Optimal. Leaf size=240 \[ \frac{x \left (c+d x^2\right )^{5/2} \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right )}{480 d^2}+\frac{c x \left (c+d x^2\right )^{3/2} \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right )}{384 d^2}+\frac{c^2 x \sqrt{c+d x^2} \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right )}{256 d^2}+\frac{c^3 \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{256 d^{5/2}}-\frac{3 b x \left (c+d x^2\right )^{7/2} (b c-4 a d)}{80 d^2}+\frac{b x \left (a+b x^2\right ) \left (c+d x^2\right )^{7/2}}{10 d} \]
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Rubi [A] time = 0.152393, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {416, 388, 195, 217, 206} \[ \frac{x \left (c+d x^2\right )^{5/2} \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right )}{480 d^2}+\frac{c x \left (c+d x^2\right )^{3/2} \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right )}{384 d^2}+\frac{c^2 x \sqrt{c+d x^2} \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right )}{256 d^2}+\frac{c^3 \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{256 d^{5/2}}-\frac{3 b x \left (c+d x^2\right )^{7/2} (b c-4 a d)}{80 d^2}+\frac{b x \left (a+b x^2\right ) \left (c+d x^2\right )^{7/2}}{10 d} \]
Antiderivative was successfully verified.
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Rule 416
Rule 388
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx &=\frac{b x \left (a+b x^2\right ) \left (c+d x^2\right )^{7/2}}{10 d}+\frac{\int \left (c+d x^2\right )^{5/2} \left (-a (b c-10 a d)-3 b (b c-4 a d) x^2\right ) \, dx}{10 d}\\ &=-\frac{3 b (b c-4 a d) x \left (c+d x^2\right )^{7/2}}{80 d^2}+\frac{b x \left (a+b x^2\right ) \left (c+d x^2\right )^{7/2}}{10 d}-\frac{(8 a d (b c-10 a d)-3 b c (b c-4 a d)) \int \left (c+d x^2\right )^{5/2} \, dx}{80 d^2}\\ &=\frac{\left (3 b^2 c^2-20 a b c d+80 a^2 d^2\right ) x \left (c+d x^2\right )^{5/2}}{480 d^2}-\frac{3 b (b c-4 a d) x \left (c+d x^2\right )^{7/2}}{80 d^2}+\frac{b x \left (a+b x^2\right ) \left (c+d x^2\right )^{7/2}}{10 d}+\frac{\left (c \left (3 b^2 c^2-20 a b c d+80 a^2 d^2\right )\right ) \int \left (c+d x^2\right )^{3/2} \, dx}{96 d^2}\\ &=\frac{c \left (3 b^2 c^2-20 a b c d+80 a^2 d^2\right ) x \left (c+d x^2\right )^{3/2}}{384 d^2}+\frac{\left (3 b^2 c^2-20 a b c d+80 a^2 d^2\right ) x \left (c+d x^2\right )^{5/2}}{480 d^2}-\frac{3 b (b c-4 a d) x \left (c+d x^2\right )^{7/2}}{80 d^2}+\frac{b x \left (a+b x^2\right ) \left (c+d x^2\right )^{7/2}}{10 d}+\frac{\left (c^2 \left (3 b^2 c^2-20 a b c d+80 a^2 d^2\right )\right ) \int \sqrt{c+d x^2} \, dx}{128 d^2}\\ &=\frac{c^2 \left (3 b^2 c^2-20 a b c d+80 a^2 d^2\right ) x \sqrt{c+d x^2}}{256 d^2}+\frac{c \left (3 b^2 c^2-20 a b c d+80 a^2 d^2\right ) x \left (c+d x^2\right )^{3/2}}{384 d^2}+\frac{\left (3 b^2 c^2-20 a b c d+80 a^2 d^2\right ) x \left (c+d x^2\right )^{5/2}}{480 d^2}-\frac{3 b (b c-4 a d) x \left (c+d x^2\right )^{7/2}}{80 d^2}+\frac{b x \left (a+b x^2\right ) \left (c+d x^2\right )^{7/2}}{10 d}+\frac{\left (c^3 \left (3 b^2 c^2-20 a b c d+80 a^2 d^2\right )\right ) \int \frac{1}{\sqrt{c+d x^2}} \, dx}{256 d^2}\\ &=\frac{c^2 \left (3 b^2 c^2-20 a b c d+80 a^2 d^2\right ) x \sqrt{c+d x^2}}{256 d^2}+\frac{c \left (3 b^2 c^2-20 