Optimal. Leaf size=202 \[ \frac{x^3 \left (8 a^2 d^2-40 a b c d+35 b^2 c^2\right )}{12 c d^3 \sqrt{c+d x^2}}-\frac{x \sqrt{c+d x^2} \left (8 a^2 d^2-40 a b c d+35 b^2 c^2\right )}{8 c d^4}+\frac{\left (8 a^2 d^2-40 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 d^{9/2}}+\frac{x^5 (b c-a d)^2}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac{b^2 x^5}{4 d^2 \sqrt{c+d x^2}} \]
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Rubi [A] time = 0.154685, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {463, 459, 288, 321, 217, 206} \[ \frac{x^3 \left (8 a^2 d^2-40 a b c d+35 b^2 c^2\right )}{12 c d^3 \sqrt{c+d x^2}}-\frac{x \sqrt{c+d x^2} \left (8 a^2 d^2-40 a b c d+35 b^2 c^2\right )}{8 c d^4}+\frac{\left (8 a^2 d^2-40 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 d^{9/2}}+\frac{x^5 (b c-a d)^2}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac{b^2 x^5}{4 d^2 \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Rule 463
Rule 459
Rule 288
Rule 321
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^4 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx &=\frac{(b c-a d)^2 x^5}{3 c d^2 \left (c+d x^2\right )^{3/2}}-\frac{\int \frac{x^4 \left (-3 a^2 d^2+5 (b c-a d)^2-3 b^2 c d x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c d^2}\\ &=\frac{(b c-a d)^2 x^5}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac{b^2 x^5}{4 d^2 \sqrt{c+d x^2}}-\frac{\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) \int \frac{x^4}{\left (c+d x^2\right )^{3/2}} \, dx}{12 c d^2}\\ &=\frac{(b c-a d)^2 x^5}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac{\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) x^3}{12 c d^3 \sqrt{c+d x^2}}+\frac{b^2 x^5}{4 d^2 \sqrt{c+d x^2}}-\frac{\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) \int \frac{x^2}{\sqrt{c+d x^2}} \, dx}{4 c d^3}\\ &=\frac{(b c-a d)^2 x^5}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac{\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) x^3}{12 c d^3 \sqrt{c+d x^2}}+\frac{b^2 x^5}{4 d^2 \sqrt{c+d x^2}}-\frac{\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) x \sqrt{c+d x^2}}{8 c d^4}+\frac{\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) \int \frac{1}{\sqrt{c+d x^2}} \, dx}{8 d^4}\\ &=\frac{(b c-a d)^2 x^5}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac{\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) x^3}{12 c d^3 \sqrt{c+d x^2}}+\frac{b^2 x^5}{4 d^2 \sqrt{c+d x^2}}-\frac{\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) x \sqrt{c+d x^2}}{8 c d^4}+\frac{\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-d x^2} \, dx,x,\frac{x}{\sqrt{c+d x^2}}\right )}{8 d^4}\\ &=\frac{(b c-a d)^2 x^5}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac{\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) x^3}{12 c d^3 \sqrt{c+d x^2}}+\frac{b^2 x^5}{4 d^2 \sqrt{c+d x^2}}-\frac{\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) x \sqrt{c+d x^2}}{8 c d^4}+\frac{\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 d^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.130485, size = 156, normalized size = 0.77 \[ \frac{x \left (-8 a^2 d^2 \left (3 c+4 d x^2\right )+8 a b d \left (15 c^2+20 c d x^2+3 d^2 x^4\right )+b^2 \left (-\left (140 c^2 d x^2+105 c^3+21 c d^2 x^4-6 d^3 x^6\right )\right )\right )}{24 d^4 \left (c+d x^2\right )^{3/2}}+\frac{\left (8 a^2 d^2-40 a b c d+35 b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{8 d^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 