Optimal. Leaf size=108 \[ -\frac{\sqrt{d+e x}}{b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{b d-a e}} \]
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Rubi [A] time = 0.0697587, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {768, 646, 63, 208} \[ -\frac{\sqrt{d+e x}}{b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{b d-a e}} \]
Antiderivative was successfully verified.
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Rule 768
Rule 646
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x) \sqrt{d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=-\frac{\sqrt{d+e x}}{b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e \int \frac{1}{\sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}} \, dx}{2 b}\\ &=-\frac{\sqrt{d+e x}}{b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (e \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right ) \sqrt{d+e x}} \, dx}{2 b \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{\sqrt{d+e x}}{b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (a b+b^2 x\right ) \operatorname{Subst}\left (\int \frac{1}{a b-\frac{b^2 d}{e}+\frac{b^2 x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{b \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{\sqrt{d+e x}}{b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2} \sqrt{b d-a e} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0896922, size = 81, normalized size = 0.75 \[ \frac{\frac{e (a+b x) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{a e-b d}}\right )}{\sqrt{a e-b d}}-\sqrt{b} \sqrt{d+e x}}{b^{3/2} \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 108, normalized size = 1. \begin{align*} -{\frac{ \left ( bx+a \right ) ^{2}}{b} \left ( -\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ) xbe-\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ) ae+\sqrt{ex+d}\sqrt{ \left ( ae-bd \right ) b} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )} \sqrt{e x + d}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.02333, size = 498, normalized size = 4.61 \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 - a^{2} b^{2} e +{\left (b^{4} d - a b^{3} e\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 - a^{2} b^{2} e +{\left (b^{4} d - a b^{3} e\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{\left (a + b x\right ) \sqrt{d + e x}}{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17479, size = 173, normalized size = 1.6 \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} b \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} - \frac{\sqrt{x e + d} e}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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