Optimal. Leaf size=101 \[ \frac {a^2 (d x)^{1+m}}{d (1+m)}+\frac {2 a b (d x)^{3+m}}{d^3 (3+m)}+\frac {\left (b^2+2 a c\right ) (d x)^{5+m}}{d^5 (5+m)}+\frac {2 b c (d x)^{7+m}}{d^7 (7+m)}+\frac {c^2 (d x)^{9+m}}{d^9 (9+m)} \]
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Rubi [A]
time = 0.04, antiderivative size = 101, 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^2 (d x)^{m+1}}{d (m+1)}+\frac {\left (2 a c+b^2\right ) (d x)^{m+5}}{d^5 (m+5)}+\frac {2 a b (d x)^{m+3}}{d^3 (m+3)}+\frac {2 b c (d x)^{m+7}}{d^7 (m+7)}+\frac {c^2 (d x)^{m+9}}{d^9 (m+9)} \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 )^2 \, dx &=\int \left (a^2 (d x)^m+\frac {2 a b (d x)^{2+m}}{d^2}+\frac {\left (b^2+2 a c\right ) (d x)^{4+m}}{d^4}+\frac {2 b c (d x)^{6+m}}{d^6}+\frac {c^2 (d x)^{8+m}}{d^8}\right ) \, dx\\ &=\frac {a^2 (d x)^{1+m}}{d (1+m)}+\frac {2 a b (d x)^{3+m}}{d^3 (3+m)}+\frac {\left (b^2+2 a c\right ) (d x)^{5+m}}{d^5 (5+m)}+\frac {2 b c (d x)^{7+m}}{d^7 (7+m)}+\frac {c^2 (d x)^{9+m}}{d^9 (9+m)}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 70, normalized size = 0.69 \begin {gather*} (d x)^m \left (\frac {a^2 x}{1+m}+\frac {2 a b x^3}{3+m}+\frac {\left (b^2+2 a c\right ) x^5}{5+m}+\frac {2 b c x^7}{7+m}+\frac {c^2 x^9}{9+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. \(300\) vs.
\(2(101)=202\).
time = 0.01, size = 301, normalized size = 2.98
method | result | size |
gosper | \(\frac {x \left (c^{2} m^{4} x^{8}+16 c^{2} m^{3} x^{8}+2 b c \,m^{4} x^{6}+86 c^{2} m^{2} x^{8}+36 b c \,m^{3} x^{6}+176 m \,x^{8} c^{2}+2 a c \,m^{4} x^{4}+b^{2} m^{4} x^{4}+208 b c \,m^{2} x^{6}+105 c^{2} x^{8}+40 a c \,m^{3} x^{4}+20 b^{2} m^{3} x^{4}+444 m \,x^{6} b c +2 a b \,m^{4} x^{2}+260 a c \,m^{2} x^{4}+130 b^{2} m^{2} x^{4}+270 b c \,x^{6}+44 a b \,m^{3} x^{2}+600 c \,x^{4} a m +300 b^{2} x^{4} m +a^{2} m^{4}+328 a b \,m^{2} x^{2}+378 c \,x^{4} a +189 b^{2} x^{4}+24 a^{2} m^{3}+916 a b \,x^{2} m +206 a^{2} m^{2}+630 a b \,x^{2}+744 a^{2} m +945 a^{2}\right ) \left (d x \right )^{m}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(301\) |
risch | \(\frac {x \left (c^{2} m^{4} x^{8}+16 c^{2} m^{3} x^{8}+2 b c \,m^{4} x^{6}+86 c^{2} m^{2} x^{8}+36 b c \,m^{3} x^{6}+176 m \,x^{8} c^{2}+2 a c \,m^{4} x^{4}+b^{2} m^{4} x^{4}+208 b c \,m^{2} x^{6}+105 c^{2} x^{8}+40 a c \,m^{3} x^{4}+20 b^{2} m^{3} x^{4}+444 m \,x^{6} b c +2 a b \,m^{4} x^{2}+260 a c \,m^{2} x^{4}+130 b^{2} m^{2} x^{4}+270 b c \,x^{6}+44 a b \,m^{3} x^{2}+600 c \,x^{4} a m +300 b^{2} x^{4} m +a^{2} m^{4}+328 a b \,m^{2} x^{2}+378 c \,x^{4} a +189 b^{2} x^{4}+24 a^{2} m^{3}+916 a b \,x^{2} m +206 a^{2} m^{2}+630 a b \,x^{2}+744 a^{2} m +945 a^{2}\right ) \left (d x \right )^{m}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(301\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 110, normalized size = 1.09 \begin {gather*} \frac {c^{2} d^{m} x^{9} x^{m}}{m + 9} + \frac {2 \, b c d^{m} x^{7} x^{m}}{m + 7} + \frac {b^{2} d^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, a c d^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, a b d^{m} x^{3} x^{m}}{m + 3} + \frac {\left (d x\right )^{m + 1} a^{2}}{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. 241 vs.
