Optimal. Leaf size=127 \[ \frac {1}{3} a^2 c^3 x^3+\frac {1}{5} a c^2 (2 b c+3 a d) x^5+\frac {1}{7} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^7+\frac {1}{9} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^9+\frac {1}{11} b d^2 (3 b c+2 a d) x^{11}+\frac {1}{13} b^2 d^3 x^{13} \]
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Rubi [A]
time = 0.05, antiderivative size = 127, 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 {1}{9} d x^9 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {1}{7} c x^7 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac {1}{3} a^2 c^3 x^3+\frac {1}{5} a c^2 x^5 (3 a d+2 b c)+\frac {1}{11} b d^2 x^{11} (2 a d+3 b c)+\frac {1}{13} b^2 d^3 x^{13} \end {gather*}
Antiderivative was successfully verified.
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Rule 459
Rubi steps
\begin {align*} \int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx &=\int \left (a^2 c^3 x^2+a c^2 (2 b c+3 a d) x^4+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^6+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^8+b d^2 (3 b c+2 a d) x^{10}+b^2 d^3 x^{12}\right ) \, dx\\ &=\frac {1}{3} a^2 c^3 x^3+\frac {1}{5} a c^2 (2 b c+3 a d) x^5+\frac {1}{7} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^7+\frac {1}{9} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^9+\frac {1}{11} b d^2 (3 b c+2 a d) x^{11}+\frac {1}{13} b^2 d^3 x^{13}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 127, normalized size = 1.00 \begin {gather*} \frac {1}{3} a^2 c^3 x^3+\frac {1}{5} a c^2 (2 b c+3 a d) x^5+\frac {1}{7} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^7+\frac {1}{9} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^9+\frac {1}{11} b d^2 (3 b c+2 a d) x^{11}+\frac {1}{13} b^2 d^3 x^{13} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 128, normalized size = 1.01
method | result | size |
norman | \(\frac {b^{2} d^{3} x^{13}}{13}+\left (\frac {2}{11} a b \,d^{3}+\frac {3}{11} b^{2} c \,d^{2}\right ) x^{11}+\left (\frac {1}{9} a^{2} d^{3}+\frac {2}{3} a b c \,d^{2}+\frac {1}{3} b^{2} c^{2} d \right ) x^{9}+\left (\frac {3}{7} a^{2} c \,d^{2}+\frac {6}{7} a b \,c^{2} d +\frac {1}{7} b^{2} c^{3}\right ) x^{7}+\left (\frac {3}{5} a^{2} c^{2} d +\frac {2}{5} a b \,c^{3}\right ) x^{5}+\frac {a^{2} c^{3} x^{3}}{3}\) | \(126\) |
default | \(\frac {b^{2} d^{3} x^{13}}{13}+\frac {\left (2 a b \,d^{3}+3 b^{2} c \,d^{2}\right ) x^{11}}{11}+\frac {\left (a^{2} d^{3}+6 a b c \,d^{2}+3 b^{2} c^{2} d \right ) x^{9}}{9}+\frac {\left (3 a^{2} c \,d^{2}+6 a b \,c^{2} d +b^{2} c^{3}\right ) x^{7}}{7}+\frac {\left (3 a^{2} c^{2} d +2 a b \,c^{3}\right ) x^{5}}{5}+\frac {a^{2} c^{3} x^{3}}{3}\) | \(128\) |
gosper | \(\frac {1}{13} b^{2} d^{3} x^{13}+\frac {2}{11} x^{11} a b \,d^{3}+\frac {3}{11} x^{11} b^{2} c \,d^{2}+\frac {1}{9} x^{9} a^{2} d^{3}+\frac {2}{3} x^{9} a b c \,d^{2}+\frac {1}{3} x^{9} b^{2} c^{2} d +\frac {3}{7} x^{7} a^{2} c \,d^{2}+\frac {6}{7} x^{7} a b \,c^{2} d +\frac {1}{7} x^{7} b^{2} c^{3}+\frac {3}{5} x^{5} a^{2} c^{2} d +\frac {2}{5} x^{5} a b \,c^{3}+\frac {1}{3} a^{2} c^{3} x^{3}\) | \(136\) |
