Optimal. Leaf size=73 \[ -\frac {A b^3}{9 x^9}-\frac {b^2 (b B+3 A c)}{7 x^7}-\frac {3 b c (b B+A c)}{5 x^5}-\frac {c^2 (3 b B+A c)}{3 x^3}-\frac {B c^3}{x} \]
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Rubi [A]
time = 0.03, antiderivative size = 73, 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}{9 x^9}-\frac {b^2 (3 A c+b B)}{7 x^7}-\frac {c^2 (A c+3 b B)}{3 x^3}-\frac {3 b c (A c+b B)}{5 x^5}-\frac {B c^3}{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 )^3}{x^{16}} \, dx &=\int \frac {\left (A+B x^2\right ) \left (b+c x^2\right )^3}{x^{10}} \, dx\\ &=\int \left (\frac {A b^3}{x^{10}}+\frac {b^2 (b B+3 A c)}{x^8}+\frac {3 b c (b B+A c)}{x^6}+\frac {c^2 (3 b B+A c)}{x^4}+\frac {B c^3}{x^2}\right ) \, dx\\ &=-\frac {A b^3}{9 x^9}-\frac {b^2 (b B+3 A c)}{7 x^7}-\frac {3 b c (b B+A c)}{5 x^5}-\frac {c^2 (3 b B+A c)}{3 x^3}-\frac {B c^3}{x}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 73, normalized size = 1.00 \begin {gather*} -\frac {A b^3}{9 x^9}-\frac {b^2 (b B+3 A c)}{7 x^7}-\frac {3 b c (b B+A c)}{5 x^5}-\frac {c^2 (3 b B+A c)}{3 x^3}-\frac {B c^3}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.46, size = 66, normalized size = 0.90
method | result | size |
default | \(-\frac {A \,b^{3}}{9 x^{9}}-\frac {b^{2} \left (3 A c +B b \right )}{7 x^{7}}-\frac {3 b c \left (A c +B b \right )}{5 x^{5}}-\frac {c^{2} \left (A c +3 B b \right )}{3 x^{3}}-\frac {B \,c^{3}}{x}\) | \(66\) |
risch | \(\frac {-B \,c^{3} x^{8}+\left (-\frac {1}{3} A \,c^{3}-B b \,c^{2}\right ) x^{6}+\left (-\frac {3}{5} A b \,c^{2}-\frac {3}{5} B \,b^{2} c \right ) x^{4}+\left (-\frac {3}{7} A \,b^{2} c -\frac {1}{7} B \,b^{3}\right ) x^{2}-\frac {A \,b^{3}}{9}}{x^{9}}\) | \(76\) |
norman | \(\frac {\left (-\frac {1}{3} A \,c^{3}-B b \,c^{2}\right ) x^{12}+\left (-\frac {3}{5} A b \,c^{2}-\frac {3}{5} B \,b^{2} c \right ) x^{10}+\left (-\frac {3}{7} A \,b^{2} c -\frac {1}{7} B \,b^{3}\right ) x^{8}-\frac {A \,b^{3} x^{6}}{9}-B \,c^{3} x^{14}}{x^{15}}\) | \(79\) |
gosper | \(-\frac {315 B \,c^{3} x^{8}+105 A \,c^{3} x^{6}+315 x^{6} B b \,c^{2}+189 A b \,c^{2} x^{4}+189 x^{4} B \,b^{2} c +135 A \,b^{2} c \,x^{2}+45 x^{2} B \,b^{3}+35 A \,b^{3}}{315 x^{9}}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 75, normalized size = 1.03 \begin {gather*} -\frac {315 \, B c^{3} x^{8} + 105 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 189 \, {\left (B b^{2} c + A b c^{2}\right )} x^{4} + 35 \, A b^{3} + 45 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}}{315 \, x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.44, size = 75, normalized size = 1.03 \begin {gather*} -\frac {315 \, B c^{3} x^{8} + 105 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 189 \, {\left (B b^{2} c + A b c^{2}\right )} x^{4} + 35 \, A b^{3} + 45 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}}{315 \, x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.46, size = 83, normalized size = 1.14 \begin {gather*} \frac {- 35 A b^{3} - 315 B c^{3} x^{8} + x^{6} \left (- 105 A c^{3} - 315 B b c^{2}\right ) + x^{4} \left (- 189 A b c^{2} - 189 B b^{2} c\right ) + x^{2} \left (- 135 A b^{2} c - 45 B b^{3}\right )}{315 x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.67, size = 79, normalized size = 1.08 \begin {gather*} -\frac {315 \, B c^{3} x^{8} + 315 \, B b c^{2} x^{6} + 105 \, A c^{3} x^{6} + 189 \, B b^{2} c x^{4} + 189 \, A b c^{2} x^{4} + 45 \, B b^{3} x^{2} + 135 \, A b^{2} c x^{2} + 35 \, A b^{3}}{315 \, x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 74, normalized size = 1.01 \begin {gather*} -\frac {x^4\,\left (\frac {3\,B\,b^2\,c}{5}+\frac {3\,A\,b\,c^2}{5}\right )+\frac {A\,b^3}{9}+x^2\,\left (\frac {B\,b^3}{7}+\frac {3\,A\,c\,b^2}{7}\right )+x^6\,\left (\frac {A\,c^3}{3}+B\,b\,c^2\right )+B\,c^3\,x^8}{x^9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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