Optimal. Leaf size=65 \[ -\frac {e (b d-a e)}{2 b^3 (a+b x)^4}-\frac {(b d-a e)^2}{5 b^3 (a+b x)^5}-\frac {e^2}{3 b^3 (a+b x)^3} \]
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Rubi [A] time = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 43} \begin {gather*} -\frac {e (b d-a e)}{2 b^3 (a+b x)^4}-\frac {(b d-a e)^2}{5 b^3 (a+b x)^5}-\frac {e^2}{3 b^3 (a+b x)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(d+e x)^2}{(a+b x)^6} \, dx\\ &=\int \left (\frac {(b d-a e)^2}{b^2 (a+b x)^6}+\frac {2 e (b d-a e)}{b^2 (a+b x)^5}+\frac {e^2}{b^2 (a+b x)^4}\right ) \, dx\\ &=-\frac {(b d-a e)^2}{5 b^3 (a+b x)^5}-\frac {e (b d-a e)}{2 b^3 (a+b x)^4}-\frac {e^2}{3 b^3 (a+b x)^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 57, normalized size = 0.88 \begin {gather*} -\frac {a^2 e^2+a b e (3 d+5 e x)+b^2 \left (6 d^2+15 d e x+10 e^2 x^2\right )}{30 b^3 (a+b x)^5} \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 {(d+e x)^2}{\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 = 0.38, size = 109, normalized size = 1.68 \begin {gather*} -\frac {10 \, b^{2} e^{2} x^{2} + 6 \, b^{2} d^{2} + 3 \, a b d e + a^{2} e^{2} + 5 \, {\left (3 \, b^{2} d e + a b e^{2}\right )} x}{30 \, {\left (b^{8} x^{5} + 5 \, a b^{7} x^{4} + 10 \, a^{2} b^{6} x^{3} + 10 \, a^{3} b^{5} x^{2} + 5 \, a^{4} b^{4} x + a^{5} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 60, normalized size = 0.92 \begin {gather*} -\frac {10 \, b^{2} x^{2} e^{2} + 15 \, b^{2} d x e + 6 \, b^{2} d^{2} + 5 \, a b x e^{2} + 3 \, a b d e + a^{2} e^{2}}{30 \, {\left (b x + a\right )}^{5} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 71, normalized size = 1.09 \begin {gather*} -\frac {e^{2}}{3 \left (b x +a \right )^{3} b^{3}}+\frac {\left (a e -b d \right ) e}{2 \left (b x +a \right )^{4} b^{3}}-\frac {a^{2} e^{2}-2 a b d e +b^{2} d^{2}}{5 \left (b x +a \right )^{5} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.41, size = 109, normalized size = 1.68 \begin {gather*} -\frac {10 \, b^{2} e^{2} x^{2} + 6 \, b^{2} d^{2} + 3 \, a b d e + a^{2} e^{2} + 5 \, {\left (3 \, b^{2} d e + a b e^{2}\right )} x}{30 \, {\left (b^{8} x^{5} + 5 \, a b^{7} x^{4} + 10 \, a^{2} b^{6} x^{3} + 10 \, a^{3} b^{5} x^{2} + 5 \, a^{4} b^{4} x + a^{5} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 107, normalized size = 1.65 \begin {gather*} -\frac {\frac {a^2\,e^2+3\,a\,b\,d\,e+6\,b^2\,d^2}{30\,b^3}+\frac {e^2\,x^2}{3\,b}+\frac {e\,x\,\left (a\,e+3\,b\,d\right )}{6\,b^2}}{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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.97, size = 116, normalized size = 1.78 \begin {gather*} \frac {- a^{2} e^{2} - 3 a b d e - 6 b^{2} d^{2} - 10 b^{2} e^{2} x^{2} + x \left (- 5 a b e^{2} - 15 b^{2} d e\right )}{30 a^{5} b^{3} + 150 a^{4} b^{4} x + 300 a^{3} b^{5} x^{2} + 300 a^{2} b^{6} x^{3} + 150 a b^{7} x^{4} + 30 b^{8} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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