Optimal. Leaf size=38 \[ -\frac {b d-a e}{3 b^2 (a+b x)^3}-\frac {e}{2 b^2 (a+b x)^2} \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {27, 43} \[ -\frac {b d-a e}{3 b^2 (a+b x)^3}-\frac {e}{2 b^2 (a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {d+e x}{(a+b x)^4} \, dx\\ &=\int \left (\frac {b d-a e}{b (a+b x)^4}+\frac {e}{b (a+b x)^3}\right ) \, dx\\ &=-\frac {b d-a e}{3 b^2 (a+b x)^3}-\frac {e}{2 b^2 (a+b x)^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.71 \[ -\frac {a e+2 b d+3 b e x}{6 b^2 (a+b x)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 50, normalized size = 1.32 \[ -\frac {3 \, b e x + 2 \, b d + a e}{6 \, {\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 27, normalized size = 0.71 \[ -\frac {3 \, b x e + 2 \, b d + a e}{6 \, {\left (b x + a\right )}^{3} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 35, normalized size = 0.92 \[ -\frac {e}{2 \left (b x +a \right )^{2} b^{2}}-\frac {-a e +b d}{3 \left (b x +a \right )^{3} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 50, normalized size = 1.32 \[ -\frac {3 \, b e x + 2 \, b d + a e}{6 \, {\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 52, normalized size = 1.37 \[ -\frac {\frac {a\,e+2\,b\,d}{6\,b^2}+\frac {e\,x}{2\,b}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.35, size = 53, normalized size = 1.39 \[ \frac {- a e - 2 b d - 3 b e x}{6 a^{3} b^{2} + 18 a^{2} b^{3} x + 18 a b^{4} x^{2} + 6 b^{5} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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