Optimal. Leaf size=46 \[ \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 (d+e x)^2 (b d-a e)} \]
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Rubi [A] time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {646, 37} \begin {gather*} \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 (d+e x)^2 (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 646
Rubi steps
\begin {align*} \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^3} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {a b+b^2 x}{(d+e x)^3} \, dx}{a b+b^2 x}\\ &=\frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 (b d-a e) (d+e x)^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 0.96 \begin {gather*} -\frac {\sqrt {(a+b x)^2} (a e+b (d+2 e x))}{2 e^2 (a+b x) (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.71, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.38, size = 38, normalized size = 0.83 \begin {gather*} -\frac {2 \, b e x + b d + a e}{2 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 44, normalized size = 0.96 \begin {gather*} -\frac {{\left (2 \, b x e \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 )}}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 41, normalized size = 0.89 \begin {gather*} -\frac {\left (2 b e x +a e +b d \right ) \sqrt {\left (b x +a \right )^{2}}}{2 \left (e x +d \right )^{2} \left (b x +a \right ) e^{2}} \end {gather*}
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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mupad [B] time = 0.57, size = 40, normalized size = 0.87 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (a\,e+b\,d+2\,b\,e\,x\right )}{2\,e^2\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.29, size = 39, normalized size = 0.85 \begin {gather*} \frac {- a e - b d - 2 b e x}{2 d^{2} e^{2} + 4 d e^{3} x + 2 e^{4} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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