Optimal. Leaf size=82 \[ a^2 c^2 x+\frac {2}{3} a c (b c+a d) x^3+\frac {1}{5} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^5+\frac {2}{7} b d (b c+a d) x^7+\frac {1}{9} b^2 d^2 x^9 \]
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Rubi [A]
time = 0.03, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {380}
\begin {gather*} \frac {1}{5} x^5 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 x+\frac {2}{7} b d x^7 (a d+b c)+\frac {2}{3} a c x^3 (a d+b c)+\frac {1}{9} b^2 d^2 x^9 \end {gather*}
Antiderivative was successfully verified.
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Rule 380
Rubi steps
\begin {align*} \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx &=\int \left (a^2 c^2+2 a c (b c+a d) x^2+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^4+2 b d (b c+a d) x^6+b^2 d^2 x^8\right ) \, dx\\ &=a^2 c^2 x+\frac {2}{3} a c (b c+a d) x^3+\frac {1}{5} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^5+\frac {2}{7} b d (b c+a d) x^7+\frac {1}{9} b^2 d^2 x^9\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 82, normalized size = 1.00 \begin {gather*} a^2 c^2 x+\frac {2}{3} a c (b c+a d) x^3+\frac {1}{5} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^5+\frac {2}{7} b d (b c+a d) x^7+\frac {1}{9} b^2 d^2 x^9 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 87, normalized size = 1.06
method | result | size |
norman | \(\frac {b^{2} d^{2} x^{9}}{9}+\left (\frac {2}{7} a b \,d^{2}+\frac {2}{7} b^{2} c d \right ) x^{7}+\left (\frac {1}{5} a^{2} d^{2}+\frac {4}{5} a b c d +\frac {1}{5} b^{2} c^{2}\right ) x^{5}+\left (\frac {2}{3} a^{2} c d +\frac {2}{3} a b \,c^{2}\right ) x^{3}+a^{2} c^{2} x\) | \(86\) |
default | \(\frac {b^{2} d^{2} x^{9}}{9}+\frac {\left (2 a b \,d^{2}+2 b^{2} c d \right ) x^{7}}{7}+\frac {\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{5}}{5}+\frac {\left (2 a^{2} c d +2 a b \,c^{2}\right ) x^{3}}{3}+a^{2} c^{2} x\) | \(87\) |
gosper | \(\frac {1}{9} b^{2} d^{2} x^{9}+\frac {2}{7} x^{7} a b \,d^{2}+\frac {2}{7} x^{7} b^{2} c d +\frac {1}{5} x^{5} a^{2} d^{2}+\frac {4}{5} x^{5} a b c d +\frac {1}{5} x^{5} b^{2} c^{2}+\frac {2}{3} x^{3} a^{2} c d +\frac {2}{3} x^{3} a b \,c^{2}+a^{2} c^{2} x\) | \(92\) |
risch | \(\frac {1}{9} b^{2} d^{2} x^{9}+\frac {2}{7} x^{7} a b \,d^{2}+\frac {2}{7} x^{7} b^{2} c d +\frac {1}{5} x^{5} a^{2} d^{2}+\frac {4}{5} x^{5} a b c d +\frac {1}{5} x^{5} b^{2} c^{2}+\frac {2}{3} x^{3} a^{2} c d +\frac {2}{3} x^{3} a b \,c^{2}+a^{2} c^{2} x\) | \(92\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 82, normalized size = 1.00 \begin {gather*} \frac {1}{9} \, b^{2} d^{2} x^{9} + \frac {2}{7} \, {\left (b^{2} c d + a b d^{2}\right )} x^{7} + \frac {1}{5} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{5} + a^{2} c^{2} x + \frac {2}{3} \, {\left (a b c^{2} + a^{2} c d\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.35, size = 82, normalized size = 1.00 \begin {gather*} \frac {1}{9} \, b^{2} d^{2} x^{9} + \frac {2}{7} \, {\left (b^{2} c d + a b d^{2}\right )} x^{7} + \frac {1}{5} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{5} + a^{2} c^{2} x + \frac {2}{3} \, {\left (a b c^{2} + a^{2} c d\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 97, normalized size = 1.18 \begin {gather*} a^{2} c^{2} x + \frac {b^{2} d^{2} x^{9}}{9} + x^{7} \cdot \left (\frac {2 a b d^{2}}{7} + \frac {2 b^{2} c d}{7}\right ) + x^{5} \left (\frac {a^{2} d^{2}}{5} + \frac {4 a b c d}{5} + \frac {b^{2} c^{2}}{5}\right ) + x^{3} \cdot \left (\frac {2 a^{2} c d}{3} + \frac {2 a b c^{2}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.78, size = 91, normalized size = 1.11 \begin {gather*} \frac {1}{9} \, b^{2} d^{2} x^{9} + \frac {2}{7} \, b^{2} c d x^{7} + \frac {2}{7} \, a b d^{2} x^{7} + \frac {1}{5} \, b^{2} c^{2} x^{5} + \frac {4}{5} \, a b c d x^{5} + \frac {1}{5} \, a^{2} d^{2} x^{5} + \frac {2}{3} \, a b c^{2} x^{3} + \frac {2}{3} \, a^{2} c d x^{3} + a^{2} c^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 75, normalized size = 0.91 \begin {gather*} x^5\,\left (\frac {a^2\,d^2}{5}+\frac {4\,a\,b\,c\,d}{5}+\frac {b^2\,c^2}{5}\right )+a^2\,c^2\,x+\frac {b^2\,d^2\,x^9}{9}+\frac {2\,a\,c\,x^3\,\left (a\,d+b\,c\right )}{3}+\frac {2\,b\,d\,x^7\,\left (a\,d+b\,c\right )}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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