Optimal. Leaf size=104 \[ \frac {a^4 (d x)^{1+m}}{d (1+m)}+\frac {4 a^3 b (d x)^{3+m}}{d^3 (3+m)}+\frac {6 a^2 b^2 (d x)^{5+m}}{d^5 (5+m)}+\frac {4 a b^3 (d x)^{7+m}}{d^7 (7+m)}+\frac {b^4 (d x)^{9+m}}{d^9 (9+m)} \]
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Rubi [A]
time = 0.05, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {28, 276}
\begin {gather*} \frac {a^4 (d x)^{m+1}}{d (m+1)}+\frac {4 a^3 b (d x)^{m+3}}{d^3 (m+3)}+\frac {6 a^2 b^2 (d x)^{m+5}}{d^5 (m+5)}+\frac {4 a b^3 (d x)^{m+7}}{d^7 (m+7)}+\frac {b^4 (d x)^{m+9}}{d^9 (m+9)} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 276
Rubi steps
\begin {align*} \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx &=\frac {\int (d x)^m \left (a b+b^2 x^2\right )^4 \, dx}{b^4}\\ &=\frac {\int \left (a^4 b^4 (d x)^m+\frac {4 a^3 b^5 (d x)^{2+m}}{d^2}+\frac {6 a^2 b^6 (d x)^{4+m}}{d^4}+\frac {4 a b^7 (d x)^{6+m}}{d^6}+\frac {b^8 (d x)^{8+m}}{d^8}\right ) \, dx}{b^4}\\ &=\frac {a^4 (d x)^{1+m}}{d (1+m)}+\frac {4 a^3 b (d x)^{3+m}}{d^3 (3+m)}+\frac {6 a^2 b^2 (d x)^{5+m}}{d^5 (5+m)}+\frac {4 a b^3 (d x)^{7+m}}{d^7 (7+m)}+\frac {b^4 (d x)^{9+m}}{d^9 (9+m)}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 73, normalized size = 0.70 \begin {gather*} x (d x)^m \left (\frac {a^4}{1+m}+\frac {4 a^3 b x^2}{3+m}+\frac {6 a^2 b^2 x^4}{5+m}+\frac {4 a b^3 x^6}{7+m}+\frac {b^4 x^8}{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. \(291\) vs.
\(2(104)=208\).
time = 0.01, size = 292, normalized size = 2.81
method | result | size |
gosper | \(\frac {\left (d x \right )^{m} \left (b^{4} m^{4} x^{8}+16 b^{4} m^{3} x^{8}+4 a \,b^{3} m^{4} x^{6}+86 b^{4} m^{2} x^{8}+72 a \,b^{3} m^{3} x^{6}+176 m \,x^{8} b^{4}+6 a^{2} b^{2} m^{4} x^{4}+416 a \,b^{3} m^{2} x^{6}+105 b^{4} x^{8}+120 a^{2} b^{2} m^{3} x^{4}+888 m \,x^{6} a \,b^{3}+4 a^{3} b \,m^{4} x^{2}+780 a^{2} b^{2} m^{2} x^{4}+540 a \,b^{3} x^{6}+88 a^{3} b \,m^{3} x^{2}+1800 m \,x^{4} a^{2} b^{2}+a^{4} m^{4}+656 a^{3} b \,m^{2} x^{2}+1134 a^{2} b^{2} x^{4}+24 a^{4} m^{3}+1832 m \,x^{2} a^{3} b +206 a^{4} m^{2}+1260 a^{3} b \,x^{2}+744 m \,a^{4}+945 a^{4}\right ) x}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(292\) |
risch | \(\frac {\left (d x \right )^{m} \left (b^{4} m^{4} x^{8}+16 b^{4} m^{3} x^{8}+4 a \,b^{3} m^{4} x^{6}+86 b^{4} m^{2} x^{8}+72 a \,b^{3} m^{3} x^{6}+176 m \,x^{8} b^{4}+6 a^{2} b^{2} m^{4} x^{4}+416 a \,b^{3} m^{2} x^{6}+105 b^{4} x^{8}+120 a^{2} b^{2} m^{3} x^{4}+888 m \,x^{6} a \,b^{3}+4 a^{3} b \,m^{4} x^{2}+780 a^{2} b^{2} m^{2} x^{4}+540 a \,b^{3} x^{6}+88 a^{3} b \,m^{3} x^{2}+1800 m \,x^{4} a^{2} b^{2}+a^{4} m^{4}+656 a^{3} b \,m^{2} x^{2}+1134 a^{2} b^{2} x^{4}+24 a^{4} m^{3}+1832 m \,x^{2} a^{3} b +206 a^{4} m^{2}+1260 a^{3} b \,x^{2}+744 m \,a^{4}+945 a^{4}\right ) x}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(292\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 100, normalized size = 0.96 \begin {gather*} \frac {b^{4} d^{m} x^{9} x^{m}}{m + 9} + \frac {4 \, a b^{3} d^{m} x^{7} x^{m}}{m + 7} + \frac {6 \, a^{2} b^{2} d^{m} x^{5} x^{m}}{m + 5} + \frac {4 \, a^{3} b d^{m} x^{3} x^{m}}{m + 3} + \frac {\left (d x\right )^{m + 1} a^{4}}{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. 253 vs.
