Optimal. Leaf size=149 \[ \frac{x \sqrt{a+b x^2} (4 b c-3 a d)}{8 c^2 \left (c+d x^2\right ) (b c-a d)}+\frac{a (4 b c-3 a d) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{3/2}}-\frac{d x \left (a+b x^2\right )^{3/2}}{4 c \left (c+d x^2\right )^2 (b c-a d)} \]
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Rubi [A] time = 0.0940901, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {382, 378, 377, 208} \[ \frac{x \sqrt{a+b x^2} (4 b c-3 a d)}{8 c^2 \left (c+d x^2\right ) (b c-a d)}+\frac{a (4 b c-3 a d) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{3/2}}-\frac{d x \left (a+b x^2\right )^{3/2}}{4 c \left (c+d x^2\right )^2 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 382
Rule 378
Rule 377
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x^2}}{\left (c+d x^2\right )^3} \, dx &=-\frac{d x \left (a+b x^2\right )^{3/2}}{4 c (b c-a d) \left (c+d x^2\right )^2}+\frac{(4 b c-3 a d) \int \frac{\sqrt{a+b x^2}}{\left (c+d x^2\right )^2} \, dx}{4 c (b c-a d)}\\ &=-\frac{d x \left (a+b x^2\right )^{3/2}}{4 c (b c-a d) \left (c+d x^2\right )^2}+\frac{(4 b c-3 a d) x \sqrt{a+b x^2}}{8 c^2 (b c-a d) \left (c+d x^2\right )}+\frac{(a (4 b c-3 a d)) \int \frac{1}{\sqrt{a+b x^2} \left (c+d x^2\right )} \, dx}{8 c^2 (b c-a d)}\\ &=-\frac{d x \left (a+b x^2\right )^{3/2}}{4 c (b c-a d) \left (c+d x^2\right )^2}+\frac{(4 b c-3 a d) x \sqrt{a+b x^2}}{8 c^2 (b c-a d) \left (c+d x^2\right )}+\frac{(a (4 b c-3 a d)) \operatorname{Subst}\left (\int \frac{1}{c-(b c-a d) x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{8 c^2 (b c-a d)}\\ &=-\frac{d x \left (a+b x^2\right )^{3/2}}{4 c (b c-a d) \left (c+d x^2\right )^2}+\frac{(4 b c-3 a d) x \sqrt{a+b x^2}}{8 c^2 (b c-a d) \left (c+d x^2\right )}+\frac{a (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.527832, size = 176, normalized size = 1.18 \[ \frac{x \left (c \left (a^2 d \left (5 c+3 d x^2\right )+a b \left (-4 c^2+3 c d x^2+3 d^2 x^4\right )-2 b^2 c x^2 \left (2 c+d x^2\right )\right )+\frac{a \left (c+d x^2\right )^2 (3 a d-4 b c) \tanh ^{-1}\left (\sqrt{\frac{x^2 (b c-a d)}{c \left (a+b x^2\right )}}\right )}{\sqrt{\frac{x^2 (b c-a d)}{c \left (a+b x^2\right )}}}\right )}{8 c^3 \sqrt{a+b x^2} \left (c+d x^2\right )^2 (a d-b c)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.022, size = 5101, normalized size = 34.2 \begin{align*} \text{output 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{b x^{2} + a}}{{\left (d x^{2} + c\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.61643, size = 1439, normalized size = 9.66 \begin{align*} \left [\frac{{\left (4 \, a b c^{3} - 3 \, a^{2} c^{2} d +{\left (4 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{4} + 2 \,{\left (4 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt{b c^{2} - a c d} \log \left (\frac{{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \,{\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} + 4 \,{\left ({\left (2 \, b c - a d\right )} x^{3} + a c x\right )} \sqrt{b c^{2} - a c d} \sqrt{b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right ) + 4 \,{\left ({\left (2 \, b^{2} c^{3} d - 5 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x^{3} +{\left (4 \, b^{2} c^{4} - 9 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt{b x^{2} + a}}{32 \,{\left (b^{2} c^{7} - 2 \, a b c^{6} d + a^{2} c^{5} d^{2} +{\left (b^{2} c^{5} d^{2} - 2 \, a b c^{4} d^{3} + a^{2} c^{3} d^{4}\right )} x^{4} + 2 \,{\left (b^{2} c^{6} d - 2 \, a b c^{5} d^{2} + a^{2} c^{4} d^{3}\right )} x^{2}\right )}}, -\frac{{\left (4 \, a b c^{3} - 3 \, a^{2} c^{2} d +{\left (4 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{4} + 2 \,{\left (4 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt{-b c^{2} + a c d} \arctan \left (\frac{\sqrt{-b c^{2} + a c d}{\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt{b x^{2} + a}}{2 \,{\left ({\left (b^{2} c^{2} - a b c d\right )} x^{3} +{\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) - 2 \,{\left ({\left (2 \, b^{2} c^{3} d - 5 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x^{3} +{\left (4 \, b^{2} c^{4} - 9 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt{b x^{2} + a}}{16 \,{\left (b^{2} c^{7} - 2 \, a b c^{6} d + a^{2} c^{5} d^{2} +{\left (b^{2} c^{5} d^{2} - 2 \, a b c^{4} d^{3} + a^{2} c^{3} d^{4}\right )} x^{4} + 2 \,{\left (b^{2} c^{6} d - 2 \, a b c^{5} d^{2} + a^{2} c^{4} d^{3}\right )} x^{2}\right )}}\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{a + b x^{2}}}{\left (c + d x^{2}\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.52096, size = 657, normalized size = 4.41 \begin{align*} -\frac{{\left (4 \, a b^{\frac{3}{2}} c - 3 \, a^{2} \sqrt{b} d\right )} \arctan \left (\frac{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt{-b^{2} c^{2} + a b c d}}\right )}{8 \, \sqrt{-b^{2} c^{2} + a b c d}{\left (b c^{3} - a c^{2} d\right )}} - \frac{4 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a b^{\frac{3}{2}} c d^{2} - 3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{2} \sqrt{b} d^{3} - 16 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} b^{\frac{7}{2}} c^{3} + 40 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a b^{\frac{5}{2}} c^{2} d - 30 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{2} b^{\frac{3}{2}} c d^{2} + 9 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{3} \sqrt{b} d^{3} - 16 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{2} b^{\frac{5}{2}} c^{2} d + 28 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{3} b^{\frac{3}{2}} c d^{2} - 9 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{4} \sqrt{b} d^{3} - 2 \, a^{4} b^{\frac{3}{2}} c d^{2} + 3 \, a^{5} \sqrt{b} d^{3}}{4 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} d + 4 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} b c - 2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a d + a^{2} d\right )}^{2}{\left (b c^{3} d - a c^{2} d^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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