Optimal. Leaf size=156 \[ \frac {a^3 (d x)^{1+m}}{d (1+m)}+\frac {3 a^2 b (d x)^{3+m}}{d^3 (3+m)}+\frac {3 a \left (b^2+a c\right ) (d x)^{5+m}}{d^5 (5+m)}+\frac {b \left (b^2+6 a c\right ) (d x)^{7+m}}{d^7 (7+m)}+\frac {3 c \left (b^2+a c\right ) (d x)^{9+m}}{d^9 (9+m)}+\frac {3 b c^2 (d x)^{11+m}}{d^{11} (11+m)}+\frac {c^3 (d x)^{13+m}}{d^{13} (13+m)} \]
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Rubi [A]
time = 0.07, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1122}
\begin {gather*} \frac {a^3 (d x)^{m+1}}{d (m+1)}+\frac {3 a^2 b (d x)^{m+3}}{d^3 (m+3)}+\frac {3 c \left (a c+b^2\right ) (d x)^{m+9}}{d^9 (m+9)}+\frac {b \left (6 a c+b^2\right ) (d x)^{m+7}}{d^7 (m+7)}+\frac {3 a \left (a c+b^2\right ) (d x)^{m+5}}{d^5 (m+5)}+\frac {3 b c^2 (d x)^{m+11}}{d^{11} (m+11)}+\frac {c^3 (d x)^{m+13}}{d^{13} (m+13)} \end {gather*}
Antiderivative was successfully verified.
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Rule 1122
Rubi steps
\begin {align*} \int (d x)^m \left (a+b x^2+c x^4\right )^3 \, dx &=\int \left (a^3 (d x)^m+\frac {3 a^2 b (d x)^{2+m}}{d^2}+\frac {3 a \left (b^2+a c\right ) (d x)^{4+m}}{d^4}+\frac {b \left (b^2+6 a c\right ) (d x)^{6+m}}{d^6}+\frac {3 c \left (b^2+a c\right ) (d x)^{8+m}}{d^8}+\frac {3 b c^2 (d x)^{10+m}}{d^{10}}+\frac {c^3 (d x)^{12+m}}{d^{12}}\right ) \, dx\\ &=\frac {a^3 (d x)^{1+m}}{d (1+m)}+\frac {3 a^2 b (d x)^{3+m}}{d^3 (3+m)}+\frac {3 a \left (b^2+a c\right ) (d x)^{5+m}}{d^5 (5+m)}+\frac {b \left (b^2+6 a c\right ) (d x)^{7+m}}{d^7 (7+m)}+\frac {3 c \left (b^2+a c\right ) (d x)^{9+m}}{d^9 (9+m)}+\frac {3 b c^2 (d x)^{11+m}}{d^{11} (11+m)}+\frac {c^3 (d x)^{13+m}}{d^{13} (13+m)}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 111, normalized size = 0.71 \begin {gather*} (d x)^m \left (\frac {a^3 x}{1+m}+\frac {3 a^2 b x^3}{3+m}+\frac {3 a \left (b^2+a c\right ) x^5}{5+m}+\frac {b \left (b^2+6 a c\right ) x^7}{7+m}+\frac {3 c \left (b^2+a c\right ) x^9}{9+m}+\frac {3 b c^2 x^{11}}{11+m}+\frac {c^3 x^{13}}{13+m}\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. \(781\) vs.
\(2(156)=312\).
