,c^2\,e^{13}\,f^3\,x^{14}\,\left (35\,c\,d^2+b\right )}{14}+c^3\,d\,e^{14}\,f^3\,x^{15}+d\,e^2\,f^3\,x^3\,\left (a^3+10\,a^2\,b\,d^2+21\,a^2\,c\,d^4+21\,a\,b^2\,d^4+72\,a\,b\,c\,d^6+55\,a\,c^2\,d^8+12\,b^3\,d^6+55\,b^2\,c\,d^8+78\,b\,c^2\,d^{10}+35\,c^3\,d^{12}\right )+\frac {3\,d^2\,e\,f^3\,x^2\,{\left (c\,d^4+b\,d^2+a\right )}^2\,\left (5\,c\,d^4+3\,b\,d^2+a\right )}{2}+c^2\,d\,e^{12}\,f^3\,x^{13}\,\left (35\,c\,d^2+3\,b\right )+3\,c\,d\,e^{10}\,f^3\,x^{11}\,\left (b^2+26\,b\,c\,d^2+91\,c^2\,d^4+a\,c\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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