Optimal. Leaf size=65 \[ -\frac {A b^3}{x}+b^2 (b B+3 A c) x+b c (b B+A c) x^3+\frac {1}{5} c^2 (3 b B+A c) x^5+\frac {1}{7} B c^3 x^7 \]
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Rubi [A]
time = 0.03, antiderivative size = 65, 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^3}{x}+b^2 x (3 A c+b B)+\frac {1}{5} c^2 x^5 (A c+3 b B)+b c x^3 (A c+b B)+\frac {1}{7} B c^3 x^7 \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 )^3}{x^8} \, dx &=\int \frac {\left (A+B x^2\right ) \left (b+c x^2\right )^3}{x^2} \, dx\\ &=\int \left (b^2 (b B+3 A c)+\frac {A b^3}{x^2}+3 b c (b B+A c) x^2+c^2 (3 b B+A c) x^4+B c^3 x^6\right ) \, dx\\ &=-\frac {A b^3}{x}+b^2 (b B+3 A c) x+b c (b B+A c) x^3+\frac {1}{5} c^2 (3 b B+A c) x^5+\frac {1}{7} B c^3 x^7\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 65, normalized size = 1.00 \begin {gather*} -\frac {A b^3}{x}+b^2 (b B+3 A c) x+b c (b B+A c) x^3+\frac {1}{5} c^2 (3 b B+A c) x^5+\frac {1}{7} B c^3 x^7 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 71, normalized size = 1.09
method | result | size |
default | \(\frac {B \,c^{3} x^{7}}{7}+\frac {A \,c^{3} x^{5}}{5}+\frac {3 B b \,c^{2} x^{5}}{5}+A b \,c^{2} x^{3}+B \,b^{2} c \,x^{3}+3 A \,b^{2} c x +B \,b^{3} x -\frac {A \,b^{3}}{x}\) | \(71\) |
risch | \(\frac {B \,c^{3} x^{7}}{7}+\frac {A \,c^{3} x^{5}}{5}+\frac {3 B b \,c^{2} x^{5}}{5}+A b \,c^{2} x^{3}+B \,b^{2} c \,x^{3}+3 A \,b^{2} c x +B \,b^{3} x -\frac {A \,b^{3}}{x}\) | \(71\) |
norman | \(\frac {\left (\frac {1}{5} A \,c^{3}+\frac {3}{5} B b \,c^{2}\right ) x^{12}+\left (A b \,c^{2}+B \,b^{2} c \right ) x^{10}+\left (3 A \,b^{2} c +B \,b^{3}\right ) x^{8}-A \,b^{3} x^{6}+\frac {B \,c^{3} x^{14}}{7}}{x^{7}}\) | \(76\) |
gosper | \(-\frac {-5 B \,c^{3} x^{8}-7 A \,c^{3} x^{6}-21 x^{6} B b \,c^{2}-35 A b \,c^{2} x^{4}-35 x^{4} B \,b^{2} c -105 A \,b^{2} c \,x^{2}-35 x^{2} B \,b^{3}+35 A \,b^{3}}{35 x}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 69, normalized size = 1.06 \begin {gather*} \frac {1}{7} \, B c^{3} x^{7} + \frac {1}{5} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{5} + {\left (B b^{2} c + A b c^{2}\right )} x^{3} - \frac {A b^{3}}{x} + {\left (B b^{3} + 3 \, A b^{2} c\right )} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.48, size = 75, normalized size = 1.15 \begin {gather*} \frac {5 \, B c^{3} x^{8} + 7 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 35 \, {\left (B b^{2} c + A b c^{2}\right )} x^{4} - 35 \, A b^{3} + 35 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}}{35 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 68, normalized size = 1.05 \begin {gather*} - \frac {A b^{3}}{x} + \frac {B c^{3} x^{7}}{7} + x^{5} \left (\frac {A c^{3}}{5} + \frac {3 B b c^{2}}{5}\right ) + x^{3} \left (A b c^{2} + B b^{2} c\right ) + x \left (3 A b^{2} c + B b^{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.77, size = 70, normalized size = 1.08 \begin {gather*} \frac {1}{7} \, B c^{3} x^{7} + \frac {3}{5} \, B b c^{2} x^{5} + \frac {1}{5} \, A c^{3} x^{5} + B b^{2} c x^{3} + A b c^{2} x^{3} + B b^{3} x + 3 \, A b^{2} c x - \frac {A b^{3}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 65, normalized size = 1.00 \begin {gather*} x\,\left (B\,b^3+3\,A\,c\,b^2\right )+x^5\,\left (\frac {A\,c^3}{5}+\frac {3\,B\,b\,c^2}{5}\right )-\frac {A\,b^3}{x}+\frac {B\,c^3\,x^7}{7}+b\,c\,x^3\,\left (A\,c+B\,b\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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