Optimal. Leaf size=92 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{4 e^2 (a+b x) (d+e x)^4}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x) (d+e x)^3} \]
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Rubi [A] time = 0.0413642, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {646, 43} \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{4 e^2 (a+b x) (d+e x)^4}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x) (d+e x)^3} \]
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} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{a b+b^2 x}{(d+e x)^5} \, 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}+\frac{b^2}{e (d+e x)^4}\right ) \, dx}{a b+b^2 x}\\ &=\frac{(b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^2 (a+b x) (d+e x)^4}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x) (d+e x)^3}\\ \end{align*}
Mathematica [A] time = 0.0180968, size = 45, normalized size = 0.49 \[ -\frac{\sqrt{(a+b x)^2} (3 a e+b (d+4 e x))}{12 e^2 (a+b x) (d+e x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 42, normalized size = 0.5 \begin{align*} -{\frac{4\,bxe+3\,ae+bd}{12\,{e}^{2} \left ( ex+d \right ) ^{4} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \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 = 1.49467, size = 128, normalized size = 1.39 \begin{align*} -\frac{4 \, b e x + b d + 3 \, a e}{12 \,{\left (e^{6} x^{4} + 4 \, d e^{5} x^{3} + 6 \, d^{2} e^{4} x^{2} + 4 \, d^{3} e^{3} x + d^{4} e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.653448, size = 65, normalized size = 0.71 \begin{align*} - \frac{3 a e + b d + 4 b e x}{12 d^{4} e^{2} + 48 d^{3} e^{3} x + 72 d^{2} e^{4} x^{2} + 48 d e^{5} x^{3} + 12 e^{6} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16593, size = 61, normalized size = 0.66 \begin{align*} -\frac{{\left (4 \, b x e \mathrm{sgn}\left (b x + a\right ) + b d \mathrm{sgn}\left (b x + a\right ) + 3 \, a e \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-2\right )}}{12 \,{\left (x e + d\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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