Optimal. Leaf size=92 \[ \frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x}}{e^2 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^2 (a+b x) \sqrt {d+e x}} \]
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Rubi [A] time = 0.04, 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 = {646, 43} \[ \frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x}}{e^2 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^2 (a+b x) \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 646
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 = 0.02, size = 45, normalized size = 0.49 \[ \frac {2 \sqrt {(a+b x)^2} (-a e+2 b d+b e x)}{e^2 (a+b x) \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 35, normalized size = 0.38 \[ \frac {2 \, {\left (b e x + 2 \, b d - a e\right )} \sqrt {e x + d}}{e^{3} x + d e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 53, normalized size = 0.58 \[ 2 \, \sqrt {x e + d} b e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) + \frac {2 \, {\left (b d \mathrm {sgn}\left (b x + a\right ) - a e \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-2\right )}}{\sqrt {x e + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 42, normalized size = 0.46 \[ -\frac {2 \left (-b e x +a e -2 b d \right ) \sqrt {\left (b x +a \right )^{2}}}{\sqrt {e x +d}\, \left (b x +a \right ) e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 25, normalized size = 0.27 \[ \frac {2 \, {\left (b e x + 2 \, b d - a e\right )}}{\sqrt {e x + d} e^{2}} \]
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 \[ \frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (\frac {2\,x}{e}-\frac {2\,a\,e-4\,b\,d}{b\,e^2}\right )}{x\,\sqrt {d+e\,x}+\frac {a\,\sqrt {d+e\,x}}{b}} \]
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 \[ \int \frac {\sqrt {\left (a + b x\right )^{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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