Optimal. Leaf size=28 \[ -\frac{(d+e x)^5}{5 (a+b x)^5 (b d-a e)} \]
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Rubi [A] time = 0.0047047, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {27, 37} \[ -\frac{(d+e x)^5}{5 (a+b x)^5 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 37
Rubi steps
\begin{align*} \int \frac{(d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{(d+e x)^4}{(a+b x)^6} \, dx\\ &=-\frac{(d+e x)^5}{5 (b d-a e) (a+b x)^5}\\ \end{align*}
Mathematica [B] time = 0.048172, size = 140, normalized size = 5. \[ -\frac{a^2 b^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+a^3 b e^3 (d+5 e x)+a^4 e^4+a b^3 e \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )+b^4 \left (10 d^2 e^2 x^2+5 d^3 e x+d^4+10 d e^3 x^3+5 e^4 x^4\right )}{5 b^5 (a+b x)^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 185, normalized size = 6.6 \begin{align*} 2\,{\frac{{e}^{3} \left ( ae-bd \right ) }{{b}^{5} \left ( bx+a \right ) ^{2}}}-{\frac{{e}^{4}{a}^{4}-4\,{a}^{3}bd{e}^{3}+6\,{d}^{2}{e}^{2}{a}^{2}{b}^{2}-4\,{d}^{3}ea{b}^{3}+{d}^{4}{b}^{4}}{5\,{b}^{5} \left ( bx+a \right ) ^{5}}}-2\,{\frac{{e}^{2} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{{b}^{5} \left ( bx+a \right ) ^{3}}}-{\frac{{e}^{4}}{{b}^{5} \left ( bx+a \right ) }}+{\frac{e \left ({a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) }{{b}^{5} \left ( bx+a \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.18791, size = 290, normalized size = 10.36 \begin{align*} -\frac{5 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + a b^{3} d^{3} e + a^{2} b^{2} d^{2} e^{2} + a^{3} b d e^{3} + a^{4} e^{4} + 10 \,{\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + 10 \,{\left (b^{4} d^{2} e^{2} + a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 5 \,{\left (b^{4} d^{3} e + a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x}{5 \,{\left (b^{10} x^{5} + 5 \, a b^{9} x^{4} + 10 \, a^{2} b^{8} x^{3} + 10 \, a^{3} b^{7} x^{2} + 5 \, a^{4} b^{6} x + a^{5} b^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.02882, size = 428, normalized size = 15.29 \begin{align*} -\frac{5 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + a b^{3} d^{3} e + a^{2} b^{2} d^{2} e^{2} + a^{3} b d e^{3} + a^{4} e^{4} + 10 \,{\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + 10 \,{\left (b^{4} d^{2} e^{2} + a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 5 \,{\left (b^{4} d^{3} e + a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x}{5 \,{\left (b^{10} x^{5} + 5 \, a b^{9} x^{4} + 10 \, a^{2} b^{8} x^{3} + 10 \, a^{3} b^{7} x^{2} + 5 \, a^{4} b^{6} x + a^{5} b^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 5.9946, size = 233, normalized size = 8.32 \begin{align*} - \frac{a^{4} e^{4} + a^{3} b d e^{3} + a^{2} b^{2} d^{2} e^{2} + a b^{3} d^{3} e + b^{4} d^{4} + 5 b^{4} e^{4} x^{4} + x^{3} \left (10 a b^{3} e^{4} + 10 b^{4} d e^{3}\right ) + x^{2} \left (10 a^{2} b^{2} e^{4} + 10 a b^{3} d e^{3} + 10 b^{4} d^{2} e^{2}\right ) + x \left (5 a^{3} b e^{4} + 5 a^{2} b^{2} d e^{3} + 5 a b^{3} d^{2} e^{2} + 5 b^{4} d^{3} e\right )}{5 a^{5} b^{5} + 25 a^{4} b^{6} x + 50 a^{3} b^{7} x^{2} + 50 a^{2} b^{8} x^{3} + 25 a b^{9} x^{4} + 5 b^{10} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19766, size = 230, normalized size = 8.21 \begin{align*} -\frac{5 \, b^{4} x^{4} e^{4} + 10 \, b^{4} d x^{3} e^{3} + 10 \, b^{4} d^{2} x^{2} e^{2} + 5 \, b^{4} d^{3} x e + b^{4} d^{4} + 10 \, a b^{3} x^{3} e^{4} + 10 \, a b^{3} d x^{2} e^{3} + 5 \, a b^{3} d^{2} x e^{2} + a b^{3} d^{3} e + 10 \, a^{2} b^{2} x^{2} e^{4} + 5 \, a^{2} b^{2} d x e^{3} + a^{2} b^{2} d^{2} e^{2} + 5 \, a^{3} b x e^{4} + a^{3} b d e^{3} + a^{4} e^{4}}{5 \,{\left (b x + a\right )}^{5} b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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