Optimal. Leaf size=175 \[ -\frac{b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 d^3}+\frac{(b c-a d)^{3/2} (a d+4 b c) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{2 c^{3/2} d^3}+\frac{b x \sqrt{a+b x^2} (2 b c-a d)}{2 c d^2}-\frac{x \left (a+b x^2\right )^{3/2} (b c-a d)}{2 c d \left (c+d x^2\right )} \]
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Rubi [A] time = 0.225999, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {413, 528, 523, 217, 206, 377, 208} \[ -\frac{b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 d^3}+\frac{(b c-a d)^{3/2} (a d+4 b c) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{2 c^{3/2} d^3}+\frac{b x \sqrt{a+b x^2} (2 b c-a d)}{2 c d^2}-\frac{x \left (a+b x^2\right )^{3/2} (b c-a d)}{2 c d \left (c+d x^2\right )} \]
Antiderivative was successfully verified.
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Rule 413
Rule 528
Rule 523
Rule 217
Rule 206
Rule 377
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^2} \, dx &=-\frac{(b c-a d) x \left (a+b x^2\right )^{3/2}}{2 c d \left (c+d x^2\right )}+\frac{\int \frac{\sqrt{a+b x^2} \left (a (b c+a d)+2 b (2 b c-a d) x^2\right )}{c+d x^2} \, dx}{2 c d}\\ &=\frac{b (2 b c-a d) x \sqrt{a+b x^2}}{2 c d^2}-\frac{(b c-a d) x \left (a+b x^2\right )^{3/2}}{2 c d \left (c+d x^2\right )}+\frac{\int \frac{-2 a \left (2 b^2 c^2-2 a b c d-a^2 d^2\right )-2 b^2 c (4 b c-5 a d) x^2}{\sqrt{a+b x^2} \left (c+d x^2\right )} \, dx}{4 c d^2}\\ &=\frac{b (2 b c-a d) x \sqrt{a+b x^2}}{2 c d^2}-\frac{(b c-a d) x \left (a+b x^2\right )^{3/2}}{2 c d \left (c+d x^2\right )}-\frac{\left (b^2 (4 b c-5 a d)\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{2 d^3}+\frac{\left ((b c-a d)^2 (4 b c+a d)\right ) \int \frac{1}{\sqrt{a+b x^2} \left (c+d x^2\right )} \, dx}{2 c d^3}\\ &=\frac{b (2 b c-a d) x \sqrt{a+b x^2}}{2 c d^2}-\frac{(b c-a d) x \left (a+b x^2\right )^{3/2}}{2 c d \left (c+d x^2\right )}-\frac{\left (b^2 (4 b c-5 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{2 d^3}+\frac{\left ((b c-a d)^2 (4 b c+a d)\right ) \operatorname{Subst}\left (\int \frac{1}{c-(b c-a d) x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{2 c d^3}\\ &=\frac{b (2 b c-a d) x \sqrt{a+b x^2}}{2 c d^2}-\frac{(b c-a d) x \left (a+b x^2\right )^{3/2}}{2 c d \left (c+d x^2\right )}-\frac{b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 d^3}+\frac{(b c-a d)^{3/2} (4 b c+a d) \tanh ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{a+b x^2}}\right )}{2 c^{3/2} d^3}\\ \end{align*}
Mathematica [A] time = 0.160223, size = 144, normalized size = 0.82 \[ \frac{d x \sqrt{a+b x^2} \left (\frac{(b c-a d)^2}{c \left (c+d x^2\right )}+b^2\right )+b^{3/2} (-(4 b c-5 a d)) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )+\frac{(a d-b c)^{3/2} (a d+4 b c) \tan ^{-1}\left (\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{c^{3/2}}}{2 d^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 7345, normalized size = 42. \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{{\left (b x^{2} + a\right )}^{\frac{5}{2}}}{{\left (d x^{2} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 6.33458, size = 2584, normalized size = 14.77 \begin{align*} \text{result too large to display} \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 )^{\frac{5}{2}}}{\left (c + d x^{2}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23954, size = 547, normalized size = 3.13 \begin{align*} \frac{\sqrt{b x^{2} + a} b^{2} x}{2 \, d^{2}} + \frac{{\left (4 \, b^{\frac{5}{2}} c - 5 \, a b^{\frac{3}{2}} d\right )} \log \left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right )}{4 \, d^{3}} - \frac{{\left (4 \, b^{\frac{7}{2}} c^{3} - 7 \, a b^{\frac{5}{2}} c^{2} d + 2 \, a^{2} b^{\frac{3}{2}} c d^{2} + a^{3} \sqrt{b} d^{3}\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 )}{2 \, \sqrt{-b^{2} c^{2} + a b c d} c d^{3}} + \frac{2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} b^{\frac{7}{2}} c^{3} - 5 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a b^{\frac{5}{2}} c^{2} d + 4 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{2} b^{\frac{3}{2}} c d^{2} -{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{3} \sqrt{b} d^{3} + a^{2} b^{\frac{5}{2}} c^{2} d - 2 \, a^{3} b^{\frac{3}{2}} c d^{2} + a^{4} \sqrt{b} d^{3}}{{\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 )} c d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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