Optimal. Leaf size=160 \[ 84 c^2 d^{10} \left (b^2-4 a c\right ) (b+2 c x)^3+252 c^2 d^{10} \left (b^2-4 a c\right )^2 (b+2 c x)-252 c^2 d^{10} \left (b^2-4 a c\right )^{5/2} \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )-\frac {9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}-\frac {d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}+\frac {252}{5} c^2 d^{10} (b+2 c x)^5 \]
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Rubi [A] time = 0.14, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {686, 692, 618, 206} \begin {gather*} 84 c^2 d^{10} \left (b^2-4 a c\right ) (b+2 c x)^3+252 c^2 d^{10} \left (b^2-4 a c\right )^2 (b+2 c x)-252 c^2 d^{10} \left (b^2-4 a c\right )^{5/2} \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )-\frac {9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}-\frac {d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}+\frac {252}{5} c^2 d^{10} (b+2 c x)^5 \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 686
Rule 692
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^{10}}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac {d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}+\left (9 c d^2\right ) \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac {d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}-\frac {9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}+\left (126 c^2 d^4\right ) \int \frac {(b d+2 c d x)^6}{a+b x+c x^2} \, dx\\ &=\frac {252}{5} c^2 d^{10} (b+2 c x)^5-\frac {d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}-\frac {9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}+\left (126 c^2 \left (b^2-4 a c\right ) d^6\right ) \int \frac {(b d+2 c d x)^4}{a+b x+c x^2} \, dx\\ &=84 c^2 \left (b^2-4 a c\right ) d^{10} (b+2 c x)^3+\frac {252}{5} c^2 d^{10} (b+2 c x)^5-\frac {d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}-\frac {9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}+\left (126 c^2 \left (b^2-4 a c\right )^2 d^8\right ) \int \frac {(b d+2 c d x)^2}{a+b x+c x^2} \, dx\\ &=252 c^2 \left (b^2-4 a c\right )^2 d^{10} (b+2 c x)+84 c^2 \left (b^2-4 a c\right ) d^{10} (b+2 c x)^3+\frac {252}{5} c^2 d^{10} (b+2 c x)^5-\frac {d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}-\frac {9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}+\left (126 c^2 \left (b^2-4 a c\right )^3 d^{10}\right ) \int \frac {1}{a+b x+c x^2} \, dx\\ &=252 c^2 \left (b^2-4 a c\right )^2 d^{10} (b+2 c x)+84 c^2 \left (b^2-4 a c\right ) d^{10} (b+2 c x)^3+\frac {252}{5} c^2 d^{10} (b+2 c x)^5-\frac {d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}-\frac {9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}-\left (252 c^2 \left (b^2-4 a c\right )^3 d^{10}\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )\\ &=252 c^2 \left (b^2-4 a c\right )^2 d^{10} (b+2 c x)+84 c^2 \left (b^2-4 a c\right ) d^{10} (b+2 c x)^3+\frac {252}{5} c^2 d^{10} (b+2 c x)^5-\frac {d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}-\frac {9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}-252 c^2 \left (b^2-4 a c\right )^{5/2} d^{10} \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\\ \end {align*}
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Mathematica [A] time = 0.11, size = 192, normalized size = 1.20 \begin {gather*} d^{10} \left (128 c^3 x \left (48 a^2 c^2-30 a b^2 c+5 b^4\right )-256 c^5 x^3 \left (4 a c-3 b^2\right )+128 b c^4 x^2 \left (5 b^2-12 a c\right )-252 c^2 \left (4 a c-b^2\right )^{5/2} \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )+\frac {17 c \left (4 a c-b^2\right )^3 (b+2 c x)}{a+x (b+c x)}-\frac {\left (b^2-4 a c\right )^4 (b+2 c x)}{2 (a+x (b+c x))^2}+512 b c^6 x^4+\frac {1024 c^7 x^5}{5}\right ) \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 {(b d+2 c d x)^{10}}{\left (a+b x+c x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.42, size = 1185, normalized size = 7.41
