Optimal. Leaf size=149 \[ \frac{x \sqrt{c+d x^2} (3 b c-4 a d)}{8 a^2 \left (a+b x^2\right ) (b c-a d)}+\frac{c (3 b c-4 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{8 a^{5/2} (b c-a d)^{3/2}}+\frac{b x \left (c+d x^2\right )^{3/2}}{4 a \left (a+b x^2\right )^2 (b c-a d)} \]
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Rubi [A] time = 0.0794741, 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, 205} \[ \frac{x \sqrt{c+d x^2} (3 b c-4 a d)}{8 a^2 \left (a+b x^2\right ) (b c-a d)}+\frac{c (3 b c-4 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{8 a^{5/2} (b c-a d)^{3/2}}+\frac{b x \left (c+d x^2\right )^{3/2}}{4 a \left (a+b x^2\right )^2 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 382
Rule 378
Rule 377
Rule 205
Rubi steps
\begin{align*} \int \frac{\sqrt{c+d x^2}}{\left (a+b x^2\right )^3} \, dx &=\frac{b x \left (c+d x^2\right )^{3/2}}{4 a (b c-a d) \left (a+b x^2\right )^2}+\frac{(3 b c-4 a d) \int \frac{\sqrt{c+d x^2}}{\left (a+b x^2\right )^2} \, dx}{4 a (b c-a d)}\\ &=\frac{(3 b c-4 a d) x \sqrt{c+d x^2}}{8 a^2 (b c-a d) \left (a+b x^2\right )}+\frac{b x \left (c+d x^2\right )^{3/2}}{4 a (b c-a d) \left (a+b x^2\right )^2}+\frac{(c (3 b c-4 a d)) \int \frac{1}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{8 a^2 (b c-a d)}\\ &=\frac{(3 b c-4 a d) x \sqrt{c+d x^2}}{8 a^2 (b c-a d) \left (a+b x^2\right )}+\frac{b x \left (c+d x^2\right )^{3/2}}{4 a (b c-a d) \left (a+b x^2\right )^2}+\frac{(c (3 b c-4 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 )}{8 a^2 (b c-a d)}\\ &=\frac{(3 b c-4 a d) x \sqrt{c+d x^2}}{8 a^2 (b c-a d) \left (a+b x^2\right )}+\frac{b x \left (c+d x^2\right )^{3/2}}{4 a (b c-a d) \left (a+b x^2\right )^2}+\frac{c (3 b c-4 a d) \tan ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{a} \sqrt{c+d x^2}}\right )}{8 a^{5/2} (b c-a d)^{3/2}}\\ \end{align*}
Mathematica [A] time = 5.17001, size = 130, normalized size = 0.87 \[ \frac{\frac{\sqrt{a} x \sqrt{c+d x^2} \left (-4 a^2 d+a b \left (5 c-2 d x^2\right )+3 b^2 c x^2\right )}{\left (a+b x^2\right )^2 (b c-a d)}+\frac{c (3 b c-4 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{(b c-a d)^{3/2}}}{8 a^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.028, size = 5177, normalized size = 34.7 \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{d x^{2} + c}}{{\left (b x^{2} + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.89617, size = 1439, normalized size = 9.66 \begin{align*} \left [-\frac{{\left (3 \, a^{2} b c^{2} - 4 \, a^{3} c d +{\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{4} + 2 \,{\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d} \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 ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt{-a b c + a^{2} d} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \,{\left ({\left (3 \, a b^{3} c^{2} - 5 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x^{3} +{\left (5 \, a^{2} b^{2} c^{2} - 9 \, a^{3} b c d + 4 \, a^{4} d^{2}\right )} x\right )} \sqrt{d x^{2} + c}}{32 \,{\left (a^{5} b^{2} c^{2} - 2 \, a^{6} b c d + a^{7} d^{2} +{\left (a^{3} b^{4} c^{2} - 2 \, a^{4} b^{3} c d + a^{5} b^{2} d^{2}\right )} x^{4} + 2 \,{\left (a^{4} b^{3} c^{2} - 2 \, a^{5} b^{2} c d + a^{6} b d^{2}\right )} x^{2}\right )}}, \frac{{\left (3 \, a^{2} b c^{2} - 4 \, a^{3} c d +{\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{4} + 2 \,{\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{2}\right )} \sqrt{a b c - a^{2} d} \arctan \left (\frac{\sqrt{a b c - a^{2} d}{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt{d x^{2} + c}}{2 \,{\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} +{\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) + 2 \,{\left ({\left (3 \, a b^{3} c^{2} - 5 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x^{3} +{\left (5 \, a^{2} b^{2} c^{2} - 9 \, a^{3} b c d + 4 \, a^{4} d^{2}\right )} x\right )} \sqrt{d x^{2} + c}}{16 \,{\left (a^{5} b^{2} c^{2} - 2 \, a^{6} b c d + a^{7} d^{2} +{\left (a^{3} b^{4} c^{2} - 2 \, a^{4} b^{3} c d + a^{5} b^{2} d^{2}\right )} x^{4} + 2 \,{\left (a^{4} b^{3} c^{2} - 2 \, a^{5} b^{2} c d + a^{6} b d^{2}\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{c + d x^{2}}}{\left (a + b x^{2}\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 3.77492, size = 657, normalized size = 4.41 \begin{align*} -\frac{{\left (3 \, b c^{2} \sqrt{d} - 4 \, a c 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 )}{8 \,{\left (a^{2} b c - a^{3} d\right )} \sqrt{a b c d - a^{2} d^{2}}} - \frac{3 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} b^{3} c^{2} \sqrt{d} - 4 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} a b^{2} c d^{\frac{3}{2}} - 9 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b^{3} c^{3} \sqrt{d} + 30 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a b^{2} c^{2} d^{\frac{3}{2}} - 40 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a^{2} b c d^{\frac{5}{2}} + 16 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a^{3} d^{\frac{7}{2}} + 9 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b^{3} c^{4} \sqrt{d} - 28 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b^{2} c^{3} d^{\frac{3}{2}} + 16 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a^{2} b c^{2} d^{\frac{5}{2}} - 3 \, b^{3} c^{5} \sqrt{d} + 2 \, a b^{2} c^{4} d^{\frac{3}{2}}}{4 \,{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b - 2 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b c + 4 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a d + b c^{2}\right )}^{2}{\left (a^{2} b^{2} c - a^{3} b d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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