Optimal. Leaf size=71 \[ -\frac{a^2 \sqrt{c+d x}}{c x}-\frac{a (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{c^{3/2}}+\frac{2 b^2 \sqrt{c+d x}}{d} \]
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Rubi [A] time = 0.0395309, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {89, 80, 63, 208} \[ -\frac{a^2 \sqrt{c+d x}}{c x}-\frac{a (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{c^{3/2}}+\frac{2 b^2 \sqrt{c+d x}}{d} \]
Antiderivative was successfully verified.
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Rule 89
Rule 80
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{x^2 \sqrt{c+d x}} \, dx &=-\frac{a^2 \sqrt{c+d x}}{c x}+\frac{\int \frac{\frac{1}{2} a (4 b c-a d)+b^2 c x}{x \sqrt{c+d x}} \, dx}{c}\\ &=\frac{2 b^2 \sqrt{c+d x}}{d}-\frac{a^2 \sqrt{c+d x}}{c x}+\frac{(a (4 b c-a d)) \int \frac{1}{x \sqrt{c+d x}} \, dx}{2 c}\\ &=\frac{2 b^2 \sqrt{c+d x}}{d}-\frac{a^2 \sqrt{c+d x}}{c x}+\frac{(a (4 b c-a d)) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{c d}\\ &=\frac{2 b^2 \sqrt{c+d x}}{d}-\frac{a^2 \sqrt{c+d x}}{c x}-\frac{a (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0496778, size = 65, normalized size = 0.92 \[ \frac{\sqrt{c+d x} \left (2 b^2 c x-a^2 d\right )}{c d x}+\frac{a (a d-4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{c^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 63, normalized size = 0.9 \begin{align*} 2\,{\frac{1}{d} \left ({b}^{2}\sqrt{dx+c}+ad \left ( -1/2\,{\frac{a\sqrt{dx+c}}{cx}}+1/2\,{\frac{ad-4\,bc}{{c}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) } \right ) \right ) } \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 = 2.42041, size = 351, normalized size = 4.94 \begin{align*} \left [-\frac{{\left (4 \, a b c d - a^{2} d^{2}\right )} \sqrt{c} x \log \left (\frac{d x + 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right ) - 2 \,{\left (2 \, b^{2} c^{2} x - a^{2} c d\right )} \sqrt{d x + c}}{2 \, c^{2} d x}, \frac{{\left (4 \, a b c d - a^{2} d^{2}\right )} \sqrt{-c} x \arctan \left (\frac{\sqrt{d x + c} \sqrt{-c}}{c}\right ) +{\left (2 \, b^{2} c^{2} x - a^{2} c d\right )} \sqrt{d x + c}}{c^{2} d x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 55.5798, size = 109, normalized size = 1.54 \begin{align*} - \frac{a^{2} \sqrt{d} \sqrt{\frac{c}{d x} + 1}}{c \sqrt{x}} + \frac{a^{2} d \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} \sqrt{x}} \right )}}{c^{\frac{3}{2}}} + \frac{4 a b \operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{c}} \sqrt{c + d x}} \right )}}{c \sqrt{- \frac{1}{c}}} + b^{2} \left (\begin{cases} \frac{x}{\sqrt{c}} & \text{for}\: d = 0 \\\frac{2 \sqrt{c + d x}}{d} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17414, size = 100, normalized size = 1.41 \begin{align*} \frac{2 \, \sqrt{d x + c} b^{2} - \frac{\sqrt{d x + c} a^{2} d}{c x} + \frac{{\left (4 \, a b c d - a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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