Optimal. Leaf size=238 \[ \frac {3 \left (b^4-7 a b^2 c+10 a^2 c^2\right ) x}{c^3 \left (b^2-4 a c\right )^2}-\frac {3 b \left (b^2-6 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )^2}+\frac {x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {x^3 \left (a \left (b^2-10 a c\right )+b \left (b^2-7 a c\right ) x\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {3 \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{5/2}}-\frac {3 b \log \left (a+b x+c x^2\right )}{2 c^4} \]
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Rubi [A]
time = 0.20, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {1354, 752, 832,
814, 648, 632, 212, 642} \begin {gather*} \frac {3 x \left (10 a^2 c^2-7 a b^2 c+b^4\right )}{c^3 \left (b^2-4 a c\right )^2}-\frac {3 \left (-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{5/2}}-\frac {3 b x^2 \left (b^2-6 a c\right )}{2 c^2 \left (b^2-4 a c\right )^2}+\frac {x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {x^3 \left (b x \left (b^2-7 a c\right )+a \left (b^2-10 a c\right )\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {3 b \log \left (a+b x+c x^2\right )}{2 c^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 752
Rule 814
Rule 832
Rule 1354
Rubi steps
\begin {align*} \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3} \, dx &=\int \frac {x^6}{\left (a+b x+c x^2\right )^3} \, dx\\ &=\frac {x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\int \frac {x^4 (10 a+2 b x)}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )}\\ &=\frac {x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {x^3 \left (a \left (b^2-10 a c\right )+b \left (b^2-7 a c\right ) x\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\int \frac {x^2 \left (6 a \left (b^2-10 a c\right )+6 b \left (b^2-6 a c\right ) x\right )}{a+b x+c x^2} \, dx}{2 c \left (b^2-4 a c\right )^2}\\ &=\frac {x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {x^3 \left (a \left (b^2-10 a c\right )+b \left (b^2-7 a c\right ) x\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\int \left (-\frac {6 \left (b^4-7 a b^2 c+10 a^2 c^2\right )}{c^2}+\frac {6 b \left (b^2-6 a c\right ) x}{c}+\frac {6 \left (a \left (b^4-7 a b^2 c+10 a^2 c^2\right )+b \left (b^2-4 a c\right )^2 x\right )}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx}{2 c \left (b^2-4 a c\right )^2}\\ &=\frac {3 \left (b^4-7 a b^2 c+10 a^2 c^2\right ) x}{c^3 \left (b^2-4 a c\right )^2}-\frac {3 b \left (b^2-6 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )^2}+\frac {x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {x^3 \left (a \left (b^2-10 a c\right )+b \left (b^2-7 a c\right ) x\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {3 \int \frac {a \left (b^4-7 a b^2 c+10 a^2 c^2\right )+b \left (b^2-4 a c\right )^2 x}{a+b x+c x^2} \, dx}{c^3 \left (b^2-4 a c\right )^2}\\ &=\frac {3 \left (b^4-7 a b^2 c+10 a^2 c^2\right ) x}{c^3 \left (b^2-4 a c\right )^2}-\frac {3 b \left (b^2-6 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )^2}+\frac {x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {x^3 \left (a \left (b^2-10 a c\right )+b \left (b^2-7 a c\right ) x\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {(3 b) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^4}+\frac {\left (3 \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c^4 \left (b^2-4 a c\right )^2}\\ &=\frac {3 \left (b^4-7 a b^2 c+10 a^2 c^2\right ) x}{c^3 \left (b^2-4 a c\right )^2}-\frac {3 b \left (b^2-6 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )^2}+\frac {x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {x^3 \left (a \left (b^2-10 a c\right )+b \left (b^2-7 a c\right ) x\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {3 b \log \left (a+b x+c x^2\right )}{2 c^4}-\frac {\left (3 \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 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 )}{c^4 \left (b^2-4 a c\right )^2}\\ &=\frac {3 \left (b^4-7 a b^2 c+10 a^2 c^2\right ) x}{c^3 \left (b^2-4 a c\right )^2}-\frac {3 b \left (b^2-6 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )^2}+\frac {x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {x^3 \left (a \left (b^2-10 a c\right )+b \left (b^2-7 a c\right ) x\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {3 \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{5/2}}-\frac {3 b \log \left (a+b x+c x^2\right )}{2 c^4}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 260, normalized size = 1.09 \begin {gather*} \frac {2 c^2 x+\frac {b^7-14 a b^5 c+61 a^2 b^3 c^2-78 a^3 b c^3-6 b^6 c x+48 a b^4 c^2 x-102 a^2 b^2 c^3 x+36 a^3 c^4 x}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac {-b^6 x+a^2 b^2 c (5 b-9 c x)-a b^4 (b-6 c x)+a^3 c^2 (-5 b+2 c x)}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}+\frac {6 c \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 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}}-3 b c \log (a+x (b+c x))}{2 c^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 401, normalized size = 1.68
method | result | size |
default | \(\frac {x}{c^{3}}-\frac {\frac {-\frac {3 \left (6 a^{3} c^{3}-17 a^{2} b^{2} c^{2}+8 a \,b^{4} c -b^{6}\right ) x^{3}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {b \left (42 a^{3} c^{3}+41 a^{2} b^{2} c^{2}-34 a \,b^{4} c +5 b^{6}\right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c}-\frac {a \left (14 a^{3} c^{3}-71 a^{2} b^{2} c^{2}+38 a \,b^{4} c -5 b^{6}\right ) x}{c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {b \,a^{2} \left (58 a^{2} c^{2}-36 a \,b^{2} c +5 b^{4}\right )}{2 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {\frac {3 \left (16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {6 \left (10 a^{3} c^{2}-7 a^{2} b^{2} c +b^{4} a -\frac {\left (16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}}{c^{3}}\) | \(401\) |
risch | \(\text {Expression too large to display}\) | \(2019\) |
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. 953 vs.
