Optimal. Leaf size=117 \[ \frac {4 b^3 (b d-a e)}{5 e^5 (d+e x)^5}-\frac {b^2 (b d-a e)^2}{e^5 (d+e x)^6}+\frac {4 b (b d-a e)^3}{7 e^5 (d+e x)^7}-\frac {(b d-a e)^4}{8 e^5 (d+e x)^8}-\frac {b^4}{4 e^5 (d+e x)^4} \]
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Rubi [A] time = 0.07, antiderivative size = 117, 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 {4 b^3 (b d-a e)}{5 e^5 (d+e x)^5}-\frac {b^2 (b d-a e)^2}{e^5 (d+e x)^6}+\frac {4 b (b d-a e)^3}{7 e^5 (d+e x)^7}-\frac {(b d-a e)^4}{8 e^5 (d+e x)^8}-\frac {b^4}{4 e^5 (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^9} \, dx &=\int \frac {(a+b x)^4}{(d+e x)^9} \, dx\\ &=\int \left (\frac {(-b d+a e)^4}{e^4 (d+e x)^9}-\frac {4 b (b d-a e)^3}{e^4 (d+e x)^8}+\frac {6 b^2 (b d-a e)^2}{e^4 (d+e x)^7}-\frac {4 b^3 (b d-a e)}{e^4 (d+e x)^6}+\frac {b^4}{e^4 (d+e x)^5}\right ) \, dx\\ &=-\frac {(b d-a e)^4}{8 e^5 (d+e x)^8}+\frac {4 b (b d-a e)^3}{7 e^5 (d+e x)^7}-\frac {b^2 (b d-a e)^2}{e^5 (d+e x)^6}+\frac {4 b^3 (b d-a e)}{5 e^5 (d+e x)^5}-\frac {b^4}{4 e^5 (d+e x)^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 144, normalized size = 1.23 \begin {gather*} -\frac {35 a^4 e^4+20 a^3 b e^3 (d+8 e x)+10 a^2 b^2 e^2 \left (d^2+8 d e x+28 e^2 x^2\right )+4 a b^3 e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+b^4 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )}{280 e^5 (d+e x)^8} \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)^9} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.39, size = 258, normalized size = 2.21 \begin {gather*} -\frac {70 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 10 \, a^{2} b^{2} d^{2} e^{2} + 20 \, a^{3} b d e^{3} + 35 \, a^{4} e^{4} + 56 \, {\left (b^{4} d e^{3} + 4 \, a b^{3} e^{4}\right )} x^{3} + 28 \, {\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + 10 \, a^{2} b^{2} e^{4}\right )} x^{2} + 8 \, {\left (b^{4} d^{3} e + 4 \, a b^{3} d^{2} e^{2} + 10 \, a^{2} b^{2} d e^{3} + 20 \, a^{3} b e^{4}\right )} x}{280 \, {\left (e^{13} x^{8} + 8 \, d e^{12} x^{7} + 28 \, d^{2} e^{11} x^{6} + 56 \, d^{3} e^{10} x^{5} + 70 \, d^{4} e^{9} x^{4} + 56 \, d^{5} e^{8} x^{3} + 28 \, d^{6} e^{7} x^{2} + 8 \, d^{7} e^{6} x + d^{8} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 174, normalized size = 1.49 \begin {gather*} -\frac {{\left (70 \, b^{4} x^{4} e^{4} + 56 \, b^{4} d x^{3} e^{3} + 28 \, b^{4} d^{2} x^{2} e^{2} + 8 \, b^{4} d^{3} x e + b^{4} d^{4} + 224 \, a b^{3} x^{3} e^{4} + 112 \, a b^{3} d x^{2} e^{3} + 32 \, a b^{3} d^{2} x e^{2} + 4 \, a b^{3} d^{3} e + 280 \, a^{2} b^{2} x^{2} e^{4} + 80 \, a^{2} b^{2} d x e^{3} + 10 \, a^{2} b^{2} d^{2} e^{2} + 160 \, a^{3} b x e^{4} + 20 \, a^{3} b d e^{3} + 35 \, a^{4} e^{4}\right )} e^{\left (-5\right )}}{280 \, {\left (x e + d\right )}^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 186, normalized size = 1.59 \begin {gather*} -\frac {b^{4}}{4 \left (e x +d \right )^{4} e^{5}}-\frac {4 \left (a e -b d \right ) b^{3}}{5 \left (e x +d \right )^{5} e^{5}}-\frac {\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}}{\left (e x +d \right )^{6} e^{5}}-\frac {4 \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}{7 \left (e x +d \right )^{7} 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}}{8 \left (e x +d \right )^{8} e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.50, size = 258, normalized size = 2.21 \begin {gather*} -\frac {70 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 10 \, a^{2} b^{2} d^{2} e^{2} + 20 \, a^{3} b d e^{3} + 35 \, a^{4} e^{4} + 56 \, {\left (b^{4} d e^{3} + 4 \, a b^{3} e^{4}\right )} x^{3} + 28 \, {\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + 10 \, a^{2} b^{2} e^{4}\right )} x^{2} + 8 \, {\left (b^{4} d^{3} e + 4 \, a b^{3} d^{2} e^{2} + 10 \, a^{2} b^{2} d e^{3} + 20 \, a^{3} b e^{4}\right )} x}{280 \, {\left (e^{13} x^{8} + 8 \, d e^{12} x^{7} + 28 \, d^{2} e^{11} x^{6} + 56 \, d^{3} e^{10} x^{5} + 70 \, d^{4} e^{9} x^{4} + 56 \, d^{5} e^{8} x^{3} + 28 \, d^{6} e^{7} x^{2} + 8 \, d^{7} e^{6} x + d^{8} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.62, size = 248, normalized size = 2.12 \begin {gather*} -\frac {\frac {35\,a^4\,e^4+20\,a^3\,b\,d\,e^3+10\,a^2\,b^2\,d^2\,e^2+4\,a\,b^3\,d^3\,e+b^4\,d^4}{280\,e^5}+\frac {b^4\,x^4}{4\,e}+\frac {b^3\,x^3\,\left (4\,a\,e+b\,d\right )}{5\,e^2}+\frac {b\,x\,\left (20\,a^3\,e^3+10\,a^2\,b\,d\,e^2+4\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{35\,e^4}+\frac {b^2\,x^2\,\left (10\,a^2\,e^2+4\,a\,b\,d\,e+b^2\,d^2\right )}{10\,e^3}}{d^8+8\,d^7\,e\,x+28\,d^6\,e^2\,x^2+56\,d^5\,e^3\,x^3+70\,d^4\,e^4\,x^4+56\,d^3\,e^5\,x^5+28\,d^2\,e^6\,x^6+8\,d\,e^7\,x^7+e^8\,x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 17.05, size = 279, normalized size = 2.38 \begin {gather*} \frac {- 35 a^{4} e^{4} - 20 a^{3} b d e^{3} - 10 a^{2} b^{2} d^{2} e^{2} - 4 a b^{3} d^{3} e - b^{4} d^{4} - 70 b^{4} e^{4} x^{4} + x^{3} \left (- 224 a b^{3} e^{4} - 56 b^{4} d e^{3}\right ) + x^{2} \left (- 280 a^{2} b^{2} e^{4} - 112 a b^{3} d e^{3} - 28 b^{4} d^{2} e^{2}\right ) + x \left (- 160 a^{3} b e^{4} - 80 a^{2} b^{2} d e^{3} - 32 a b^{3} d^{2} e^{2} - 8 b^{4} d^{3} e\right )}{280 d^{8} e^{5} + 2240 d^{7} e^{6} x + 7840 d^{6} e^{7} x^{2} + 15680 d^{5} e^{8} x^{3} + 19600 d^{4} e^{9} x^{4} + 15680 d^{3} e^{10} x^{5} + 7840 d^{2} e^{11} x^{6} + 2240 d e^{12} x^{7} + 280 e^{13} x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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