Optimal. Leaf size=26 \[ -\frac {A b}{x}+(b B+A c) x+\frac {1}{3} B c x^3 \]
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Rubi [A]
time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1598, 459}
\begin {gather*} x (A c+b B)-\frac {A b}{x}+\frac {1}{3} B c 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 )}{x^4} \, dx &=\int \frac {\left (A+B x^2\right ) \left (b+c x^2\right )}{x^2} \, dx\\ &=\int \left (b B \left (1+\frac {A c}{b B}\right )+\frac {A b}{x^2}+B c x^2\right ) \, dx\\ &=-\frac {A b}{x}+(b B+A c) x+\frac {1}{3} B c x^3\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 26, normalized size = 1.00 \begin {gather*} -\frac {A b}{x}+(b B+A c) x+\frac {1}{3} B c x^3 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 24, normalized size = 0.92
method | result | size |
default | \(\frac {B c \,x^{3}}{3}+A c x +b B x -\frac {A b}{x}\) | \(24\) |
risch | \(\frac {B c \,x^{3}}{3}+A c x +b B x -\frac {A b}{x}\) | \(24\) |
norman | \(\frac {\left (A c +B b \right ) x^{4}-A b \,x^{2}+\frac {B c \,x^{6}}{3}}{x^{3}}\) | \(31\) |
gosper | \(-\frac {-B c \,x^{4}-3 A c \,x^{2}-3 b B \,x^{2}+3 A b}{3 x}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 24, normalized size = 0.92 \begin {gather*} \frac {1}{3} \, B c x^{3} + {\left (B b + A c\right )} x - \frac {A b}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.14, size = 28, normalized size = 1.08 \begin {gather*} \frac {B c x^{4} + 3 \, {\left (B b + A c\right )} x^{2} - 3 \, A b}{3 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 20, normalized size = 0.77 \begin {gather*} - \frac {A b}{x} + \frac {B c x^{3}}{3} + x \left (A c + B b\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.74, size = 23, normalized size = 0.88 \begin {gather*} \frac {1}{3} \, B c x^{3} + B b x + A c x - \frac {A b}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 24, normalized size = 0.92 \begin {gather*} x\,\left (A\,c+B\,b\right )-\frac {A\,b}{x}+\frac {B\,c\,x^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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