Optimal. Leaf size=151 \[ \frac {a^2 c^3 x^{1+m}}{1+m}+\frac {a c^2 (2 b c+3 a d) x^{3+m}}{3+m}+\frac {c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{5+m}}{5+m}+\frac {d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{7+m}}{7+m}+\frac {b d^2 (3 b c+2 a d) x^{9+m}}{9+m}+\frac {b^2 d^3 x^{11+m}}{11+m} \]
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Rubi [A]
time = 0.06, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {459}
\begin {gather*} \frac {c x^{m+5} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )}{m+5}+\frac {d x^{m+7} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )}{m+7}+\frac {a^2 c^3 x^{m+1}}{m+1}+\frac {a c^2 x^{m+3} (3 a d+2 b c)}{m+3}+\frac {b d^2 x^{m+9} (2 a d+3 b c)}{m+9}+\frac {b^2 d^3 x^{m+11}}{m+11} \end {gather*}
Antiderivative was successfully verified.
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Rule 459
Rubi steps
\begin {align*} \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx &=\int \left (a^2 c^3 x^m+a c^2 (2 b c+3 a d) x^{2+m}+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{4+m}+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{6+m}+b d^2 (3 b c+2 a d) x^{8+m}+b^2 d^3 x^{10+m}\right ) \, dx\\ &=\frac {a^2 c^3 x^{1+m}}{1+m}+\frac {a c^2 (2 b c+3 a d) x^{3+m}}{3+m}+\frac {c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{5+m}}{5+m}+\frac {d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{7+m}}{7+m}+\frac {b d^2 (3 b c+2 a d) x^{9+m}}{9+m}+\frac {b^2 d^3 x^{11+m}}{11+m}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 141, normalized size = 0.93 \begin {gather*} x^m \left (\frac {a^2 c^3 x}{1+m}+\frac {a c^2 (2 b c+3 a d) x^3}{3+m}+\frac {c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^5}{5+m}+\frac {d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^7}{7+m}+\frac {b d^2 (3 b c+2 a d) x^9}{9+m}+\frac {b^2 d^3 x^{11}}{11+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. \(974\) vs.
\(2(151)=302\).
time = 0.09, size = 975, normalized size = 6.46
method | result | size |
