Optimal. Leaf size=94 \[ -\frac {2 (b d-a e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^2 (a+b x)}+\frac {2 b (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x)} \]
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Rubi [A]
time = 0.02, antiderivative size = 94, 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} (d+e x)^{3/2}}{3 e^2 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)}{e^2 (a+b 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}}{\sqrt {d+e x}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {a b+b^2 x}{\sqrt {d+e x}} \, 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 \sqrt {d+e x}}+\frac {b^2 \sqrt {d+e x}}{e}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {2 (b d-a e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^2 (a+b x)}+\frac {2 b (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 47, normalized size = 0.50 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \sqrt {d+e x} (-2 b d+3 a e+b e x)}{3 e^2 (a+b 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 = 0.48, size = 32, normalized size = 0.34
method | result | size |
default | \(\frac {2 \,\mathrm {csgn}\left (b x +a \right ) \sqrt {e x +d}\, \left (b e x +3 a e -2 b d \right )}{3 e^{2}}\) | \(32\) |
gosper | \(\frac {2 \sqrt {e x +d}\, \left (b e x +3 a e -2 b d \right ) \sqrt {\left (b x +a \right )^{2}}}{3 e^{2} \left (b x +a \right )}\) | \(42\) |
risch | \(\frac {2 \sqrt {e x +d}\, \left (b e x +3 a e -2 b d \right ) \sqrt {\left (b x +a \right )^{2}}}{3 e^{2} \left (b x +a \right )}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 46, normalized size = 0.49 \begin {gather*} \frac {2 \, {\left (b x^{2} e^{2} - 2 \, b d^{2} + 3 \, a d e - {\left (b d e - 3 \, a e^{2}\right )} x\right )} e^{\left (-2\right )}}{3 \, \sqrt {x e + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.05, size = 28, normalized size = 0.30 \begin {gather*} -\frac {2}{3} \, {\left (2 \, b d - {\left (b x + 3 \, a\right )} e\right )} \sqrt {x e + d} e^{\left (-2\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 {\sqrt {\left (a + b x\right )^{2}}}{\sqrt {d + e x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.18, size = 52, normalized size = 0.55 \begin {gather*} \frac {2}{3} \, {\left ({\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} b e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 3 \, \sqrt {x e + d} a \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.75, size = 81, normalized size = 0.86 \begin {gather*} \frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (\frac {2\,x^2}{3}-\frac {4\,b\,d^2-6\,a\,d\,e}{3\,b\,e^2}+\frac {x\,\left (6\,a\,e^2-2\,b\,d\,e\right )}{3\,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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