\(2 (104) = 208\).
time = 0.35, size = 253, normalized size = 2.43 \begin {gather*} \frac {{\left ({\left (b^{4} m^{4} + 16 \, b^{4} m^{3} + 86 \, b^{4} m^{2} + 176 \, b^{4} m + 105 \, b^{4}\right )} x^{9} + 4 \, {\left (a b^{3} m^{4} + 18 \, a b^{3} m^{3} + 104 \, a b^{3} m^{2} + 222 \, a b^{3} m + 135 \, a b^{3}\right )} x^{7} + 6 \, {\left (a^{2} b^{2} m^{4} + 20 \, a^{2} b^{2} m^{3} + 130 \, a^{2} b^{2} m^{2} + 300 \, a^{2} b^{2} m + 189 \, a^{2} b^{2}\right )} x^{5} + 4 \, {\left (a^{3} b m^{4} + 22 \, a^{3} b m^{3} + 164 \, a^{3} b m^{2} + 458 \, a^{3} b m + 315 \, a^{3} b\right )} x^{3} + {\left (a^{4} m^{4} + 24 \, a^{4} m^{3} + 206 \, a^{4} m^{2} + 744 \, a^{4} m + 945 \, a^{4}\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. 1278 vs.
\(2 (94) = 188\).
time = 0.51, size = 1278, normalized size = 12.29 \begin {gather*} \begin {cases} \frac {- \frac {a^{4}}{8 x^{8}} - \frac {2 a^{3} b}{3 x^{6}} - \frac {3 a^{2} b^{2}}{2 x^{4}} - \frac {2 a b^{3}}{x^{2}} + b^{4} \log {\left (x \right )}}{d^{9}} & \text {for}\: m = -9 \\\frac {- \frac {a^{4}}{6 x^{6}} - \frac {a^{3} b}{x^{4}} - \frac {3 a^{2} b^{2}}{x^{2}} + 4 a b^{3} \log {\left (x \right )} + \frac {b^{4} x^{2}}{2}}{d^{7}} & \text {for}\: m = -7 \\\frac {- \frac {a^{4}}{4 x^{4}} - \frac {2 a^{3} b}{x^{2}} + 6 a^{2} b^{2} \log {\left (x \right )} + 2 a b^{3} x^{2} + \frac {b^{4} x^{4}}{4}}{d^{5}} & \text {for}\: m = -5 \\\frac {- \frac {a^{4}}{2 x^{2}} + 4 a^{3} b \log {\left (x \right )} + 3 a^{2} b^{2} x^{2} + a b^{3} x^{4} + \frac {b^{4} x^{6}}{6}}{d^{3}} & \text {for}\: m = -3 \\\frac {a^{4} \log {\left (x \right )} + 2 a^{3} b x^{2} + \frac {3 a^{2} b^{2} x^{4}}{2} + \frac {2 a b^{3} x^{6}}{3} + \frac {b^{4} x^{8}}{8}}{d} & \text {for}\: m = -1 \\\frac {a^{4} 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^{4} 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^{4} 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^{4} m x \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {945 a^{4} x \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {4 a^{3} 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 {88 a^{3} 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 {656 a^{3} 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 {1832 a^{3} b m x^{3} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {1260 a^{3} b x^{3} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {6 a^{2} 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 {120 a^{2} 