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 460, normalized size = 2.88 \begin {gather*} \frac {252 \, {\left (b^{6} c^{2} d^{10} - 12 \, a b^{4} c^{3} d^{10} + 48 \, a^{2} b^{2} c^{4} d^{10} - 64 \, a^{3} c^{5} d^{10}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c}} - \frac {68 \, b^{6} c^{3} d^{10} x^{3} - 816 \, a b^{4} c^{4} d^{10} x^{3} + 3264 \, a^{2} b^{2} c^{5} d^{10} x^{3} - 4352 \, a^{3} c^{6} d^{10} x^{3} + 102 \, b^{7} c^{2} d^{10} x^{2} - 1224 \, a b^{5} c^{3} d^{10} x^{2} + 4896 \, a^{2} b^{3} c^{4} d^{10} x^{2} - 6528 \, a^{3} b c^{5} d^{10} x^{2} + 36 \, b^{8} c d^{10} x - 372 \, a b^{6} c^{2} d^{10} x + 1008 \, a^{2} b^{4} c^{3} d^{10} x + 576 \, a^{3} b^{2} c^{4} d^{10} x - 3840 \, a^{4} c^{5} d^{10} x + b^{9} d^{10} + 18 \, a b^{7} c d^{10} - 312 \, a^{2} b^{5} c^{2} d^{10} + 1376 \, a^{3} b^{3} c^{3} d^{10} - 1920 \, a^{4} b c^{4} d^{10}}{2 \, {\left (c x^{2} + b x + a\right )}^{2}} + \frac {128 \, {\left (8 \, c^{22} d^{10} x^{5} + 20 \, b c^{21} d^{10} x^{4} + 30 \, b^{2} c^{20} d^{10} x^{3} - 40 \, a c^{21} d^{10} x^{3} + 25 \, b^{3} c^{19} d^{10} x^{2} - 60 \, a b c^{20} d^{10} x^{2} + 25 \, b^{4} c^{18} d^{10} x - 150 \, a b^{2} c^{19} d^{10} x + 240 \, a^{2} c^{20} d^{10} x\right )}}{5 \, c^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 751, normalized size = 4.69 \begin {gather*} \frac {2176 a^{3} c^{6} d^{10} x^{3}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {1632 a^{2} b^{2} c^{5} d^{10} x^{3}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {408 a \,b^{4} c^{4} d^{10} x^{3}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {34 b^{6} c^{3} d^{10} x^{3}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {1024 c^{7} d^{10} x^{5}}{5}+\frac {3264 a^{3} b \,c^{5} d^{10} x^{2}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {2448 a^{2} b^{3} c^{4} d^{10} x^{2}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {612 a \,b^{5} c^{3} d^{10} x^{2}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {51 b^{7} c^{2} d^{10} x^{2}}{\left (c \,x^{2}+b x +a \right )^{2}}+512 b \,c^{6} d^{10} x^{4}+\frac {1920 a^{4} c^{5} d^{10} x}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {288 a^{3} b^{2} c^{4} d^{10} x}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {504 a^{2} b^{4} c^{3} d^{10} x}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {186 a \,b^{6} c^{2} d^{10} x}{\left (c \,x^{2}+b x +a \right )^{2}}-1024 a \,c^{6} d^{10} x^{3}-\frac {18 b^{8} c \,d^{10} x}{\left (c \,x^{2}+b x +a \right )^{2}}+768 b^{2} c^{5} d^{10} x^{3}+\frac {960 a^{4} b \,c^{4} d^{10}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {688 a^{3} b^{3} c^{3} d^{10}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {16128 a^{3} c^{5} d^{10} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}+\frac {156 a^{2} b^{5} c^{2} d^{10}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {12096 a^{2} b^{2} c^{4} d^{10} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}-\frac {9 a \,b^{7} c \,d^{10}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {3024 a \,b^{4} c^{3} d^{10} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}-1536 a b \,c^{5} d^{10} x^{2}-\frac {b^{9} d^{10}}{2 \left (c \,x^{2}+b x +a \right )^{2}}+\frac {252 b^{6} c^{2} d^{10} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}+640 b^{3} c^{4} d^{10} x^{2}+6144 a^{2} c^{5} d^{10} x -3840 a \,b^{2} c^{4} d^{10} x +640 