Optimal. Leaf size=54 \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{3/2}}+\frac{x (b c-a d)}{a b \sqrt{a+b x^2}} \]
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Rubi [A] time = 0.0172689, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {385, 217, 206} \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{3/2}}+\frac{x (b c-a d)}{a b \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 385
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{c+d x^2}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac{(b c-a d) x}{a b \sqrt{a+b x^2}}+\frac{d \int \frac{1}{\sqrt{a+b x^2}} \, dx}{b}\\ &=\frac{(b c-a d) x}{a b \sqrt{a+b x^2}}+\frac{d \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{b}\\ &=\frac{(b c-a d) x}{a b \sqrt{a+b x^2}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0575813, size = 70, normalized size = 1.3 \[ \frac{a^{3/2} d \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )+\sqrt{b} x (b c-a d)}{a b^{3/2} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 54, normalized size = 1. \begin{align*} -{\frac{dx}{b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{d\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{cx}{a}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \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 = 1.545, size = 367, normalized size = 6.8 \begin{align*} \left [\frac{2 \,{\left (b^{2} c - a b d\right )} \sqrt{b x^{2} + a} x +{\left (a b d x^{2} + a^{2} d\right )} \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right )}{2 \,{\left (a b^{3} x^{2} + a^{2} b^{2}\right )}}, \frac{{\left (b^{2} c - a b d\right )} \sqrt{b x^{2} + a} x -{\left (a b d x^{2} + a^{2} d\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{a b^{3} x^{2} + a^{2} b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.70417, size = 60, normalized size = 1.11 \begin{align*} d \left (\frac{\operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{b^{\frac{3}{2}}} - \frac{x}{\sqrt{a} b \sqrt{1 + \frac{b x^{2}}{a}}}\right ) + \frac{c x}{a^{\frac{3}{2}} \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 = 1.1755, size = 68, normalized size = 1.26 \begin{align*} -\frac{d \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{b^{\frac{3}{2}}} + \frac{{\left (b c - a d\right )} x}{\sqrt{b x^{2} + a} a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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