Optimal. Leaf size=81 \[ -\frac {a^2 c^2}{x}+2 a c (b c+a d) x+\frac {1}{3} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^3+\frac {2}{5} b d (b c+a d) x^5+\frac {1}{7} b^2 d^2 x^7 \]
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Rubi [A]
time = 0.03, antiderivative size = 81, 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}{3} x^3 \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac {a^2 c^2}{x}+\frac {2}{5} b d x^5 (a d+b c)+2 a c x (a d+b c)+\frac {1}{7} b^2 d^2 x^7 \end {gather*}
Antiderivative was successfully verified.
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Rule 459
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^2} \, dx &=\int \left (2 a c (b c+a d)+\frac {a^2 c^2}{x^2}+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^2+2 b d (b c+a d) x^4+b^2 d^2 x^6\right ) \, dx\\ &=-\frac {a^2 c^2}{x}+2 a c (b c+a d) x+\frac {1}{3} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^3+\frac {2}{5} b d (b c+a d) x^5+\frac {1}{7} b^2 d^2 x^7\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 81, normalized size = 1.00 \begin {gather*} -\frac {a^2 c^2}{x}+2 a c (b c+a d) x+\frac {1}{3} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^3+\frac {2}{5} b d (b c+a d) x^5+\frac {1}{7} b^2 d^2 x^7 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 91, normalized size = 1.12
method | result | size |
norman | \(\frac {\frac {b^{2} d^{2} x^{8}}{7}+\left (\frac {2}{5} a b \,d^{2}+\frac {2}{5} b^{2} c d \right ) x^{6}+\left (\frac {1}{3} a^{2} d^{2}+\frac {4}{3} a b c d +\frac {1}{3} b^{2} c^{2}\right ) x^{4}+\left (2 a^{2} c d +2 a b \,c^{2}\right ) x^{2}-a^{2} c^{2}}{x}\) | \(90\) |
default | \(\frac {b^{2} d^{2} x^{7}}{7}+\frac {2 a b \,d^{2} x^{5}}{5}+\frac {2 b^{2} c d \,x^{5}}{5}+\frac {a^{2} d^{2} x^{3}}{3}+\frac {4 a b c d \,x^{3}}{3}+\frac {b^{2} c^{2} x^{3}}{3}+2 a^{2} c d x +2 a b \,c^{2} x -\frac {a^{2} c^{2}}{x}\) | \(91\) |
risch | \(\frac {b^{2} d^{2} x^{7}}{7}+\frac {2 a b \,d^{2} x^{5}}{5}+\frac {2 b^{2} c d \,x^{5}}{5}+\frac {a^{2} d^{2} x^{3}}{3}+\frac {4 a b c d \,x^{3}}{3}+\frac {b^{2} c^{2} x^{3}}{3}+2 a^{2} c d x +2 a b \,c^{2} x -\frac {a^{2} c^{2}}{x}\) | \(91\) |
gosper | \(-\frac {-15 b^{2} d^{2} x^{8}-42 a b \,d^{2} x^{6}-42 b^{2} c d \,x^{6}-35 a^{2} d^{2} x^{4}-140 a b c d \,x^{4}-35 b^{2} c^{2} x^{4}-210 a^{2} c d \,x^{2}-210 a b \,c^{2} x^{2}+105 a^{2} c^{2}}{105 x}\) | \(97\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 83, normalized size = 1.02 \begin {gather*} \frac {1}{7} \, b^{2} d^{2} x^{7} + \frac {2}{5} \, {\left (b^{2} c d + a b d^{2}\right )} x^{5} + \frac {1}{3} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{3} - \frac {a^{2} c^{2}}{x} + 2 \, {\left (a b c^{2} + a^{2} c d\right )} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.92, size = 87, normalized size = 1.07 \begin {gather*} \frac {15 \, b^{2} d^{2} x^{8} + 42 \, {\left (b^{2} c d + a b d^{2}\right )} x^{6} + 35 \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} - 105 \, a^{2} c^{2} + 210 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2}}{105 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 92, normalized size = 1.14 \begin {gather*} - \frac {a^{2} c^{2}}{x} + \frac {b^{2} d^{2} x^{7}}{7} + x^{5} \cdot \left (\frac {2 a b d^{2}}{5} + \frac {2 b^{2} c d}{5}\right ) + x^{3} \left (\frac {a^{2} d^{2}}{3} + \frac {4 a b c d}{3} + \frac {b^{2} c^{2}}{3}\right ) + x \left (2 a^{2} c d + 2 a b c^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.99, size = 90, normalized size = 1.11 \begin {gather*} \frac {1}{7} \, b^{2} d^{2} x^{7} + \frac {2}{5} \, b^{2} c d x^{5} + \frac {2}{5} \, a b d^{2} x^{5} + \frac {1}{3} \, b^{2} c^{2} x^{3} + \frac {4}{3} \, a b c d x^{3} + \frac {1}{3} \, a^{2} d^{2} x^{3} + 2 \, a b c^{2} x + 2 \, a^{2} c d x - \frac {a^{2} c^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 76, normalized size = 0.94 \begin {gather*} x^3\,\left (\frac {a^2\,d^2}{3}+\frac {4\,a\,b\,c\,d}{3}+\frac {b^2\,c^2}{3}\right )-\frac {a^2\,c^2}{x}+\frac {b^2\,d^2\,x^7}{7}+2\,a\,c\,x\,\left (a\,d+b\,c\right )+\frac {2\,b\,d\,x^5\,\left (a\,d+b\,c\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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