Optimal. Leaf size=76 \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{\sqrt{b} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x}}{(a+b x) (b c-a d)} \]
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Rubi [A] time = 0.0269979, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {51, 63, 208} \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{\sqrt{b} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x}}{(a+b x) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^2 \sqrt{c+d x}} \, dx &=-\frac{\sqrt{c+d x}}{(b c-a d) (a+b x)}-\frac{d \int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx}{2 (b c-a d)}\\ &=-\frac{\sqrt{c+d x}}{(b c-a d) (a+b x)}-\frac{\operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{b c-a d}\\ &=-\frac{\sqrt{c+d x}}{(b c-a d) (a+b x)}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{\sqrt{b} (b c-a d)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0685772, size = 76, normalized size = 1. \[ \frac{d \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{a d-b c}}\right )}{\sqrt{b} (a d-b c)^{3/2}}-\frac{\sqrt{c+d x}}{(a+b x) (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 77, normalized size = 1. \begin{align*}{\frac{d}{ \left ( ad-bc \right ) \left ( bdx+ad \right ) }\sqrt{dx+c}}+{\frac{d}{ad-bc}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) 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 [B] time = 2.1204, size = 603, normalized size = 7.93 \begin{align*} \left [-\frac{\sqrt{b^{2} c - a b d}{\left (b d x + a d\right )} \log \left (\frac{b d x + 2 \, b c - a d - 2 \, \sqrt{b^{2} c - a b d} \sqrt{d x + c}}{b x + a}\right ) + 2 \,{\left (b^{2} c - a b d\right )} \sqrt{d x + c}}{2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x\right )}}, -\frac{\sqrt{-b^{2} c + a b d}{\left (b d x + a d\right )} \arctan \left (\frac{\sqrt{-b^{2} c + a b d} \sqrt{d x + c}}{b d x + b c}\right ) +{\left (b^{2} c - a b d\right )} \sqrt{d x + c}}{a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [A] time = 1.07563, size = 117, normalized size = 1.54 \begin{align*} -\frac{d \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d}{\left (b c - a d\right )}} - \frac{\sqrt{d x + c} d}{{\left ({\left (d x + c\right )} b - b c + a d\right )}{\left (b c - a d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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