Optimal. Leaf size=26 \[ -\frac {a A}{3 x^3}-\frac {A b+a B}{x}+b B x \]
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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {459}
\begin {gather*} -\frac {a B+A b}{x}-\frac {a A}{3 x^3}+b B x \end {gather*}
Antiderivative was successfully verified.
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Rule 459
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^4} \, dx &=\int \left (b B+\frac {a A}{x^4}+\frac {A b+a B}{x^2}\right ) \, dx\\ &=-\frac {a A}{3 x^3}-\frac {A b+a B}{x}+b B x\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 27, normalized size = 1.04 \begin {gather*} -\frac {a A}{3 x^3}+\frac {-A b-a B}{x}+b B x \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 25, normalized size = 0.96
method | result | size |
default | \(b B x -\frac {a A}{3 x^{3}}-\frac {A b +B a}{x}\) | \(25\) |
risch | \(b B x +\frac {\left (-A b -B a \right ) x^{2}-\frac {A a}{3}}{x^{3}}\) | \(28\) |
norman | \(\frac {b B \,x^{4}+\left (-A b -B a \right ) x^{2}-\frac {A a}{3}}{x^{3}}\) | \(29\) |
gosper | \(-\frac {-3 b B \,x^{4}+3 A b \,x^{2}+3 B a \,x^{2}+A a}{3 x^{3}}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 26, normalized size = 1.00 \begin {gather*} B b x - \frac {3 \, {\left (B a + A b\right )} x^{2} + A a}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.18, size = 29, normalized size = 1.12 \begin {gather*} \frac {3 \, B b x^{4} - 3 \, {\left (B a + A b\right )} x^{2} - A a}{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 b x + \frac {- A a + x^{2} \left (- 3 A b - 3 B a\right )}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.04, size = 28, normalized size = 1.08 \begin {gather*} B b x - \frac {3 \, B a x^{2} + 3 \, A b x^{2} + A a}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 26, normalized size = 1.00 \begin {gather*} B\,b\,x-\frac {\left (A\,b+B\,a\right )\,x^2+\frac {A\,a}{3}}{x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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