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 = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {(a+b x)^5}{5 (d+e x)^5 (b d-a e)} \end {gather*}
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*}
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Mathematica [B] time = 0.05, size = 140, normalized size = 5.00 \begin {gather*} -\frac {a^4 e^4+a^3 b e^3 (d+5 e x)+a^2 b^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+a b^3 e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+b^4 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )}{5 e^5 (d+e 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 {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^6} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.39, size = 215, normalized size = 7.68 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 170, normalized size = 6.07 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 186, normalized size = 6.64 \begin {gather*} -\frac {b^{4}}{\left (e x +d \right ) e^{5}}-\frac {2 \left (a e -b d \right ) b^{3}}{\left (e x +d \right )^{2} e^{5}}-\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}}{\left (e x +d \right )^{3} e^{5}}-\frac {\left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) b}{\left (e x +d \right )^{4} e^{5}}-\frac {e^{4} a^{4}-4 d \,e^{3} a^{3} b +6 d^{2} e^{2} b^{2} a^{2}-4 d^{3} a \,b^{3} e +b^{4} d^{4}}{5 \left (e x +d \right )^{5} e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.47, size = 215, normalized size = 7.68 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 203, normalized size = 7.25 \begin {gather*} -\frac {\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\,e^5}+\frac {b^4\,x^4}{e}+\frac {2\,b^3\,x^3\,\left (a\,e+b\,d\right )}{e^2}+\frac {b\,x\,\left (a^3\,e^3+a^2\,b\,d\,e^2+a\,b^2\,d^2\,e+b^3\,d^3\right )}{e^4}+\frac {2\,b^2\,x^2\,\left (a^2\,e^2+a\,b\,d\,e+b^2\,d^2\right )}{e^3}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.13, size = 236, normalized size = 8.43 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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