Optimal. Leaf size=124 \[ -\frac {A}{4 a^3 x^4}+\frac {3 A b-a B}{2 a^4 x^2}+\frac {b (A b-a B)}{4 a^3 \left (a+b x^2\right )^2}+\frac {b (3 A b-2 a B)}{2 a^4 \left (a+b x^2\right )}+\frac {3 b (2 A b-a B) \log (x)}{a^5}-\frac {3 b (2 A b-a B) \log \left (a+b x^2\right )}{2 a^5} \]
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Rubi [A]
time = 0.09, antiderivative size = 124, 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, 78}
\begin {gather*} -\frac {3 b (2 A b-a B) \log \left (a+b x^2\right )}{2 a^5}+\frac {3 b \log (x) (2 A b-a B)}{a^5}+\frac {b (3 A b-2 a B)}{2 a^4 \left (a+b x^2\right )}+\frac {3 A b-a B}{2 a^4 x^2}+\frac {b (A b-a B)}{4 a^3 \left (a+b x^2\right )^2}-\frac {A}{4 a^3 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^3} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {A+B x}{x^3 (a+b x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {A}{a^3 x^3}+\frac {-3 A b+a B}{a^4 x^2}-\frac {3 b (-2 A b+a B)}{a^5 x}+\frac {b^2 (-A b+a B)}{a^3 (a+b x)^3}+\frac {b^2 (-3 A b+2 a B)}{a^4 (a+b x)^2}+\frac {3 b^2 (-2 A b+a B)}{a^5 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {A}{4 a^3 x^4}+\frac {3 A b-a B}{2 a^4 x^2}+\frac {b (A b-a B)}{4 a^3 \left (a+b x^2\right )^2}+\frac {b (3 A b-2 a B)}{2 a^4 \left (a+b x^2\right )}+\frac {3 b (2 A b-a B) \log (x)}{a^5}-\frac {3 b (2 A b-a B) \log \left (a+b x^2\right )}{2 a^5}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 108, normalized size = 0.87 \begin {gather*} \frac {-\frac {a^2 A}{x^4}-\frac {2 a (-3 A b+a B)}{x^2}+\frac {a^2 b (A b-a B)}{\left (a+b x^2\right )^2}+\frac {2 a b (3 A b-2 a B)}{a+b x^2}+12 b (2 A b-a B) \log (x)+6 b (-2 A b+a B) \log \left (a+b x^2\right )}{4 a^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 123, normalized size = 0.99
method | result | size |
default | \(-\frac {b^{2} \left (-\frac {a \left (3 A b -2 B a \right )}{b \left (b \,x^{2}+a \right )}+\frac {\left (6 A b -3 B a \right ) \ln \left (b \,x^{2}+a \right )}{b}-\frac {a^{2} \left (A b -B a \right )}{2 b \left (b \,x^{2}+a \right )^{2}}\right )}{2 a^{5}}-\frac {A}{4 a^{3} x^{4}}-\frac {-3 A b +B a}{2 a^{4} x^{2}}+\frac {3 b \left (2 A b -B a \right ) \ln \left (x \right )}{a^{5}}\) | \(123\) |
norman | \(\frac {-\frac {A}{4 a}+\frac {\left (2 A b -B a \right ) x^{2}}{2 a^{2}}-\frac {b \left (6 b^{2} A -3 a b B \right ) x^{6}}{a^{4}}-\frac {b^{2} \left (18 b^{2} A -9 a b B \right ) x^{8}}{4 a^{5}}}{x^{4} \left (b \,x^{2}+a \right )^{2}}+\frac {3 b \left (2 A b -B a \right ) \ln \left (x \right )}{a^{5}}-\frac {3 b \left (2 A b -B a \right ) \ln \left (b \,x^{2}+a \right )}{2 a^{5}}\) | \(123\) |
