Optimal. Leaf size=39 \[ -\frac {2 (a+b x)}{e \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x}} \]
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Rubi [A] time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {2 (a+b x)}{e \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 32
Rule 770
Rubi steps
\begin {align*} \int \frac {a+b x}{(d+e x)^{3/2} \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)^{3/2}} \, 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)^{3/2}} \, dx}{b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {2 (a+b x)}{e \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 0.77 \begin {gather*} -\frac {2 (a+b x)}{e \sqrt {(a+b x)^2} \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 9.59, size = 42, normalized size = 1.08 \begin {gather*} \frac {2 (-a e-b e x)}{e^2 \sqrt {d+e x} \sqrt {\frac {(a e+b e x)^2}{e^2}}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 20, normalized size = 0.51 \begin {gather*} -\frac {2 \, \sqrt {e x + d}}{e^{2} x + d e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 18, normalized size = 0.46 \begin {gather*} -\frac {2 \, e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right )}{\sqrt {x e + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 27, normalized size = 0.69 \begin {gather*} -\frac {2 \left (b x +a \right )}{\sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}\, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.80, size = 20, normalized size = 0.51 \begin {gather*} -\frac {2 \, \sqrt {e x + d}}{e^{2} x + d e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.50, size = 41, normalized size = 1.05 \begin {gather*} -\frac {2\,\sqrt {{\left (a+b\,x\right )}^2}}{b\,e\,\left (x\,\sqrt {d+e\,x}+\frac {a\,\sqrt {d+e\,x}}{b}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b x}{\left (d + e x\right )^{\frac {3}{2}} \sqrt {\left (a + b x\right )^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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