Optimal. Leaf size=306 \[ -\frac {3 \left (2 b^4-13 a b^2 c+16 a^2 c^2\right )}{2 a^3 \left (b^2-4 a c\right )^2 x^2}+\frac {3 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{a^4 \left (b^2-4 a c\right )^2 x}+\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )^2}+\frac {4 b^4-25 a b^2 c+24 a^2 c^2+2 b c \left (2 b^2-11 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x+c x^2\right )}+\frac {3 b \left (2 b^6-21 a b^4 c+70 a^2 b^2 c^2-70 a^3 c^3\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^5 \left (b^2-4 a c\right )^{5/2}}+\frac {3 \left (2 b^2-a c\right ) \log (x)}{a^5}-\frac {3 \left (2 b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^5} \]
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Rubi [A]
time = 0.29, antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {754, 836, 814,
648, 632, 212, 642} \begin {gather*} -\frac {3 \left (2 b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^5}+\frac {3 \log (x) \left (2 b^2-a c\right )}{a^5}+\frac {3 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{a^4 x \left (b^2-4 a c\right )^2}+\frac {24 a^2 c^2+2 b c x \left (2 b^2-11 a c\right )-25 a b^2 c+4 b^4}{2 a^2 x^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {3 \left (16 a^2 c^2-13 a b^2 c+2 b^4\right )}{2 a^3 x^2 \left (b^2-4 a c\right )^2}+\frac {3 b \left (-70 a^3 c^3+70 a^2 b^2 c^2-21 a b^4 c+2 b^6\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^5 \left (b^2-4 a c\right )^{5/2}}+\frac {-2 a c+b^2+b c x}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 754
Rule 814
Rule 836
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b x+c x^2\right )^3} \, dx &=\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )^2}-\frac {\int \frac {-4 \left (b^2-3 a c\right )-5 b c x}{x^3 \left (a+b x+c x^2\right )^2} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )^2}+\frac {4 b^4-25 a b^2 c+24 a^2 c^2+2 b c \left (2 b^2-11 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x+c x^2\right )}+\frac {\int \frac {6 \left (2 b^4-13 a b^2 c+16 a^2 c^2\right )+6 b c \left (2 b^2-11 a c\right ) x}{x^3 \left (a+b x+c x^2\right )} \, dx}{2 a^2 \left (b^2-4 a c\right )^2}\\ &=\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )^2}+\frac {4 b^4-25 a b^2 c+24 a^2 c^2+2 b c \left (2 b^2-11 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x+c x^2\right )}+\frac {\int \left (\frac {6 \left (2 b^4-13 a b^2 c+16 a^2 c^2\right )}{a x^3}+\frac {6 b \left (2 b^2-9 a c\right ) \left (-b^2+3 a c\right )}{a^2 x^2}-\frac {6 \left (-2 b^2+a c\right ) \left (-b^2+4 a c\right )^2}{a^3 x}+\frac {6 \left (-b \left (2 b^6-19 a b^4 c+55 a^2 b^2 c^2-43 a^3 c^3\right )-c \left (b^2-4 a c\right )^2 \left (2 b^2-a c\right ) x\right )}{a^3 \left (a+b x+c x^2\right )}\right ) \, dx}{2 a^2 \left (b^2-4 a c\right )^2}\\ &=-\frac {3 \left (2 b^4-13 a b^2 c+16 a^2 c^2\right )}{2 a^3 \left (b^2-4 a c\right )^2 x^2}+\frac {3 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{a^4 \left (b^2-4 a c\right )^2 x}+\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )^2}+\frac {4 b^4-25 a b^2 c+24 a^2 c^2+2 b c \left (2 b^2-11 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x+c x^2\right )}+\frac {3 \left (2 b^2-a c\right ) \log (x)}{a^5}+\frac {3 \int \frac {-b \left (2 b^6-19 a b^4 c+55 a^2 b^2 c^2-43 a^3 c^3\right )-c \left (b^2-4 a c\right )^2 \left (2 b^2-a c\right ) x}{a+b x+c x^2} \, dx}{a^5 \left (b^2-4 a c\right )^2}\\ &=-\frac {3 \left (2 b^4-13 a b^2 c+16 a^2 c^2\right )}{2 a^3 \left (b^2-4 a c\right )^2 x^2}+\frac {3 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{a^4 \left (b^2-4 a