Optimal. Leaf size=194 \[ -\frac{\sqrt{b c-a d} \left (3 a^2 d^2+4 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{8 c^{5/2} d^3}+\frac{b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{d^3}-\frac{x \sqrt{a+b x^2} (b c-a d) (3 a d+4 b c)}{8 c^2 d^2 \left (c+d x^2\right )}-\frac{x \left (a+b x^2\right )^{3/2} (b c-a d)}{4 c d \left (c+d x^2\right )^2} \]
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Rubi [A] time = 0.194419, antiderivative size = 194, 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, 526, 523, 217, 206, 377, 208} \[ -\frac{\sqrt{b c-a d} \left (3 a^2 d^2+4 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{8 c^{5/2} d^3}+\frac{b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{d^3}-\frac{x \sqrt{a+b x^2} (b c-a d) (3 a d+4 b c)}{8 c^2 d^2 \left (c+d x^2\right )}-\frac{x \left (a+b x^2\right )^{3/2} (b c-a d)}{4 c d \left (c+d x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 413
Rule 526
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 )^3} \, dx &=-\frac{(b c-a d) x \left (a+b x^2\right )^{3/2}}{4 c d \left (c+d x^2\right )^2}+\frac{\int \frac{\sqrt{a+b x^2} \left (a (b c+3 a d)+4 b^2 c x^2\right )}{\left (c+d x^2\right )^2} \, dx}{4 c d}\\ &=-\frac{(b c-a d) x \left (a+b x^2\right )^{3/2}}{4 c d \left (c+d x^2\right )^2}-\frac{(b c-a d) (4 b c+3 a d) x \sqrt{a+b x^2}}{8 c^2 d^2 \left (c+d x^2\right )}-\frac{\int \frac{-a \left (4 b^2 c^2+a d (b c+3 a d)\right )-8 b^3 c^2 x^2}{\sqrt{a+b x^2} \left (c+d x^2\right )} \, dx}{8 c^2 d^2}\\ &=-\frac{(b c-a d) x \left (a+b x^2\right )^{3/2}}{4 c d \left (c+d x^2\right )^2}-\frac{(b c-a d) (4 b c+3 a d) x \sqrt{a+b x^2}}{8 c^2 d^2 \left (c+d x^2\right )}+\frac{b^3 \int \frac{1}{\sqrt{a+b x^2}} \, dx}{d^3}-\frac{\left ((b c-a d) \left (8 b^2 c^2+4 a b c d+3 a^2 d^2\right )\right ) \int \frac{1}{\sqrt{a+b x^2} \left (c+d x^2\right )} \, dx}{8 c^2 d^3}\\ &=-\frac{(b c-a d) x \left (a+b x^2\right )^{3/2}}{4 c d \left (c+d x^2\right )^2}-\frac{(b c-a d) (4 b c+3 a d) x \sqrt{a+b x^2}}{8 c^2 d^2 \left (c+d x^2\right )}+\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{d^3}-\frac{\left ((b c-a d) \left (8 b^2 c^2+4 a b c d+3 a^2 d^2\right )\right ) \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 d^3}\\ &=-\frac{(b c-a d) x \left (a+b x^2\right )^{3/2}}{4 c d \left (c+d x^2\right )^2}-\frac{(b c-a d) (4 b c+3 a d) x \sqrt{a+b x^2}}{8 c^2 d^2 \left (c+d x^2\right )}+\frac{b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{d^3}-\frac{\sqrt{b c-a d} \left (8 b^2 c^2+4 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{a+b x^2}}\right )}{8 c^{5/2} d^3}\\ \end{align*}
Mathematica [A] time = 0.193342, size = 184, normalized size = 0.95 \[ \frac{\frac{\left (a^2 b c d^2+3 a^3 d^3+4 a b^2 c^2 d-8 b^3 c^3\right ) \tan ^{-1}\left (\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{c^{5/2} \sqrt{a d-b c}}+8 b^{5/2} \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )+\frac{d x \sqrt{a+b x^2} (a d-b c) \left (a d \left (5 c+3 d x^2\right )+2 b c \left (2 c+3 d x^2\right )\right )}{c^2 \left (c+d x^2\right )^2}}{8 d^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.026, size = 14133, normalized size = 72.9 \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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 4.26576, size = 3125, normalized size = 16.11 \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 )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23433, size = 890, normalized size = 4.59 \begin{align*} -\frac{b^{\frac{5}{2}} \log \left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right )}{2 \, d^{3}} + \frac{{\left (8 \, b^{\frac{7}{2}} c^{3} - 4 \, a b^{\frac{5}{2}} c^{2} d - a^{2} b^{\frac{3}{2}} c d^{2} - 3 \, 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 )}{8 \, \sqrt{-b^{2} c^{2} + a b c d} c^{2} d^{3}} - \frac{16 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} b^{\frac{7}{2}} c^{3} d - 20 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a b^{\frac{5}{2}} c^{2} d^{2} +{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{2} b^{\frac{3}{2}} c d^{3} + 3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{3} \sqrt{b} d^{4} + 48 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} b^{\frac{9}{2}} c^{4} - 72 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a b^{\frac{7}{2}} c^{3} d + 18 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{2} b^{\frac{5}{2}} c^{2} d^{2} + 15 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{3} b^{\frac{3}{2}} c d^{3} - 9 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{4} \sqrt{b} d^{4} + 32 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{2} b^{\frac{7}{2}} c^{3} d - 28 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{3} b^{\frac{5}{2}} c^{2} d^{2} - 13 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{4} b^{\frac{3}{2}} c d^{3} + 9 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{5} \sqrt{b} d^{4} + 6 \, a^{4} b^{\frac{5}{2}} c^{2} d^{2} - 3 \, a^{5} b^{\frac{3}{2}} c d^{3} - 3 \, a^{6} \sqrt{b} d^{4}}{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} c^{2} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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