Optimal. Leaf size=48 \[ -\frac {A b^2}{3 x^3}-\frac {b (b B+2 A c)}{x}+c (2 b B+A c) x+\frac {1}{3} B c^2 x^3 \]
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Rubi [A]
time = 0.02, antiderivative size = 48, 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}{3 x^3}+c x (A c+2 b B)-\frac {b (2 A c+b B)}{x}+\frac {1}{3} B c^2 x^3 \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^8} \, dx &=\int \frac {\left (A+B x^2\right ) \left (b+c x^2\right )^2}{x^4} \, dx\\ &=\int \left (c (2 b B+A c)+\frac {A b^2}{x^4}+\frac {b (b B+2 A c)}{x^2}+B c^2 x^2\right ) \, dx\\ &=-\frac {A b^2}{3 x^3}-\frac {b (b B+2 A c)}{x}+c (2 b B+A c) x+\frac {1}{3} B c^2 x^3\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 50, normalized size = 1.04 \begin {gather*} -\frac {A b^2}{3 x^3}+\frac {-b^2 B-2 A b c}{x}+c (2 b B+A c) x+\frac {1}{3} B c^2 x^3 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 46, normalized size = 0.96
method | result | size |
default | \(\frac {B \,c^{2} x^{3}}{3}+A \,c^{2} x +2 b B c x -\frac {A \,b^{2}}{3 x^{3}}-\frac {b \left (2 A c +B b \right )}{x}\) | \(46\) |
risch | \(\frac {B \,c^{2} x^{3}}{3}+A \,c^{2} x +2 b B c x +\frac {\left (-2 A b c -b^{2} B \right ) x^{2}-\frac {b^{2} A}{3}}{x^{3}}\) | \(50\) |
gosper | \(-\frac {-B \,c^{2} x^{6}-3 A \,c^{2} x^{4}-6 x^{4} b B c +6 A b c \,x^{2}+3 b^{2} B \,x^{2}+b^{2} A}{3 x^{3}}\) | \(55\) |
norman | \(\frac {\left (A \,c^{2}+2 b B c \right ) x^{8}+\left (-2 A b c -b^{2} B \right ) x^{6}-\frac {A \,b^{2} x^{4}}{3}+\frac {B \,c^{2} x^{10}}{3}}{x^{7}}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 50, normalized size = 1.04 \begin {gather*} \frac {1}{3} \, B c^{2} x^{3} + {\left (2 \, B b c + A c^{2}\right )} x - \frac {A b^{2} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} x^{2}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.71, size = 52, normalized size = 1.08 \begin {gather*} \frac {B c^{2} x^{6} + 3 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} - A b^{2} - 3 \, {\left (B b^{2} + 2 \, A b c\right )} x^{2}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 51, normalized size = 1.06 \begin {gather*} \frac {B c^{2} x^{3}}{3} + x \left (A c^{2} + 2 B b c\right ) + \frac {- A b^{2} + x^{2} \left (- 6 A b c - 3 B b^{2}\right )}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.60, size = 50, normalized size = 1.04 \begin {gather*} \frac {1}{3} \, B c^{2} x^{3} + 2 \, B b c x + A c^{2} x - \frac {3 \, B b^{2} x^{2} + 6 \, A b c x^{2} + A b^{2}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 50, normalized size = 1.04 \begin {gather*} x\,\left (A\,c^2+2\,B\,b\,c\right )-\frac {x^2\,\left (B\,b^2+2\,A\,c\,b\right )+\frac {A\,b^2}{3}}{x^3}+\frac {B\,c^2\,x^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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