Optimal. Leaf size=59 \[ \frac {\sqrt {b} (b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2}}+\frac {b c-a d}{a^2 x}-\frac {c}{3 a x^3} \]
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Rubi [A] time = 0.04, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {453, 325, 205} \[ \frac {b c-a d}{a^2 x}+\frac {\sqrt {b} (b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2}}-\frac {c}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 205
Rule 325
Rule 453
Rubi steps
\begin {align*} \int \frac {c+d x^2}{x^4 \left (a+b x^2\right )} \, dx &=-\frac {c}{3 a x^3}-\frac {(3 b c-3 a d) \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx}{3 a}\\ &=-\frac {c}{3 a x^3}+\frac {b c-a d}{a^2 x}+\frac {(b (b c-a d)) \int \frac {1}{a+b x^2} \, dx}{a^2}\\ &=-\frac {c}{3 a x^3}+\frac {b c-a d}{a^2 x}+\frac {\sqrt {b} (b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 60, normalized size = 1.02 \[ -\frac {\sqrt {b} (a d-b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2}}+\frac {b c-a d}{a^2 x}-\frac {c}{3 a x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 136, normalized size = 2.31 \[ \left [-\frac {3 \, {\left (b c - a d\right )} x^{3} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) - 6 \, {\left (b c - a d\right )} x^{2} + 2 \, a c}{6 \, a^{2} x^{3}}, \frac {3 \, {\left (b c - a d\right )} x^{3} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 3 \, {\left (b c - a d\right )} x^{2} - a c}{3 \, a^{2} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 57, normalized size = 0.97 \[ \frac {{\left (b^{2} c - a b d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {3 \, b c x^{2} - 3 \, a d x^{2} - a c}{3 \, a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 72, normalized size = 1.22 \[ -\frac {b d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a}+\frac {b^{2} c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{2}}-\frac {d}{a x}+\frac {b c}{a^{2} x}-\frac {c}{3 a \,x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.33, size = 56, normalized size = 0.95 \[ \frac {{\left (b^{2} c - a b d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {3 \, {\left (b c - a d\right )} x^{2} - a c}{3 \, a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 53, normalized size = 0.90 \[ -\frac {\frac {c}{3\,a}+\frac {x^2\,\left (a\,d-b\,c\right )}{a^2}}{x^3}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (a\,d-b\,c\right )}{a^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.41, size = 129, normalized size = 2.19 \[ \frac {\sqrt {- \frac {b}{a^{5}}} \left (a d - b c\right ) \log {\left (- \frac {a^{3} \sqrt {- \frac {b}{a^{5}}} \left (a d - b c\right )}{a b d - b^{2} c} + x \right )}}{2} - \frac {\sqrt {- \frac {b}{a^{5}}} \left (a d - b c\right ) \log {\left (\frac {a^{3} \sqrt {- \frac {b}{a^{5}}} \left (a d - b c\right )}{a b d - b^{2} c} + x \right )}}{2} + \frac {- a c + x^{2} \left (- 3 a d + 3 b c\right )}{3 a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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