time = 0.02, size = 782, normalized size = 5.01
method | result | size |
gosper | \(\frac {x \left (c^{3} m^{6} x^{12}+36 c^{3} m^{5} x^{12}+3 b \,c^{2} m^{6} x^{10}+505 c^{3} m^{4} x^{12}+114 b \,c^{2} m^{5} x^{10}+3480 c^{3} m^{3} x^{12}+3 a \,c^{2} m^{6} x^{8}+3 b^{2} c \,m^{6} x^{8}+1665 b \,c^{2} m^{4} x^{10}+12139 c^{3} m^{2} x^{12}+120 a \,c^{2} m^{5} x^{8}+120 b^{2} c \,m^{5} x^{8}+11820 b \,c^{2} m^{3} x^{10}+19524 m \,x^{12} c^{3}+6 a b c \,m^{6} x^{6}+1839 a \,c^{2} m^{4} x^{8}+b^{3} m^{6} x^{6}+1839 b^{2} c \,m^{4} x^{8}+42117 b \,c^{2} m^{2} x^{10}+10395 c^{3} x^{12}+252 a b c \,m^{5} x^{6}+13584 a \,c^{2} m^{3} x^{8}+42 b^{3} m^{5} x^{6}+13584 b^{2} c \,m^{3} x^{8}+68706 m \,x^{10} b \,c^{2}+3 a^{2} c \,m^{6} x^{4}+3 a \,b^{2} m^{6} x^{4}+4074 a b c \,m^{4} x^{6}+49881 a \,c^{2} m^{2} x^{8}+679 b^{3} m^{4} x^{6}+49881 b^{2} c \,m^{2} x^{8}+36855 b \,c^{2} x^{10}+132 a^{2} c \,m^{5} x^{4}+132 a \,b^{2} m^{5} x^{4}+31752 a b c \,m^{3} x^{6}+83064 x^{8} c^{2} a m +5292 b^{3} m^{3} x^{6}+83064 x^{8} b^{2} c m +3 a^{2} b \,m^{6} x^{2}+2259 a^{2} c \,m^{4} x^{4}+2259 a \,b^{2} m^{4} x^{4}+122010 a b c \,m^{2} x^{6}+45045 x^{8} c^{2} a +20335 b^{3} m^{2} x^{6}+45045 x^{8} b^{2} c +138 a^{2} b \,m^{5} x^{2}+18840 a^{2} c \,m^{3} x^{4}+18840 a \,b^{2} m^{3} x^{4}+209916 a b c \,x^{6} m +34986 b^{3} x^{6} m +a^{3} m^{6}+2505 a^{2} b \,m^{4} x^{2}+77937 a^{2} c \,m^{2} x^{4}+77937 a \,b^{2} m^{2} x^{4}+115830 a b c \,x^{6}+19305 b^{3} x^{6}+48 a^{3} m^{5}+22620 a^{2} b \,m^{3} x^{2}+142308 x^{4} a^{2} c m +142308 a \,b^{2} x^{4} m +925 a^{3} m^{4}+104277 a^{2} b \,m^{2} x^{2}+81081 x^{4} a^{2} c +81081 a \,b^{2} x^{4}+9120 a^{3} m^{3}+219162 a^{2} b \,x^{2} m +48259 a^{3} m^{2}+135135 a^{2} b \,x^{2}+129072 a^{3} m +135135 a^{3}\right ) \left (d x \right )^{m}}{\left (13+m \right ) \left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(782\) |
risch | \(\frac {x \left (c^{3} m^{6} x^{12}+36 c^{3} m^{5} x^{12}+3 b \,c^{2} m^{6} x^{10}+505 c^{3} m^{4} x^{12}+114 b \,c^{2} m^{5} x^{10}+3480 c^{3} m^{3} x^{12}+3 a \,c^{2} m^{6} x^{8}+3 b^{2} c \,m^{6} x^{8}+1665 b \,c^{2} m^{4} x^{10}+12139 c^{3} m^{2} x^{12}+120 a \,c^{2} m^{5} x^{8}+120 b^{2} c \,m^{5} x^{8}+11820 b \,c^{2} m^{3} x^{10}+19524 m \,x^{12} c^{3}+6 a b c \,m^{6} x^{6}+1839 a \,c^{2} m^{4} x^{8}+b^{3} m^{6} x^{6}+1839 b^{2} c \,m^{4} x^{8}+42117 b \,c^{2} m^{2} x^{10}+10395 c^{3} x^{12}+252 a b c \,m^{5} x^{6}+13584 a \,c^{2} m^{3} x^{8}+42 b^{3} m^{5} x^{6}+13584 b^{2} c \,m^{3} x^{8}+68706 m \,x^{10} b \,c^{2}+3 a^{2} c \,m^{6} x^{4}+3 a \,b^{2} m^{6} x^{4}+4074 a b c \,m^{4} x^{6}+49881 a \,c^{2} m^{2} x^{8}+679 b^{3} m^{4} x^{6}+49881 b^{2} c \,m^{2} x^{8}+36855 b \,c^{2} x^{10}+132 a^{2} c \,m^{5} x^{4}+132 a \,b^{2} m^{5} x^{4}+31752 a b c \,m^{3} x^{6}+83064 x^{8} c^{2} a m +5292 b^{3} m^{3} x^{6}+83064 x^{8} b^{2} c m +3 a^{2} b \,m^{6} x^{2}+2259 a^{2} c \,m^{4} x^{4}+2259 a \,b^{2} m^{4} x^{4}+122010 a b c \,m^{2} x^{6}+45045 x^{8} c^{2} a +20335 b^{3} m^{2} x^{6}+45045 x^{8} b^{2} c +138 a^{2} b \,m^{5} x^{2}+18840 a^{2} c \,m^{3} x^{4}+18840 a \,b^{2} m^{3} x^{4}+209916 a b c \,x^{6} m +34986 b^{3} x^{6} m +a^{3} m^{6}+2505 a^{2} b \,m^{4} x^{2}+77937 a^{2} c \,m^{2} x^{4}+77937 a \,b^{2} m^{2} x^{4}+115830 a b c \,x^{6}+19305 b^{3} x^{6}+48 a^{3} m^{5}+22620 a^{2} b \,m^{3} x^{2}+142308 x^{4} a^{2} c m +142308 a \,b^{2} x^{4} m +925 a^{3} m^{4}+104277 a^{2} b \,m^{2} x^{2}+81081 x^{4} a^{2} c +81081 a \,b^{2} x^{4}+9120 a^{3} m^{3}+219162 a^{2} b \,x^{2} m +48259 a^{3} m^{2}+135135 a^{2} b \,x^{2}+129072 a^{3} m +135135 a^{3}\right ) \left (d x \right )^{m}}{\left (13+m \right ) \left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(782\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 195, normalized size = 1.25 \begin {gather*} \frac {c^{3} d^{m} x^{13} x^{m}}{m + 13} + \frac {3 \, b c^{2} d^{m} x^{11} x^{m}}{m + 11} + \frac {3 \, b^{2} c d^{m} x^{9} x^{m}}{m + 9} + \frac {3 \, a c^{2} d^{m} x^{9} x^{m}}{m + 9} + \frac {b^{3} d^{m} x^{7} x^{m}}{m + 7} + \frac {6 \, a b c d^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, a b^{2} d^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, a^{2} c d^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, a^{2} b d^{m} x^{3} x^{m}}{m + 3} + \frac {\left (d x\right )^{m + 1} a^{3}}{d {\left (m + 1\right )}} \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. 594 vs.
\(2 (156) = 312\).
time = 0.35, size = 594, normalized size = 3.81 \begin {gather*} \frac {{\left ({\left (c^{3} m^{6} + 36 \, c^{3} m^{5} + 505 \, c^{3} m^{4} + 3480 \, c^{3} m^{3} + 12139 \, c^{3} m^{2} + 19524 \, c^{3} m + 10395 \, c^{3}\right )} x^{13} + 3 \, {\left (b c^{2} m^{6} + 38 \, b c^{2} m^{5} + 555 \, b c^{2} m^{4} + 3940 \, b c^{2} m^{3} + 14039 \, b c^{2} m^{2} + 22902 \, b c^{2} m + 12285 \, b c^{2}\right )} x^{11} + 3 \, {\left ({\left (b^{2} c + a c^{2}\right )} m^{6} + 40 \, {\left (b^{2} c + a c^{2}\right )} m^{5} + 613 \, {\left (b^{2} c + a c^{2}\right )} m^{4} + 4528 \, {\left (b^{2} c + a c^{2}\right )} m^{3} + 15015 \, b^{2} c + 15015 \, a c^{2} + 16627 \, {\left (b^{2} c + a c^{2}\right )} m^{2} + 27688 \, {\left (b^{2} c + a c^{2}\right )} m\right )} x^{9} + {\left ({\left (b^{3} + 6 \, a b c\right )} m^{6} + 42 \, {\left (b^{3} + 6 \, a b c\right )} m^{5} + 679 \, {\left (b^{3} + 6 \, a b c\right )} m^{4} + 5292 \, {\left (b^{3} + 6 \, a b c\right )} m^{3} + 19305 \, b^{3} + 115830 \, a b c + 20335 \, {\left (b^{3} + 6 \, a b c\right )} m^{2} + 34986 \, {\left (b^{3} + 6 \, a b c\right )} m\right )} x^{7} + 3 \, {\left ({\left (a b^{2} + a^{2} c\right )} m^{6} + 44 \, {\left (a b^{2} + a^{2} c\right )} m^{5} + 753 \, {\left (a b^{2} + a^{2} c\right )} m^{4} + 6280 \, {\left (a b^{2} + a^{2} c\right )} m^{3} + 27027 \, a b^{2} + 27027 \, a^{2} c + 25979 \, {\left (a b^{2} + a^{2} c\right )} m^{2} + 47436 \, {\left (a b^{2} + a^{2} c\right )} m\right )} x^{5} + 3 \, {\left (a^{2} b m^{6} + 46 \, a^{2} b m^{5} + 835 \, a^{2} b m^{4} + 7540 \, a^{2} b m^{3} + 34759 \, a^{2} b m^{2} + 73054 \, a^{2} b m + 45045 \, a^{2} b\right )} x^{3} + {\left (a^{3} m^{6} + 48 \, a^{3} m^{5} + 925 \, a^{3} m^{4} + 9120 \, a^{3} m^{3} + 48259 \, a^{3} m^{2} + 129072 \, a^{3} m + 135135 \, a^{3}\right )} x\right )} \left (d x\right )^{m}}{m^{7} + 49 \, m^{6} + 973 \, m^{5} + 10045 \, m^{4} + 57379 \, m^{3} + 177331 \, m^{2} + 264207 \, m + 135135} \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. 4332 vs.