c\right )^2 x}+\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )^2}+\frac {4 b^4-25 a b^2 c+24 a^2 c^2+2 b c \left (2 b^2-11 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x+c x^2\right )}+\frac {3 \left (2 b^2-a c\right ) \log (x)}{a^5}-\frac {\left (3 \left (2 b^2-a c\right )\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 a^5}-\frac {\left (3 b \left (2 b^6-21 a b^4 c+70 a^2 b^2 c^2-70 a^3 c^3\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 a^5 \left (b^2-4 a c\right )^2}\\ &=-\frac {3 \left (2 b^4-13 a b^2 c+16 a^2 c^2\right )}{2 a^3 \left (b^2-4 a c\right )^2 x^2}+\frac {3 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{a^4 \left (b^2-4 a c\right )^2 x}+\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )^2}+\frac {4 b^4-25 a b^2 c+24 a^2 c^2+2 b c \left (2 b^2-11 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x+c x^2\right )}+\frac {3 \left (2 b^2-a c\right ) \log (x)}{a^5}-\frac {3 \left (2 b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^5}+\frac {\left (3 b \left (2 b^6-21 a b^4 c+70 a^2 b^2 c^2-70 a^3 c^3\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{a^5 \left (b^2-4 a c\right )^2}\\ &=-\frac {3 \left (2 b^4-13 a b^2 c+16 a^2 c^2\right )}{2 a^3 \left (b^2-4 a c\right )^2 x^2}+\frac {3 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{a^4 \left (b^2-4 a c\right )^2 x}+\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )^2}+\frac {4 b^4-25 a b^2 c+24 a^2 c^2+2 b c \left (2 b^2-11 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x+c x^2\right )}+\frac {3 b \left (2 b^6-21 a b^4 c+70 a^2 b^2 c^2-70 a^3 c^3\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^5 \left (b^2-4 a c\right )^{5/2}}+\frac {3 \left (2 b^2-a c\right ) \log (x)}{a^5}-\frac {3 \left (2 b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^5}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 269, normalized size = 0.88 \begin {gather*} \frac {-\frac {a^2}{x^2}+\frac {6 a b}{x}+\frac {a^2 \left (b^4-4 a b^2 c+2 a^2 c^2+b^3 c x-3 a b c^2 x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}+\frac {a \left (6 b^6-47 a b^4 c+97 a^2 b^2 c^2-32 a^3 c^3+6 b^5 c x-42 a b^3 c^2 x+66 a^2 b c^3 x\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}-\frac {6 b \left (2 b^6-21 a b^4 c+70 a^2 b^2 c^2-70 a^3 c^3\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2}}+6 \left (2 b^2-a c\right ) \log (x)+3 \left (-2 b^2+a c\right ) \log (a+x (b+c x))}{2 a^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.95, size = 462, normalized size = 1.51
method | result | size |
default | \(\frac {\frac {\frac {3 a b \,c^{2} \left (11 a^{2} c^{2}-7 a c \,b^{2}+b^{4}\right ) x^{3}}{16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}}-\frac {c a \left (32 a^{3} c^{3}-163 a^{2} b^{2} c^{2}+89 a \,b^{4} c -12 b^{6}\right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}+\frac {a b \left (23 a^{3} c^{3}+24 a^{2} b^{2} c^{2}-20 a \,b^{4} c +3 b^{6}\right ) x}{16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}}-\frac {a^{2} \left (40 a^{3} c^{3}-115 a^{2} b^{2} c^{2}+55 a \,b^{4} c -7 b^{6}\right )}{2 \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {\frac {3 \left (16 a^{3} c^{4}-40 a^{2} b^{2} c^{3}+17 a \,b^{4} c^{2}-2 b^{6} c \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {6 \left (43 a^{3} b \,c^{3}-55 a^{2} b^{3} c^{2}+19 a \,b^{5} c -2 b^{7}-\frac {\left (16 a^{3} c^{4}-40 a^{2} b^{2} c^{3}+17 a \,b^{4} c^{2}-2 b^{6} c \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}}}{a^{5}}-\frac {1}{2 a^{3} x^{2}}+\frac {\left (-3 a c +6 b^{2}\right ) \ln \left (x \right )}{a^{5}}+\frac {3 b}{a^{4} x}\) | \(462\) |