risch | \(\frac {1}{13} b^{2} d^{3} x^{13}+\frac {2}{11} x^{11} a b \,d^{3}+\frac {3}{11} x^{11} b^{2} c \,d^{2}+\frac {1}{9} x^{9} a^{2} d^{3}+\frac {2}{3} x^{9} a b c \,d^{2}+\frac {1}{3} x^{9} b^{2} c^{2} d +\frac {3}{7} x^{7} a^{2} c \,d^{2}+\frac {6}{7} x^{7} a b \,c^{2} d +\frac {1}{7} x^{7} b^{2} c^{3}+\frac {3}{5} x^{5} a^{2} c^{2} d +\frac {2}{5} x^{5} a b \,c^{3}+\frac {1}{3} a^{2} c^{3} x^{3}\) | \(136\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 127, normalized size = 1.00 \begin {gather*} \frac {1}{13} \, b^{2} d^{3} x^{13} + \frac {1}{11} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{11} + \frac {1}{9} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{9} + \frac {1}{3} \, a^{2} c^{3} x^{3} + \frac {1}{7} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{7} + \frac {1}{5} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.65, size = 127, normalized size = 1.00 \begin {gather*} \frac {1}{13} \, b^{2} d^{3} x^{13} + \frac {1}{11} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{11} + \frac {1}{9} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{9} + \frac {1}{3} \, a^{2} c^{3} x^{3} + \frac {1}{7} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{7} + \frac {1}{5} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 143, normalized size = 1.13 \begin {gather*} \frac {a^{2} c^{3} x^{3}}{3} + \frac {b^{2} d^{3} x^{13}}{13} + x^{11} \cdot \left (\frac {2 a b d^{3}}{11} + \frac {3 b^{2} c d^{2}}{11}\right ) + x^{9} \left (\frac {a^{2} d^{3}}{9} + \frac {2 a b c d^{2}}{3} + \frac {b^{2} c^{2} d}{3}\right ) + x^{7} \cdot \left (\frac {3 a^{2} c d^{2}}{7} + \frac {6 a b c^{2} d}{7} + \frac {b^{2} c^{3}}{7}\right ) + x^{5} \cdot \left (\frac {3 a^{2} c^{2} d}{5} + \frac {2 a b c^{3}}{5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.75, size = 135, normalized size = 1.06 \begin {gather*} \frac {1}{13} \, b^{2} d^{3} x^{13} + \frac {3}{11} \, b^{2} c d^{2} x^{11} + \frac {2}{11} \, a b d^{3} x^{11} + \frac {1}{3} \, b^{2} c^{2} d x^{9} + \frac {2}{3} \, a b c d^{2} x^{9} + \frac {1}{9} \, a^{2} d^{3} x^{9} + \frac {1}{7} \, b^{2} c^{3} x^{7} + \frac {6}{7} \, a b c^{2} d x^{7} + \frac {3}{7} \, a^{2} c d^{2} x^{7} + \frac {2}{5} \, a b c^{3} x^{5} + \frac {3}{5} \, a^{2} c^{2} d x^{5} + \frac {1}{3} \, a^{2} c^{3} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 119, normalized size = 0.94 \begin {gather*} x^7\,\left (\frac {3\,a^2\,c\,d^2}{7}+\frac {6\,a\,b\,c^2\,d}{7}+\frac {b^2\,c^3}{7}\right )+x^9\,\left (\frac {a^2\,d^3}{9}+\frac {2\,a\,b\,c\,d^2}{3}+\frac {b^2\,c^2\,d}{3}\right )+\frac {a^2\,c^3\,x^3}{3}+\frac {b^2\,d^3\,x^{13}}{13}+\frac {a\,c^2\,x^5\,\left (3\,a\,d+2\,b\,c\right )}{5}+\frac {b\,d^2\,x^{11}\,\left (2\,a\,d+3\,b\,c\right )}{11} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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