Optimal. Leaf size=69 \[ -\frac {A b^3}{3 x^3}-\frac {b^2 (b B+3 A c)}{x}+3 b c (b B+A c) x+\frac {1}{3} c^2 (3 b B+A c) x^3+\frac {1}{5} B c^3 x^5 \]
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Rubi [A]
time = 0.03, antiderivative size = 69, 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}{3 x^3}-\frac {b^2 (3 A c+b B)}{x}+\frac {1}{3} c^2 x^3 (A c+3 b B)+3 b c x (A c+b B)+\frac {1}{5} B c^3 x^5 \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^{10}} \, dx &=\int \frac {\left (A+B x^2\right ) \left (b+c x^2\right )^3}{x^4} \, dx\\ &=\int \left (3 b c (b B+A c)+\frac {A b^3}{x^4}+\frac {b^2 (b B+3 A c)}{x^2}+c^2 (3 b B+A c) x^2+B c^3 x^4\right ) \, dx\\ &=-\frac {A b^3}{3 x^3}-\frac {b^2 (b B+3 A c)}{x}+3 b c (b B+A c) x+\frac {1}{3} c^2 (3 b B+A c) x^3+\frac {1}{5} B c^3 x^5\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 71, normalized size = 1.03 \begin {gather*} -\frac {A b^3}{3 x^3}+\frac {-b^3 B-3 A b^2 c}{x}+3 b c (b B+A c) x+\frac {1}{3} c^2 (3 b B+A c) x^3+\frac {1}{5} B c^3 x^5 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 70, normalized size = 1.01
method | result | size |
default | \(\frac {B \,c^{3} x^{5}}{5}+\frac {A \,c^{3} x^{3}}{3}+B b \,c^{2} x^{3}+3 A b \,c^{2} x +3 B \,b^{2} c x -\frac {A \,b^{3}}{3 x^{3}}-\frac {b^{2} \left (3 A c +B b \right )}{x}\) | \(70\) |
risch | \(\frac {B \,c^{3} x^{5}}{5}+\frac {A \,c^{3} x^{3}}{3}+B b \,c^{2} x^{3}+3 A b \,c^{2} x +3 B \,b^{2} c x +\frac {\left (-3 A \,b^{2} c -B \,b^{3}\right ) x^{2}-\frac {A \,b^{3}}{3}}{x^{3}}\) | \(74\) |
norman | \(\frac {\left (\frac {1}{3} A \,c^{3}+B b \,c^{2}\right ) x^{12}+\left (3 A b \,c^{2}+3 B \,b^{2} c \right ) x^{10}+\left (-3 A \,b^{2} c -B \,b^{3}\right ) x^{8}-\frac {A \,b^{3} x^{6}}{3}+\frac {B \,c^{3} x^{14}}{5}}{x^{9}}\) | \(78\) |
gosper | \(-\frac {-3 B \,c^{3} x^{8}-5 A \,c^{3} x^{6}-15 x^{6} B b \,c^{2}-45 A b \,c^{2} x^{4}-45 x^{4} B \,b^{2} c +45 A \,b^{2} c \,x^{2}+15 x^{2} B \,b^{3}+5 A \,b^{3}}{15 x^{3}}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 73, normalized size = 1.06 \begin {gather*} \frac {1}{5} \, B c^{3} x^{5} + \frac {1}{3} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} x - \frac {A b^{3} + 3 \, {\left (B b^{3} + 3 \, A b^{2} 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 = 1.68, size = 75, normalized size = 1.09 \begin {gather*} \frac {3 \, B c^{3} x^{8} + 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 45 \, {\left (B b^{2} c + A b c^{2}\right )} x^{4} - 5 \, A b^{3} - 15 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}}{15 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 75, normalized size = 1.09 \begin {gather*} \frac {B c^{3} x^{5}}{5} + x^{3} \left (\frac {A c^{3}}{3} + B b c^{2}\right ) + x \left (3 A b c^{2} + 3 B b^{2} c\right ) + \frac {- A b^{3} + x^{2} \left (- 9 A b^{2} c - 3 B b^{3}\right )}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.71, size = 74, normalized size = 1.07 \begin {gather*} \frac {1}{5} \, B c^{3} x^{5} + B b c^{2} x^{3} + \frac {1}{3} \, A c^{3} x^{3} + 3 \, B b^{2} c x + 3 \, A b c^{2} x - \frac {3 \, B b^{3} x^{2} + 9 \, A b^{2} c x^{2} + A b^{3}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 68, normalized size = 0.99 \begin {gather*} x^3\,\left (\frac {A\,c^3}{3}+B\,b\,c^2\right )-\frac {\frac {A\,b^3}{3}+x^2\,\left (B\,b^3+3\,A\,c\,b^2\right )}{x^3}+\frac {B\,c^3\,x^5}{5}+3\,b\,c\,x\,\left (A\,c+B\,b\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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