risch | \(\frac {\frac {3 b \,c^{2} \left (27 a^{2} c^{2}-15 a c \,b^{2}+2 b^{4}\right ) x^{5}}{a^{4} \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}-\frac {3 c \left (16 a^{3} c^{3}-121 a^{2} b^{2} c^{2}+62 a \,b^{4} c -8 b^{6}\right ) x^{4}}{2 \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right ) a^{4}}+\frac {b \left (103 a^{3} c^{3}+32 a^{2} b^{2} c^{2}-39 a \,b^{4} c +6 b^{6}\right ) x^{3}}{a^{4} \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}-\frac {\left (72 a^{3} c^{3}-307 a^{2} b^{2} c^{2}+145 a \,b^{4} c -18 b^{6}\right ) x^{2}}{2 a^{3} \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}+\frac {2 b x}{a^{2}}-\frac {1}{2 a}}{x^{2} \left (c \,x^{2}+b x +a \right )^{2}}-\frac {3 c \ln \left (x \right )}{a^{4}}+\frac {6 b^{2} \ln \left (x \right )}{a^{5}}+3 \left (\munderset {\textit {\_R} =\RootOf \left (\left (1024 c^{5} a^{10}-1280 a^{9} b^{2} c^{4}+640 a^{8} b^{4} c^{3}-160 a^{7} b^{6} c^{2}+20 a^{6} b^{8} c -a^{5} b^{10}\right ) \textit {\_Z}^{2}+\left (-1024 c^{6} a^{6}+3328 a^{5} b^{2} c^{5}-3200 b^{4} c^{4} a^{4}+1440 b^{6} c^{3} a^{3}-340 b^{8} c^{2} a^{2}+41 b^{10} c a -2 b^{12}\right ) \textit {\_Z} +256 a^{2} c^{7}-119 a \,b^{2} c^{6}+14 b^{4} c^{5}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (1536 a^{13} c^{5}-2048 a^{12} b^{2} c^{4}+1088 a^{11} b^{4} c^{3}-288 a^{10} b^{6} c^{2}+38 a^{9} b^{8} c -2 a^{8} b^{10}\right ) \textit {\_R}^{2}+\left (-768 a^{9} c^{6}+1872 a^{8} b^{2} c^{5}-1368 a^{7} b^{4} c^{4}+445 a^{6} b^{6} c^{3}-68 a^{5} b^{8} c^{2}+4 a^{4} b^{10} c \right ) \textit {\_R} +729 a^{4} b^{2} c^{6}-810 a^{3} b^{4} c^{5}+333 a^{2} b^{6} c^{4}-60 a \,b^{8} c^{3}+4 b^{10} c^{2}\right ) x +\left (-256 a^{13} b \,c^{4}+256 a^{12} b^{3} c^{3}-96 a^{11} b^{5} c^{2}+16 a^{10} b^{7} c -a^{9} b^{9}\right ) \textit {\_R}^{2}+\left (-688 a^{9} b \,c^{5}+1224 a^{8} b^{3} c^{4}-787 a^{7} b^{5} c^{3}+239 a^{6} b^{7} c^{2}-35 a^{5} b^{9} c +2 a^{4} b^{11}\right ) \textit {\_R} -432 a^{5} b \,c^{6}+1320 a^{4} b^{3} c^{5}-1091 a^{3} b^{5} c^{4}+389 a^{2} b^{7} c^{3}-64 a \,b^{9} c^{2}+4 b^{11} c \right )\right )\) | \(805\) |
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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1324 vs.
\(2 (294) = 588\).
time = 6.67, size = 2669, normalized size = 8.72 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.80, size = 410, normalized size = 1.34 \begin {gather*} -\frac {3 \, {\left (2 \, b^{7} - 21 \, a b^{5} c + 70 \, a^{2} b^{3} c^{2} - 70 \, a^{3} b c^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{5} b^{4} - 8 \, a^{6} b^{2} c + 16 \, a^{7} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {12 \, b^{5} c^{2} x^{5} - 90 \, a b^{3} c^{3} x^{5} + 162 \, a^{2} b c^{4} x^{5} + 24 \, b^{6} c x^{4} - 186 \, a b^{4} c^{2} x^{4} + 363 \, a^{2} b^{2} c^{3} x^{4} - 48 \, a^{3} c^{4} x^{4} + 12 \, b^{7} x^{3} - 78 \, a b^{5} c x^{3} + 64 \, a^{2} b^{3} c^{2} x^{3} + 206 \, a^{3} b c^{3} x^{3} + 18 \, a b^{6} x^{2} - 145 \, a^{2} b^{4} c x^{2} + 307 \, a^{3} b^{2} c^{2} x^{2} - 72 \, a^{4} c^{3} x^{2} + 4 \, a^{2} b^{5} x - 32 \, a^{3} b^{3} c x + 64 \, a^{4} b c^{2} x - a^{3} b^{4} + 8 \, a^{4} b^{2} c - 16 \, a^{5} c^{2}}{2 \, {\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )} {\left (c x^{3} + b x^{2} + a x\right )}^{2}} - \frac {3 \, {\left (2 \, b^{2} - a c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, a^{5}} + \frac {3 \, {\left (2 \, b^{2} - a c\right )} \log \left ({\left | x \right |}\right )}{a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.31, size = 1404, normalized size = 4.59 \begin {gather*} \frac {\frac {2\,b\,x}{a^2}-\frac {1}{2\,a}+\frac {x^2\,\left (-72\,a^3\,c^3+307\,a^2\,b^2\,c^2-145\,a\,b^4\,c+18\,b^6\right )}{2\,a^3\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {3\,x^4\,\left (-16\,a^3\,c^4+121\,a^2\,b^2\,c^3-62\,a\,b^4\,c^2+8\,b^6\,c\right )}{2\,a^4\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {b\,x^3\,\left (103\,a^3\,c^3+32\,a^2\,b^2\,c^2-39\,a\,b^4\,c+6\,b^6\right )}{a^4\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {3\,b\,c^2\,x^5\,\left (27\,a^2\,c^2-15\,a\,b^2\,c+2\,b^4\right )}{a^4\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{x^4\,\left (b^2+2\,a\,c\right )+a^2\,x^2+c^2\,x^6+2\,a\,b\,x^3+2\,b\,c\,x^5}-\frac {3\,\ln \left (x\right )\,\left (a\,c-2\,b^2\right )}{a^5}+\frac {3\,\ln \left (4\,a\,b^{12}+4\,b^{13}\,x+1536\,a^7\,c^6-4\,a\,b^7\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-80\,a^2\,b^{10}\,c-4\,b^8\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+645\,a^3\,b^8\,c^2-2643\,a^4\,b^6\,c^3+5640\,a^5\,b^4\,c^4-5552\,a^6\,b^2\,c^5+36\,a^2\,b^5\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+59\,a^4\,b\,c^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+682\,a^2\,b^9\,c^2\,x-2913\,a^3\,b^7\,c^3\,x+6606\,a^4\,b^5\,c^4\,x-7232\,a^5\,b^3\,c^5\,x-48\,a^4\,c^4\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-82\,a\,b^{11}\,c\,x-95\,a^3\,b^3\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+2656\,a^6\,b\,c^6\,x+42\,a\,b^6\,c\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-146\,a^2\,b^4\,c^2\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+179\,a^3\,b^2\,c^3\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}\right )\,\left (2\,b^{12}+1024\,a^6\,c^6-2\,b^7\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+340\,a^2\,b^8\,c^2-1440\,a^3\,b^6\,c^3+3200\,a^4\,b^4\,c^4-3328\,a^5\,b^2\,c^5-41\,a\,b^{10}\,c+70\,a^3\,b\,c^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-70\,a^2\,b^3\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+21\,a\,b^5\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}\right )}{2\,a^5\,{\left (4\,a\,c-b^2\right )}^5}+\frac {3\,\ln \left (4\,a\,b^{12}+4\,b^{13}\,x+1536\,a^7\,c^6+4\,a\,b^7\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-80\,a^2\,b^{10}\,c+4\,b^8\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+645\,a^3\,b^8\,c^2-2643\,a^4\,b^6\,c^3+5640\,a^5\,b^4\,c^4-5552\,a^6\,b^2\,c^5-36\,a^2\,b^5\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-59\,a^4\,b\,c^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+682\,a^2\,b^9\,c^2\,x-2913\,a^3\,b^7\,c^3\,x+6606\,a^4\,b^5\,c^4\,x-7232\,a^5\,b^3\,c^5\,x+48\,a^4\,c^4\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-82\,a\,b^{11}\,c\,x+95\,a^3\,b^3\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+2656\,a^6\,b\,c^6\,x-42\,a\,b^6\,c\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+146\,a^2\,b^4\,c^2\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-179\,a^3\,b^2\,c^3\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}\right )\,\left (2\,b^{12}+1024\,a^6\,c^6+2\,b^7\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+340\,a^2\,b^8\,c^2-1440\,a^3\,b^6\,c^3+3200\,a^4\,b^4\,c^4-3328\,a^5\,b^2\,c^5-41\,a\,b^{10}\,c-70\,a^3\,b\,c^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+70\,a^2\,b^3\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-21\,a\,b^5\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}\right )}{2\,a^5\,{\left (4\,a\,c-b^2\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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