Optimal. Leaf size=101 \[ \frac {\left (b^2-4 a c\right )^3}{1408 c^4 d^{12} (b+2 c x)^{11}}-\frac {\left (b^2-4 a c\right )^2}{384 c^4 d^{12} (b+2 c x)^9}+\frac {3 \left (b^2-4 a c\right )}{896 c^4 d^{12} (b+2 c x)^7}-\frac {1}{640 c^4 d^{12} (b+2 c x)^5} \]
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Rubi [A]
time = 0.06, 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 = {697}
\begin {gather*} \frac {\left (b^2-4 a c\right )^3}{1408 c^4 d^{12} (b+2 c x)^{11}}-\frac {\left (b^2-4 a c\right )^2}{384 c^4 d^{12} (b+2 c x)^9}+\frac {3 \left (b^2-4 a c\right )}{896 c^4 d^{12} (b+2 c x)^7}-\frac {1}{640 c^4 d^{12} (b+2 c x)^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{12}} \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^3}{64 c^3 d^{12} (b+2 c x)^{12}}+\frac {3 \left (-b^2+4 a c\right )^2}{64 c^3 d^{12} (b+2 c x)^{10}}+\frac {3 \left (-b^2+4 a c\right )}{64 c^3 d^{12} (b+2 c x)^8}+\frac {1}{64 c^3 d^{12} (b+2 c x)^6}\right ) \, dx\\ &=\frac {\left (b^2-4 a c\right )^3}{1408 c^4 d^{12} (b+2 c x)^{11}}-\frac {\left (b^2-4 a c\right )^2}{384 c^4 d^{12} (b+2 c x)^9}+\frac {3 \left (b^2-4 a c\right )}{896 c^4 d^{12} (b+2 c x)^7}-\frac {1}{640 c^4 d^{12} (b+2 c x)^5}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 79, normalized size = 0.78 \begin {gather*} \frac {105 \left (b^2-4 a c\right )^3-385 \left (b^2-4 a c\right )^2 (b+2 c x)^2+495 \left (b^2-4 a c\right ) (b+2 c x)^4-231 (b+2 c x)^6}{147840 c^4 d^{12} (b+2 c x)^{11}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.64, size = 121, normalized size = 1.20
method | result | size |
default | \(\frac {-\frac {48 a^{2} c^{2}-24 a c \,b^{2}+3 b^{4}}{1152 c^{4} \left (2 c x +b \right )^{9}}-\frac {12 a c -3 b^{2}}{896 c^{4} \left (2 c x +b \right )^{7}}-\frac {1}{640 c^{4} \left (2 c x +b \right )^{5}}-\frac {64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}{1408 c^{4} \left (2 c x +b \right )^{11}}}{d^{12}}\) | \(121\) |
risch | \(\frac {-\frac {c^{2} x^{6}}{10}-\frac {3 b c \,x^{5}}{10}+\left (-\frac {3 a c}{14}-\frac {9 b^{2}}{28}\right ) x^{4}-\frac {b \left (3 a c +b^{2}\right ) x^{3}}{7 c}-\frac {\left (7 a^{2} c^{2}+10 a c \,b^{2}+b^{4}\right ) x^{2}}{42 c^{2}}-\frac {b \left (70 a^{2} c^{2}+10 a c \,b^{2}+b^{4}\right ) x}{420 c^{3}}-\frac {420 a^{3} c^{3}+70 a^{2} b^{2} c^{2}+10 a \,b^{4} c +b^{6}}{9240 c^{4}}}{d^{12} \left (2 c x +b \right )^{11}}\) | \(148\) |
gosper | \(-\frac {924 c^{6} x^{6}+2772 b \,c^{5} x^{5}+1980 a \,c^{5} x^{4}+2970 b^{2} c^{4} x^{4}+3960 a b \,c^{4} x^{3}+1320 b^{3} c^{3} x^{3}+1540 a^{2} c^{4} x^{2}+2200 a \,b^{2} c^{3} x^{2}+220 b^{4} c^{2} x^{2}+1540 a^{2} b \,c^{3} x +220 a \,b^{3} c^{2} x +22 b^{5} c x +420 a^{3} c^{3}+70 a^{2} b^{2} c^{2}+10 a \,b^{4} c +b^{6}}{9240 \left (2 c x +b \right )^{11} d^{12} c^{4}}\) | \(166\) |
