Optimal. Leaf size=92 \[ \frac {2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^2 (a+b x) \sqrt {d+e x}}+\frac {2 b \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^2 (a+b x)} \]
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Rubi [A]
time = 0.03, antiderivative size = 92, normalized size of antiderivative = 1.00, 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 = {660, 45}
\begin {gather*} \frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x}}{e^2 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^2 (a+b x) \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 660
Rubi steps
\begin {align*} \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {a b+b^2 x}{(d+e x)^{3/2}} \, 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)^{3/2}}+\frac {b^2}{e \sqrt {d+e x}}\right ) \, dx}{a b+b^2 x}\\ &=\frac {2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^2 (a+b x) \sqrt {d+e x}}+\frac {2 b \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^2 (a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 48, normalized size = 0.52 \begin {gather*} -\frac {2 \sqrt {(a+b x)^2} (-b d+a e-b (d+e x))}{e^2 (a+b x) \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
2.
time = 0.50, size = 32, normalized size = 0.35
method | result | size |
default | \(-\frac {2 \,\mathrm {csgn}\left (b x +a \right ) \left (-b e x +a e -2 b d \right )}{e^{2} \sqrt {e x +d}}\) | \(32\) |
gosper | \(-\frac {2 \left (-b e x +a e -2 b d \right ) \sqrt {\left (b x +a \right )^{2}}}{\sqrt {e x +d}\, e^{2} \left (b x +a \right )}\) | \(42\) |
risch | \(\frac {2 b \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{e^{2} \left (b x +a \right )}-\frac {2 \left (a e -b d \right ) \sqrt {\left (b x +a \right )^{2}}}{e^{2} \sqrt {e x +d}\, \left (b x +a \right )}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 27, normalized size = 0.29 \begin {gather*} \frac {2 \, {\left (b x e + 2 \, b d - a e\right )} e^{\left (-2\right )}}{\sqrt {x e + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.05, size = 36, normalized size = 0.39 \begin {gather*} \frac {2 \, {\left (2 \, b d + {\left (b x - a\right )} e\right )} \sqrt {x e + d}}{x e^{3} + d e^{2}} \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 {\sqrt {\left (a + b x\right )^{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.70, size = 53, normalized size = 0.58 \begin {gather*} 2 \, \sqrt {x e + d} b e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) + \frac {2 \, {\left (b d \mathrm {sgn}\left (b x + a\right ) - a e \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-2\right )}}{\sqrt {x e + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.85, size = 58, normalized size = 0.63 \begin {gather*} \frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (\frac {2\,x}{e}-\frac {2\,a\,e-4\,b\,d}{b\,e^2}\right )}{x\,\sqrt {d+e\,x}+\frac {a\,\sqrt {d+e\,x}}{b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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