Optimal. Leaf size=126 \[ \frac {28}{3} c d^8 \left (b^2-4 a c\right ) (b+2 c x)^3+28 c d^8 \left (b^2-4 a c\right )^2 (b+2 c x)-28 c d^8 \left (b^2-4 a c\right )^{5/2} \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )-\frac {d^8 (b+2 c x)^7}{a+b x+c x^2}+\frac {28}{5} c d^8 (b+2 c x)^5 \]
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Rubi [A] time = 0.10, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 6, 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*} \frac {28}{3} c d^8 \left (b^2-4 a c\right ) (b+2 c x)^3+28 c d^8 \left (b^2-4 a c\right )^2 (b+2 c x)-28 c d^8 \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 {d^8 (b+2 c x)^7}{a+b x+c x^2}+\frac {28}{5} c d^8 (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)^8}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac {d^8 (b+2 c x)^7}{a+b x+c x^2}+\left (14 c d^2\right ) \int \frac {(b d+2 c d x)^6}{a+b x+c x^2} \, dx\\ &=\frac {28}{5} c d^8 (b+2 c x)^5-\frac {d^8 (b+2 c x)^7}{a+b x+c x^2}+\left (14 c \left (b^2-4 a c\right ) d^4\right ) \int \frac {(b d+2 c d x)^4}{a+b x+c x^2} \, dx\\ &=\frac {28}{3} c \left (b^2-4 a c\right ) d^8 (b+2 c x)^3+\frac {28}{5} c d^8 (b+2 c x)^5-\frac {d^8 (b+2 c x)^7}{a+b x+c x^2}+\left (14 c \left (b^2-4 a c\right )^2 d^6\right ) \int \frac {(b d+2 c d x)^2}{a+b x+c x^2} \, dx\\ &=28 c \left (b^2-4 a c\right )^2 d^8 (b+2 c x)+\frac {28}{3} c \left (b^2-4 a c\right ) d^8 (b+2 c x)^3+\frac {28}{5} c d^8 (b+2 c x)^5-\frac {d^8 (b+2 c x)^7}{a+b x+c x^2}+\left (14 c \left (b^2-4 a c\right )^3 d^8\right ) \int \frac {1}{a+b x+c x^2} \, dx\\ &=28 c \left (b^2-4 a c\right )^2 d^8 (b+2 c x)+\frac {28}{3} c \left (b^2-4 a c\right ) d^8 (b+2 c x)^3+\frac {28}{5} c d^8 (b+2 c x)^5-\frac {d^8 (b+2 c x)^7}{a+b x+c x^2}-\left (28 c \left (b^2-4 a c\right )^3 d^8\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )\\ &=28 c \left (b^2-4 a c\right )^2 d^8 (b+2 c x)+\frac {28}{3} c \left (b^2-4 a c\right ) d^8 (b+2 c x)^3+\frac {28}{5} c d^8 (b+2 c x)^5-\frac {d^8 (b+2 c x)^7}{a+b x+c x^2}-28 c \left (b^2-4 a c\right )^{5/2} d^8 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\\ \end {align*}
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Mathematica [A] time = 0.08, size = 155, normalized size = 1.23 \begin {gather*} d^8 \left (32 c^2 x \left (24 a^2 c^2-16 a b^2 c+3 b^4\right )-\frac {512}{3} c^4 x^3 \left (a c-b^2\right )+128 b c^3 x^2 \left (b^2-2 a c\right )-\frac {\left (b^2-4 a c\right )^3 (b+2 c x)}{a+x (b+c x)}-28 c \left (4 a c-b^2\right )^{5/2} \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )+128 b c^5 x^4+\frac {256 c^6 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)^8}{\left (a+b x+c x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.42, size = 753, normalized size = 5.98 \begin {gather*} \left [\frac {768 \, c^{7} d^{8} x^{7} + 2688 \, b c^{6} d^{8} x^{6} + 896 \, {\left (5 \, b^{2} c^{5} - 2 \, a c^{6}\right )} d^{8} x^{5} + 4480 \, {\left (b^{3} c^{4} - a b c^{5}\right )} d^{8} x^{4} + 1120 \, {\left (3 \, b^{4} c^{3} - 8 \, a b^{2} c^{4} + 8 \, a^{2} c^{5}\right )} d^{8} x^{3} + 