Optimal. Leaf size=90 \[ \frac {\sqrt {b} (5 b c-3 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2}}+\frac {b x (b c-a d)}{2 a^3 \left (a+b x^2\right )}+\frac {2 b c-a d}{a^3 x}-\frac {c}{3 a^2 x^3} \]
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Rubi [A] time = 0.11, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {456, 1261, 205} \begin {gather*} \frac {b x (b c-a d)}{2 a^3 \left (a+b x^2\right )}+\frac {2 b c-a d}{a^3 x}+\frac {\sqrt {b} (5 b c-3 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2}}-\frac {c}{3 a^2 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 456
Rule 1261
Rubi steps
\begin {align*} \int \frac {c+d x^2}{x^4 \left (a+b x^2\right )^2} \, dx &=\frac {b (b c-a d) x}{2 a^3 \left (a+b x^2\right )}-\frac {1}{2} b \int \frac {-\frac {2 c}{a b}+\frac {2 (b c-a d) x^2}{a^2 b}-\frac {(b c-a d) x^4}{a^3}}{x^4 \left (a+b x^2\right )} \, dx\\ &=\frac {b (b c-a d) x}{2 a^3 \left (a+b x^2\right )}-\frac {1}{2} b \int \left (-\frac {2 c}{a^2 b x^4}-\frac {2 (-2 b c+a d)}{a^3 b x^2}+\frac {-5 b c+3 a d}{a^3 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac {c}{3 a^2 x^3}+\frac {2 b c-a d}{a^3 x}+\frac {b (b c-a d) x}{2 a^3 \left (a+b x^2\right )}+\frac {(b (5 b c-3 a d)) \int \frac {1}{a+b x^2} \, dx}{2 a^3}\\ &=-\frac {c}{3 a^2 x^3}+\frac {2 b c-a d}{a^3 x}+\frac {b (b c-a d) x}{2 a^3 \left (a+b x^2\right )}+\frac {\sqrt {b} (5 b c-3 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 90, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {b} (3 a d-5 b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2}}-\frac {b x (a d-b c)}{2 a^3 \left (a+b x^2\right )}+\frac {2 b c-a d}{a^3 x}-\frac {c}{3 a^2 x^3} \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 {c+d x^2}{x^4 \left (a+b x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.02, size = 250, normalized size = 2.78 \begin {gather*} \left [\frac {6 \, {\left (5 \, b^{2} c - 3 \, a b d\right )} x^{4} - 4 \, a^{2} c + 4 \, {\left (5 \, a b c - 3 \, a^{2} d\right )} x^{2} - 3 \, {\left ({\left (5 \, b^{2} c - 3 \, a b d\right )} x^{5} + {\left (5 \, a b c - 3 \, a^{2} d\right )} x^{3}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{12 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )}}, \frac {3 \, {\left (5 \, b^{2} c - 3 \, a b d\right )} x^{4} - 2 \, a^{2} c + 2 \, {\left (5 \, a b c - 3 \, a^{2} d\right )} x^{2} + 3 \, {\left ({\left (5 \, b^{2} c - 3 \, a b d\right )} x^{5} + {\left (5 \, a b c - 3 \, a^{2} d\right )} x^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{6 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 86, normalized size = 0.96 \begin {gather*} \frac {{\left (5 \, b^{2} c - 3 \, a b d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{3}} + \frac {b^{2} c x - a b d x}{2 \, {\left (b x^{2} + a\right )} a^{3}} + \frac {6 \, b c x^{2} - 3 \, a d x^{2} - a c}{3 \, a^{3} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 110, normalized size = 1.22 \begin {gather*} -\frac {b d x}{2 \left (b \,x^{2}+a \right ) a^{2}}-\frac {3 b d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a^{2}}+\frac {b^{2} c x}{2 \left (b \,x^{2}+a \right ) a^{3}}+\frac {5 b^{2} c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a^{3}}-\frac {d}{a^{2} x}+\frac {2 b c}{a^{3} x}-\frac {c}{3 a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.36, size = 93, normalized size = 1.03 \begin {gather*} \frac {3 \, {\left (5 \, b^{2} c - 3 \, a b d\right )} x^{4} - 2 \, a^{2} c + 2 \, {\left (5 \, a b c - 3 \, a^{2} d\right )} x^{2}}{6 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )}} + \frac {{\left (5 \, b^{2} c - 3 \, a b d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 84, normalized size = 0.93 \begin {gather*} -\frac {\frac {c}{3\,a}+\frac {x^2\,\left (3\,a\,d-5\,b\,c\right )}{3\,a^2}+\frac {b\,x^4\,\left (3\,a\,d-5\,b\,c\right )}{2\,a^3}}{b\,x^5+a\,x^3}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (3\,a\,d-5\,b\,c\right )}{2\,a^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.59, size = 184, normalized size = 2.04 \begin {gather*} \frac {\sqrt {- \frac {b}{a^{7}}} \left (3 a d - 5 b c\right ) \log {\left (- \frac {a^{4} \sqrt {- \frac {b}{a^{7}}} \left (3 a d - 5 b c\right )}{3 a b d - 5 b^{2} c} + x \right )}}{4} - \frac {\sqrt {- \frac {b}{a^{7}}} \left (3 a d - 5 b c\right ) \log {\left (\frac {a^{4} \sqrt {- \frac {b}{a^{7}}} \left (3 a d - 5 b c\right )}{3 a b d - 5 b^{2} c} + x \right )}}{4} + \frac {- 2 a^{2} c + x^{4} \left (- 9 a b d + 15 b^{2} c\right ) + x^{2} \left (- 6 a^{2} d + 10 a b c\right )}{6 a^{4} x^{3} + 6 a^{3} b x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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