Optimal. Leaf size=69 \[ -\frac{b d-a e}{2 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.0218046, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {640, 607} \[ -\frac{b d-a e}{2 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 640
Rule 607
Rubi steps
\begin{align*} \int \frac{d+e x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=-\frac{e}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (2 b^2 d-2 a b e\right ) \int \frac{1}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx}{2 b^2}\\ &=-\frac{e}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b d-a e}{2 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.014965, size = 39, normalized size = 0.57 \[ \frac{-a e-b (d+2 e x)}{2 b^2 (a+b x) \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.154, size = 32, normalized size = 0.5 \begin{align*} -{\frac{ \left ( bx+a \right ) \left ( 2\,bxe+ae+bd \right ) }{2\,{b}^{2}} \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 [A] time = 1.01296, size = 85, normalized size = 1.23 \begin{align*} -\frac{e}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac{d}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{a e}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}} b{\left (x + \frac{a}{b}\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53212, size = 81, normalized size = 1.17 \begin{align*} -\frac{2 \, b e x + b d + a e}{2 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\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{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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