Optimal. Leaf size=76 \[ \frac{e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} (b d-a e)^{3/2}}-\frac{\sqrt{d+e x}}{(a+b x) (b d-a e)} \]
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Rubi [A] time = 0.0444152, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {27, 51, 63, 208} \[ \frac{e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} (b d-a e)^{3/2}}-\frac{\sqrt{d+e x}}{(a+b x) (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac{1}{(a+b x)^2 \sqrt{d+e x}} \, dx\\ &=-\frac{\sqrt{d+e x}}{(b d-a e) (a+b x)}-\frac{e \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{2 (b d-a e)}\\ &=-\frac{\sqrt{d+e x}}{(b d-a e) (a+b x)}-\frac{\operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{b d-a e}\\ &=-\frac{\sqrt{d+e x}}{(b d-a e) (a+b x)}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} (b d-a e)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0556035, size = 76, normalized size = 1. \[ \frac{e \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{a e-b d}}\right )}{\sqrt{b} (a e-b d)^{3/2}}-\frac{\sqrt{d+e x}}{(a+b x) (b d-a e)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.196, size = 77, normalized size = 1. \begin{align*}{\frac{e}{ \left ( ae-bd \right ) \left ( bxe+ae \right ) }\sqrt{ex+d}}+{\frac{e}{ae-bd}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \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 = 1.99259, size = 603, normalized size = 7.93 \begin{align*} \left [-\frac{\sqrt{b^{2} d - a b e}{\left (b e x + a e\right )} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{b^{2} d - a b e} \sqrt{e x + d}}{b x + a}\right ) + 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{2 \,{\left (a b^{3} d^{2} - 2 \, a^{2} b^{2} d e + a^{3} b e^{2} +{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} x\right )}}, -\frac{\sqrt{-b^{2} d + a b e}{\left (b e x + a e\right )} \arctan \left (\frac{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}{b e x + b d}\right ) +{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{a b^{3} d^{2} - 2 \, a^{2} b^{2} d e + a^{3} b e^{2} +{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x\right )^{2} \sqrt{d + e x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1706, size = 131, normalized size = 1.72 \begin{align*} -\frac{\arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e}{\sqrt{-b^{2} d + a b e}{\left (b d - a e\right )}} - \frac{\sqrt{x e + d} e}{{\left ({\left (x e + d\right )} b - b d + a e\right )}{\left (b d - a e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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