b^{4} c^{3} d^{10} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.59, size = 695, normalized size = 4.34 \begin {gather*} x\,\left (\frac {3\,b\,\left (1024\,c^4\,d^{10}\,\left (b^3+6\,a\,c\,b\right )-15360\,b^3\,c^4\,d^{10}-\frac {3\,b\,\left (3072\,c^5\,d^{10}\,\left (b^2+a\,c\right )-5376\,b^2\,c^5\,d^{10}\right )}{c}+6144\,b\,c^4\,d^{10}\,\left (b^2+a\,c\right )\right )}{c}+\frac {3\,\left (3072\,c^5\,d^{10}\,\left (b^2+a\,c\right )-5376\,b^2\,c^5\,d^{10}\right )\,\left (b^2+a\,c\right )}{c^2}+13440\,b^4\,c^3\,d^{10}-3072\,a\,c^4\,d^{10}\,\left (b^2+a\,c\right )-2048\,b\,c^3\,d^{10}\,\left (b^3+6\,a\,c\,b\right )\right )-x^2\,\left (512\,c^4\,d^{10}\,\left (b^3+6\,a\,c\,b\right )-7680\,b^3\,c^4\,d^{10}-\frac {3\,b\,\left (3072\,c^5\,d^{10}\,\left (b^2+a\,c\right )-5376\,b^2\,c^5\,d^{10}\right )}{2\,c}+3072\,b\,c^4\,d^{10}\,\left (b^2+a\,c\right )\right )-x^3\,\left (1024\,c^5\,d^{10}\,\left (b^2+a\,c\right )-1792\,b^2\,c^5\,d^{10}\right )-\frac {x^2\,\left (-3264\,a^3\,b\,c^5\,d^{10}+2448\,a^2\,b^3\,c^4\,d^{10}-612\,a\,b^5\,c^3\,d^{10}+51\,b^7\,c^2\,d^{10}\right )-x^3\,\left (2176\,a^3\,c^6\,d^{10}-1632\,a^2\,b^2\,c^5\,d^{10}+408\,a\,b^4\,c^4\,d^{10}-34\,b^6\,c^3\,d^{10}\right )+\frac {b^9\,d^{10}}{2}+x\,\left (-1920\,a^4\,c^5\,d^{10}+288\,a^3\,b^2\,c^4\,d^{10}+504\,a^2\,b^4\,c^3\,d^{10}-186\,a\,b^6\,c^2\,d^{10}+18\,b^8\,c\,d^{10}\right )-960\,a^4\,b\,c^4\,d^{10}-156\,a^2\,b^5\,c^2\,d^{10}+688\,a^3\,b^3\,c^3\,d^{10}+9\,a\,b^7\,c\,d^{10}}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3}+\frac {1024\,c^7\,d^{10}\,x^5}{5}+512\,b\,c^6\,d^{10}\,x^4-252\,c^2\,d^{10}\,\mathrm {atan}\left (\frac {126\,b\,c^2\,d^{10}\,{\left (4\,a\,c-b^2\right )}^{5/2}+252\,c^3\,d^{10}\,x\,{\left (4\,a\,c-b^2\right )}^{5/2}}{8064\,a^3\,c^5\,d^{10}-6048\,a^2\,b^2\,c^4\,d^{10}+1512\,a\,b^4\,c^3\,d^{10}-126\,b^6\,c^2\,d^{10}}\right )\,{\left (4\,a\,c-b^2\right )}^{5/2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 8.18, size = 660, normalized size = 4.12 \begin {gather*} 512 b c^{6} d^{10} x^{4} + \frac {1024 c^{7} d^{10} x^{5}}{5} + 126 c^{2} d^{10} \sqrt {- \left (4 a c - b^{2}\right )^{5}} \log {\left (x + \frac {2016 a^{2} b c^{4} d^{10} - 1008 a b^{3} c^{3} d^{10} + 126 b^{5} c^{2} d^{10} - 126 c^{2} d^{10} \sqrt {- \left (4 a c - b^{2}\right )^{5}}}{4032 a^{2} c^{5} d^{10} - 2016 a b^{2} c^{4} d^{10} + 252 b^{4} c^{3} d^{10}} \right )} - 126 c^{2} d^{10} \sqrt {- \left (4 a c - b^{2}\right )^{5}} \log {\left (x + \frac {2016 a^{2} b c^{4} d^{10} - 1008 a b^{3} c^{3} d^{10} + 126 b^{5} c^{2} d^{10} + 126 c^{2} d^{10} \sqrt {- \left (4 a c - b^{2}\right )^{5}}}{4032 a^{2} c^{5} d^{10} - 2016 a b^{2} c^{4} d^{10} + 252 b^{4} c^{3} d^{10}} \right )} + x^{3} \left (- 1024 a c^{6} d^{10} + 768 b^{2} c^{5} d^{10}\right ) + x^{2} \left (- 1536 a b c^{5} d^{10} + 640 b^{3} c^{4} d^{10}\right ) + x \left (6144 a^{2} c^{5} d^{10} - 3840 a b^{2} c^{4} d^{10} + 640 b^{4} c^{3} d^{10}\right ) + \frac {1920 a^{4} b c^{4} d^{10} - 1376 a^{3} b^{3} c^{3} d^{10} + 312 a^{2} b^{5} c^{2} d^{10} - 18 a b^{7} c d^{10} - b^{9} d^{10} + x^{3} \left (4352 a^{3} c^{6} d^{10} - 3264 a^{2} b^{2} c^{5} d^{10} + 816 a b^{4} c^{4} d^{10} - 68 b^{6} c^{3} d^{10}\right ) + x^{2} \left (6528 a^{3} b c^{5} d^{10} - 4896 a^{2} b^{3} c^{4} d^{10} + 1224 a b^{5} c^{3} d^{10} - 102 b^{7} c^{2} d^{10}\right ) + x \left (3840 a^{4} c^{5} d^{10} - 576 a^{3} b^{2} c^{4} d^{10} - 1008 a^{2} b^{4} c^{3} d^{10} + 372 a b^{6} c^{2} d^{10} - 36 b^{8} c d^{10}\right )}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \left (4 a c + 2 b^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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