Optimal. Leaf size=53 \[ -\frac {A b^2}{7 x^7}-\frac {b (b B+2 A c)}{5 x^5}-\frac {c (2 b B+A c)}{3 x^3}-\frac {B c^2}{x} \]
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Rubi [A]
time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1598, 459}
\begin {gather*} -\frac {A b^2}{7 x^7}-\frac {b (2 A c+b B)}{5 x^5}-\frac {c (A c+2 b B)}{3 x^3}-\frac {B c^2}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 459
Rule 1598
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^2}{x^{12}} \, dx &=\int \frac {\left (A+B x^2\right ) \left (b+c x^2\right )^2}{x^8} \, dx\\ &=\int \left (\frac {A b^2}{x^8}+\frac {b (b B+2 A c)}{x^6}+\frac {c (2 b B+A c)}{x^4}+\frac {B c^2}{x^2}\right ) \, dx\\ &=-\frac {A b^2}{7 x^7}-\frac {b (b B+2 A c)}{5 x^5}-\frac {c (2 b B+A c)}{3 x^3}-\frac {B c^2}{x}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 59, normalized size = 1.11 \begin {gather*} -\frac {A b^2}{7 x^7}+\frac {-b^2 B-2 A b c}{5 x^5}+\frac {-2 b B c-A c^2}{3 x^3}-\frac {B c^2}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 48, normalized size = 0.91
method | result | size |
default | \(-\frac {A \,b^{2}}{7 x^{7}}-\frac {b \left (2 A c +B b \right )}{5 x^{5}}-\frac {c \left (A c +2 B b \right )}{3 x^{3}}-\frac {B \,c^{2}}{x}\) | \(48\) |
risch | \(\frac {-B \,c^{2} x^{6}+\left (-\frac {1}{3} A \,c^{2}-\frac {2}{3} b B c \right ) x^{4}+\left (-\frac {2}{5} A b c -\frac {1}{5} b^{2} B \right ) x^{2}-\frac {b^{2} A}{7}}{x^{7}}\) | \(53\) |
gosper | \(-\frac {105 B \,c^{2} x^{6}+35 A \,c^{2} x^{4}+70 x^{4} b B c +42 A b c \,x^{2}+21 b^{2} B \,x^{2}+15 b^{2} A}{105 x^{7}}\) | \(56\) |
norman | \(\frac {\left (-\frac {1}{3} A \,c^{2}-\frac {2}{3} b B c \right ) x^{8}+\left (-\frac {2}{5} A b c -\frac {1}{5} b^{2} B \right ) x^{6}-\frac {A \,b^{2} x^{4}}{7}-B \,c^{2} x^{10}}{x^{11}}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 53, normalized size = 1.00 \begin {gather*} -\frac {105 \, B c^{2} x^{6} + 35 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + 15 \, A b^{2} + 21 \, {\left (B b^{2} + 2 \, A b c\right )} x^{2}}{105 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.69, size = 53, normalized size = 1.00 \begin {gather*} -\frac {105 \, B c^{2} x^{6} + 35 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + 15 \, A b^{2} + 21 \, {\left (B b^{2} + 2 \, A b c\right )} x^{2}}{105 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.57, size = 58, normalized size = 1.09 \begin {gather*} \frac {- 15 A b^{2} - 105 B c^{2} x^{6} + x^{4} \left (- 35 A c^{2} - 70 B b c\right ) + x^{2} \left (- 42 A b c - 21 B b^{2}\right )}{105 x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.60, size = 55, normalized size = 1.04 \begin {gather*} -\frac {105 \, B c^{2} x^{6} + 70 \, B b c x^{4} + 35 \, A c^{2} x^{4} + 21 \, B b^{2} x^{2} + 42 \, A b c x^{2} + 15 \, A b^{2}}{105 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 52, normalized size = 0.98 \begin {gather*} -\frac {x^2\,\left (\frac {B\,b^2}{5}+\frac {2\,A\,c\,b}{5}\right )+x^4\,\left (\frac {A\,c^2}{3}+\frac {2\,B\,b\,c}{3}\right )+\frac {A\,b^2}{7}+B\,c^2\,x^6}{x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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