Optimal. Leaf size=108 \[ \frac{\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{5/2}}+\frac{3 d x \sqrt{a+b x^2} (2 b c-a d)}{8 b^2}+\frac{d x \sqrt{a+b x^2} \left (c+d x^2\right )}{4 b} \]
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Rubi [A] time = 0.0556046, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {416, 388, 217, 206} \[ \frac{\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{5/2}}+\frac{3 d x \sqrt{a+b x^2} (2 b c-a d)}{8 b^2}+\frac{d x \sqrt{a+b x^2} \left (c+d x^2\right )}{4 b} \]
Antiderivative was successfully verified.
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Rule 416
Rule 388
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (c+d x^2\right )^2}{\sqrt{a+b x^2}} \, dx &=\frac{d x \sqrt{a+b x^2} \left (c+d x^2\right )}{4 b}+\frac{\int \frac{c (4 b c-a d)+3 d (2 b c-a d) x^2}{\sqrt{a+b x^2}} \, dx}{4 b}\\ &=\frac{3 d (2 b c-a d) x \sqrt{a+b x^2}}{8 b^2}+\frac{d x \sqrt{a+b x^2} \left (c+d x^2\right )}{4 b}-\frac{(3 a d (2 b c-a d)-2 b c (4 b c-a d)) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{8 b^2}\\ &=\frac{3 d (2 b c-a d) x \sqrt{a+b x^2}}{8 b^2}+\frac{d x \sqrt{a+b x^2} \left (c+d x^2\right )}{4 b}-\frac{(3 a d (2 b c-a d)-2 b c (4 b c-a d)) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{8 b^2}\\ &=\frac{3 d (2 b c-a d) x \sqrt{a+b x^2}}{8 b^2}+\frac{d x \sqrt{a+b x^2} \left (c+d x^2\right )}{4 b}+\frac{\left (8 b^2 c^2-8 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{5/2}}\\ \end{align*}
Mathematica [C] time = 2.35633, size = 160, normalized size = 1.48 \[ \frac{x \sqrt{\frac{b x^2}{a}+1} \left (-2 b x^2 \left (c+d x^2\right )^2 \text{HypergeometricPFQ}\left (\left \{\frac{3}{2},\frac{3}{2},2\right \},\left \{1,\frac{9}{2}\right \},-\frac{b x^2}{a}\right )-4 b x^2 \left (2 c^2+3 c d x^2+d^2 x^4\right ) \, _2F_1\left (\frac{3}{2},\frac{3}{2};\frac{9}{2};-\frac{b x^2}{a}\right )+7 a \left (15 c^2+10 c d x^2+3 d^2 x^4\right ) \, _2F_1\left (\frac{1}{2},\frac{1}{2};\frac{7}{2};-\frac{b x^2}{a}\right )\right )}{105 a \sqrt{a+b x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.006, size = 131, normalized size = 1.2 \begin{align*}{\frac{{d}^{2}{x}^{3}}{4\,b}\sqrt{b{x}^{2}+a}}-{\frac{3\,a{d}^{2}x}{8\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,{a}^{2}{d}^{2}}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}+{\frac{cdx}{b}\sqrt{b{x}^{2}+a}}-{cda\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}}+{{c}^{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 = 1.66686, size = 440, normalized size = 4.07 \begin{align*} \left [\frac{{\left (8 \, b^{2} c^{2} - 8 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \,{\left (2 \, b^{2} d^{2} x^{3} +{\left (8 \, b^{2} c d - 3 \, a b d^{2}\right )} x\right )} \sqrt{b x^{2} + a}}{16 \, b^{3}}, -\frac{{\left (8 \, b^{2} c^{2} - 8 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (2 \, b^{2} d^{2} x^{3} +{\left (8 \, b^{2} c d - 3 \, a b d^{2}\right )} x\right )} \sqrt{b x^{2} + a}}{8 \, b^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.26663, size = 238, normalized size = 2.2 \begin{align*} - \frac{3 a^{\frac{3}{2}} d^{2} x}{8 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{\sqrt{a} c d x \sqrt{1 + \frac{b x^{2}}{a}}}{b} - \frac{\sqrt{a} d^{2} x^{3}}{8 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 a^{2} d^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{5}{2}}} - \frac{a c d \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{b^{\frac{3}{2}}} + c^{2} \left (\begin{cases} \frac{\sqrt{- \frac{a}{b}} \operatorname{asin}{\left (x \sqrt{- \frac{b}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge b < 0 \\\frac{\sqrt{\frac{a}{b}} \operatorname{asinh}{\left (x \sqrt{\frac{b}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge b > 0 \\\frac{\sqrt{- \frac{a}{b}} \operatorname{acosh}{\left (x \sqrt{- \frac{b}{a}} \right )}}{\sqrt{- a}} & \text{for}\: b > 0 \wedge a < 0 \end{cases}\right ) + \frac{d^{2} x^{5}}{4 \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 = 1.12174, size = 122, normalized size = 1.13 \begin{align*} \frac{1}{8} \, \sqrt{b x^{2} + a}{\left (\frac{2 \, d^{2} x^{2}}{b} + \frac{8 \, b^{2} c d - 3 \, a b d^{2}}{b^{3}}\right )} x - \frac{{\left (8 \, b^{2} c^{2} - 8 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{8 \, b^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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