a b c d+80 a^2 d^2\right ) x \left (c+d x^2\right )^{3/2}}{384 d^2}+\frac{\left (3 b^2 c^2-20 a b c d+80 a^2 d^2\right ) x \left (c+d x^2\right )^{5/2}}{480 d^2}-\frac{3 b (b c-4 a d) x \left (c+d x^2\right )^{7/2}}{80 d^2}+\frac{b x \left (a+b x^2\right ) \left (c+d x^2\right )^{7/2}}{10 d}+\frac{\left (c^3 \left (3 b^2 c^2-20 a b c d+80 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-d x^2} \, dx,x,\frac{x}{\sqrt{c+d x^2}}\right )}{256 d^2}\\ &=\frac{c^2 \left (3 b^2 c^2-20 a b c d+80 a^2 d^2\right ) x \sqrt{c+d x^2}}{256 d^2}+\frac{c \left (3 b^2 c^2-20 a b c d+80 a^2 d^2\right ) x \left (c+d x^2\right )^{3/2}}{384 d^2}+\frac{\left (3 b^2 c^2-20 a b c d+80 a^2 d^2\right ) x \left (c+d x^2\right )^{5/2}}{480 d^2}-\frac{3 b (b c-4 a d) x \left (c+d x^2\right )^{7/2}}{80 d^2}+\frac{b x \left (a+b x^2\right ) \left (c+d x^2\right )^{7/2}}{10 d}+\frac{c^3 \left (3 b^2 c^2-20 a b c d+80 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{256 d^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.122641, size = 192, normalized size = 0.8 \[ \frac{\sqrt{d} x \sqrt{c+d x^2} \left (80 a^2 d^2 \left (33 c^2+26 c d x^2+8 d^2 x^4\right )+20 a b d \left (118 c^2 d x^2+15 c^3+136 c d^2 x^4+48 d^3 x^6\right )+b^2 \left (744 c^2 d^2 x^4+30 c^3 d x^2-45 c^4+1008 c d^3 x^6+384 d^4 x^8\right )\right )+15 c^3 \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{3840 d^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 308, normalized size = 1.3 \begin{align*}{\frac{{b}^{2}{x}^{3}}{10\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{3\,{b}^{2}cx}{80\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}+{\frac{{b}^{2}{c}^{2}x}{160\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{{b}^{2}{c}^{3}x}{128\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{b}^{2}{c}^{4}x}{256\,{d}^{2}}\sqrt{d{x}^{2}+c}}+{\frac{3\,{b}^{2}{c}^{5}}{256}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{5}{2}}}}+{\frac{abx}{4\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{abcx}{24\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{5\,ab{c}^{2}x}{96\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{5\,ab{c}^{3}x}{64\,d}\sqrt{d{x}^{2}+c}}-{\frac{5\,ab{c}^{4}}{64}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{3}{2}}}}+{\frac{{a}^{2}x}{6} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{a}^{2}cx}{24} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}{c}^{2}x}{16}\sqrt{d{x}^{2}+c}}+{\frac{5\,{a}^{2}{c}^{3}}{16}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){\frac{1}{\sqrt{d}}}} \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 = 1.91466, size = 963, normalized size = 4.01 \begin{align*} \left [\frac{15 \,{\left (3 \, b^{2} c^{5} - 20 \, a b c^{4} d + 80 \, a^{2} c^{3} d^{2}\right )} \sqrt{d} \log \left (-2 \, d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{d} x - c\right ) + 2 \,{\left (384 \, b^{2} d^{5} x^{9} + 48 \,{\left (21 \, b^{2} c d^{4} + 20 \, a b d^{5}\right )} x^{7} + 8 \,{\left (93 \, b^{2} c^{2} d^{3} + 340 \, a b c d^{4} + 80 \, a^{2} d^{5}\right )} x^{5} + 10 \,{\left (3 \, b^{2} c^{3} d^{2} + 236 \, a b c^{2} d^{3} + 208 \, a^{2} c d^{4}\right )} x^{3} - 15 \,{\left (3 \, b^{2} c^{4} d - 20 \, a b c^{3} d^{2} - 176 \, a^{2} c^{2} d^{3}\right )} x\right )} \sqrt{d x^{2} + c}}{7680 \, d^{3}}, -\frac{15 \,{\left (3 \, b^{2} c^{5} - 20 \, a b c^{4} d + 80 \, a^{2} c^{3} d^{2}\right )} \sqrt{-d} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right ) -{\left (384 \, b^{2} d^{5} x^{9} + 48 \,{\left (21 \, b^{2} c d^{4} + 20 \, a b d^{5}\right )} x^{7} + 8 \,{\left (93 \, b^{2} c^{2} d^{3} + 340 \, a b c d^{4} + 80 \, a^{2} d^{5}\right )} x^{5} + 10 \,{\left (3 \, b^{2} c^{3} d^{2} + 236 \, a b c^{2} d^{3} + 208 \, a^{2} c d^{4}\right )} x^{3} - 15 \,{\left (3 \, b^{2} c^{4} d - 20 \, a b c^{3} d^{2} - 176 \, a^{2} c^{2} d^{3}\right )} x\right )} \sqrt{d x^{2} + c}}{3840 \, d^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 50.0722, size = 537, normalized size = 2.24 \begin{align*} \frac{a^{2} c^{\frac{5}{2}} x \sqrt{1 + \frac{d x^{2}}{c}}}{2} + \frac{3 a^{2} c^{\frac{5}{2}} x}{16 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{35 a^{2} c^{\frac{3}{2}} d x^{3}}{48 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{17 a^{2} \sqrt{c} d^{2} x^{5}}{24 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{5 a^{2} c^{3} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{16 \sqrt{d}} + \frac{a^{2} d^{3} x^{7}}{6 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{5 a b c^{\frac{7}{2}} x}{64 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{133 a b c^{\frac{5}{2}} x^{3}}{192 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{127 a b c^{\frac{3}{2}} d x^{5}}{96 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{23 a b \sqrt{c} d^{2} x^{7}}{24 \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{5 a b c^{4} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{64 d^{\frac{3}{2}}} + \frac{a b d^{3} x^{9}}{4 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{3 b^{2} c^{\frac{9}{2}} x}{256 d^{2} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{b^{2} c^{\frac{7}{2}} x^{3}}{256 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{129 b^{2} c^{\frac{5}{2}} x^{5}}{640 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{73 b^{2} c^{\frac{3}{2}} d x^{7}}{160 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{29 b^{2} \sqrt{c} d^{2} x^{9}}{80 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 b^{2} c^{5} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{256 d^{\frac{5}{2}}} + \frac{b^{2} d^{3} x^{11}}{10 \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 = 1.16371, size = 298, normalized size = 1.24 \begin{align*} \frac{1}{3840} \,{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, b^{2} d^{2} x^{2} + \frac{21 \, b^{2} c d^{9} + 20 \, a b d^{10}}{d^{8}}\right )} x^{2} + \frac{93 \, b^{2} c^{2} d^{8} + 340 \, a b c d^{9} + 80 \, a^{2} d^{10}}{d^{8}}\right )} x^{2} + \frac{5 \,{\left (3 \, b^{2} c^{3} d^{7} + 236 \, a b c^{2} d^{8} + 208 \, a^{2} c d^{9}\right )}}{d^{8}}\right )} x^{2} - \frac{15 \,{\left (3 \, b^{2} c^{4} d^{6} - 20 \, a b c^{3} d^{7} - 176 \, a^{2} c^{2} d^{8}\right )}}{d^{8}}\right )} \sqrt{d x^{2} + c} x - \frac{{\left (3 \, b^{2} c^{5} - 20 \, a b c^{4} d + 80 \, a^{2} c^{3} d^{2}\right )} \log \left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right )}{256 \, d^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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