Optimal. Leaf size=38 \[ -\frac {b d-a e}{5 b^2 (a+b x)^5}-\frac {e}{4 b^2 (a+b x)^4} \]
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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} \[ -\frac {b d-a e}{5 b^2 (a+b x)^5}-\frac {e}{4 b^2 (a+b x)^4} \]
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 )^3} \, dx &=\int \frac {d+e x}{(a+b x)^6} \, dx\\ &=\int \left (\frac {b d-a e}{b (a+b x)^6}+\frac {e}{b (a+b x)^5}\right ) \, dx\\ &=-\frac {b d-a e}{5 b^2 (a+b x)^5}-\frac {e}{4 b^2 (a+b x)^4}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.71 \[ -\frac {a e+4 b d+5 b e x}{20 b^2 (a+b x)^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.98, size = 72, normalized size = 1.89 \[ -\frac {5 \, b e x + 4 \, b d + a e}{20 \, {\left (b^{7} x^{5} + 5 \, a b^{6} x^{4} + 10 \, a^{2} b^{5} x^{3} + 10 \, a^{3} b^{4} x^{2} + 5 \, a^{4} b^{3} x + a^{5} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 27, normalized size = 0.71 \[ -\frac {5 \, b x e + 4 \, b d + a e}{20 \, {\left (b x + a\right )}^{5} b^{2}} \]
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 \[ -\frac {e}{4 \left (b x +a \right )^{4} b^{2}}-\frac {-a e +b d}{5 \left (b x +a \right )^{5} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.42, size = 72, normalized size = 1.89 \[ -\frac {5 \, b e x + 4 \, b d + a e}{20 \, {\left (b^{7} x^{5} + 5 \, a b^{6} x^{4} + 10 \, a^{2} b^{5} x^{3} + 10 \, a^{3} b^{4} x^{2} + 5 \, a^{4} b^{3} x + a^{5} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 74, normalized size = 1.95 \[ -\frac {\frac {a\,e+4\,b\,d}{20\,b^2}+\frac {e\,x}{4\,b}}{a^5+5\,a^4\,b\,x+10\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3+5\,a\,b^4\,x^4+b^5\,x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.53, size = 76, normalized size = 2.00 \[ \frac {- a e - 4 b d - 5 b e x}{20 a^{5} b^{2} + 100 a^{4} b^{3} x + 200 a^{3} b^{4} x^{2} + 200 a^{2} b^{5} x^{3} + 100 a b^{6} x^{4} + 20 b^{7} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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