norman | \(\frac {\frac {a^{3} x}{b d}+\frac {\left (60 c^{2} a^{3}+10 c \,a^{2} b^{2}+a \,b^{4}\right ) x^{3}}{b^{3} d}+\frac {\left (20 c \,a^{3}+3 a^{2} b^{2}\right ) x^{2}}{2 b^{2} d}+\frac {\left (960 a^{3} c^{3}+160 a^{2} b^{2} c^{2}+22 a \,b^{4} c +b^{6}\right ) x^{4}}{4 b^{4} d}+\frac {128 c^{6} \left (420 a^{3} c^{3}+70 a^{2} b^{2} c^{2}+10 a \,b^{4} c +b^{6}\right ) x^{10}}{105 b^{10} d}+\frac {64 c^{5} \left (420 a^{3} c^{3}+70 a^{2} b^{2} c^{2}+10 a \,b^{4} c +b^{6}\right ) x^{9}}{21 b^{9} d}+\frac {32 c^{4} \left (420 a^{3} c^{3}+70 a^{2} b^{2} c^{2}+10 a \,b^{4} c +b^{6}\right ) x^{8}}{7 b^{8} d}+\frac {32 c^{3} \left (420 a^{3} c^{3}+70 a^{2} b^{2} c^{2}+10 a \,b^{4} c +b^{6}\right ) x^{7}}{7 b^{7} d}+\frac {c \left (6720 a^{3} c^{3}+1120 a^{2} b^{2} c^{2}+160 a \,b^{4} c +13 b^{6}\right ) x^{5}}{10 b^{5} d}+\frac {c^{2} \left (13440 a^{3} c^{3}+2240 a^{2} b^{2} c^{2}+320 a \,b^{4} c +31 b^{6}\right ) x^{6}}{10 b^{6} d}+\frac {256 c^{7} \left (420 a^{3} c^{3}+70 a^{2} b^{2} c^{2}+10 a \,b^{4} c +b^{6}\right ) x^{11}}{1155 b^{11} d}}{d^{11} \left (2 c x +b \right )^{11}}\) | \(435\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 306 vs.
\(2 (93) = 186\).
time = 0.31, size = 306, normalized size = 3.03 \begin {gather*} -\frac {924 \, c^{6} x^{6} + 2772 \, b c^{5} x^{5} + b^{6} + 10 \, a b^{4} c + 70 \, a^{2} b^{2} c^{2} + 420 \, a^{3} c^{3} + 990 \, {\left (3 \, b^{2} c^{4} + 2 \, a c^{5}\right )} x^{4} + 1320 \, {\left (b^{3} c^{3} + 3 \, a b c^{4}\right )} x^{3} + 220 \, {\left (b^{4} c^{2} + 10 \, a b^{2} c^{3} + 7 \, a^{2} c^{4}\right )} x^{2} + 22 \, {\left (b^{5} c + 10 \, a b^{3} c^{2} + 70 \, a^{2} b c^{3}\right )} x}{9240 \, {\left (2048 \, c^{15} d^{12} x^{11} + 11264 \, b c^{14} d^{12} x^{10} + 28160 \, b^{2} c^{13} d^{12} x^{9} + 42240 \, b^{3} c^{12} d^{12} x^{8} + 42240 \, b^{4} c^{11} d^{12} x^{7} + 29568 \, b^{5} c^{10} d^{12} x^{6} + 14784 \, b^{6} c^{9} d^{12} x^{5} + 5280 \, b^{7} c^{8} d^{12} x^{4} + 1320 \, b^{8} c^{7} d^{12} x^{3} + 220 \, b^{9} c^{6} d^{12} x^{2} + 22 \, b^{10} c^{5} d^{12} x + b^{11} c^{4} d^{12}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 306 vs.
\(2 (93) = 186\).
time = 1.83, size = 306, normalized size = 3.03 \begin {gather*} -\frac {924 \, c^{6} x^{6} + 2772 \, b c^{5} x^{5} + b^{6} + 10 \, a b^{4} c + 70 \, a^{2} b^{2} c^{2} + 420 \, a^{3} c^{3} + 990 \, {\left (3 \, b^{2} c^{4} + 2 \, a c^{5}\right )} x^{4} + 1320 \, {\left (b^{3} c^{3} + 3 \, a b c^{4}\right )} x^{3} + 220 \, {\left (b^{4} c^{2} + 10 \, a b^{2} c^{3} + 7 \, a^{2} c^{4}\right )} x^{2} + 22 \, {\left (b^{5} c + 10 \, a b^{3} c^{2} + 70 \, a^{2} b c^{3}\right )} x}{9240 \, {\left (2048 \, c^{15} d^{12} x^{11} + 11264 \, b c^{14} d^{12} x^{10} + 28160 \, b^{2} c^{13} d^{12} x^{9} + 42240 \, b^{3} c^{12} d^{12} x^{8} + 42240 \, b^{4} c^{11} d^{12} x^{7} + 29568 \, b^{5} c^{10} d^{12} x^{6} + 14784 \, b^{6} c^{9} d^{12} x^{5} + 5280 \, b^{7} c^{8} d^{12} x^{4} + 1320 \, b^{8} c^{7} d^{12} x^{3} + 220 \, b^{9} c^{6} d^{12} x^{2} + 22 \, b^{10} c^{5} d^{12} x + b^{11} c^{4} d^{12}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 328 vs.
