Optimal. Leaf size=101 \[ \frac {\left (b^2-4 a c\right )^3}{1280 c^4 d^{11} (b+2 c x)^{10}}-\frac {3 \left (b^2-4 a c\right )^2}{1024 c^4 d^{11} (b+2 c x)^8}+\frac {b^2-4 a c}{256 c^4 d^{11} (b+2 c x)^6}-\frac {1}{512 c^4 d^{11} (b+2 c x)^4} \]
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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} \[ \frac {\left (b^2-4 a c\right )^3}{1280 c^4 d^{11} (b+2 c x)^{10}}-\frac {3 \left (b^2-4 a c\right )^2}{1024 c^4 d^{11} (b+2 c x)^8}+\frac {b^2-4 a c}{256 c^4 d^{11} (b+2 c x)^6}-\frac {1}{512 c^4 d^{11} (b+2 c x)^4} \]
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)^{11}} \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^3}{64 c^3 d^{11} (b+2 c x)^{11}}+\frac {3 \left (-b^2+4 a c\right )^2}{64 c^3 d^{11} (b+2 c x)^9}+\frac {3 \left (-b^2+4 a c\right )}{64 c^3 d^{11} (b+2 c x)^7}+\frac {1}{64 c^3 d^{11} (b+2 c x)^5}\right ) \, dx\\ &=\frac {\left (b^2-4 a c\right )^3}{1280 c^4 d^{11} (b+2 c x)^{10}}-\frac {3 \left (b^2-4 a c\right )^2}{1024 c^4 d^{11} (b+2 c x)^8}+\frac {b^2-4 a c}{256 c^4 d^{11} (b+2 c x)^6}-\frac {1}{512 c^4 d^{11} (b+2 c x)^4}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 79, normalized size = 0.78 \[ \frac {20 \left (b^2-4 a c\right ) (b+2 c x)^4-15 \left (b^2-4 a c\right )^2 (b+2 c x)^2+4 \left (b^2-4 a c\right )^3-10 (b+2 c x)^6}{5120 c^4 d^{11} (b+2 c x)^{10}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.09, size = 294, normalized size = 2.91 \[ -\frac {640 \, c^{6} x^{6} + 1920 \, b c^{5} x^{5} + b^{6} + 8 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} + 256 \, a^{3} c^{3} + 160 \, {\left (13 \, b^{2} c^{4} + 8 \, a c^{5}\right )} x^{4} + 320 \, {\left (3 \, b^{3} c^{3} + 8 \, a b c^{4}\right )} x^{3} + 60 \, {\left (3 \, b^{4} c^{2} + 24 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 20 \, {\left (b^{5} c + 8 \, a b^{3} c^{2} + 48 \, a^{2} b c^{3}\right )} x}{5120 \, {\left (1024 \, c^{14} d^{11} x^{10} + 5120 \, b c^{13} d^{11} x^{9} + 11520 \, b^{2} c^{12} d^{11} x^{8} + 15360 \, b^{3} c^{11} d^{11} x^{7} + 13440 \, b^{4} c^{10} d^{11} x^{6} + 8064 \, b^{5} c^{9} d^{11} x^{5} + 3360 \, b^{6} c^{8} d^{11} x^{4} + 960 \, b^{7} c^{7} d^{11} x^{3} + 180 \, b^{8} c^{6} d^{11} x^{2} + 20 \, b^{9} c^{5} d^{11} x + b^{10} c^{4} d^{11}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 165, normalized size = 1.63 \[ -\frac {640 \, c^{6} x^{6} + 1920 \, b c^{5} x^{5} + 2080 \, b^{2} c^{4} x^{4} + 1280 \, a c^{5} x^{4} + 960 \, b^{3} c^{3} x^{3} + 2560 \, a b c^{4} x^{3} + 180 \, b^{4} c^{2} x^{2} + 1440 \, a b^{2} c^{3} x^{2} + 960 \, a^{2} c^{4} x^{2} + 20 \, b^{5} c x + 160 \, a b^{3} c^{2} x + 960 \, a^{2} b c^{3} x + b^{6} + 8 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} + 256 \, a^{3} c^{3}}{5120 \, {\left (2 \, c x + b\right )}^{10} c^{4} d^{11}} \]
