Optimal. Leaf size=92 \[ \frac {(b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x) (d+e x)^3}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^2 (a+b x) (d+e x)^2} \]
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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 = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 45}
\begin {gather*} \frac {\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}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^2 (a+b x) (d+e x)^2} \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)^4} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {a b+b^2 x}{(d+e x)^4} \, 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)^4}+\frac {b^2}{e (d+e x)^3}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(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}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^2 (a+b x) (d+e x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 45, normalized size = 0.49 \begin {gather*} -\frac {\sqrt {(a+b x)^2} (2 a e+b (d+3 e x))}{6 e^2 (a+b x) (d+e x)^3} \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 = 0.45, size = 32, normalized size = 0.35
method | result | size |
default | \(-\frac {\mathrm {csgn}\left (b x +a \right ) \left (3 b e x +2 a e +b d \right )}{6 e^{2} \left (e x +d \right )^{3}}\) | \(32\) |
gosper | \(-\frac {\left (3 b e x +2 a e +b d \right ) \sqrt {\left (b x +a \right )^{2}}}{6 e^{2} \left (e x +d \right )^{3} \left (b x +a \right )}\) | \(42\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b x}{2 e}-\frac {2 a e +b d}{6 e^{2}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{3}}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.71, size = 48, normalized size = 0.52 \begin {gather*} -\frac {b d + {\left (3 \, b x + 2 \, a\right )} e}{6 \, {\left (x^{3} e^{5} + 3 \, d x^{2} e^{4} + 3 \, d^{2} x e^{3} + d^{3} e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.17, size = 53, normalized size = 0.58 \begin {gather*} \frac {- 2 a e - b d - 3 b e x}{6 d^{3} e^{2} + 18 d^{2} e^{3} x + 18 d e^{4} x^{2} + 6 e^{5} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.01, size = 45, normalized size = 0.49 \begin {gather*} -\frac {{\left (3 \, b x e \mathrm {sgn}\left (b x + a\right ) + b d \mathrm {sgn}\left (b x + a\right ) + 2 \, a e \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-2\right )}}{6 \, {\left (x e + d\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.57, size = 41, normalized size = 0.45 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (2\,a\,e+b\,d+3\,b\,e\,x\right )}{6\,e^2\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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