Optimal. Leaf size=51 \[ -\frac {a^2 A}{6 x^6}-\frac {a (2 A b+a B)}{4 x^4}-\frac {b (A b+2 a B)}{2 x^2}+b^2 B \log (x) \]
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Rubi [A]
time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {457, 77}
\begin {gather*} -\frac {a^2 A}{6 x^6}-\frac {a (a B+2 A b)}{4 x^4}-\frac {b (2 a B+A b)}{2 x^2}+b^2 B \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 457
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^7} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^2 (A+B x)}{x^4} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {a^2 A}{x^4}+\frac {a (2 A b+a B)}{x^3}+\frac {b (A b+2 a B)}{x^2}+\frac {b^2 B}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^2 A}{6 x^6}-\frac {a (2 A b+a B)}{4 x^4}-\frac {b (A b+2 a B)}{2 x^2}+b^2 B \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 54, normalized size = 1.06 \begin {gather*} -\frac {6 A b^2 x^4+6 a b x^2 \left (A+2 B x^2\right )+a^2 \left (2 A+3 B x^2\right )}{12 x^6}+b^2 B \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 46, normalized size = 0.90
method | result | size |
default | \(-\frac {a^{2} A}{6 x^{6}}-\frac {a \left (2 A b +B a \right )}{4 x^{4}}-\frac {b \left (A b +2 B a \right )}{2 x^{2}}+b^{2} B \ln \left (x \right )\) | \(46\) |
norman | \(\frac {\left (-\frac {1}{2} b^{2} A -a b B \right ) x^{4}+\left (-\frac {1}{2} a b A -\frac {1}{4} a^{2} B \right ) x^{2}-\frac {a^{2} A}{6}}{x^{6}}+b^{2} B \ln \left (x \right )\) | \(52\) |
risch | \(\frac {\left (-\frac {1}{2} b^{2} A -a b B \right ) x^{4}+\left (-\frac {1}{2} a b A -\frac {1}{4} a^{2} B \right ) x^{2}-\frac {a^{2} A}{6}}{x^{6}}+b^{2} B \ln \left (x \right )\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 55, normalized size = 1.08 \begin {gather*} \frac {1}{2} \, B b^{2} \log \left (x^{2}\right ) - \frac {6 \, {\left (2 \, B a b + A b^{2}\right )} x^{4} + 2 \, A a^{2} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}}{12 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.89, size = 55, normalized size = 1.08 \begin {gather*} \frac {12 \, B b^{2} x^{6} \log \left (x\right ) - 6 \, {\left (2 \, B a b + A b^{2}\right )} x^{4} - 2 \, A a^{2} - 3 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}}{12 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.52, size = 56, normalized size = 1.10 \begin {gather*} B b^{2} \log {\left (x \right )} + \frac {- 2 A a^{2} + x^{4} \left (- 6 A b^{2} - 12 B a b\right ) + x^{2} \left (- 6 A a b - 3 B a^{2}\right )}{12 x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.02, size = 66, normalized size = 1.29 \begin {gather*} \frac {1}{2} \, B b^{2} \log \left (x^{2}\right ) - \frac {11 \, B b^{2} x^{6} + 12 \, B a b x^{4} + 6 \, A b^{2} x^{4} + 3 \, B a^{2} x^{2} + 6 \, A a b x^{2} + 2 \, A a^{2}}{12 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 51, normalized size = 1.00 \begin {gather*} B\,b^2\,\ln \left (x\right )-\frac {x^2\,\left (\frac {B\,a^2}{4}+\frac {A\,b\,a}{2}\right )+x^4\,\left (\frac {A\,b^2}{2}+B\,a\,b\right )+\frac {A\,a^2}{6}}{x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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