\(2 (143) = 286\).
time = 1.05, size = 4332, normalized size = 27.77 \begin {gather*} \text {Too large to display} \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. 1132 vs.
\(2 (156) = 312\).
time = 3.21, size = 1132, normalized size = 7.26 \begin {gather*} \frac {\left (d x\right )^{m} c^{3} m^{6} x^{13} + 36 \, \left (d x\right )^{m} c^{3} m^{5} x^{13} + 3 \, \left (d x\right )^{m} b c^{2} m^{6} x^{11} + 505 \, \left (d x\right )^{m} c^{3} m^{4} x^{13} + 114 \, \left (d x\right )^{m} b c^{2} m^{5} x^{11} + 3480 \, \left (d x\right )^{m} c^{3} m^{3} x^{13} + 3 \, \left (d x\right )^{m} b^{2} c m^{6} x^{9} + 3 \, \left (d x\right )^{m} a c^{2} m^{6} x^{9} + 1665 \, \left (d x\right )^{m} b c^{2} m^{4} x^{11} + 12139 \, \left (d x\right )^{m} c^{3} m^{2} x^{13} + 120 \, \left (d x\right )^{m} b^{2} c m^{5} x^{9} + 120 \, \left (d x\right )^{m} a c^{2} m^{5} x^{9} + 11820 \, \left (d x\right )^{m} b c^{2} m^{3} x^{11} + 19524 \, \left (d x\right )^{m} c^{3} m x^{13} + \left (d x\right )^{m} b^{3} m^{6} x^{7} + 6 \, \left (d x\right )^{m} a b c m^{6} x^{7} + 1839 \, \left (d x\right )^{m} b^{2} c m^{4} x^{9} + 1839 \, \left (d x\right )^{m} a c^{2} m^{4} x^{9} + 42117 \, \left (d x\right )^{m} b c^{2} m^{2} x^{11} + 10395 \, \left (d x\right )^{m} c^{3} x^{13} + 42 \, \left (d x\right )^{m} b^{3} m^{5} x^{7} + 252 \, \left (d x\right )^{m} a b c m^{5} x^{7} + 13584 \, \left (d x\right )^{m} b^{2} c m^{3} x^{9} + 13584 \, \left (d x\right )^{m} a c^{2} m^{3} x^{9} + 68706 \, \left (d x\right )^{m} b c^{2} m x^{11} + 3 \, \left (d x\right )^{m} a b^{2} m^{6} x^{5} + 3 \, \left (d x\right )^{m} a^{2} c m^{6} x^{5} + 679 \, \left (d x\right )^{m} b^{3} m^{4} x^{7} + 4074 \, \left (d x\right )^{m} a b c m^{4} x^{7} + 49881 \, \left (d x\right )^{m} b^{2} c m^{2} x^{9} + 49881 \, \left (d x\right )^{m} a c^{2} m^{2} x^{9} + 36855 \, \left (d x\right )^{m} b c^{2} x^{11} + 132 \, \left (d x\right )^{m} a b^{2} m^{5} x^{5} + 132 \, \left (d x\right )^{m} a^{2} c m^{5} x^{5} + 5292 \, \left (d x\right )^{m} b^{3} m^{3} x^{7} + 31752 \, \left (d x\right )^{m} a b c m^{3} x^{7} + 83064 \, \left (d x\right )^{m} b^{2} c m x^{9} + 83064 \, \left (d x\right )^{m} a c^{2} m x^{9} + 3 \, \left (d x\right )^{m} a^{2} b m^{6} x^{3} + 2259 \, \left (d x\right )^{m} a b^{2} m^{4} x^{5} + 2259 \, \left (d x\right )^{m} a^{2} c m^{4} x^{5} + 20335 \, \left (d x\right )^{m} b^{3} m^{2} x^{7} + 122010 \, \left (d x\right )^{m} a b c m^{2} x^{7} + 45045 \, \left (d x\right )^{m} b^{2} c x^{9} + 45045 \, \left (d x\right )^{m} a c^{2} x^{9} + 138 \, \left (d x\right )^{m} a^{2} b m^{5} x^{3} + 18840 \, \left (d x\right )^{m} a b^{2} m^{3} x^{5} + 18840 \, \left (d x\right )^{m} a^{2} c m^{3} x^{5} + 34986 \, \left (d x\right )^{m} b^{3} m x^{7} + 209916 \, \left (d x\right )^{m} a b c m x^{7} + \left (d x\right )^{m} a^{3} m^{6} x + 2505 \, \left (d x\right )^{m} a^{2} b m^{4} x^{3} + 77937 \, \left (d x\right )^{m} a b^{2} m^{2} x^{5} + 77937 \, \left (d x\right )^{m} a^{2} c m^{2} x^{5} + 19305 \, \left (d x\right )^{m} b^{3} x^{7} + 115830 \, \left (d x\right )^{m} a b c x^{7} + 48 \, \left (d x\right )^{m} a^{3} m^{5} x + 22620 \, \left (d x\right )^{m} a^{2} b m^{3} x^{3} + 142308 \, \left (d x\right )^{m} a b^{2} m x^{5} + 142308 \, \left (d x\right )^{m} a^{2} c m x^{5} + 925 \, \left (d x\right )^{m} a^{3} m^{4} x + 104277 \, \left (d x\right )^{m} a^{2} b m^{2} x^{3} + 81081 \, \left (d x\right )^{m} a b^{2} x^{5} + 81081 \, \left (d x\right )^{m} a^{2} c x^{5} + 9120 \, \left (d x\right )^{m} a^{3} m^{3} x + 219162 \, \left (d x\right )^{m} a^{2} b m x^{3} + 48259 \, \left (d x\right )^{m} a^{3} m^{2} x + 135135 \, \left (d x\right )^{m} a^{2} b x^{3} + 129072 \, \left (d x\right )^{m} a^{3} m x + 135135 \, \left (d x\right )^{m} a^{3} x}{m^{7} + 49 \, m^{6} + 973 \, m^{5} + 10045 \, m^{4} + 57379 \, m^{3} + 177331 \, m^{2} + 264207 \, m + 135135} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.83, size = 546, normalized size = 3.50 \begin {gather*} \frac {a^3\,x\,{\left (d\,x\right )}^m\,\left (m^6+48\,m^5+925\,m^4+9120\,m^3+48259\,m^2+129072\,m+135135\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {c^3\,x^{13}\,{\left (d\,x\right )}^m\,\left (m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {3\,a^2\,b\,x^3\,{\left (d\,x\right )}^m\,\left (m^6+46\,m^5+835\,m^4+7540\,m^3+34759\,m^2+73054\,m+45045\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {3\,b\,c^2\,x^{11}\,{\left (d\,x\right )}^m\,\left (m^6+38\,m^5+555\,m^4+3940\,m^3+14039\,m^2+22902\,m+12285\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {3\,a\,x^5\,{\left (d\,x\right )}^m\,\left (b^2+a\,c\right )\,\left (m^6+44\,m^5+753\,m^4+6280\,m^3+25979\,m^2+47436\,m+27027\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {b\,x^7\,{\left (d\,x\right )}^m\,\left (b^2+6\,a\,c\right )\,\left (m^6+42\,m^5+679\,m^4+5292\,m^3+20335\,m^2+34986\,m+19305\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {3\,c\,x^9\,{\left (d\,x\right )}^m\,\left (b^2+a\,c\right )\,\left (m^6+40\,m^5+613\,m^4+4528\,m^3+16627\,m^2+27688\,m+15015\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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