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 \[ \frac {-\frac {1}{512 \left (2 c x +b \right )^{4} c^{4}}-\frac {48 a^{2} c^{2}-24 a \,b^{2} c +3 b^{4}}{1024 \left (2 c x +b \right )^{8} c^{4}}-\frac {12 a c -3 b^{2}}{768 \left (2 c x +b \right )^{6} c^{4}}-\frac {64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}{1280 \left (2 c x +b \right )^{10} c^{4}}}{d^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.66, size = 294, normalized size = 2.91 \[ -\frac {640 \, c^{6} x^{6} + 1920 \, b c^{5} x^{5} + b^{6} + 8 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} + 256 \, a^{3} c^{3} + 160 \, {\left (13 \, b^{2} c^{4} + 8 \, a c^{5}\right )} x^{4} + 320 \, {\left (3 \, b^{3} c^{3} + 8 \, a b c^{4}\right )} x^{3} + 60 \, {\left (3 \, b^{4} c^{2} + 24 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 20 \, {\left (b^{5} c + 8 \, a b^{3} c^{2} + 48 \, a^{2} b c^{3}\right )} x}{5120 \, {\left (1024 \, c^{14} d^{11} x^{10} + 5120 \, b c^{13} d^{11} x^{9} + 11520 \, b^{2} c^{12} d^{11} x^{8} + 15360 \, b^{3} c^{11} d^{11} x^{7} + 13440 \, b^{4} c^{10} d^{11} x^{6} + 8064 \, b^{5} c^{9} d^{11} x^{5} + 3360 \, b^{6} c^{8} d^{11} x^{4} + 960 \, b^{7} c^{7} d^{11} x^{3} + 180 \, b^{8} c^{6} d^{11} x^{2} + 20 \, b^{9} c^{5} d^{11} x + b^{10} c^{4} d^{11}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.65, size = 282, normalized size = 2.79 \[ -\frac {\frac {256\,a^3\,c^3+48\,a^2\,b^2\,c^2+8\,a\,b^4\,c+b^6}{5120\,c^4}+x^4\,\left (\frac {13\,b^2}{32}+\frac {a\,c}{4}\right )+\frac {c^2\,x^6}{8}+\frac {3\,x^2\,\left (16\,a^2\,c^2+24\,a\,b^2\,c+3\,b^4\right )}{256\,c^2}+\frac {3\,b\,c\,x^5}{8}+\frac {x^3\,\left (3\,b^3+8\,a\,c\,b\right )}{16\,c}+\frac {b\,x\,\left (48\,a^2\,c^2+8\,a\,b^2\,c+b^4\right )}{256\,c^3}}{b^{10}\,d^{11}+20\,b^9\,c\,d^{11}\,x+180\,b^8\,c^2\,d^{11}\,x^2+960\,b^7\,c^3\,d^{11}\,x^3+3360\,b^6\,c^4\,d^{11}\,x^4+8064\,b^5\,c^5\,d^{11}\,x^5+13440\,b^4\,c^6\,d^{11}\,x^6+15360\,b^3\,c^7\,d^{11}\,x^7+11520\,b^2\,c^8\,d^{11}\,x^8+5120\,b\,c^9\,d^{11}\,x^9+1024\,c^{10}\,d^{11}\,x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 18.29, size = 313, normalized size = 3.10 \[ \frac {- 256 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} - 8 a b^{4} c - b^{6} - 1920 b c^{5} x^{5} - 640 c^{6} x^{6} + x^{4} \left (- 1280 a c^{5} - 2080 b^{2} c^{4}\right ) + x^{3} \left (- 2560 a b c^{4} - 960 b^{3} c^{3}\right ) + x^{2} \left (- 960 a^{2} c^{4} - 1440 a b^{2} c^{3} - 180 b^{4} c^{2}\right ) + x \left (- 960 a^{2} b c^{3} - 160 a b^{3} c^{2} - 20 b^{5} c\right )}{5120 b^{10} c^{4} d^{11} + 102400 b^{9} c^{5} d^{11} x + 921600 b^{8} c^{6} d^{11} x^{2} + 4915200 b^{7} c^{7} d^{11} x^{3} + 17203200 b^{6} c^{8} d^{11} x^{4} + 41287680 b^{5} c^{9} d^{11} x^{5} + 68812800 b^{4} c^{10} d^{11} x^{6} + 78643200 b^{3} c^{11} d^{11} x^{7} + 58982400 b^{2} c^{12} d^{11} x^{8} + 26214400 b c^{13} d^{11} x^{9} + 5242880 c^{14} d^{11} x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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