Optimal. Leaf size=281 \[ \frac{c^2 x^3 \sqrt{c+d x^2} \left (40 a^2 d^2+b c (5 b c-24 a d)\right )}{512 d^2}+\frac{c^3 x \sqrt{c+d x^2} \left (40 a^2 d^2+b c (5 b c-24 a d)\right )}{1024 d^3}-\frac{c^4 \left (40 a^2 d^2+b c (5 b c-24 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{1024 d^{7/2}}+\frac{x^3 \left (c+d x^2\right )^{5/2} \left (40 a^2 d^2+b c (5 b c-24 a d)\right )}{320 d^2}+\frac{c x^3 \left (c+d x^2\right )^{3/2} \left (40 a^2 d^2+b c (5 b c-24 a d)\right )}{384 d^2}-\frac{b x^3 \left (c+d x^2\right )^{7/2} (5 b c-24 a d)}{120 d^2}+\frac{b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d} \]
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Rubi [A] time = 0.257678, antiderivative size = 278, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {464, 459, 279, 321, 217, 206} \[ \frac{c^2 x^3 \sqrt{c+d x^2} \left (40 a^2 d^2+b c (5 b c-24 a d)\right )}{512 d^2}+\frac{c^3 x \sqrt{c+d x^2} \left (40 a^2 d^2+b c (5 b c-24 a d)\right )}{1024 d^3}-\frac{c^4 \left (40 a^2 d^2+b c (5 b c-24 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{1024 d^{7/2}}+\frac{1}{320} x^3 \left (c+d x^2\right )^{5/2} \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right )+\frac{c x^3 \left (c+d x^2\right )^{3/2} \left (40 a^2 d^2+b c (5 b c-24 a d)\right )}{384 d^2}-\frac{b x^3 \left (c+d x^2\right )^{7/2} (5 b c-24 a d)}{120 d^2}+\frac{b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d} \]
Antiderivative was successfully verified.
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Rule 464
Rule 459
Rule 279
Rule 321
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx &=\frac{b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d}+\frac{\int x^2 \left (c+d x^2\right )^{5/2} \left (12 a^2 d-b (5 b c-24 a d) x^2\right ) \, dx}{12 d}\\ &=-\frac{b (5 b c-24 a d) x^3 \left (c+d x^2\right )^{7/2}}{120 d^2}+\frac{b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d}+\frac{1}{40} \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right ) \int x^2 \left (c+d x^2\right )^{5/2} \, dx\\ &=\frac{1}{320} \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{5/2}-\frac{b (5 b c-24 a d) x^3 \left (c+d x^2\right )^{7/2}}{120 d^2}+\frac{b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d}+\frac{1}{64} \left (c \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right )\right ) \int x^2 \left (c+d x^2\right )^{3/2} \, dx\\ &=\frac{1}{384} c \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{3/2}+\frac{1}{320} \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{5/2}-\frac{b (5 b c-24 a d) x^3 \left (c+d x^2\right )^{7/2}}{120 d^2}+\frac{b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d}+\frac{1}{128} \left (c^2 \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right )\right ) \int x^2 \sqrt{c+d x^2} \, dx\\ &=\frac{1}{512} c^2 \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right ) x^3 \sqrt{c+d x^2}+\frac{1}{384} c \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{3/2}+\frac{1}{320} \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{5/2}-\frac{b (5 b c-24 a d) x^3 \left (c+d x^2\right )^{7/2}}{120 d^2}+\frac{b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d}+\frac{1}{512} \left (c^3 \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right )\right ) \int \frac{x^2}{\sqrt{c+d x^2}} \, dx\\ &=\frac{c^3 \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right ) x \sqrt{c+d x^2}}{1024 d}+\frac{1}{512} c^2 \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right ) x^3 \sqrt{c+d x^2}+\frac{1}{384} c \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{3/2}+\frac{1}{320} \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{5/2}-\frac{b (5 b c-24 a d) x^3 \left (c+d x^2\right )^{7/2}}{120 d^2}+\frac{b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d}-\frac{\left (c^4 \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right )\right ) \int \frac{1}{\sqrt{c+d x^2}} \, dx}{1024 d}\\ &=\frac{c^3 \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right ) x \sqrt{c+d x^2}}{1024 d}+\frac{1}{512} c^2 \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right ) x^3 \sqrt{c+d x^2}+\frac{1}{384} c \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{3/2}+\frac{1}{320} \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{5/2}-\frac{b (5 b c-24 a d) x^3 \left (c+d x^2\right )^{7/2}}{120 d^2}+\frac{b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d}-\frac{\left (c^4 \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-d x^2} \, dx,x,\frac{x}{\sqrt{c+d x^2}}\right )}{1024 d}\\ &=\frac{c^3 \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right ) x \sqrt{c+d x^2}}{1024 d}+\frac{1}{512} c^2 \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right ) x^3 \sqrt{c+d x^2}+\frac{1}{384} c \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{3/2}+\frac{1}{320} \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{5/2}-\frac{b (5 b c-24 a d) x^3 \left (c+d x^2\right )^{7/2}}{120 d^2}+\frac{b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d}-\frac{c^4 \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{1024 d^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.146804, size = 226, normalized size = 0.8 \[ \frac{\sqrt{d} x \sqrt{c+d x^2} \left (40 a^2 d^2 \left (118 c^2 d x^2+15 c^3+136 c d^2 x^4+48 d^3 x^6\right )+24 a b d \left (248 c^2 d^2 x^4+10 c^3 d x^2-15 c^4+336 c d^3 x^6+128 d^4 x^8\right )+5 b^2 \left (432 c^2 d^3 x^6+8 c^3 d^2 x^4-10 c^4 d x^2+15 c^5+640 c d^4 x^8+256 d^5 x^{10}\right )\right )-15 c^4 \left (40 a^2 d^2-24 a b c d+5 b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{15360 d^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 383, normalized size = 1.4 \begin{align*}{\frac{{b}^{2}{x}^{5}}{12\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{{b}^{2}c{x}^{3}}{24\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}+{\frac{{b}^{2}{c}^{2}x}{64\,{d}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{{b}^{2}{c}^{3}x}{384\,{d}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{b}^{2}{c}^{4}x}{1536\,{d}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{b}^{2}{c}^{5}x}{1024\,{d}^{3}}\sqrt{d{x}^{2}+c}}-{\frac{5\,{b}^{2}{c}^{6}}{1024}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{7}{2}}}}+{\frac{ab{x}^{3}}{5\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{3\,abcx}{40\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}+{\frac{ab{c}^{2}x}{80\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{ab{c}^{3}x}{64\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{3\,ab{c}^{4}x}{128\,{d}^{2}}\sqrt{d{x}^{2}+c}}+{\frac{3\,ab{c}^{5}}{128}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{5}{2}}}}+{\frac{{a}^{2}x}{8\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{{a}^{2}cx}{48\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{a}^{2}{c}^{2}x}{192\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{a}^{2}{c}^{3}x}{128\,d}\sqrt{d{x}^{2}+c}}-{\frac{5\,{a}^{2}{c}^{4}}{128}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{3}{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 = 2.72963, size = 1131, normalized size = 4.02 \begin{align*} \left [\frac{15 \,{\left (5 \, b^{2} c^{6} - 24 \, a b c^{5} d + 40 \, a^{2} c^{4} d^{2}\right )} \sqrt{d} \log \left (-2 \, d x^{2} + 2 \, \sqrt{d x^{2} + c} \sqrt{d} x - c\right ) + 2 \,{\left (1280 \, b^{2} d^{6} x^{11} + 128 \,{\left (25 \, b^{2} c d^{5} + 24 \, a b d^{6}\right )} x^{9} + 48 \,{\left (45 \, b^{2} c^{2} d^{4} + 168 \, a b c d^{5} + 40 \, a^{2} d^{6}\right )} x^{7} + 8 \,{\left (5 \, b^{2} c^{3} d^{3} + 744 \, a b c^{2} d^{4} + 680 \, a^{2} c d^{5}\right )} x^{5} - 10 \,{\left (5 \, b^{2} c^{4} d^{2} - 24 \, a b c^{3} d^{3} - 472 \, a^{2} c^{2} d^{4}\right )} x^{3} + 15 \,{\left (5 \, b^{2} c^{5} d - 24 \, a b c^{4} d^{2} + 40 \, a^{2} c^{3} d^{3}\right )} x\right )} \sqrt{d x^{2} + c}}{30720 \, d^{4}}, \frac{15 \,{\left (5 \, b^{2} c^{6} - 24 \, a b c^{5} d + 40 \, a^{2} c^{4} d^{2}\right )} \sqrt{-d} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right ) +{\left (1280 \, b^{2} d^{6} x^{11} + 128 \,{\left (25 \, b^{2} c d^{5} + 24 \, a b d^{6}\right )} x^{9} + 48 \,{\left (45 \, b^{2} c^{2} d^{4} + 168 \, a b c d^{5} + 40 \, a^{2} d^{6}\right )} x^{7} + 8 \,{\left (5 \, b^{2} c^{3} d^{3} + 744 \, a b c^{2} d^{4} + 680 \, a^{2} c d^{5}\right )} x^{5} - 10 \,{\left (5 \, b^{2} c^{4} d^{2} - 24 \, a b c^{3} d^{3} - 472 \, a^{2} c^{2} d^{4}\right )} x^{3} + 15 \,{\left (5 \, b^{2} c^{5} d - 24 \, a b c^{4} d^{2} + 40 \, a^{2} c^{3} d^{3}\right )} x\right )} \sqrt{d x^{2} + c}}{15360 \, d^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 77.3702, size = 602, normalized size = 2.14 \begin{align*} \frac{5 a^{2} c^{\frac{7}{2}} x}{128 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{133 a^{2} c^{\frac{5}{2}} x^{3}}{384 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{127 a^{2} c^{\frac{3}{2}} d x^{5}}{192 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{23 a^{2} \sqrt{c} d^{2} x^{7}}{48 \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{5 a^{2} c^{4} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{128 d^{\frac{3}{2}}} + \frac{a^{2} d^{3} x^{9}}{8 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{3 a b c^{\frac{9}{2}} x}{128 d^{2} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{a b c^{\frac{7}{2}} x^{3}}{128 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{129 a b c^{\frac{5}{2}} x^{5}}{320 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{73 a b c^{\frac{3}{2}} d x^{7}}{80 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{29 a b \sqrt{c} d^{2} x^{9}}{40 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 a b c^{5} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{128 d^{\frac{5}{2}}} + \frac{a b d^{3} x^{11}}{5 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{5 b^{2} c^{\frac{11}{2}} x}{1024 d^{3} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{5 b^{2} c^{\frac{9}{2}} x^{3}}{3072 d^{2} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{b^{2} c^{\frac{7}{2}} x^{5}}{1536 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{55 b^{2} c^{\frac{5}{2}} x^{7}}{384 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{67 b^{2} c^{\frac{3}{2}} d x^{9}}{192 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{7 b^{2} \sqrt{c} d^{2} x^{11}}{24 \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{5 b^{2} c^{6} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{1024 d^{\frac{7}{2}}} + \frac{b^{2} d^{3} x^{13}}{12 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1303, size = 358, normalized size = 1.27 \begin{align*} \frac{1}{15360} \,{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, b^{2} d^{2} x^{2} + \frac{25 \, b^{2} c d^{11} + 24 \, a b d^{12}}{d^{10}}\right )} x^{2} + \frac{3 \,{\left (45 \, b^{2} c^{2} d^{10} + 168 \, a b c d^{11} + 40 \, a^{2} d^{12}\right )}}{d^{10}}\right )} x^{2} + \frac{5 \, b^{2} c^{3} d^{9} + 744 \, a b c^{2} d^{10} + 680 \, a^{2} c d^{11}}{d^{10}}\right )} x^{2} - \frac{5 \,{\left (5 \, b^{2} c^{4} d^{8} - 24 \, a b c^{3} d^{9} - 472 \, a^{2} c^{2} d^{10}\right )}}{d^{10}}\right )} x^{2} + \frac{15 \,{\left (5 \, b^{2} c^{5} d^{7} - 24 \, a b c^{4} d^{8} + 40 \, a^{2} c^{3} d^{9}\right )}}{d^{10}}\right )} \sqrt{d x^{2} + c} x + \frac{{\left (5 \, b^{2} c^{6} - 24 \, a b c^{5} d + 40 \, a^{2} c^{4} d^{2}\right )} \log \left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right )}{1024 \, d^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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