risch | \(\frac {x \left (b^{2} d^{3} m^{5} x^{10}+25 b^{2} d^{3} m^{4} x^{10}+2 a b \,d^{3} m^{5} x^{8}+3 b^{2} c \,d^{2} m^{5} x^{8}+230 b^{2} d^{3} m^{3} x^{10}+54 a b \,d^{3} m^{4} x^{8}+81 b^{2} c \,d^{2} m^{4} x^{8}+950 b^{2} d^{3} m^{2} x^{10}+a^{2} d^{3} m^{5} x^{6}+6 a b c \,d^{2} m^{5} x^{6}+524 a b \,d^{3} m^{3} x^{8}+3 b^{2} c^{2} d \,m^{5} x^{6}+786 b^{2} c \,d^{2} m^{3} x^{8}+1689 m \,x^{10} b^{2} d^{3}+29 a^{2} d^{3} m^{4} x^{6}+174 a b c \,d^{2} m^{4} x^{6}+2244 a b \,d^{3} m^{2} x^{8}+87 b^{2} c^{2} d \,m^{4} x^{6}+3366 b^{2} c \,d^{2} m^{2} x^{8}+945 b^{2} d^{3} x^{10}+3 a^{2} c \,d^{2} m^{5} x^{4}+302 a^{2} d^{3} m^{3} x^{6}+6 a b \,c^{2} d \,m^{5} x^{4}+1812 a b c \,d^{2} m^{3} x^{6}+4082 a b \,d^{3} x^{8} m +b^{2} c^{3} m^{5} x^{4}+906 b^{2} c^{2} d \,m^{3} x^{6}+6123 b^{2} c \,d^{2} x^{8} m +93 a^{2} c \,d^{2} m^{4} x^{4}+1366 a^{2} d^{3} m^{2} x^{6}+186 a b \,c^{2} d \,m^{4} x^{4}+8196 a b c \,d^{2} m^{2} x^{6}+2310 a b \,d^{3} x^{8}+31 b^{2} c^{3} m^{4} x^{4}+4098 b^{2} c^{2} d \,m^{2} x^{6}+3465 b^{2} c \,d^{2} x^{8}+3 a^{2} c^{2} d \,m^{5} x^{2}+1050 a^{2} c \,d^{2} m^{3} x^{4}+2577 a^{2} d^{3} x^{6} m +2 a b \,c^{3} m^{5} x^{2}+2100 a b \,c^{2} d \,m^{3} x^{4}+15462 a b c \,d^{2} x^{6} m +350 b^{2} c^{3} m^{3} x^{4}+7731 b^{2} c^{2} d \,x^{6} m +99 a^{2} c^{2} d \,m^{4} x^{2}+5190 a^{2} c \,d^{2} m^{2} x^{4}+1485 a^{2} d^{3} x^{6}+66 a b \,c^{3} m^{4} x^{2}+10380 a b \,c^{2} d \,m^{2} x^{4}+8910 a b c \,d^{2} x^{6}+1730 b^{2} c^{3} m^{2} x^{4}+4455 b^{2} c^{2} d \,x^{6}+a^{2} c^{3} m^{5}+1218 a^{2} c^{2} d \,m^{3} x^{2}+10467 a^{2} c \,d^{2} x^{4} m +812 a b \,c^{3} m^{3} x^{2}+20934 a b \,c^{2} d \,x^{4} m +3489 b^{2} c^{3} x^{4} m +35 a^{2} c^{3} m^{4}+6786 a^{2} c^{2} d \,m^{2} x^{2}+6237 a^{2} c \,d^{2} x^{4}+4524 a b \,c^{3} m^{2} x^{2}+12474 a b \,c^{2} d \,x^{4}+2079 b^{2} c^{3} x^{4}+470 a^{2} c^{3} m^{3}+16059 a^{2} c^{2} d \,x^{2} m +10706 a b \,c^{3} x^{2} m +3010 a^{2} c^{3} m^{2}+10395 a^{2} c^{2} d \,x^{2}+6930 a b \,c^{3} x^{2}+9129 a^{2} c^{3} m +10395 a^{2} c^{3}\right ) x^{m}}{\left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(975\) |
gosper | \(\frac {x^{1+m} \left (b^{2} d^{3} m^{5} x^{10}+25 b^{2} d^{3} m^{4} x^{10}+2 a b \,d^{3} m^{5} x^{8}+3 b^{2} c \,d^{2} m^{5} x^{8}+230 b^{2} d^{3} m^{3} x^{10}+54 a b \,d^{3} m^{4} x^{8}+81 b^{2} c \,d^{2} m^{4} x^{8}+950 b^{2} d^{3} m^{2} x^{10}+a^{2} d^{3} m^{5} x^{6}+6 a b c \,d^{2} m^{5} x^{6}+524 a b \,d^{3} m^{3} x^{8}+3 b^{2} c^{2} d \,m^{5} x^{6}+786 b^{2} c \,d^{2} m^{3} x^{8}+1689 m \,x^{10} b^{2} d^{3}+29 a^{2} d^{3} m^{4} x^{6}+174 a b c \,d^{2} m^{4} x^{6}+2244 