Optimal. Leaf size=122 \[ \frac {2}{5} d^8 \left (b^2-4 a c\right ) (b+2 c x)^5+\frac {2}{3} d^8 \left (b^2-4 a c\right )^2 (b+2 c x)^3+2 d^8 \left (b^2-4 a c\right )^3 (b+2 c x)-2 d^8 \left (b^2-4 a c\right )^{7/2} \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )+\frac {2}{7} d^8 (b+2 c x)^7 \]
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Rubi [A] time = 0.15, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {692, 618, 206} \begin {gather*} \frac {2}{5} d^8 \left (b^2-4 a c\right ) (b+2 c x)^5+\frac {2}{3} d^8 \left (b^2-4 a c\right )^2 (b+2 c x)^3+2 d^8 \left (b^2-4 a c\right )^3 (b+2 c x)-2 d^8 \left (b^2-4 a c\right )^{7/2} \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )+\frac {2}{7} d^8 (b+2 c x)^7 \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 692
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^8}{a+b x+c x^2} \, dx &=\frac {2}{7} d^8 (b+2 c x)^7+\left (\left (b^2-4 a c\right ) d^2\right ) \int \frac {(b d+2 c d x)^6}{a+b x+c x^2} \, dx\\ &=\frac {2}{5} \left (b^2-4 a c\right ) d^8 (b+2 c x)^5+\frac {2}{7} d^8 (b+2 c x)^7+\left (\left (b^2-4 a c\right )^2 d^4\right ) \int \frac {(b d+2 c d x)^4}{a+b x+c x^2} \, dx\\ &=\frac {2}{3} \left (b^2-4 a c\right )^2 d^8 (b+2 c x)^3+\frac {2}{5} \left (b^2-4 a c\right ) d^8 (b+2 c x)^5+\frac {2}{7} d^8 (b+2 c x)^7+\left (\left (b^2-4 a c\right )^3 d^6\right ) \int \frac {(b d+2 c d x)^2}{a+b x+c x^2} \, dx\\ &=2 \left (b^2-4 a c\right )^3 d^8 (b+2 c x)+\frac {2}{3} \left (b^2-4 a c\right )^2 d^8 (b+2 c x)^3+\frac {2}{5} \left (b^2-4 a c\right ) d^8 (b+2 c x)^5+\frac {2}{7} d^8 (b+2 c x)^7+\left (\left (b^2-4 a c\right )^4 d^8\right ) \int \frac {1}{a+b x+c x^2} \, dx\\ &=2 \left (b^2-4 a c\right )^3 d^8 (b+2 c x)+\frac {2}{3} \left (b^2-4 a c\right )^2 d^8 (b+2 c x)^3+\frac {2}{5} \left (b^2-4 a c\right ) d^8 (b+2 c x)^5+\frac {2}{7} d^8 (b+2 c x)^7-\left (2 \left (b^2-4 a c\right )^4 d^8\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )\\ &=2 \left (b^2-4 a c\right )^3 d^8 (b+2 c x)+\frac {2}{3} \left (b^2-4 a c\right )^2 d^8 (b+2 c x)^3+\frac {2}{5} \left (b^2-4 a c\right ) d^8 (b+2 c x)^5+\frac {2}{7} d^8 (b+2 c x)^7-2 \left (b^2-4 a c\right )^{7/2} 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 = 0.11, size = 188, normalized size = 1.54 \begin {gather*} d^8 \left (\frac {16}{105} c x \left (112 b^2 c^2 \left (15 a^2-10 a c x^2+12 c^2 x^4\right )+840 b c^3 x \left (a^2-a c x^2+c^2 x^4\right )+16 c^3 \left (-105 a^3+35 a^2 c x^2-21 a c^2 x^4+15 c^3 x^6\right )+70 b^4 c \left (11 c x^2-9 a\right )+420 b^3 c^2 x \left (3 c x^2-2 a\right )+105 b^6+315 b^5 c x\right )+2 \left (4 a c-b^2\right )^{7/2} \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )\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}{a+b x+c x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.43, size = 536, normalized size = 4.39 \begin {gather*} \left [\frac {256}{7} \, c^{7} d^{8} x^{7} + 128 \, b c^{6} d^{8} x^{6} + \frac {256}{5} \, {\left (4 \, b^{2} c^{5} - a c^{6}\right )} d^{8} x^{5} + 64 \, {\left (3 \, b^{3} c^{4} - 2 \, a b c^{5}\right )} d^{8} x^{4} + \frac {32}{3} \, {\left (11 \, b^{4} c^{3} - 16 \, a b^{2} c^{4} + 8 \, a^{2} c^{5}\right )} d^{8} x^{3} + 16 \, {\left (3 \, b^{5} c^{2} - 8 \, a b^{3} c^{3} + 8 \, a^{2} b c^{4}\right )} d^{8} x^{2} - {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {b^{2} - 4 \, a c} d^{8} \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 ) + 16 \, {\left (b^{6} c - 6 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3} - 16 \, a^{3} c^{4}\right )} d^{8} x, \frac {256}{7} \, c^{7} d^{8} x^{7} + 128 \, b c^{6} d^{8} x^{6} + \frac {256}{5} \, {\left (4 \, b^{2} c^{5} - a c^{6}\right )} d^{8} x^{5} + 64 \, {\left (3 \, b^{3} c^{4} - 2 \, a b c^{5}\right )} d^{8} x^{4} + \frac {32}{3} \, {\left (11 \, b^{4} c^{3} - 16 \, a b^{2} c^{4} + 8 \, a^{2} c^{5}\right )} d^{8} x^{3} + 16 \, {\left (3 \, b^{5} c^{2} - 8 \, a b^{3} c^{3} + 8 \, a^{2} b c^{4}\right )} d^{8} x^{2} - 2 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-b^{2} + 4 \, a c} d^{8} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 16 \, {\left (b^{6} c - 6 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3} - 16 \, a^{3} c^{4}\right )} d^{8} x\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 313, normalized size = 2.57 \begin {gather*} \frac {2 \, {\left (b^{8} d^{8} - 16 \, a b^{6} c d^{8} + 96 \, a^{2} b^{4} c^{2} d^{8} - 256 \, a^{3} b^{2} c^{3} d^{8} + 256 \, a^{4} 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 {16 \, {\left (240 \, c^{14} d^{8} x^{7} + 840 \, b c^{13} d^{8} x^{6} + 1344 \, b^{2} c^{12} d^{8} x^{5} - 336 \, a c^{13} d^{8} x^{5} + 1260 \, b^{3} c^{11} d^{8} x^{4} - 840 \, a b c^{12} d^{8} x^{4} + 770 \, b^{4} c^{10} d^{8} x^{3} - 1120 \, a b^{2} c^{11} d^{8} x^{3} + 560 \, a^{2} c^{12} d^{8} x^{3} + 315 \, b^{5} c^{9} d^{8} x^{2} - 840 \, a b^{3} c^{10} d^{8} x^{2} + 840 \, a^{2} b c^{11} d^{8} x^{2} + 105 \, b^{6} c^{8} d^{8} x - 630 \, a b^{4} c^{9} d^{8} x + 1680 \, a^{2} b^{2} c^{10} d^{8} x - 1680 \, a^{3} c^{11} d^{8} x\right )}}{105 \, c^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 432, normalized size = 3.54 \begin {gather*} \frac {256 c^{7} d^{8} x^{7}}{7}+128 b \,c^{6} d^{8} x^{6}-\frac {256 a \,c^{6} d^{8} x^{5}}{5}+\frac {1024 b^{2} c^{5} d^{8} x^{5}}{5}-128 a b \,c^{5} d^{8} x^{4}+192 b^{3} c^{4} d^{8} x^{4}+\frac {256 a^{2} c^{5} d^{8} x^{3}}{3}-\frac {512 a \,b^{2} c^{4} d^{8} x^{3}}{3}+\frac {352 b^{4} c^{3} d^{8} x^{3}}{3}+\frac {512 a^{4} 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 {512 a^{3} 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 {192 a^{2} 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}}}+128 a^{2} b \,c^{4} d^{8} x^{2}-\frac {32 a \,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 a \,b^{3} c^{3} d^{8} x^{2}+\frac {2 b^{8} d^{8} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}+48 b^{5} c^{2} d^{8} x^{2}-256 a^{3} c^{4} d^{8} x +256 a^{2} b^{2} c^{3} d^{8} x -96 a \,b^{4} c^{2} d^{8} x +16 b^{6} c \,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 = 0.47, size = 761, normalized size = 6.24 \begin {gather*} x^3\,\left (\frac {1120\,b^4\,c^3\,d^8}{3}-\frac {b\,\left (1792\,b^3\,c^4\,d^8+\frac {b\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}-768\,a\,b\,c^5\,d^8\right )}{3\,c}+\frac {a\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{3\,c}\right )-x^5\,\left (\frac {256\,a\,c^6\,d^8}{5}-\frac {1024\,b^2\,c^5\,d^8}{5}\right )-x^2\,\left (\frac {a\,\left (1792\,b^3\,c^4\,d^8+\frac {b\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}-768\,a\,b\,c^5\,d^8\right )}{2\,c}+\frac {b\,\left (1120\,b^4\,c^3\,d^8-\frac {b\,\left (1792\,b^3\,c^4\,d^8+\frac {b\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}-768\,a\,b\,c^5\,d^8\right )}{c}+\frac {a\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}\right )}{2\,c}-224\,b^5\,c^2\,d^8\right )+x^4\,\left (448\,b^3\,c^4\,d^8+\frac {b\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{4\,c}-192\,a\,b\,c^5\,d^8\right )+x\,\left (112\,b^6\,c\,d^8-\frac {a\,\left (1120\,b^4\,c^3\,d^8-\frac {b\,\left (1792\,b^3\,c^4\,d^8+\frac {b\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}-768\,a\,b\,c^5\,d^8\right )}{c}+\frac {a\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}\right )}{c}+\frac {b\,\left (\frac {a\,\left (1792\,b^3\,c^4\,d^8+\frac {b\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}-768\,a\,b\,c^5\,d^8\right )}{c}+\frac {b\,\left (1120\,b^4\,c^3\,d^8-\frac {b\,\left (1792\,b^3\,c^4\,d^8+\frac {b\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}-768\,a\,b\,c^5\,d^8\right )}{c}+\frac {a\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}\right )}{c}-448\,b^5\,c^2\,d^8\right )}{c}\right )+2\,d^8\,\mathrm {atan}\left (\frac {b\,d^8\,{\left (4\,a\,c-b^2\right )}^{7/2}+2\,c\,d^8\,x\,{\left (4\,a\,c-b^2\right )}^{7/2}}{256\,a^4\,c^4\,d^8-256\,a^3\,b^2\,c^3\,d^8+96\,a^2\,b^4\,c^2\,d^8-16\,a\,b^6\,c\,d^8+b^8\,d^8}\right )\,{\left (4\,a\,c-b^2\right )}^{7/2}+\frac {256\,c^7\,d^8\,x^7}{7}+128\,b\,c^6\,d^8\,x^6 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.05, size = 502, normalized size = 4.11 \begin {gather*} 128 b c^{6} d^{8} x^{6} + \frac {256 c^{7} d^{8} x^{7}}{7} - d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{7}} \log {\left (x + \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} - d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{7}}}{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 )} + d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{7}} \log {\left (x + \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} + d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{7}}}{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 )} + x^{5} \left (- \frac {256 a c^{6} d^{8}}{5} + \frac {1024 b^{2} c^{5} d^{8}}{5}\right ) + x^{4} \left (- 128 a b c^{5} d^{8} + 192 b^{3} c^{4} d^{8}\right ) + x^{3} \left (\frac {256 a^{2} c^{5} d^{8}}{3} - \frac {512 a b^{2} c^{4} d^{8}}{3} + \frac {352 b^{4} c^{3} d^{8}}{3}\right ) + x^{2} \left (128 a^{2} b c^{4} d^{8} - 128 a b^{3} c^{3} d^{8} + 48 b^{5} c^{2} d^{8}\right ) + x \left (- 256 a^{3} c^{4} d^{8} + 256 a^{2} b^{2} c^{3} d^{8} - 96 a b^{4} c^{2} d^{8} + 16 b^{6} c d^{8}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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