255, normalized size = 1.3 \begin{align*}{\frac{{b}^{2}{x}^{7}}{4\,d} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}-{\frac{7\,{b}^{2}c{x}^{5}}{8\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}-{\frac{35\,{x}^{3}{b}^{2}{c}^{2}}{24\,{d}^{3}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}-{\frac{35\,{b}^{2}{c}^{2}x}{8\,{d}^{4}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{35\,{b}^{2}{c}^{2}}{8}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{9}{2}}}}+{\frac{ab{x}^{5}}{d} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}+{\frac{5\,abc{x}^{3}}{3\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}+5\,{\frac{abcx}{{d}^{3}\sqrt{d{x}^{2}+c}}}-5\,{\frac{abc\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) }{{d}^{7/2}}}-{\frac{{a}^{2}{x}^{3}}{3\,d} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}-{\frac{{a}^{2}x}{{d}^{2}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{{a}^{2}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{5}{2}}}} \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 = 1.58405, size = 1127, normalized size = 5.58 \begin{align*} \left [\frac{3 \,{\left (35 \, b^{2} c^{4} - 40 \, a b c^{3} d + 8 \, a^{2} c^{2} d^{2} +{\left (35 \, b^{2} c^{2} d^{2} - 40 \, a b c d^{3} + 8 \, a^{2} d^{4}\right )} x^{4} + 2 \,{\left (35 \, b^{2} c^{3} d - 40 \, a b c^{2} d^{2} + 8 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt{d} \log \left (-2 \, d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{d} x - c\right ) + 2 \,{\left (6 \, b^{2} d^{4} x^{7} - 3 \,{\left (7 \, b^{2} c d^{3} - 8 \, a b d^{4}\right )} x^{5} - 4 \,{\left (35 \, b^{2} c^{2} d^{2} - 40 \, a b c d^{3} + 8 \, a^{2} d^{4}\right )} x^{3} - 3 \,{\left (35 \, b^{2} c^{3} d - 40 \, a b c^{2} d^{2} + 8 \, a^{2} c d^{3}\right )} x\right )} \sqrt{d x^{2} + c}}{48 \,{\left (d^{7} x^{4} + 2 \, c d^{6} x^{2} + c^{2} d^{5}\right )}}, -\frac{3 \,{\left (35 \, b^{2} c^{4} - 40 \, a b c^{3} d + 8 \, a^{2} c^{2} d^{2} +{\left (35 \, b^{2} c^{2} d^{2} - 40 \, a b c d^{3} + 8 \, a^{2} d^{4}\right )} x^{4} + 2 \,{\left (35 \, b^{2} c^{3} d - 40 \, a b c^{2} d^{2} + 8 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt{-d} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right ) -{\left (6 \, b^{2} d^{4} x^{7} - 3 \,{\left (7 \, b^{2} c d^{3} - 8 \, a b d^{4}\right )} x^{5} - 4 \,{\left (35 \, b^{2} c^{2} d^{2} - 40 \, a b c d^{3} + 8 \, a^{2} d^{4}\right )} x^{3} - 3 \,{\left (35 \, b^{2} c^{3} d - 40 \, a b c^{2} d^{2} + 8 \, a^{2} c d^{3}\right )} x\right )} \sqrt{d x^{2} + c}}{24 \,{\left (d^{7} x^{4} + 2 \, c d^{6} x^{2} + c^{2} d^{5}\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{x^{4} \left (a + b x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1684, size = 257, normalized size = 1.27 \begin{align*} \frac{{\left ({\left (3 \,{\left (\frac{2 \, b^{2} x^{2}}{d} - \frac{7 \, b^{2} c^{2} d^{5} - 8 \, a b c d^{6}}{c d^{7}}\right )} x^{2} - \frac{4 \,{\left (35 \, b^{2} c^{3} d^{4} - 40 \, a b c^{2} d^{5} + 8 \, a^{2} c d^{6}\right )}}{c d^{7}}\right )} x^{2} - \frac{3 \,{\left (35 \, b^{2} c^{4} d^{3} - 40 \, a b c^{3} d^{4} + 8 \, a^{2} c^{2} d^{5}\right )}}{c d^{7}}\right )} x}{24 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}}} - \frac{{\left (35 \, b^{2} c^{2} - 40 \, a b c d + 8 \, a^{2} d^{2}\right )} \log \left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right )}{8 \, d^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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