a b \,d^{3} m^{2} x^{8}+87 b^{2} c^{2} d \,m^{4} x^{6}+3366 b^{2} c \,d^{2} m^{2} x^{8}+945 b^{2} d^{3} x^{10}+3 a^{2} c \,d^{2} m^{5} x^{4}+302 a^{2} d^{3} m^{3} x^{6}+6 a b \,c^{2} d \,m^{5} x^{4}+1812 a b c \,d^{2} m^{3} x^{6}+4082 a b \,d^{3} x^{8} m +b^{2} c^{3} m^{5} x^{4}+906 b^{2} c^{2} d \,m^{3} x^{6}+6123 b^{2} c \,d^{2} x^{8} m +93 a^{2} c \,d^{2} m^{4} x^{4}+1366 a^{2} d^{3} m^{2} x^{6}+186 a b \,c^{2} d \,m^{4} x^{4}+8196 a b c \,d^{2} m^{2} x^{6}+2310 a b \,d^{3} x^{8}+31 b^{2} c^{3} m^{4} x^{4}+4098 b^{2} c^{2} d \,m^{2} x^{6}+3465 b^{2} c \,d^{2} x^{8}+3 a^{2} c^{2} d \,m^{5} x^{2}+1050 a^{2} c \,d^{2} m^{3} x^{4}+2577 a^{2} d^{3} x^{6} m +2 a b \,c^{3} m^{5} x^{2}+2100 a b \,c^{2} d \,m^{3} x^{4}+15462 a b c \,d^{2} x^{6} m +350 b^{2} c^{3} m^{3} x^{4}+7731 b^{2} c^{2} d \,x^{6} m +99 a^{2} c^{2} d \,m^{4} x^{2}+5190 a^{2} c \,d^{2} m^{2} x^{4}+1485 a^{2} d^{3} x^{6}+66 a b \,c^{3} m^{4} x^{2}+10380 a b \,c^{2} d \,m^{2} x^{4}+8910 a b c \,d^{2} x^{6}+1730 b^{2} c^{3} m^{2} x^{4}+4455 b^{2} c^{2} d \,x^{6}+a^{2} c^{3} m^{5}+1218 a^{2} c^{2} d \,m^{3} x^{2}+10467 a^{2} c \,d^{2} x^{4} m +812 a b \,c^{3} m^{3} x^{2}+20934 a b \,c^{2} d \,x^{4} m +3489 b^{2} c^{3} x^{4} m +35 a^{2} c^{3} m^{4}+6786 a^{2} c^{2} d \,m^{2} x^{2}+6237 a^{2} c \,d^{2} x^{4}+4524 a b \,c^{3} m^{2} x^{2}+12474 a b \,c^{2} d \,x^{4}+2079 b^{2} c^{3} x^{4}+470 a^{2} c^{3} m^{3}+16059 a^{2} c^{2} d \,x^{2} m +10706 a b \,c^{3} x^{2} m +3010 a^{2} c^{3} m^{2}+10395 a^{2} c^{2} d \,x^{2}+6930 a b \,c^{3} x^{2}+9129 a^{2} c^{3} m +10395 a^{2} c^{3}\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 )}\) | \(976\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 215, normalized size = 1.42 \begin {gather*} \frac {b^{2} d^{3} x^{m + 11}}{m + 11} + \frac {3 \, b^{2} c d^{2} x^{m + 9}}{m + 9} + \frac {2 \, a b d^{3} x^{m + 9}}{m + 9} + \frac {3 \, b^{2} c^{2} d x^{m + 7}}{m + 7} + \frac {6 \, a b c d^{2} x^{m + 7}}{m + 7} + \frac {a^{2} d^{3} x^{m + 7}}{m + 7} + \frac {b^{2} c^{3} x^{m + 5}}{m + 5} + \frac {6 \, a b c^{2} d x^{m + 5}}{m + 5} + \frac {3 \, a^{2} c d^{2} x^{m + 5}}{m + 5} + \frac {2 \, a b c^{3} x^{m + 3}}{m + 3} + \frac {3 \, a^{2} c^{2} d x^{m + 3}}{m + 3} + \frac {a^{2} c^{3} x^{m + 1}}{m + 1} \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.
\(2 (151) = 302\).
