Optimal. Leaf size=59 \[ \frac{b c-a d}{a^2 x}+\frac{\sqrt{b} (b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2}}-\frac{c}{3 a x^3} \]
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Rubi [A] time = 0.0359807, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, 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 453
Rule 325
Rule 205
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*}
Mathematica [A] time = 0.050795, size = 60, normalized size = 1.02 \[ \frac{b c-a d}{a^2 x}-\frac{\sqrt{b} (a d-b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2}}-\frac{c}{3 a x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 72, normalized size = 1.2 \begin{align*} -{\frac{c}{3\,a{x}^{3}}}-{\frac{d}{ax}}+{\frac{bc}{{a}^{2}x}}-{\frac{bd}{a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{b}^{2}c}{{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51599, size = 294, normalized size = 4.98 \begin{align*} \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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.560699, size = 129, normalized size = 2.19 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20487, size = 77, normalized size = 1.31 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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