\(2 (101) = 202\).
time = 0.33, size = 241, normalized size = 2.39 \begin {gather*} \frac {{\left ({\left (c^{2} m^{4} + 16 \, c^{2} m^{3} + 86 \, c^{2} m^{2} + 176 \, c^{2} m + 105 \, c^{2}\right )} x^{9} + 2 \, {\left (b c m^{4} + 18 \, b c m^{3} + 104 \, b c m^{2} + 222 \, b c m + 135 \, b c\right )} x^{7} + {\left ({\left (b^{2} + 2 \, a c\right )} m^{4} + 20 \, {\left (b^{2} + 2 \, a c\right )} m^{3} + 130 \, {\left (b^{2} + 2 \, a c\right )} m^{2} + 189 \, b^{2} + 378 \, a c + 300 \, {\left (b^{2} + 2 \, a c\right )} m\right )} x^{5} + 2 \, {\left (a b m^{4} + 22 \, a b m^{3} + 164 \, a b m^{2} + 458 \, a b m + 315 \, a b\right )} x^{3} + {\left (a^{2} m^{4} + 24 \, a^{2} m^{3} + 206 \, a^{2} m^{2} + 744 \, a^{2} m + 945 \, a^{2}\right )} x\right )} \left (d x\right )^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \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. 1435 vs.
\(2 (90) = 180\).
time = 0.53, size = 1435, normalized size = 14.21 \begin {gather*} \begin {cases} \frac {- \frac {a^{2}}{8 x^{8}} - \frac {a b}{3 x^{6}} - \frac {a c}{2 x^{4}} - \frac {b^{2}}{4 x^{4}} - \frac {b c}{x^{2}} + c^{2} \log {\left (x \right )}}{d^{9}} & \text {for}\: m = -9 \\\frac {- \frac {a^{2}}{6 x^{6}} - \frac {a b}{2 x^{4}} - \frac {a c}{x^{2}} - \frac {b^{2}}{2 x^{2}} + 2 b c \log {\left (x \right )} + \frac {c^{2} x^{2}}{2}}{d^{7}} & \text {for}\: m = -7 \\\frac {- \frac {a^{2}}{4 x^{4}} - \frac {a b}{x^{2}} + 2 a c \log {\left (x \right )} + b^{2} \log {\left (x \right )} + b c x^{2} + \frac {c^{2} x^{4}}{4}}{d^{5}} & \text {for}\: m = -5 \\\frac {- \frac {a^{2}}{2 x^{2}} + 2 a b \log {\left (x \right )} + a c x^{2} + \frac {b^{2} x^{2}}{2} + \frac {b c x^{4}}{2} + \frac {c^{2} x^{6}}{6}}{d^{3}} & \text {for}\: m = -3 \\\frac {a^{2} \log {\left (x \right )} + a b x^{2} + \frac {a c x^{4}}{2} + \frac {b^{2} x^{4}}{4} + \frac {b c x^{6}}{3} + \frac {c^{2} x^{8}}{8}}{d} & \text {for}\: m = -1 \\\frac {a^{2} m^{4} x \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {24 a^{2} m^{3} x \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {206 a^{2} m^{2} x \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {744 a^{2} m x \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {945 a^{2} x \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {2 a b m^{4} x^{3} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {44 a b m^{3} x^{3} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {328 a b m^{2} x^{3} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {916 a b m x^{3} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {630 a b x^{3} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {2 a c m^{4} x^{5} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {40 a c m^{3} x^{5} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {260 a c m^{2} x^{5} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {600 a c m x^{5} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {378 a c x^{5} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {b^{2} m^{4} x^{5} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {20 b^{2} m^{3} x^{5} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {130 b^{2} m^{2} x^{5} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {300 b^{2} m x^{5} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {189 b^{2} x^{5} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {2 b c m^{4} x^{7} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {36 b c m^{3} x^{7} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {208 b c m^{2} x^{7} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {444 b c m x^{7} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {270 b c x^{7} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {c^{2} m^{4} x^{9} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {16 c^{2} m^{3} x^{9} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {86 c^{2} m^{2} x^{9} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {176 c^{2} m x^{9} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {105 c^{2} x^{9} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} & \text {otherwise} \end {cases} \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. 449 vs.