time = 0.91, size = 773, normalized size = 5.12 \begin {gather*} \frac {{\left ({\left (b^{2} d^{3} m^{5} + 25 \, b^{2} d^{3} m^{4} + 230 \, b^{2} d^{3} m^{3} + 950 \, b^{2} d^{3} m^{2} + 1689 \, b^{2} d^{3} m + 945 \, b^{2} d^{3}\right )} x^{11} + {\left ({\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} m^{5} + 3465 \, b^{2} c d^{2} + 2310 \, a b d^{3} + 27 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} m^{4} + 262 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} m^{3} + 1122 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} m^{2} + 2041 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} m\right )} x^{9} + {\left ({\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} m^{5} + 4455 \, b^{2} c^{2} d + 8910 \, a b c d^{2} + 1485 \, a^{2} d^{3} + 29 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} m^{4} + 302 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} m^{3} + 1366 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} m^{2} + 2577 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} m\right )} x^{7} + {\left ({\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} m^{5} + 2079 \, b^{2} c^{3} + 12474 \, a b c^{2} d + 6237 \, a^{2} c d^{2} + 31 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} m^{4} + 350 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} m^{3} + 1730 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} m^{2} + 3489 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} m\right )} x^{5} + {\left ({\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} m^{5} + 6930 \, a b c^{3} + 10395 \, a^{2} c^{2} d + 33 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} m^{4} + 406 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} m^{3} + 2262 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} m^{2} + 5353 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} m\right )} x^{3} + {\left (a^{2} c^{3} m^{5} + 35 \, a^{2} c^{3} m^{4} + 470 \, a^{2} c^{3} m^{3} + 3010 \, a^{2} c^{3} m^{2} + 9129 \, a^{2} c^{3} m + 10395 \, a^{2} c^{3}\right )} x\right )} x^{m}}{m^{6} + 36 \, m^{5} + 505 \, m^{4} + 3480 \, m^{3} + 12139 \, m^{2} + 19524 \, m + 10395} \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. 4345 vs.
\(2 (144) = 288\).
time = 0.91, size = 4345, normalized size = 28.77 \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. 1192 vs.
\(2 (151) = 302\).
time = 0.61, size = 1192, normalized size = 7.89 \begin {gather*} \frac {b^{2} d^{3} m^{5} x^{11} x^{m} + 25 \, b^{2} d^{3} m^{4} x^{11} x^{m} + 3 \, b^{2} c d^{2} m^{5} x^{9} x^{m} + 2 \, a b d^{3} m^{5} x^{9} x^{m} + 230 \, b^{2} d^{3} m^{3} x^{11} x^{m} + 81 \, b^{2} c d^{2} m^{4} x^{9} x^{m} + 54 \, a b d^{3} m^{4} x^{9} x^{m} + 950 \, b^{2} d^{3} m^{2} x^{11} x^{m} + 3 \, b^{2} c^{2} d m^{5} x^{7} x^{m} + 6 \, a b c d^{2} m^{5} x^{7} x^{m} + a^{2} d^{3} m^{5} x^{7} x^{m} + 786 \, b^{2} c d^{2} m^{3} x^{9} x^{m} + 524 \, a b d^{3} m^{3} x^{9} x^{m} + 1689 \, b^{2} d^{3} m x^{11} x^{m} + 87 \, b^{2} c^{2} d m^{4} x^{7} x^{m} + 174 \, a b c d^{2} m^{4} x^{7} x^{m} + 29 \, a^{2} d^{3} m^{4} x^{7} x^{m} + 3366 \, b^{2} c d^{2} m^{2} x^{9} x^{m} + 