\(2 (100) = 200\).
time = 9.60, size = 328, normalized size = 3.25 \begin {gather*} \frac {- 420 a^{3} c^{3} - 70 a^{2} b^{2} c^{2} - 10 a b^{4} c - b^{6} - 2772 b c^{5} x^{5} - 924 c^{6} x^{6} + x^{4} \left (- 1980 a c^{5} - 2970 b^{2} c^{4}\right ) + x^{3} \left (- 3960 a b c^{4} - 1320 b^{3} c^{3}\right ) + x^{2} \left (- 1540 a^{2} c^{4} - 2200 a b^{2} c^{3} - 220 b^{4} c^{2}\right ) + x \left (- 1540 a^{2} b c^{3} - 220 a b^{3} c^{2} - 22 b^{5} c\right )}{9240 b^{11} c^{4} d^{12} + 203280 b^{10} c^{5} d^{12} x + 2032800 b^{9} c^{6} d^{12} x^{2} + 12196800 b^{8} c^{7} d^{12} x^{3} + 48787200 b^{7} c^{8} d^{12} x^{4} + 136604160 b^{6} c^{9} d^{12} x^{5} + 273208320 b^{5} c^{10} d^{12} x^{6} + 390297600 b^{4} c^{11} d^{12} x^{7} + 390297600 b^{3} c^{12} d^{12} x^{8} + 260198400 b^{2} c^{13} d^{12} x^{9} + 104079360 b c^{14} d^{12} x^{10} + 18923520 c^{15} d^{12} x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.57, size = 165, normalized size = 1.63 \begin {gather*} -\frac {924 \, c^{6} x^{6} + 2772 \, b c^{5} x^{5} + 2970 \, b^{2} c^{4} x^{4} + 1980 \, a c^{5} x^{4} + 1320 \, b^{3} c^{3} x^{3} + 3960 \, a b c^{4} x^{3} + 220 \, b^{4} c^{2} x^{2} + 2200 \, a b^{2} c^{3} x^{2} + 1540 \, a^{2} c^{4} x^{2} + 22 \, b^{5} c x + 220 \, a b^{3} c^{2} x + 1540 \, a^{2} b c^{3} x + b^{6} + 10 \, a b^{4} c + 70 \, a^{2} b^{2} c^{2} + 420 \, a^{3} c^{3}}{9240 \, {\left (2 \, c x + b\right )}^{11} c^{4} d^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.78, size = 292, normalized size = 2.89 \begin {gather*} -\frac {\frac {420\,a^3\,c^3+70\,a^2\,b^2\,c^2+10\,a\,b^4\,c+b^6}{9240\,c^4}+x^4\,\left (\frac {9\,b^2}{28}+\frac {3\,a\,c}{14}\right )+\frac {c^2\,x^6}{10}+\frac {x^3\,\left (b^3+3\,a\,c\,b\right )}{7\,c}+\frac {x^2\,\left (7\,a^2\,c^2+10\,a\,b^2\,c+b^4\right )}{42\,c^2}+\frac {3\,b\,c\,x^5}{10}+\frac {b\,x\,\left (70\,a^2\,c^2+10\,a\,b^2\,c+b^4\right )}{420\,c^3}}{b^{11}\,d^{12}+22\,b^{10}\,c\,d^{12}\,x+220\,b^9\,c^2\,d^{12}\,x^2+1320\,b^8\,c^3\,d^{12}\,x^3+5280\,b^7\,c^4\,d^{12}\,x^4+14784\,b^6\,c^5\,d^{12}\,x^5+29568\,b^5\,c^6\,d^{12}\,x^6+42240\,b^4\,c^7\,d^{12}\,x^7+42240\,b^3\,c^8\,d^{12}\,x^8+28160\,b^2\,c^9\,d^{12}\,x^9+11264\,b\,c^{10}\,d^{12}\,x^{10}+2048\,c^{11}\,d^{12}\,x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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