Optimal. Leaf size=105 \[ \frac{\sqrt{c+d x^2} (3 b c-a d)}{3 a^2 c x}+\frac{b \sqrt{b c-a d} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{5/2}}-\frac{\sqrt{c+d x^2}}{3 a x^3} \]
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Rubi [A] time = 0.119622, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {475, 583, 12, 377, 205} \[ \frac{\sqrt{c+d x^2} (3 b c-a d)}{3 a^2 c x}+\frac{b \sqrt{b c-a d} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{5/2}}-\frac{\sqrt{c+d x^2}}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 475
Rule 583
Rule 12
Rule 377
Rule 205
Rubi steps
\begin{align*} \int \frac{\sqrt{c+d x^2}}{x^4 \left (a+b x^2\right )} \, dx &=-\frac{\sqrt{c+d x^2}}{3 a x^3}+\frac{\int \frac{-3 b c+a d-2 b d x^2}{x^2 \left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{3 a}\\ &=-\frac{\sqrt{c+d x^2}}{3 a x^3}+\frac{(3 b c-a d) \sqrt{c+d x^2}}{3 a^2 c x}-\frac{\int -\frac{3 b c (b c-a d)}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{3 a^2 c}\\ &=-\frac{\sqrt{c+d x^2}}{3 a x^3}+\frac{(3 b c-a d) \sqrt{c+d x^2}}{3 a^2 c x}+\frac{(b (b c-a d)) \int \frac{1}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{a^2}\\ &=-\frac{\sqrt{c+d x^2}}{3 a x^3}+\frac{(3 b c-a d) \sqrt{c+d x^2}}{3 a^2 c x}+\frac{(b (b c-a d)) \operatorname{Subst}\left (\int \frac{1}{a-(-b c+a d) x^2} \, dx,x,\frac{x}{\sqrt{c+d x^2}}\right )}{a^2}\\ &=-\frac{\sqrt{c+d x^2}}{3 a x^3}+\frac{(3 b c-a d) \sqrt{c+d x^2}}{3 a^2 c x}+\frac{b \sqrt{b c-a d} \tan ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{5/2}}\\ \end{align*}
Mathematica [A] time = 5.1209, size = 93, normalized size = 0.89 \[ \frac{\sqrt{c+d x^2} \left (3 b c x^2-a \left (c+d x^2\right )\right )}{3 a^2 c x^3}+\frac{b \sqrt{b c-a d} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 1059, normalized size = 10.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d x^{2} + c}}{{\left (b x^{2} + a\right )} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68074, size = 678, normalized size = 6.46 \begin{align*} \left [\frac{3 \, b c x^{3} \sqrt{-\frac{b c - a d}{a}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \,{\left (a^{2} c x -{\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt{d x^{2} + c} \sqrt{-\frac{b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \,{\left ({\left (3 \, b c - a d\right )} x^{2} - a c\right )} \sqrt{d x^{2} + c}}{12 \, a^{2} c x^{3}}, \frac{3 \, b c x^{3} \sqrt{\frac{b c - a d}{a}} \arctan \left (\frac{{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt{d x^{2} + c} \sqrt{\frac{b c - a d}{a}}}{2 \,{\left ({\left (b c d - a d^{2}\right )} x^{3} +{\left (b c^{2} - a c d\right )} x\right )}}\right ) + 2 \,{\left ({\left (3 \, b c - a d\right )} x^{2} - a c\right )} \sqrt{d x^{2} + c}}{6 \, a^{2} c x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c + d x^{2}}}{x^{4} \left (a + b x^{2}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.18278, size = 290, normalized size = 2.76 \begin{align*} -\frac{{\left (b^{2} c \sqrt{d} - a b d^{\frac{3}{2}}\right )} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{\sqrt{a b c d - a^{2} d^{2}} a^{2}} - \frac{2 \,{\left (3 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b c \sqrt{d} - 3 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a d^{\frac{3}{2}} - 6 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b c^{2} \sqrt{d} + 3 \, b c^{3} \sqrt{d} - a c^{2} d^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )}^{3} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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