\(2 (101) = 202\).
time = 3.74, size = 449, normalized size = 4.45 \begin {gather*} \frac {\left (d x\right )^{m} c^{2} m^{4} x^{9} + 16 \, \left (d x\right )^{m} c^{2} m^{3} x^{9} + 2 \, \left (d x\right )^{m} b c m^{4} x^{7} + 86 \, \left (d x\right )^{m} c^{2} m^{2} x^{9} + 36 \, \left (d x\right )^{m} b c m^{3} x^{7} + 176 \, \left (d x\right )^{m} c^{2} m x^{9} + \left (d x\right )^{m} b^{2} m^{4} x^{5} + 2 \, \left (d x\right )^{m} a c m^{4} x^{5} + 208 \, \left (d x\right )^{m} b c m^{2} x^{7} + 105 \, \left (d x\right )^{m} c^{2} x^{9} + 20 \, \left (d x\right )^{m} b^{2} m^{3} x^{5} + 40 \, \left (d x\right )^{m} a c m^{3} x^{5} + 444 \, \left (d x\right )^{m} b c m x^{7} + 2 \, \left (d x\right )^{m} a b m^{4} x^{3} + 130 \, \left (d x\right )^{m} b^{2} m^{2} x^{5} + 260 \, \left (d x\right )^{m} a c m^{2} x^{5} + 270 \, \left (d x\right )^{m} b c x^{7} + 44 \, \left (d x\right )^{m} a b m^{3} x^{3} + 300 \, \left (d x\right )^{m} b^{2} m x^{5} + 600 \, \left (d x\right )^{m} a c m x^{5} + \left (d x\right )^{m} a^{2} m^{4} x + 328 \, \left (d x\right )^{m} a b m^{2} x^{3} + 189 \, \left (d x\right )^{m} b^{2} x^{5} + 378 \, \left (d x\right )^{m} a c x^{5} + 24 \, \left (d x\right )^{m} a^{2} m^{3} x + 916 \, \left (d x\right )^{m} a b m x^{3} + 206 \, \left (d x\right )^{m} a^{2} m^{2} x + 630 \, \left (d x\right )^{m} a b x^{3} + 744 \, \left (d x\right )^{m} a^{2} m x + 945 \, \left (d x\right )^{m} a^{2} x}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.58, size = 260, normalized size = 2.57 \begin {gather*} {\left (d\,x\right )}^m\,\left (\frac {c^2\,x^9\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {x^5\,\left (b^2+2\,a\,c\right )\,\left (m^4+20\,m^3+130\,m^2+300\,m+189\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {a^2\,x\,\left (m^4+24\,m^3+206\,m^2+744\,m+945\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {2\,a\,b\,x^3\,\left (m^4+22\,m^3+164\,m^2+458\,m+315\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {2\,b\,c\,x^7\,\left (m^4+18\,m^3+104\,m^2+222\,m+135\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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