Optimal. Leaf size=28 \[ \frac{(a+b x)^5}{5 (d+e x)^5 (b d-a e)} \]
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Rubi [A] time = 0.004715, 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} \[ \frac{(a+b x)^5}{5 (d+e x)^5 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 37
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^6} \, dx &=\int \frac{(a+b x)^4}{(d+e x)^6} \, dx\\ &=\frac{(a+b x)^5}{5 (b d-a e) (d+e x)^5}\\ \end{align*}
Mathematica [B] time = 0.0535618, size = 140, normalized size = 5. \[ -\frac{a^2 b^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+a^3 b e^3 (d+5 e x)+a^4 e^4+a b^3 e \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )+b^4 \left (10 d^2 e^2 x^2+5 d^3 e x+d^4+10 d e^3 x^3+5 e^4 x^4\right )}{5 e^5 (d+e x)^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 186, normalized size = 6.6 \begin{align*} -2\,{\frac{{b}^{2} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{b \left ({a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) }{{e}^{5} \left ( ex+d \right ) ^{4}}}-2\,{\frac{{b}^{3} \left ( ae-bd \right ) }{{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{{b}^{4}}{{e}^{5} \left ( ex+d \right ) }}-{\frac{{a}^{4}{e}^{4}-4\,{a}^{3}bd{e}^{3}+6\,{d}^{2}{e}^{2}{b}^{2}{a}^{2}-4\,{d}^{3}ea{b}^{3}+{b}^{4}{d}^{4}}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.16982, 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 (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.92392, 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 (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 6.00661, 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 d^{5} e^{5} + 25 d^{4} e^{6} x + 50 d^{3} e^{7} x^{2} + 50 d^{2} e^{8} x^{3} + 25 d e^{9} x^{4} + 5 e^{10} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.171, size = 230, normalized size = 8.21 \begin{align*} -\frac{{\left (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}\right )} e^{\left (-5\right )}}{5 \,{\left (x e + d\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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