Optimal. Leaf size=47 \[ \frac{2 a}{b^2 d \left (a+b \sqrt{c+d x}\right )}+\frac{2 \log \left (a+b \sqrt{c+d x}\right )}{b^2 d} \]
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Rubi [A] time = 0.0327996, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {247, 190, 43} \[ \frac{2 a}{b^2 d \left (a+b \sqrt{c+d x}\right )}+\frac{2 \log \left (a+b \sqrt{c+d x}\right )}{b^2 d} \]
Antiderivative was successfully verified.
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Rule 247
Rule 190
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \sqrt{c+d x}\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b \sqrt{x}\right )^2} \, dx,x,c+d x\right )}{d}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{x}{(a+b x)^2} \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^2}+\frac{1}{b (a+b x)}\right ) \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=\frac{2 a}{b^2 d \left (a+b \sqrt{c+d x}\right )}+\frac{2 \log \left (a+b \sqrt{c+d x}\right )}{b^2 d}\\ \end{align*}
Mathematica [A] time = 0.0335396, size = 40, normalized size = 0.85 \[ \frac{2 \left (\frac{a}{a+b \sqrt{c+d x}}+\log \left (a+b \sqrt{c+d x}\right )\right )}{b^2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 142, normalized size = 3. \begin{align*} -2\,{\frac{{a}^{2}}{ \left ({b}^{2}dx+{b}^{2}c-{a}^{2} \right ){b}^{2}d}}+{\frac{\ln \left ({b}^{2}dx+{b}^{2}c-{a}^{2} \right ) }{{b}^{2}d}}+{\frac{a}{{b}^{2}d} \left ( -a+b\sqrt{dx+c} \right ) ^{-1}}-{\frac{1}{{b}^{2}d}\ln \left ( -a+b\sqrt{dx+c} \right ) }+{\frac{a}{{b}^{2}d} \left ( a+b\sqrt{dx+c} \right ) ^{-1}}+{\frac{1}{{b}^{2}d}\ln \left ( a+b\sqrt{dx+c} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01556, size = 58, normalized size = 1.23 \begin{align*} \frac{2 \,{\left (\frac{a}{\sqrt{d x + c} b^{3} + a b^{2}} + \frac{\log \left (\sqrt{d x + c} b + a\right )}{b^{2}}\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71093, size = 154, normalized size = 3.28 \begin{align*} \frac{2 \,{\left (\sqrt{d x + c} a b - a^{2} +{\left (b^{2} d x + b^{2} c - a^{2}\right )} \log \left (\sqrt{d x + c} b + a\right )\right )}}{b^{4} d^{2} x +{\left (b^{4} c - a^{2} b^{2}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.918834, size = 124, normalized size = 2.64 \begin{align*} \begin{cases} \frac{x}{a^{2}} & \text{for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\\frac{x}{\left (a + b \sqrt{c}\right )^{2}} & \text{for}\: d = 0 \\\frac{2 a \log{\left (\frac{a}{b} + \sqrt{c + d x} \right )}}{a b^{2} d + b^{3} d \sqrt{c + d x}} + \frac{2 a}{a b^{2} d + b^{3} d \sqrt{c + d x}} + \frac{2 b \sqrt{c + d x} \log{\left (\frac{a}{b} + \sqrt{c + d x} \right )}}{a b^{2} d + b^{3} d \sqrt{c + d x}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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