Optimal. Leaf size=26 \[ -\frac {A b}{3 x^3}-\frac {b B+A c}{x}+B c x \]
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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*} -\frac {A c+b B}{x}-\frac {A b}{3 x^3}+B c 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 )}{x^6} \, dx &=\int \frac {\left (A+B x^2\right ) \left (b+c x^2\right )}{x^4} \, dx\\ &=\int \left (B c+\frac {A b}{x^4}+\frac {b B+A c}{x^2}\right ) \, dx\\ &=-\frac {A b}{3 x^3}-\frac {b B+A c}{x}+B c x\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 27, normalized size = 1.04 \begin {gather*} -\frac {A b}{3 x^3}+\frac {-b B-A c}{x}+B c x \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 25, normalized size = 0.96
method | result | size |
default | \(B c x -\frac {A b}{3 x^{3}}-\frac {A c +B b}{x}\) | \(25\) |
risch | \(B c x +\frac {\left (-A c -B b \right ) x^{2}-\frac {A b}{3}}{x^{3}}\) | \(28\) |
gosper | \(-\frac {-3 B c \,x^{4}+3 A c \,x^{2}+3 b B \,x^{2}+A b}{3 x^{3}}\) | \(31\) |
norman | \(\frac {\left (-A c -B b \right ) x^{4}+B c \,x^{6}-\frac {A b \,x^{2}}{3}}{x^{5}}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 26, normalized size = 1.00 \begin {gather*} B c x - \frac {3 \, {\left (B b + A c\right )} x^{2} + A b}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.55, size = 29, normalized size = 1.12 \begin {gather*} \frac {3 \, B c x^{4} - 3 \, {\left (B b + A c\right )} x^{2} - A b}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 27, normalized size = 1.04 \begin {gather*} B c x + \frac {- A b + x^{2} \left (- 3 A c - 3 B b\right )}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.62, size = 28, normalized size = 1.08 \begin {gather*} B c x - \frac {3 \, B b x^{2} + 3 \, A c x^{2} + A b}{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 = 26, normalized size = 1.00 \begin {gather*} B\,c\,x-\frac {\left (A\,c+B\,b\right )\,x^2+\frac {A\,b}{3}}{x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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