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 {780 a^{2} 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 {1800 a^{2} 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 {1134 a^{2} b^{2} x^{5} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {4 a b^{3} m^{4} x^{7} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {72 a b^{3} m^{3} x^{7} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {416 a b^{3} m^{2} x^{7} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {888 a b^{3} m x^{7} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {540 a b^{3} x^{7} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {b^{4} 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 b^{4} 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 b^{4} 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 b^{4} m x^{9} \left (d x\right )^{m}}{m^{5} + 25 m^{4} + 230 m^{3} + 950 m^{2} + 1689 m + 945} + \frac {105 b^{4} 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. 415 vs.
\(2 (104) = 208\).
time = 2.58, size = 415, normalized size = 3.99 \begin {gather*} \frac {\left (d x\right )^{m} b^{4} m^{4} x^{9} + 16 \, \left (d x\right )^{m} b^{4} m^{3} x^{9} + 4 \, \left (d x\right )^{m} a b^{3} m^{4} x^{7} + 86 \, \left (d x\right )^{m} b^{4} m^{2} x^{9} + 72 \, \left (d x\right )^{m} a b^{3} m^{3} x^{7} + 176 \, \left (d x\right )^{m} b^{4} m x^{9} + 6 \, \left (d x\right )^{m} a^{2} b^{2} m^{4} x^{5} + 416 \, \left (d x\right )^{m} a b^{3} m^{2} x^{7} + 105 \, \left (d x\right )^{m} b^{4} x^{9} + 120 \, \left (d x\right )^{m} a^{2} b^{2} m^{3} x^{5} + 888 \, \left (d x\right )^{m} a b^{3} m x^{7} + 4 \, \left (d x\right )^{m} a^{3} b m^{4} x^{3} + 780 \, \left (d x\right )^{m} a^{2} b^{2} m^{2} x^{5} + 540 \, \left (d x\right )^{m} a b^{3} x^{7} + 88 \, \left (d x\right )^{m} a^{3} b m^{3} x^{3} + 1800 \, \left (d x\right )^{m} a^{2} b^{2} m x^{5} + \left (d x\right )^{m} a^{4} m^{4} x + 656 \, \left (d x\right )^{m} a^{3} b m^{2} x^{3} + 1134 \, \left (d x\right )^{m} a^{2} b^{2} x^{5} + 24 \, \left (d x\right )^{m} a^{4} m^{3} x + 1832 \, \left (d x\right )^{m} a^{3} b m x^{3} + 206 \, \left (d x\right )^{m} a^{4} m^{2} x + 1260 \, \left (d x\right )^{m} a^{3} b x^{3} + 744 \, \left (d x\right )^{m} a^{4} m x + 945 \, \left (d x\right )^{m} a^{4} 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 = 4.51, size = 263, normalized size = 2.53 \begin {gather*} {\left (d\,x\right )}^m\,\left (\frac {b^4\,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 {a^4\,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 {4\,a\,b^3\,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}+\frac {4\,a^3\,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 {6\,a^2\,b^2\,x^5\,\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}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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