risch | \(\frac {\frac {3 b^{2} \left (2 A b -B a \right ) x^{6}}{2 a^{4}}+\frac {9 b \left (2 A b -B a \right ) x^{4}}{4 a^{3}}+\frac {\left (2 A b -B a \right ) x^{2}}{2 a^{2}}-\frac {A}{4 a}}{x^{4} \left (b \,x^{2}+a \right )^{2}}+\frac {6 b^{2} \ln \left (x \right ) A}{a^{5}}-\frac {3 b \ln \left (x \right ) B}{a^{4}}-\frac {3 b^{2} \ln \left (b \,x^{2}+a \right ) A}{a^{5}}+\frac {3 b \ln \left (b \,x^{2}+a \right ) B}{2 a^{4}}\) | \(129\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 137, normalized size = 1.10 \begin {gather*} -\frac {6 \, {\left (B a b^{2} - 2 \, A b^{3}\right )} x^{6} + 9 \, {\left (B a^{2} b - 2 \, A a b^{2}\right )} x^{4} + A a^{3} + 2 \, {\left (B a^{3} - 2 \, A a^{2} b\right )} x^{2}}{4 \, {\left (a^{4} b^{2} x^{8} + 2 \, a^{5} b x^{6} + a^{6} x^{4}\right )}} + \frac {3 \, {\left (B a b - 2 \, A b^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, a^{5}} - \frac {3 \, {\left (B a b - 2 \, A b^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 229 vs.
\(2 (111) = 222\).
time = 1.06, size = 229, normalized size = 1.85 \begin {gather*} -\frac {6 \, {\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{6} + A a^{4} + 9 \, {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{4} + 2 \, {\left (B a^{4} - 2 \, A a^{3} b\right )} x^{2} - 6 \, {\left ({\left (B a b^{3} - 2 \, A b^{4}\right )} x^{8} + 2 \, {\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{6} + {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{4}\right )} \log \left (b x^{2} + a\right ) + 12 \, {\left ({\left (B a b^{3} - 2 \, A b^{4}\right )} x^{8} + 2 \, {\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{6} + {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{4}\right )} \log \left (x\right )}{4 \, {\left (a^{5} b^{2} x^{8} + 2 \, a^{6} b x^{6} + a^{7} x^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.72, size = 136, normalized size = 1.10 \begin {gather*} \frac {- A a^{3} + x^{6} \cdot \left (12 A b^{3} - 6 B a b^{2}\right ) + x^{4} \cdot \left (18 A a b^{2} - 9 B a^{2} b\right ) + x^{2} \cdot \left (4 A a^{2} b - 2 B a^{3}\right )}{4 a^{6} x^{4} + 8 a^{5} b x^{6} + 4 a^{4} b^{2} x^{8}} - \frac {3 b \left (- 2 A b + B a\right ) \log {\left (x \right )}}{a^{5}} + \frac {3 b \left (- 2 A b + B a\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.80, size = 133, normalized size = 1.07 \begin {gather*} -\frac {3 \, {\left (B a b - 2 \, A b^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{5}} + \frac {3 \, {\left (B a b^{2} - 2 \, A b^{3}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{5} b} - \frac {6 \, B a b^{2} x^{6} - 12 \, A b^{3} x^{6} + 9 \, B a^{2} b x^{4} - 18 \, A a b^{2} x^{4} + 2 \, B a^{3} x^{2} - 4 \, A a^{2} b x^{2} + A a^{3}}{4 \, {\left (b x^{4} + a x^{2}\right )}^{2} a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 131, normalized size = 1.06 \begin {gather*} \frac {\frac {x^2\,\left (2\,A\,b-B\,a\right )}{2\,a^2}-\frac {A}{4\,a}+\frac {3\,b^2\,x^6\,\left (2\,A\,b-B\,a\right )}{2\,a^4}+\frac {9\,b\,x^4\,\left (2\,A\,b-B\,a\right )}{4\,a^3}}{a^2\,x^4+2\,a\,b\,x^6+b^2\,x^8}-\frac {\ln \left (b\,x^2+a\right )\,\left (6\,A\,b^2-3\,B\,a\,b\right )}{2\,a^5}+\frac {\ln \left (x\right )\,\left (6\,A\,b^2-3\,B\,a\,b\right )}{a^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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