Optimal. Leaf size=101 \[ \frac {\left (b^2-4 a c\right )^3}{896 c^4 d^8 (b+2 c x)^7}-\frac {3 \left (b^2-4 a c\right )^2}{640 c^4 d^8 (b+2 c x)^5}+\frac {b^2-4 a c}{128 c^4 d^8 (b+2 c x)^3}-\frac {1}{128 c^4 d^8 (b+2 c x)} \]
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Rubi [A] time = 0.08, antiderivative size = 101, 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 )^3}{896 c^4 d^8 (b+2 c x)^7}-\frac {3 \left (b^2-4 a c\right )^2}{640 c^4 d^8 (b+2 c x)^5}+\frac {b^2-4 a c}{128 c^4 d^8 (b+2 c x)^3}-\frac {1}{128 c^4 d^8 (b+2 c x)} \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 )^3}{(b d+2 c d x)^8} \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^3}{64 c^3 d^8 (b+2 c x)^8}+\frac {3 \left (-b^2+4 a c\right )^2}{64 c^3 d^8 (b+2 c x)^6}+\frac {3 \left (-b^2+4 a c\right )}{64 c^3 d^8 (b+2 c x)^4}+\frac {1}{64 c^3 d^8 (b+2 c x)^2}\right ) \, dx\\ &=\frac {\left (b^2-4 a c\right )^3}{896 c^4 d^8 (b+2 c x)^7}-\frac {3 \left (b^2-4 a c\right )^2}{640 c^4 d^8 (b+2 c x)^5}+\frac {b^2-4 a c}{128 c^4 d^8 (b+2 c x)^3}-\frac {1}{128 c^4 d^8 (b+2 c x)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 79, normalized size = 0.78 \begin {gather*} \frac {35 \left (b^2-4 a c\right ) (b+2 c x)^4-21 \left (b^2-4 a c\right )^2 (b+2 c x)^2+5 \left (b^2-4 a c\right )^3-35 (b+2 c x)^6}{4480 c^4 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 )^3}{(b d+2 c d x)^8} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.39, size = 248, normalized size = 2.46 \begin {gather*} -\frac {140 \, c^{6} x^{6} + 420 \, b c^{5} x^{5} + b^{6} + 2 \, a b^{4} c + 6 \, a^{2} b^{2} c^{2} + 20 \, a^{3} c^{3} + 70 \, {\left (7 \, b^{2} c^{4} + 2 \, a c^{5}\right )} x^{4} + 280 \, {\left (b^{3} c^{3} + a b c^{4}\right )} x^{3} + 84 \, {\left (b^{4} c^{2} + 2 \, a b^{2} c^{3} + a^{2} c^{4}\right )} x^{2} + 14 \, {\left (b^{5} c + 2 \, a b^{3} c^{2} + 6 \, a^{2} b c^{3}\right )} x}{280 \, {\left (128 \, c^{11} d^{8} x^{7} + 448 \, b c^{10} d^{8} x^{6} + 672 \, b^{2} c^{9} d^{8} x^{5} + 560 \, b^{3} c^{8} d^{8} x^{4} + 280 \, b^{4} c^{7} d^{8} x^{3} + 84 \, b^{5} c^{6} d^{8} x^{2} + 14 \, b^{6} c^{5} d^{8} x + b^{7} c^{4} d^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 165, normalized size = 1.63 \begin {gather*} -\frac {140 \, c^{6} x^{6} + 420 \, b c^{5} x^{5} + 490 \, b^{2} c^{4} x^{4} + 140 \, a c^{5} x^{4} + 280 \, b^{3} c^{3} x^{3} + 280 \, a b c^{4} x^{3} + 84 \, b^{4} c^{2} x^{2} + 168 \, a b^{2} c^{3} x^{2} + 84 \, a^{2} c^{4} x^{2} + 14 \, b^{5} c x + 28 \, a b^{3} c^{2} x + 84 \, a^{2} b c^{3} x + b^{6} + 2 \, a b^{4} c + 6 \, a^{2} b^{2} c^{2} + 20 \, a^{3} c^{3}}{280 \, {\left (2 \, c x + b\right )}^{7} c^{4} d^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 121, normalized size = 1.20 \begin {gather*} \frac {-\frac {12 a c -3 b^{2}}{384 \left (2 c x +b \right )^{3} c^{4}}-\frac {64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}{896 \left (2 c x +b \right )^{7} c^{4}}-\frac {48 a^{2} c^{2}-24 a \,b^{2} c +3 b^{4}}{640 \left (2 c x +b \right )^{5} c^{4}}-\frac {1}{128 \left (2 c x +b \right ) c^{4}}}{d^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.55, size = 248, normalized size = 2.46 \begin {gather*} -\frac {140 \, c^{6} x^{6} + 420 \, b c^{5} x^{5} + b^{6} + 2 \, a b^{4} c + 6 \, a^{2} b^{2} c^{2} + 20 \, a^{3} c^{3} + 70 \, {\left (7 \, b^{2} c^{4} + 2 \, a c^{5}\right )} x^{4} + 280 \, {\left (b^{3} c^{3} + a b c^{4}\right )} x^{3} + 84 \, {\left (b^{4} c^{2} + 2 \, a b^{2} c^{3} + a^{2} c^{4}\right )} x^{2} + 14 \, {\left (b^{5} c + 2 \, a b^{3} c^{2} + 6 \, a^{2} b c^{3}\right )} x}{280 \, {\left (128 \, c^{11} d^{8} x^{7} + 448 \, b c^{10} d^{8} x^{6} + 672 \, b^{2} c^{9} d^{8} x^{5} + 560 \, b^{3} c^{8} d^{8} x^{4} + 280 \, b^{4} c^{7} d^{8} x^{3} + 84 \, b^{5} c^{6} d^{8} x^{2} + 14 \, b^{6} c^{5} d^{8} x + b^{7} c^{4} d^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.53, size = 233, normalized size = 2.31 \begin {gather*} -\frac {\frac {20\,a^3\,c^3+6\,a^2\,b^2\,c^2+2\,a\,b^4\,c+b^6}{280\,c^4}+x^4\,\left (\frac {7\,b^2}{4}+\frac {a\,c}{2}\right )+\frac {c^2\,x^6}{2}+\frac {x^3\,\left (b^3+a\,c\,b\right )}{c}+\frac {3\,x^2\,\left (a^2\,c^2+2\,a\,b^2\,c+b^4\right )}{10\,c^2}+\frac {3\,b\,c\,x^5}{2}+\frac {b\,x\,\left (6\,a^2\,c^2+2\,a\,b^2\,c+b^4\right )}{20\,c^3}}{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 = 4.63, size = 267, normalized size = 2.64 \begin {gather*} \frac {- 20 a^{3} c^{3} - 6 a^{2} b^{2} c^{2} - 2 a b^{4} c - b^{6} - 420 b c^{5} x^{5} - 140 c^{6} x^{6} + x^{4} \left (- 140 a c^{5} - 490 b^{2} c^{4}\right ) + x^{3} \left (- 280 a b c^{4} - 280 b^{3} c^{3}\right ) + x^{2} \left (- 84 a^{2} c^{4} - 168 a b^{2} c^{3} - 84 b^{4} c^{2}\right ) + x \left (- 84 a^{2} b c^{3} - 28 a b^{3} c^{2} - 14 b^{5} c\right )}{280 b^{7} c^{4} d^{8} + 3920 b^{6} c^{5} d^{8} x + 23520 b^{5} c^{6} d^{8} x^{2} + 78400 b^{4} c^{7} d^{8} x^{3} + 156800 b^{3} c^{8} d^{8} x^{4} + 188160 b^{2} c^{9} d^{8} x^{5} + 125440 b c^{10} d^{8} x^{6} + 35840 c^{11} d^{8} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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