2244 \, a b d^{3} m^{2} x^{9} x^{m} + 945 \, b^{2} d^{3} x^{11} x^{m} + b^{2} c^{3} m^{5} x^{5} x^{m} + 6 \, a b c^{2} d m^{5} x^{5} x^{m} + 3 \, a^{2} c d^{2} m^{5} x^{5} x^{m} + 906 \, b^{2} c^{2} d m^{3} x^{7} x^{m} + 1812 \, a b c d^{2} m^{3} x^{7} x^{m} + 302 \, a^{2} d^{3} m^{3} x^{7} x^{m} + 6123 \, b^{2} c d^{2} m x^{9} x^{m} + 4082 \, a b d^{3} m x^{9} x^{m} + 31 \, b^{2} c^{3} m^{4} x^{5} x^{m} + 186 \, a b c^{2} d m^{4} x^{5} x^{m} + 93 \, a^{2} c d^{2} m^{4} x^{5} x^{m} + 4098 \, b^{2} c^{2} d m^{2} x^{7} x^{m} + 8196 \, a b c d^{2} m^{2} x^{7} x^{m} + 1366 \, a^{2} d^{3} m^{2} x^{7} x^{m} + 3465 \, b^{2} c d^{2} x^{9} x^{m} + 2310 \, a b d^{3} x^{9} x^{m} + 2 \, a b c^{3} m^{5} x^{3} x^{m} + 3 \, a^{2} c^{2} d m^{5} x^{3} x^{m} + 350 \, b^{2} c^{3} m^{3} x^{5} x^{m} + 2100 \, a b c^{2} d m^{3} x^{5} x^{m} + 1050 \, a^{2} c d^{2} m^{3} x^{5} x^{m} + 7731 \, b^{2} c^{2} d m x^{7} x^{m} + 15462 \, a b c d^{2} m x^{7} x^{m} + 2577 \, a^{2} d^{3} m x^{7} x^{m} + 66 \, a b c^{3} m^{4} x^{3} x^{m} + 99 \, a^{2} c^{2} d m^{4} x^{3} x^{m} + 1730 \, b^{2} c^{3} m^{2} x^{5} x^{m} + 10380 \, a b c^{2} d m^{2} x^{5} x^{m} + 5190 \, a^{2} c d^{2} m^{2} x^{5} x^{m} + 4455 \, b^{2} c^{2} d x^{7} x^{m} + 8910 \, a b c d^{2} x^{7} x^{m} + 1485 \, a^{2} d^{3} x^{7} x^{m} + a^{2} c^{3} m^{5} x x^{m} + 812 \, a b c^{3} m^{3} x^{3} x^{m} + 1218 \, a^{2} c^{2} d m^{3} x^{3} x^{m} + 3489 \, b^{2} c^{3} m x^{5} x^{m} + 20934 \, a b c^{2} d m x^{5} x^{m} + 10467 \, a^{2} c d^{2} m x^{5} x^{m} + 35 \, a^{2} c^{3} m^{4} x x^{m} + 4524 \, a b c^{3} m^{2} x^{3} x^{m} + 6786 \, a^{2} c^{2} d m^{2} x^{3} x^{m} + 2079 \, b^{2} c^{3} x^{5} x^{m} + 12474 \, a b c^{2} d x^{5} x^{m} + 6237 \, a^{2} c d^{2} x^{5} x^{m} + 470 \, a^{2} c^{3} m^{3} x x^{m} + 10706 \, a b c^{3} m x^{3} x^{m} + 16059 \, a^{2} c^{2} d m x^{3} x^{m} + 3010 \, a^{2} c^{3} m^{2} x x^{m} + 6930 \, a b c^{3} x^{3} x^{m} + 10395 \, a^{2} c^{2} d x^{3} x^{m} + 9129 \, a^{2} c^{3} m x x^{m} + 10395 \, a^{2} c^{3} x x^{m}}{m^{6} + 36 \, m^{5} + 505 \, m^{4} + 3480 \, m^{3} + 12139 \, m^{2} + 19524 \, m + 10395} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.35, size = 443, normalized size = 2.93 \begin {gather*} \frac {a^2\,c^3\,x\,x^m\,\left (m^5+35\,m^4+470\,m^3+3010\,m^2+9129\,m+10395\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {c\,x^m\,x^5\,\left (3\,a^2\,d^2+6\,a\,b\,c\,d+b^2\,c^2\right )\,\left (m^5+31\,m^4+350\,m^3+1730\,m^2+3489\,m+2079\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {d\,x^m\,x^7\,\left (a^2\,d^2+6\,a\,b\,c\,d+3\,b^2\,c^2\right )\,\left (m^5+29\,m^4+302\,m^3+1366\,m^2+2577\,m+1485\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {b^2\,d^3\,x^m\,x^{11}\,\left (m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {a\,c^2\,x^m\,x^3\,\left (3\,a\,d+2\,b\,c\right )\,\left (m^5+33\,m^4+406\,m^3+2262\,m^2+5353\,m+3465\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {b\,d^2\,x^m\,x^9\,\left (2\,a\,d+3\,b\,c\right )\,\left (m^5+27\,m^4+262\,m^3+1122\,m^2+2041\,m+1155\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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