Optimal. Leaf size=69 \[ \frac{2}{\sqrt{c+d x} (b c-a d)}-\frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{3/2}} \]
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Rubi [A] time = 0.0273585, antiderivative size = 69, 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} \[ \frac{2}{\sqrt{c+d x} (b c-a d)}-\frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(a+b x) (c+d x)^{3/2}} \, dx &=\frac{2}{(b c-a d) \sqrt{c+d x}}+\frac{b \int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx}{b c-a d}\\ &=\frac{2}{(b c-a d) \sqrt{c+d x}}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{d (b c-a d)}\\ &=\frac{2}{(b c-a d) \sqrt{c+d x}}-\frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.009799, size = 46, normalized size = 0.67 \[ -\frac{2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{b (c+d x)}{b c-a d}\right )}{\sqrt{c+d x} (a d-b c)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 68, normalized size = 1. \begin{align*} -2\,{\frac{b}{ \left ( ad-bc \right ) \sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-2\,{\frac{1}{ \left ( ad-bc \right ) \sqrt{dx+c}}} \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 = 2.10208, size = 456, normalized size = 6.61 \begin{align*} \left [-\frac{{\left (d x + c\right )} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b d x + 2 \, b c - a d + 2 \,{\left (b c - a d\right )} \sqrt{d x + c} \sqrt{\frac{b}{b c - a d}}}{b x + a}\right ) - 2 \, \sqrt{d x + c}}{b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x}, -\frac{2 \,{\left ({\left (d x + c\right )} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{{\left (b c - a d\right )} \sqrt{d x + c} \sqrt{-\frac{b}{b c - a d}}}{b d x + b c}\right ) - \sqrt{d x + c}\right )}}{b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.06421, size = 60, normalized size = 0.87 \begin{align*} - \frac{2}{\sqrt{c + d x} \left (a d - b c\right )} - \frac{2 \operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{a d - b c}{b}}} \right )}}{\sqrt{\frac{a d - b c}{b}} \left (a d - b c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06208, size = 93, normalized size = 1.35 \begin{align*} \frac{2 \, b \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{2}{{\left (b c - a d\right )} \sqrt{d x + c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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