Optimal. Leaf size=94 \[ \frac {2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x) (d+e x)^{3/2}}-\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2}}{e^2 (a+b x) \sqrt {d+e x}} \]
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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 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^2 (a+b x) (d+e x)^{3/2}}-\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2}}{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)^{5/2}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {a b+b^2 x}{(d+e x)^{5/2}} \, 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)^{5/2}}+\frac {b^2}{e (d+e x)^{3/2}}\right ) \, dx}{a b+b^2 x}\\ &=\frac {2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x) (d+e x)^{3/2}}-\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2}}{e^2 (a+b x) \sqrt {d+e 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} (2 b d+a e+3 b e x)}{3 e^2 (a+b x) (d+e x)^{3/2}} \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 | \(-\frac {2 \,\mathrm {csgn}\left (b x +a \right ) \left (3 b e x +a e +2 b d \right )}{3 e^{2} \left (e x +d \right )^{\frac {3}{2}}}\) | \(32\) |
gosper | \(-\frac {2 \left (3 b e x +a e +2 b d \right ) \sqrt {\left (b x +a \right )^{2}}}{3 \left (e x +d \right )^{\frac {3}{2}} e^{2} \left (b x +a \right )}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 36, normalized size = 0.38 \begin {gather*} -\frac {2 \, {\left (3 \, b x e + 2 \, b d + a e\right )}}{3 \, {\left (x e^{3} + d e^{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 = 2.79, size = 45, normalized size = 0.48 \begin {gather*} -\frac {2 \, {\left (2 \, b d + {\left (3 \, b x + a\right )} e\right )} \sqrt {x e + d}}{3 \, {\left (x^{2} e^{4} + 2 \, d x e^{3} + d^{2} e^{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}}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.20, size = 48, normalized size = 0.51 \begin {gather*} -\frac {2 \, {\left (3 \, {\left (x e + d\right )} b \mathrm {sgn}\left (b x + a\right ) - b d \mathrm {sgn}\left (b x + a\right ) + a e \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-2\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.90, size = 95, normalized size = 1.01 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (\frac {2\,x}{e^2}+\frac {2\,a\,e+4\,b\,d}{3\,b\,e^3}\right )}{x^2\,\sqrt {d+e\,x}+\frac {a\,d\,\sqrt {d+e\,x}}{b\,e}+\frac {x\,\left (3\,a\,e^3+3\,b\,d\,e^2\right )\,\sqrt {d+e\,x}}{3\,b\,e^3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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