Optimal. Leaf size=97 \[ -\frac{a^2}{3 c x^3 \sqrt{c+d x^2}}+\frac{x \left (3 b^2 c^2-4 a d (3 b c-2 a d)\right )}{3 c^3 \sqrt{c+d x^2}}-\frac{2 a (3 b c-2 a d)}{3 c^2 x \sqrt{c+d x^2}} \]
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Rubi [A] time = 0.0720084, antiderivative size = 98, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {462, 453, 191} \[ \frac{x \left (8 a^2 d^2-12 a b c d+3 b^2 c^2\right )}{3 c^3 \sqrt{c+d x^2}}-\frac{a^2}{3 c x^3 \sqrt{c+d x^2}}-\frac{2 a (3 b c-2 a d)}{3 c^2 x \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Rule 462
Rule 453
Rule 191
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )^{3/2}} \, dx &=-\frac{a^2}{3 c x^3 \sqrt{c+d x^2}}+\frac{\int \frac{2 a (3 b c-2 a d)+3 b^2 c x^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx}{3 c}\\ &=-\frac{a^2}{3 c x^3 \sqrt{c+d x^2}}-\frac{2 a (3 b c-2 a d)}{3 c^2 x \sqrt{c+d x^2}}-\frac{1}{3} \left (-3 b^2+\frac{4 a d (3 b c-2 a d)}{c^2}\right ) \int \frac{1}{\left (c+d x^2\right )^{3/2}} \, dx\\ &=-\frac{a^2}{3 c x^3 \sqrt{c+d x^2}}-\frac{2 a (3 b c-2 a d)}{3 c^2 x \sqrt{c+d x^2}}+\frac{\left (3 b^2-\frac{4 a d (3 b c-2 a d)}{c^2}\right ) x}{3 c \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [A] time = 0.0236606, size = 74, normalized size = 0.76 \[ \frac{a^2 \left (-c^2+4 c d x^2+8 d^2 x^4\right )-6 a b c x^2 \left (c+2 d x^2\right )+3 b^2 c^2 x^4}{3 c^3 x^3 \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 77, normalized size = 0.8 \begin{align*} -{\frac{-8\,{a}^{2}{d}^{2}{x}^{4}+12\,abcd{x}^{4}-3\,{b}^{2}{c}^{2}{x}^{4}-4\,{a}^{2}cd{x}^{2}+6\,a{c}^{2}b{x}^{2}+{a}^{2}{c}^{2}}{3\,{x}^{3}{c}^{3}}{\frac{1}{\sqrt{d{x}^{2}+c}}}} \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 = 1.36997, size = 173, normalized size = 1.78 \begin{align*} \frac{{\left ({\left (3 \, b^{2} c^{2} - 12 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} - a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 2 \, a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{3 \,{\left (c^{3} d x^{5} + c^{4} 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{\left (a + b x^{2}\right )^{2}}{x^{4} \left (c + d x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13776, size = 269, normalized size = 2.77 \begin{align*} \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{\sqrt{d x^{2} + c} c^{3}} + \frac{2 \,{\left (6 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a b c \sqrt{d} - 3 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a^{2} d^{\frac{3}{2}} - 12 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b c^{2} \sqrt{d} + 12 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a^{2} c d^{\frac{3}{2}} + 6 \, a b c^{3} \sqrt{d} - 5 \, a^{2} c^{2} d^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )}^{3} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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