Optimal. Leaf size=94 \[ \frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^2 (a+b x) (d+e x)^{3/2}}-\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2}}{e^2 (a+b x) \sqrt{d+e x}} \]
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Rubi [A] time = 0.0365931, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {646, 43} \[ \frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^2 (a+b x) (d+e x)^{3/2}}-\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2}}{e^2 (a+b x) \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 646
Rule 43
Rubi steps
\begin{align*} \int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^{5/2}} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{a b+b^2 x}{(d+e x)^{5/2}} \, dx}{a b+b^2 x}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (-\frac{b (b d-a e)}{e (d+e x)^{5/2}}+\frac{b^2}{e (d+e x)^{3/2}}\right ) \, dx}{a b+b^2 x}\\ &=\frac{2 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x) (d+e x)^{3/2}}-\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2}}{e^2 (a+b x) \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.0232539, size = 47, normalized size = 0.5 \[ -\frac{2 \sqrt{(a+b x)^2} (a e+2 b d+3 b e x)}{3 e^2 (a+b x) (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 42, normalized size = 0.5 \begin{align*} -{\frac{6\,bxe+2\,ae+4\,bd}{3\, \left ( bx+a \right ){e}^{2}}\sqrt{ \left ( bx+a \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19077, size = 47, normalized size = 0.5 \begin{align*} -\frac{2 \,{\left (3 \, b e x + 2 \, b d + a e\right )}}{3 \,{\left (e^{3} x + d e^{2}\right )} \sqrt{e x + d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55682, size = 103, normalized size = 1.1 \begin{align*} -\frac{2 \,{\left (3 \, b e x + 2 \, b d + a e\right )} \sqrt{e x + d}}{3 \,{\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13638, size = 65, normalized size = 0.69 \begin{align*} -\frac{2 \,{\left (3 \,{\left (x e + d\right )} b \mathrm{sgn}\left (b x + a\right ) - b d \mathrm{sgn}\left (b x + a\right ) + a e \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-2\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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