Optimal. Leaf size=73 \[ -\frac {\left (b^2-4 a c\right )^2}{224 c^3 d^8 (b+2 c x)^7}+\frac {b^2-4 a c}{80 c^3 d^8 (b+2 c x)^5}-\frac {1}{96 c^3 d^8 (b+2 c x)^3} \]
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Rubi [A] time = 0.06, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {683} \begin {gather*} -\frac {\left (b^2-4 a c\right )^2}{224 c^3 d^8 (b+2 c x)^7}+\frac {b^2-4 a c}{80 c^3 d^8 (b+2 c x)^5}-\frac {1}{96 c^3 d^8 (b+2 c x)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^8} \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^2}{16 c^2 d^8 (b+2 c x)^8}+\frac {-b^2+4 a c}{8 c^2 d^8 (b+2 c x)^6}+\frac {1}{16 c^2 d^8 (b+2 c x)^4}\right ) \, dx\\ &=-\frac {\left (b^2-4 a c\right )^2}{224 c^3 d^8 (b+2 c x)^7}+\frac {b^2-4 a c}{80 c^3 d^8 (b+2 c x)^5}-\frac {1}{96 c^3 d^8 (b+2 c x)^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 59, normalized size = 0.81 \begin {gather*} \frac {42 \left (b^2-4 a c\right ) (b+2 c x)^2-15 \left (b^2-4 a c\right )^2-35 (b+2 c x)^4}{3360 c^3 d^8 (b+2 c x)^7} \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+b x+c x^2\right )^2}{(b d+2 c d x)^8} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.40, size = 176, normalized size = 2.41 \begin {gather*} -\frac {70 \, c^{4} x^{4} + 140 \, b c^{3} x^{3} + b^{4} + 6 \, a b^{2} c + 30 \, a^{2} c^{2} + 84 \, {\left (b^{2} c^{2} + a c^{3}\right )} x^{2} + 14 \, {\left (b^{3} c + 6 \, a b c^{2}\right )} x}{420 \, {\left (128 \, c^{10} d^{8} x^{7} + 448 \, b c^{9} d^{8} x^{6} + 672 \, b^{2} c^{8} d^{8} x^{5} + 560 \, b^{3} c^{7} d^{8} x^{4} + 280 \, b^{4} c^{6} d^{8} x^{3} + 84 \, b^{5} c^{5} d^{8} x^{2} + 14 \, b^{6} c^{4} d^{8} x + b^{7} c^{3} d^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 87, normalized size = 1.19 \begin {gather*} -\frac {70 \, c^{4} x^{4} + 140 \, b c^{3} x^{3} + 84 \, b^{2} c^{2} x^{2} + 84 \, a c^{3} x^{2} + 14 \, b^{3} c x + 84 \, a b c^{2} x + b^{4} + 6 \, a b^{2} c + 30 \, a^{2} c^{2}}{420 \, {\left (2 \, c x + b\right )}^{7} c^{3} d^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 74, normalized size = 1.01 \begin {gather*} \frac {-\frac {1}{96 \left (2 c x +b \right )^{3} c^{3}}-\frac {16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}{224 \left (2 c x +b \right )^{7} c^{3}}-\frac {4 a c -b^{2}}{80 \left (2 c x +b \right )^{5} c^{3}}}{d^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.53, size = 176, normalized size = 2.41 \begin {gather*} -\frac {70 \, c^{4} x^{4} + 140 \, b c^{3} x^{3} + b^{4} + 6 \, a b^{2} c + 30 \, a^{2} c^{2} + 84 \, {\left (b^{2} c^{2} + a c^{3}\right )} x^{2} + 14 \, {\left (b^{3} c + 6 \, a b c^{2}\right )} x}{420 \, {\left (128 \, c^{10} d^{8} x^{7} + 448 \, b c^{9} d^{8} x^{6} + 672 \, b^{2} c^{8} d^{8} x^{5} + 560 \, b^{3} c^{7} d^{8} x^{4} + 280 \, b^{4} c^{6} d^{8} x^{3} + 84 \, b^{5} c^{5} d^{8} x^{2} + 14 \, b^{6} c^{4} d^{8} x + b^{7} c^{3} d^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 168, normalized size = 2.30 \begin {gather*} -\frac {\frac {30\,a^2\,c^2+6\,a\,b^2\,c+b^4}{420\,c^3}+\frac {b\,x^3}{3}+\frac {c\,x^4}{6}+\frac {x^2\,\left (b^2+a\,c\right )}{5\,c}+\frac {b\,x\,\left (b^2+6\,a\,c\right )}{30\,c^2}}{b^7\,d^8+14\,b^6\,c\,d^8\,x+84\,b^5\,c^2\,d^8\,x^2+280\,b^4\,c^3\,d^8\,x^3+560\,b^3\,c^4\,d^8\,x^4+672\,b^2\,c^5\,d^8\,x^5+448\,b\,c^6\,d^8\,x^6+128\,c^7\,d^8\,x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.00, size = 189, normalized size = 2.59 \begin {gather*} \frac {- 30 a^{2} c^{2} - 6 a b^{2} c - b^{4} - 140 b c^{3} x^{3} - 70 c^{4} x^{4} + x^{2} \left (- 84 a c^{3} - 84 b^{2} c^{2}\right ) + x \left (- 84 a b c^{2} - 14 b^{3} c\right )}{420 b^{7} c^{3} d^{8} + 5880 b^{6} c^{4} d^{8} x + 35280 b^{5} c^{5} d^{8} x^{2} + 117600 b^{4} c^{6} d^{8} x^{3} + 235200 b^{3} c^{7} d^{8} x^{4} + 282240 b^{2} c^{8} d^{8} x^{5} + 188160 b c^{9} d^{8} x^{6} + 53760 c^{10} d^{8} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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