Optimal. Leaf size=241 \[ \frac{x \left (a+b x^2\right )^{5/2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{480 b^2}+\frac{a x \left (a+b x^2\right )^{3/2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{384 b^2}+\frac{a^2 x \sqrt{a+b x^2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{256 b^2}+\frac{a^3 \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 b^{5/2}}+\frac{3 d x \left (a+b x^2\right )^{7/2} (4 b c-a d)}{80 b^2}+\frac{d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b} \]
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Rubi [A] time = 0.14837, antiderivative size = 241, 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} \[ \frac{x \left (a+b x^2\right )^{5/2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{480 b^2}+\frac{a x \left (a+b x^2\right )^{3/2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{384 b^2}+\frac{a^2 x \sqrt{a+b x^2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{256 b^2}+\frac{a^3 \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 b^{5/2}}+\frac{3 d x \left (a+b x^2\right )^{7/2} (4 b c-a d)}{80 b^2}+\frac{d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b} \]
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 )^{5/2} \left (c+d x^2\right )^2 \, dx &=\frac{d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}+\frac{\int \left (a+b x^2\right )^{5/2} \left (c (10 b c-a d)+3 d (4 b c-a d) x^2\right ) \, dx}{10 b}\\ &=\frac{3 d (4 b c-a d) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac{d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}-\frac{(3 a d (4 b c-a d)-8 b c (10 b c-a d)) \int \left (a+b x^2\right )^{5/2} \, dx}{80 b^2}\\ &=\frac{\left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{480 b^2}+\frac{3 d (4 b c-a d) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac{d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}+\frac{\left (a \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right )\right ) \int \left (a+b x^2\right )^{3/2} \, dx}{96 b^2}\\ &=\frac{a \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{384 b^2}+\frac{\left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{480 b^2}+\frac{3 d (4 b c-a d) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac{d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}+\frac{\left (a^2 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right )\right ) \int \sqrt{a+b x^2} \, dx}{128 b^2}\\ &=\frac{a^2 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \sqrt{a+b x^2}}{256 b^2}+\frac{a \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{384 b^2}+\frac{\left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{480 b^2}+\frac{3 d (4 b c-a d) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac{d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}+\frac{\left (a^3 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right )\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{256 b^2}\\ &=\frac{a^2 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \sqrt{a+b x^2}}{256 b^2}+\frac{a \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{384 b^2}+\frac{\left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{480 b^2}+\frac{3 d (4 b c-a d) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac{d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}+\frac{\left (a^3 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{256 b^2}\\ &=\frac{a^2 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \sqrt{a+b x^2}}{256 b^2}+\frac{a \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{384 b^2}+\frac{\left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{480 b^2}+\frac{3 d (4 b c-a d) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac{d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}+\frac{a^3 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 b^{5/2}}\\ \end{align*}
Mathematica [C] time = 2.62425, size = 158, normalized size = 0.66 \[ \frac{a x \sqrt{a+b x^2} \left (10 b x^2 \left (c+d x^2\right )^2 \text{HypergeometricPFQ}\left (\left \{-\frac{3}{2},\frac{3}{2},2\right \},\left \{1,\frac{9}{2}\right \},-\frac{b x^2}{a}\right )+20 b x^2 \left (2 c^2+3 c d x^2+d^2 x^4\right ) \, _2F_1\left (-\frac{3}{2},\frac{3}{2};\frac{9}{2};-\frac{b x^2}{a}\right )+7 a \left (15 c^2+10 c d x^2+3 d^2 x^4\right ) \, _2F_1\left (-\frac{5}{2},\frac{1}{2};\frac{7}{2};-\frac{b x^2}{a}\right )\right )}{105 \sqrt{\frac{b x^2}{a}+1}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.007, size = 308, normalized size = 1.3 \begin{align*}{\frac{{d}^{2}{x}^{3}}{10\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{3\,a{d}^{2}x}{80\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{{a}^{2}{d}^{2}x}{160\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{d}^{2}{a}^{3}x}{128\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{d}^{2}{a}^{4}x}{256\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,{d}^{2}{a}^{5}}{256}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}+{\frac{cdx}{4\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{cdax}{24\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,cd{a}^{2}x}{96\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,cd{a}^{3}x}{64\,b}\sqrt{b{x}^{2}+a}}-{\frac{5\,cd{a}^{4}}{64}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{{c}^{2}x}{6} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,a{c}^{2}x}{24} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}{c}^{2}x}{16}\sqrt{b{x}^{2}+a}}+{\frac{5\,{c}^{2}{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}} \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 = 2.9962, size = 963, normalized size = 4. \begin{align*} \left [\frac{15 \,{\left (80 \, a^{3} b^{2} c^{2} - 20 \, a^{4} b c d + 3 \, a^{5} d^{2}\right )} \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \,{\left (384 \, b^{5} d^{2} x^{9} + 48 \,{\left (20 \, b^{5} c d + 21 \, a b^{4} d^{2}\right )} x^{7} + 8 \,{\left (80 \, b^{5} c^{2} + 340 \, a b^{4} c d + 93 \, a^{2} b^{3} d^{2}\right )} x^{5} + 10 \,{\left (208 \, a b^{4} c^{2} + 236 \, a^{2} b^{3} c d + 3 \, a^{3} b^{2} d^{2}\right )} x^{3} + 15 \,{\left (176 \, a^{2} b^{3} c^{2} + 20 \, a^{3} b^{2} c d - 3 \, a^{4} b d^{2}\right )} x\right )} \sqrt{b x^{2} + a}}{7680 \, b^{3}}, -\frac{15 \,{\left (80 \, a^{3} b^{2} c^{2} - 20 \, a^{4} b c d + 3 \, a^{5} d^{2}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (384 \, b^{5} d^{2} x^{9} + 48 \,{\left (20 \, b^{5} c d + 21 \, a b^{4} d^{2}\right )} x^{7} + 8 \,{\left (80 \, b^{5} c^{2} + 340 \, a b^{4} c d + 93 \, a^{2} b^{3} d^{2}\right )} x^{5} + 10 \,{\left (208 \, a b^{4} c^{2} + 236 \, a^{2} b^{3} c d + 3 \, a^{3} b^{2} d^{2}\right )} x^{3} + 15 \,{\left (176 \, a^{2} b^{3} c^{2} + 20 \, a^{3} b^{2} c d - 3 \, a^{4} b d^{2}\right )} x\right )} \sqrt{b x^{2} + a}}{3840 \, b^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 50.966, size = 537, normalized size = 2.23 \begin{align*} - \frac{3 a^{\frac{9}{2}} d^{2} x}{256 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 a^{\frac{7}{2}} c d x}{64 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{a^{\frac{7}{2}} d^{2} x^{3}}{256 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{a^{\frac{5}{2}} c^{2} x \sqrt{1 + \frac{b x^{2}}{a}}}{2} + \frac{3 a^{\frac{5}{2}} c^{2} x}{16 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{133 a^{\frac{5}{2}} c d x^{3}}{192 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{129 a^{\frac{5}{2}} d^{2} x^{5}}{640 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{35 a^{\frac{3}{2}} b c^{2} x^{3}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{127 a^{\frac{3}{2}} b c d x^{5}}{96 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{73 a^{\frac{3}{2}} b d^{2} x^{7}}{160 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{17 \sqrt{a} b^{2} c^{2} x^{5}}{24 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{23 \sqrt{a} b^{2} c d x^{7}}{24 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{29 \sqrt{a} b^{2} d^{2} x^{9}}{80 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 a^{5} d^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{256 b^{\frac{5}{2}}} - \frac{5 a^{4} c d \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{64 b^{\frac{3}{2}}} + \frac{5 a^{3} c^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 \sqrt{b}} + \frac{b^{3} c^{2} x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{b^{3} c d x^{9}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{b^{3} d^{2} x^{11}}{10 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17603, 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{20 \, b^{10} c d + 21 \, a b^{9} d^{2}}{b^{8}}\right )} x^{2} + \frac{80 \, b^{10} c^{2} + 340 \, a b^{9} c d + 93 \, a^{2} b^{8} d^{2}}{b^{8}}\right )} x^{2} + \frac{5 \,{\left (208 \, a b^{9} c^{2} + 236 \, a^{2} b^{8} c d + 3 \, a^{3} b^{7} d^{2}\right )}}{b^{8}}\right )} x^{2} + \frac{15 \,{\left (176 \, a^{2} b^{8} c^{2} + 20 \, a^{3} b^{7} c d - 3 \, a^{4} b^{6} d^{2}\right )}}{b^{8}}\right )} \sqrt{b x^{2} + a} x - \frac{{\left (80 \, a^{3} b^{2} c^{2} - 20 \, a^{4} b c d + 3 \, a^{5} d^{2}\right )} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{256 \, b^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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