Optimal. Leaf size=149 \[ \frac{x \sqrt{a+b x^2} \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{16 b^2}+\frac{a \left (a^2 d^2-4 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{5/2}}+\frac{d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{24 b^2}+\frac{d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b} \]
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Rubi [A] time = 0.0879085, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ \frac{x \sqrt{a+b x^2} \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{16 b^2}+\frac{a \left (a^2 d^2-4 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{5/2}}+\frac{d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{24 b^2}+\frac{d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b} \]
Antiderivative was successfully verified.
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Rule 416
Rule 388
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \sqrt{a+b x^2} \left (c+d x^2\right )^2 \, dx &=\frac{d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}+\frac{\int \sqrt{a+b x^2} \left (c (6 b c-a d)+d (8 b c-3 a d) x^2\right ) \, dx}{6 b}\\ &=\frac{d (8 b c-3 a d) x \left (a+b x^2\right )^{3/2}}{24 b^2}+\frac{d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}+\frac{\left (8 b^2 c^2-4 a b c d+a^2 d^2\right ) \int \sqrt{a+b x^2} \, dx}{8 b^2}\\ &=\frac{\left (8 b^2 c^2-4 a b c d+a^2 d^2\right ) x \sqrt{a+b x^2}}{16 b^2}+\frac{d (8 b c-3 a d) x \left (a+b x^2\right )^{3/2}}{24 b^2}+\frac{d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}+\frac{\left (a \left (8 b^2 c^2-4 a b c d+a^2 d^2\right )\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{16 b^2}\\ &=\frac{\left (8 b^2 c^2-4 a b c d+a^2 d^2\right ) x \sqrt{a+b x^2}}{16 b^2}+\frac{d (8 b c-3 a d) x \left (a+b x^2\right )^{3/2}}{24 b^2}+\frac{d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}+\frac{\left (a \left (8 b^2 c^2-4 a b c d+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 )}{16 b^2}\\ &=\frac{\left (8 b^2 c^2-4 a b c d+a^2 d^2\right ) x \sqrt{a+b x^2}}{16 b^2}+\frac{d (8 b c-3 a d) x \left (a+b x^2\right )^{3/2}}{24 b^2}+\frac{d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}+\frac{a \left (8 b^2 c^2-4 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{5/2}}\\ \end{align*}
Mathematica [C] time = 2.47574, size = 160, normalized size = 1.07 \[ \frac{x \sqrt{a+b x^2} \left (2 b x^2 \left (c+d x^2\right )^2 \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},\frac{3}{2},2\right \},\left \{1,\frac{9}{2}\right \},-\frac{b x^2}{a}\right )+4 b x^2 \left (2 c^2+3 c d x^2+d^2 x^4\right ) \, _2F_1\left (\frac{1}{2},\frac{3}{2};\frac{9}{2};-\frac{b x^2}{a}\right )+7 a \left (15 c^2+10 c d x^2+3 d^2 x^4\right ) \, _2F_1\left (-\frac{1}{2},\frac{1}{2};\frac{7}{2};-\frac{b x^2}{a}\right )\right )}{105 a \sqrt{\frac{b x^2}{a}+1}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.006, size = 190, normalized size = 1.3 \begin{align*}{\frac{{d}^{2}{x}^{3}}{6\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{a{d}^{2}x}{8\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}{d}^{2}x}{16\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{{a}^{3}{d}^{2}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}+{\frac{cdx}{2\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{acdx}{4\,b}\sqrt{b{x}^{2}+a}}-{\frac{cd{a}^{2}}{4}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{{c}^{2}x}{2}\sqrt{b{x}^{2}+a}}+{\frac{{c}^{2}a}{2}\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 = 1.81612, size = 585, normalized size = 3.93 \begin{align*} \left [\frac{3 \,{\left (8 \, a b^{2} c^{2} - 4 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \,{\left (8 \, b^{3} d^{2} x^{5} + 2 \,{\left (12 \, b^{3} c d + a b^{2} d^{2}\right )} x^{3} + 3 \,{\left (8 \, b^{3} c^{2} + 4 \, a b^{2} c d - a^{2} b d^{2}\right )} x\right )} \sqrt{b x^{2} + a}}{96 \, b^{3}}, -\frac{3 \,{\left (8 \, a b^{2} c^{2} - 4 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (8 \, b^{3} d^{2} x^{5} + 2 \,{\left (12 \, b^{3} c d + a b^{2} d^{2}\right )} x^{3} + 3 \,{\left (8 \, b^{3} c^{2} + 4 \, a b^{2} c d - a^{2} b d^{2}\right )} x\right )} \sqrt{b x^{2} + a}}{48 \, b^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 10.1787, size = 291, normalized size = 1.95 \begin{align*} - \frac{a^{\frac{5}{2}} d^{2} x}{16 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{a^{\frac{3}{2}} c d x}{4 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{a^{\frac{3}{2}} d^{2} x^{3}}{48 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{\sqrt{a} c^{2} x \sqrt{1 + \frac{b x^{2}}{a}}}{2} + \frac{3 \sqrt{a} c d x^{3}}{4 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 \sqrt{a} d^{2} x^{5}}{24 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{a^{3} d^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 b^{\frac{5}{2}}} - \frac{a^{2} c d \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{4 b^{\frac{3}{2}}} + \frac{a c^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 \sqrt{b}} + \frac{b c d x^{5}}{2 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{b d^{2} x^{7}}{6 \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 = 1.14444, size = 174, normalized size = 1.17 \begin{align*} \frac{1}{48} \,{\left (2 \,{\left (4 \, d^{2} x^{2} + \frac{12 \, b^{4} c d + a b^{3} d^{2}}{b^{4}}\right )} x^{2} + \frac{3 \,{\left (8 \, b^{4} c^{2} + 4 \, a b^{3} c d - a^{2} b^{2} d^{2}\right )}}{b^{4}}\right )} \sqrt{b x^{2} + a} x - \frac{{\left (8 \, a b^{2} c^{2} - 4 \, a^{2} b c d + a^{3} d^{2}\right )} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{16 \, b^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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