Optimal. Leaf size=92 \[ \frac {(a B+3 A b) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{3/2}}+\frac {x (a B+3 A b)}{8 a^2 b \left (a+b x^2\right )}+\frac {x (A b-a B)}{4 a b \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.03, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {385, 199, 205} \begin {gather*} \frac {(a B+3 A b) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{3/2}}+\frac {x (a B+3 A b)}{8 a^2 b \left (a+b x^2\right )}+\frac {x (A b-a B)}{4 a b \left (a+b x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 385
Rubi steps
\begin {align*} \int \frac {A+B x^2}{\left (a+b x^2\right )^3} \, dx &=\frac {(A b-a B) x}{4 a b \left (a+b x^2\right )^2}+\frac {(3 A b+a B) \int \frac {1}{\left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=\frac {(A b-a B) x}{4 a b \left (a+b x^2\right )^2}+\frac {(3 A b+a B) x}{8 a^2 b \left (a+b x^2\right )}+\frac {(3 A b+a B) \int \frac {1}{a+b x^2} \, dx}{8 a^2 b}\\ &=\frac {(A b-a B) x}{4 a b \left (a+b x^2\right )^2}+\frac {(3 A b+a B) x}{8 a^2 b \left (a+b x^2\right )}+\frac {(3 A b+a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 84, normalized size = 0.91 \begin {gather*} \frac {(a B+3 A b) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{3/2}}+\frac {x \left (a^2 (-B)+a b \left (5 A+B x^2\right )+3 A b^2 x^2\right )}{8 a^2 b \left (a+b x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x^2}{\left (a+b x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.44, size = 300, normalized size = 3.26 \begin {gather*} \left [\frac {2 \, {\left (B a^{2} b^{2} + 3 \, A a b^{3}\right )} x^{3} - {\left ({\left (B a b^{2} + 3 \, A b^{3}\right )} x^{4} + B a^{3} + 3 \, A a^{2} b + 2 \, {\left (B a^{2} b + 3 \, A a b^{2}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) - 2 \, {\left (B a^{3} b - 5 \, A a^{2} b^{2}\right )} x}{16 \, {\left (a^{3} b^{4} x^{4} + 2 \, a^{4} b^{3} x^{2} + a^{5} b^{2}\right )}}, \frac {{\left (B a^{2} b^{2} + 3 \, A a b^{3}\right )} x^{3} + {\left ({\left (B a b^{2} + 3 \, A b^{3}\right )} x^{4} + B a^{3} + 3 \, A a^{2} b + 2 \, {\left (B a^{2} b + 3 \, A a b^{2}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - {\left (B a^{3} b - 5 \, A a^{2} b^{2}\right )} x}{8 \, {\left (a^{3} b^{4} x^{4} + 2 \, a^{4} b^{3} x^{2} + a^{5} b^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 78, normalized size = 0.85 \begin {gather*} \frac {{\left (B a + 3 \, A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2} b} + \frac {B a b x^{3} + 3 \, A b^{2} x^{3} - B a^{2} x + 5 \, A a b x}{8 \, {\left (b x^{2} + a\right )}^{2} a^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 90, normalized size = 0.98 \begin {gather*} \frac {3 A \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, a^{2}}+\frac {B \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, a b}+\frac {\frac {\left (3 A b +B a \right ) x^{3}}{8 a^{2}}+\frac {\left (5 A b -B a \right ) x}{8 a b}}{\left (b \,x^{2}+a \right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.52, size = 92, normalized size = 1.00 \begin {gather*} \frac {{\left (B a b + 3 \, A b^{2}\right )} x^{3} - {\left (B a^{2} - 5 \, A a b\right )} x}{8 \, {\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )}} + \frac {{\left (B a + 3 \, A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 82, normalized size = 0.89 \begin {gather*} \frac {\frac {x^3\,\left (3\,A\,b+B\,a\right )}{8\,a^2}+\frac {x\,\left (5\,A\,b-B\,a\right )}{8\,a\,b}}{a^2+2\,a\,b\,x^2+b^2\,x^4}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (3\,A\,b+B\,a\right )}{8\,a^{5/2}\,b^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.55, size = 150, normalized size = 1.63 \begin {gather*} - \frac {\sqrt {- \frac {1}{a^{5} b^{3}}} \left (3 A b + B a\right ) \log {\left (- a^{3} b \sqrt {- \frac {1}{a^{5} b^{3}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{a^{5} b^{3}}} \left (3 A b + B a\right ) \log {\left (a^{3} b \sqrt {- \frac {1}{a^{5} b^{3}}} + x \right )}}{16} + \frac {x^{3} \left (3 A b^{2} + B a b\right ) + x \left (5 A a b - B a^{2}\right )}{8 a^{4} b + 16 a^{3} b^{2} x^{2} + 8 a^{2} b^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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