480 \, {\left (3 \, b^{5} c^{2} - 12 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{8} x^{2} - 30 \, {\left (b^{6} c - 60 \, a b^{4} c^{2} + 304 \, a^{2} b^{2} c^{3} - 448 \, a^{3} c^{4}\right )} d^{8} x - 15 \, {\left (b^{7} - 12 \, a b^{5} c + 48 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} d^{8} + 210 \, {\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{8} x^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{8} x + {\left (a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3}\right )} d^{8}\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right )}{15 \, {\left (c x^{2} + b x + a\right )}}, \frac {768 \, c^{7} d^{8} x^{7} + 2688 \, b c^{6} d^{8} x^{6} + 896 \, {\left (5 \, b^{2} c^{5} - 2 \, a c^{6}\right )} d^{8} x^{5} + 4480 \, {\left (b^{3} c^{4} - a b c^{5}\right )} d^{8} x^{4} + 1120 \, {\left (3 \, b^{4} c^{3} - 8 \, a b^{2} c^{4} + 8 \, a^{2} c^{5}\right )} d^{8} x^{3} + 480 \, {\left (3 \, b^{5} c^{2} - 12 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{8} x^{2} - 30 \, {\left (b^{6} c - 60 \, a b^{4} c^{2} + 304 \, a^{2} b^{2} c^{3} - 448 \, a^{3} c^{4}\right )} d^{8} x - 15 \, {\left (b^{7} - 12 \, a b^{5} c + 48 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} d^{8} - 420 \, {\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{8} x^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{8} x + {\left (a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3}\right )} d^{8}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right )}{15 \, {\left (c x^{2} + b x + a\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 308, normalized size = 2.44 \begin {gather*} \frac {28 \, {\left (b^{6} c d^{8} - 12 \, a b^{4} c^{2} d^{8} + 48 \, a^{2} b^{2} c^{3} d^{8} - 64 \, a^{3} c^{4} d^{8}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, b^{6} c d^{8} x - 24 \, a b^{4} c^{2} d^{8} x + 96 \, a^{2} b^{2} c^{3} d^{8} x - 128 \, a^{3} c^{4} d^{8} x + b^{7} d^{8} - 12 \, a b^{5} c d^{8} + 48 \, a^{2} b^{3} c^{2} d^{8} - 64 \, a^{3} b c^{3} d^{8}}{c x^{2} + b x + a} + \frac {32 \, {\left (24 \, c^{16} d^{8} x^{5} + 60 \, b c^{15} d^{8} x^{4} + 80 \, b^{2} c^{14} d^{8} x^{3} - 80 \, a c^{15} d^{8} x^{3} + 60 \, b^{3} c^{13} d^{8} x^{2} - 120 \, a b c^{14} d^{8} x^{2} + 45 \, b^{4} c^{12} d^{8} x - 240 \, a b^{2} c^{13} d^{8} x + 360 \, a^{2} c^{14} d^{8} x\right )}}{15 \, c^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 479, normalized size = 3.80 \begin {gather*} \frac {256 c^{6} d^{8} x^{5}}{5}+128 b \,c^{5} d^{8} x^{4}-\frac {512 a \,c^{5} d^{8} x^{3}}{3}+\frac {512 b^{2} c^{4} d^{8} x^{3}}{3}+\frac {128 a^{3} c^{4} d^{8} x}{c \,x^{2}+b x +a}-\frac {1792 a^{3} c^{4} d^{8} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}-\frac {96 a^{2} b^{2} c^{3} d^{8} x}{c \,x^{2}+b x +a}+\frac {1344 a^{2} b^{2} c^{3} d^{8} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}+\frac {24 a \,b^{4} c^{2} d^{8} x}{c \,x^{2}+b x +a}-\frac {336 a \,b^{4} c^{2} d^{8} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}-256 a b \,c^{4} d^{8} x^{2}-\frac {2 b^{6} c \,d^{8} x}{c \,x^{2}+b x +a}+\frac {28 b^{6} c \,d^{8} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}+128 b^{3} c^{3} d^{8} x^{2}+\frac {64 a^{3} b \,c^{3} d^{8}}{c \,x^{2}+b x +a}-\frac {48 a^{2} b^{3} c^{2} d^{8}}{c \,x^{2}+b x +a}+768 a^{2} c^{4} d^{8} x +\frac {12 a \,b^{5} c \,d^{8}}{c \,x^{2}+b x +a}-512 a \,b^{2} c^{3} d^{8} x -\frac {b^{7} d^{8}}{c \,x^{2}+b x +a}+96 b^{4} c^{2} d^{8} 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 = 0.12, size = 508, normalized size = 4.03 \begin {gather*} x^2\,\left (896\,b^3\,c^3\,d^8+\frac {b\,\left (256\,c^4\,d^8\,\left (b^2+2\,a\,c\right )-768\,b^2\,c^4\,d^8\right )}{c}-256\,b\,c^3\,d^8\,\left (b^2+2\,a\,c\right )-256\,a\,b\,c^4\,d^8\right )-\frac {x\,\left (-128\,a^3\,c^4\,d^8+96\,a^2\,b^2\,c^3\,d^8-24\,a\,b^4\,c^2\,d^8+2\,b^6\,c\,d^8\right )+b^7\,d^8-64\,a^3\,b\,c^3\,d^8+48\,a^2\,b^3\,c^2\,d^8-12\,a\,b^5\,c\,d^8}{c\,x^2+b\,x+a}-x^3\,\left (\frac {256\,c^4\,d^8\,\left (b^2+2\,a\,c\right )}{3}-256\,b^2\,c^4\,d^8\right )-x\,\left (\frac {2\,b\,\left (1792\,b^3\,c^3\,d^8+\frac {2\,b\,\left (256\,c^4\,d^8\,\left (b^2+2\,a\,c\right )-768\,b^2\,c^4\,d^8\right )}{c}-512\,b\,c^3\,d^8\,\left (b^2+2\,a\,c\right )-512\,a\,b\,c^4\,d^8\right )}{c}-\frac {\left (256\,c^4\,d^8\,\left (b^2+2\,a\,c\right )-768\,b^2\,c^4\,d^8\right )\,\left (b^2+2\,a\,c\right )}{c^2}+256\,a^2\,c^4\,d^8-1120\,b^4\,c^2\,d^8+1024\,a\,b^2\,c^3\,d^8\right )+\frac {256\,c^6\,d^8\,x^5}{5}+28\,c\,d^8\,\mathrm {atan}\left (\frac {28\,c^2\,d^8\,x\,{\left (4\,a\,c-b^2\right )}^{5/2}+14\,b\,c\,d^8\,{\left (4\,a\,c-b^2\right )}^{5/2}}{-896\,a^3\,c^4\,d^8+672\,a^2\,b^2\,c^3\,d^8-168\,a\,b^4\,c^2\,d^8+14\,b^6\,c\,d^8}\right )\,{\left (4\,a\,c-b^2\right )}^{5/2}+128\,b\,c^5\,d^8\,x^4 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.34, size = 476, normalized size = 3.78 \begin {gather*} 128 b c^{5} d^{8} x^{4} + \frac {256 c^{6} d^{8} x^{5}}{5} + 14 c d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{5}} \log {\left (x + \frac {224 a^{2} b c^{3} d^{8} - 112 a b^{3} c^{2} d^{8} + 14 b^{5} c d^{8} - 14 c d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{5}}}{448 a^{2} c^{4} d^{8} - 224 a b^{2} c^{3} d^{8} + 28 b^{4} c^{2} d^{8}} \right )} - 14 c d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{5}} \log {\left (x + \frac {224 a^{2} b c^{3} d^{8} - 112 a b^{3} c^{2} d^{8} + 14 b^{5} c d^{8} + 14 c d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{5}}}{448 a^{2} c^{4} d^{8} - 224 a b^{2} c^{3} d^{8} + 28 b^{4} c^{2} d^{8}} \right )} + x^{3} \left (- \frac {512 a c^{5} d^{8}}{3} + \frac {512 b^{2} c^{4} d^{8}}{3}\right ) + x^{2} \left (- 256 a b c^{4} d^{8} + 128 b^{3} c^{3} d^{8}\right ) + x \left (768 a^{2} c^{4} d^{8} - 512 a b^{2} c^{3} d^{8} + 96 b^{4} c^{2} d^{8}\right ) + \frac {64 a^{3} b c^{3} d^{8} - 48 a^{2} b^{3} c^{2} d^{8} + 12 a b^{5} c d^{8} - b^{7} d^{8} + x \left (128 a^{3} c^{4} d^{8} - 96 a^{2} b^{2} c^{3} d^{8} + 24 a b^{4} c^{2} d^{8} - 2 b^{6} c d^{8}\right )}{a + b x + c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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