Optimal. Leaf size=38 \[ -\frac {b d-a e}{4 b^2 (a+b x)^4}-\frac {e}{3 b^2 (a+b x)^3} \]
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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 = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {27, 43} \begin {gather*} -\frac {b d-a e}{4 b^2 (a+b x)^4}-\frac {e}{3 b^2 (a+b x)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {(a+b x) (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)^5} \, dx\\ &=\int \left (\frac {b d-a e}{b (a+b x)^5}+\frac {e}{b (a+b x)^4}\right ) \, dx\\ &=-\frac {b d-a e}{4 b^2 (a+b x)^4}-\frac {e}{3 b^2 (a+b x)^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.71 \begin {gather*} -\frac {a e+3 b d+4 b e x}{12 b^2 (a+b x)^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) (d+e x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 61, normalized size = 1.61 \begin {gather*} -\frac {4 \, b e x + 3 \, b d + a e}{12 \, {\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 27, normalized size = 0.71 \begin {gather*} -\frac {4 \, b x e + 3 \, b d + a e}{12 \, {\left (b x + a\right )}^{4} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 35, normalized size = 0.92 \begin {gather*} -\frac {e}{3 \left (b x +a \right )^{3} b^{2}}-\frac {-a e +b d}{4 \left (b x +a \right )^{4} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 61, normalized size = 1.61 \begin {gather*} -\frac {4 \, b e x + 3 \, b d + a e}{12 \, {\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.01, size = 63, normalized size = 1.66 \begin {gather*} -\frac {\frac {a\,e+3\,b\,d}{12\,b^2}+\frac {e\,x}{3\,b}}{a^4+4\,a^3\,b\,x+6\,a^2\,b^2\,x^2+4\,a\,b^3\,x^3+b^4\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.46, size = 65, normalized size = 1.71 \begin {gather*} \frac {- a e - 3 b d - 4 b e x}{12 a^{4} b^{2} + 48 a^{3} b^{3} x + 72 a^{2} b^{4} x^{2} + 48 a b^{5} x^{3} + 12 b^{6} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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