Optimal. Leaf size=41 \[ -\frac{2 (a+b x)}{5 e \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}} \]
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Rubi [A] time = 0.0285665, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {770, 21, 32} \[ -\frac{2 (a+b x)}{5 e \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 770
Rule 21
Rule 32
Rubi steps
\begin{align*} \int \frac{a+b x}{(d+e x)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}} \, dx &=\frac{\left (a b+b^2 x\right ) \int \frac{a+b x}{\left (a b+b^2 x\right ) (d+e x)^{7/2}} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (a b+b^2 x\right ) \int \frac{1}{(d+e x)^{7/2}} \, dx}{b \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{2 (a+b x)}{5 e (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0171365, size = 32, normalized size = 0.78 \[ -\frac{2 (a+b x)}{5 e \sqrt{(a+b x)^2} (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 27, normalized size = 0.7 \begin{align*} -{\frac{2\,bx+2\,a}{5\,e} \left ( ex+d \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21639, size = 57, normalized size = 1.39 \begin{align*} -\frac{2 \, \sqrt{e x + d}}{5 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.969213, size = 89, normalized size = 2.17 \begin{align*} -\frac{2 \, \sqrt{e x + d}}{5 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\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 = 1.13856, size = 24, normalized size = 0.59 \begin{align*} -\frac{2 \, e^{\left (-1\right )} \mathrm{sgn}\left (b x + a\right )}{5 \,{\left (x e + d\right )}^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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