Optimal. Leaf size=58 \[ \frac{e (d+e x)^4}{20 (a+b x)^4 (b d-a e)^2}-\frac{(d+e x)^4}{5 (a+b x)^5 (b d-a e)} \]
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Rubi [A] time = 0.0110566, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {27, 45, 37} \[ \frac{e (d+e x)^4}{20 (a+b x)^4 (b d-a e)^2}-\frac{(d+e x)^4}{5 (a+b x)^5 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{(d+e x)^3}{(a+b x)^6} \, dx\\ &=-\frac{(d+e x)^4}{5 (b d-a e) (a+b x)^5}-\frac{e \int \frac{(d+e x)^3}{(a+b x)^5} \, dx}{5 (b d-a e)}\\ &=-\frac{(d+e x)^4}{5 (b d-a e) (a+b x)^5}+\frac{e (d+e x)^4}{20 (b d-a e)^2 (a+b x)^4}\\ \end{align*}
Mathematica [A] time = 0.0359294, size = 97, normalized size = 1.67 \[ -\frac{a^2 b e^2 (2 d+5 e x)+a^3 e^3+a b^2 e \left (3 d^2+10 d e x+10 e^2 x^2\right )+b^3 \left (15 d^2 e x+4 d^3+20 d e^2 x^2+10 e^3 x^3\right )}{20 b^4 (a+b x)^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 121, normalized size = 2.1 \begin{align*} -{\frac{{e}^{3}}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}-{\frac{-{a}^{3}{e}^{3}+3\,{a}^{2}bd{e}^{2}-3\,a{b}^{2}{d}^{2}e+{b}^{3}{d}^{3}}{5\,{b}^{4} \left ( bx+a \right ) ^{5}}}+{\frac{{e}^{2} \left ( ae-bd \right ) }{{b}^{4} \left ( bx+a \right ) ^{3}}}-{\frac{3\,e \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{4\,{b}^{4} \left ( bx+a \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.16924, size = 216, normalized size = 3.72 \begin{align*} -\frac{10 \, b^{3} e^{3} x^{3} + 4 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e + 2 \, a^{2} b d e^{2} + a^{3} e^{3} + 10 \,{\left (2 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 5 \,{\left (3 \, b^{3} d^{2} e + 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{20 \,{\left (b^{9} x^{5} + 5 \, a b^{8} x^{4} + 10 \, a^{2} b^{7} x^{3} + 10 \, a^{3} b^{6} x^{2} + 5 \, a^{4} b^{5} x + a^{5} b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.80828, size = 328, normalized size = 5.66 \begin{align*} -\frac{10 \, b^{3} e^{3} x^{3} + 4 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e + 2 \, a^{2} b d e^{2} + a^{3} e^{3} + 10 \,{\left (2 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 5 \,{\left (3 \, b^{3} d^{2} e + 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{20 \,{\left (b^{9} x^{5} + 5 \, a b^{8} x^{4} + 10 \, a^{2} b^{7} x^{3} + 10 \, a^{3} b^{6} x^{2} + 5 \, a^{4} b^{5} x + a^{5} b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.68908, size = 170, normalized size = 2.93 \begin{align*} - \frac{a^{3} e^{3} + 2 a^{2} b d e^{2} + 3 a b^{2} d^{2} e + 4 b^{3} d^{3} + 10 b^{3} e^{3} x^{3} + x^{2} \left (10 a b^{2} e^{3} + 20 b^{3} d e^{2}\right ) + x \left (5 a^{2} b e^{3} + 10 a b^{2} d e^{2} + 15 b^{3} d^{2} e\right )}{20 a^{5} b^{4} + 100 a^{4} b^{5} x + 200 a^{3} b^{6} x^{2} + 200 a^{2} b^{7} x^{3} + 100 a b^{8} x^{4} + 20 b^{9} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1305, size = 147, normalized size = 2.53 \begin{align*} -\frac{10 \, b^{3} x^{3} e^{3} + 20 \, b^{3} d x^{2} e^{2} + 15 \, b^{3} d^{2} x e + 4 \, b^{3} d^{3} + 10 \, a b^{2} x^{2} e^{3} + 10 \, a b^{2} d x e^{2} + 3 \, a b^{2} d^{2} e + 5 \, a^{2} b x e^{3} + 2 \, a^{2} b d e^{2} + a^{3} e^{3}}{20 \,{\left (b x + a\right )}^{5} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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