Optimal. Leaf size=69 \[ \frac{2}{\sqrt{d+e x} (b d-a e)}-\frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{3/2}} \]
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Rubi [A] time = 0.0406016, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {27, 51, 63, 208} \[ \frac{2}{\sqrt{d+e x} (b d-a e)}-\frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 27
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac{1}{(a+b x) (d+e x)^{3/2}} \, dx\\ &=\frac{2}{(b d-a e) \sqrt{d+e x}}+\frac{b \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{b d-a e}\\ &=\frac{2}{(b d-a e) \sqrt{d+e x}}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{e (b d-a e)}\\ &=\frac{2}{(b d-a e) \sqrt{d+e x}}-\frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.010785, size = 46, normalized size = 0.67 \[ -\frac{2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right )}{\sqrt{d+e x} (a e-b d)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 68, normalized size = 1. \begin{align*} -2\,{\frac{b}{ \left ( ae-bd \right ) \sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-2\,{\frac{1}{ \left ( ae-bd \right ) \sqrt{ex+d}}} \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 = 1.06135, size = 456, normalized size = 6.61 \begin{align*} \left [-\frac{{\left (e x + d\right )} \sqrt{\frac{b}{b d - a e}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \,{\left (b d - a e\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}}}{b x + a}\right ) - 2 \, \sqrt{e x + d}}{b d^{2} - a d e +{\left (b d e - a e^{2}\right )} x}, -\frac{2 \,{\left ({\left (e x + d\right )} \sqrt{-\frac{b}{b d - a e}} \arctan \left (-\frac{{\left (b d - a e\right )} \sqrt{e x + d} \sqrt{-\frac{b}{b d - a e}}}{b e x + b d}\right ) - \sqrt{e x + d}\right )}}{b d^{2} - a d e +{\left (b d e - a e^{2}\right )} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 42.5096, size = 60, normalized size = 0.87 \begin{align*} - \frac{2}{\sqrt{d + e x} \left (a e - b d\right )} - \frac{2 \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{a e - b d}{b}}} \right )}}{\sqrt{\frac{a e - b d}{b}} \left (a e - b d\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18672, size = 101, normalized size = 1.46 \begin{align*} \frac{2 \, b \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e}{\left (b d - a e\right )}} + \frac{2}{{\left (b d - a e\right )} \sqrt{x e + d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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