\(2 (228) = 456\).
time = 0.38, size = 1926, normalized size = 8.09 \begin {gather*} \text {Too large to display} \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. 1714 vs.
\(2 (236) = 472\).
time = 3.13, size = 1714, normalized size = 7.20 \begin {gather*} \left (- \frac {3 b}{2 c^{4}} - \frac {3 \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (20 a^{3} c^{3} - 30 a^{2} b^{2} c^{2} + 10 a b^{4} c - b^{6}\right )}{2 c^{4} \cdot \left (1024 a^{5} c^{5} - 1280 a^{4} b^{2} c^{4} + 640 a^{3} b^{4} c^{3} - 160 a^{2} b^{6} c^{2} + 20 a b^{8} c - b^{10}\right )}\right ) \log {\left (x + \frac {- 66 a^{3} b c^{2} - 64 a^{3} c^{6} \left (- \frac {3 b}{2 c^{4}} - \frac {3 \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (20 a^{3} c^{3} - 30 a^{2} b^{2} c^{2} + 10 a b^{4} c - b^{6}\right )}{2 c^{4} \cdot \left (1024 a^{5} c^{5} - 1280 a^{4} b^{2} c^{4} + 640 a^{3} b^{4} c^{3} - 160 a^{2} b^{6} c^{2} + 20 a b^{8} c - b^{10}\right )}\right ) + 27 a^{2} b^{3} c + 48 a^{2} b^{2} c^{5} \left (- \frac {3 b}{2 c^{4}} - \frac {3 \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (20 a^{3} c^{3} - 30 a^{2} b^{2} c^{2} + 10 a b^{4} c - b^{6}\right )}{2 c^{4} \cdot \left (1024 a^{5} c^{5} - 1280 a^{4} b^{2} c^{4} + 640 a^{3} b^{4} c^{3} - 160 a^{2} b^{6} c^{2} + 20 a b^{8} c - b^{10}\right )}\right ) - 3 a b^{5} - 12 a b^{4} c^{4} \left (- \frac {3 b}{2 c^{4}} - \frac {3 \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (20 a^{3} c^{3} - 30 a^{2} b^{2} c^{2} + 10 a b^{4} c - b^{6}\right )}{2 c^{4} \cdot \left (1024 a^{5} c^{5} - 1280 a^{4} b^{2} c^{4} + 640 a^{3} b^{4} c^{3} - 160 a^{2} b^{6} c^{2} + 20 a b^{8} c - b^{10}\right )}\right ) + b^{6} c^{3} \left (- \frac {3 b}{2 c^{4}} - \frac {3 \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (20 a^{3} c^{3} - 30 a^{2} b^{2} c^{2} + 10 a b^{4} c - b^{6}\right )}{2 c^{4} \cdot \left (1024 a^{5} c^{5} - 1280 a^{4} b^{2} c^{4} + 640 a^{3} b^{4} c^{3} - 160 a^{2} b^{6} c^{2} + 20 a b^{8} c - b^{10}\right )}\right )}{60 a^{3} c^{3} - 90 a^{2} b^{2} c^{2} + 30 a b^{4} c - 3 b^{6}} \right )} + \left (- \frac {3 b}{2 c^{4}} + \frac {3 \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (20 a^{3} c^{3} - 30 a^{2} b^{2} c^{2} + 10 a b^{4} c - b^{6}\right )}{2 c^{4} \cdot \left (1024 a^{5} c^{5} - 1280 a^{4} b^{2} c^{4} + 640 a^{3} b^{4} c^{3} - 160 a^{2} b^{6} c^{2} + 20 a b^{8} c - b^{10}\right )}\right ) \log {\left (x + \frac {- 66 a^{3} b c^{2} - 64 a^{3} c^{6} \left (- \frac {3 b}{2 c^{4}} + \frac {3 \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (20 a^{3} c^{3} - 30 a^{2} b^{2} c^{2} + 10 a b^{4} c - b^{6}\right )}{2 c^{4} \cdot \left (1024 a^{5} c^{5} - 1280 a^{4} b^{2} c^{4} + 640 a^{3} b^{4} c^{3} - 160 a^{2} b^{6} c^{2} + 20 a b^{8} c - b^{10}\right )}\right ) + 27 a^{2} b^{3} c + 48 a^{2} b^{2} c^{5} \left (- \frac {3 b}{2 c^{4}} + \frac {3 \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (20 a^{3} c^{3} - 30 a^{2} b^{2} c^{2} + 10 a b^{4} c - b^{6}\right )}{2 c^{4} \cdot \left (1024 a^{5} c^{5} - 1280 a^{4} b^{2} c^{4} + 640 a^{3} b^{4} c^{3} - 160 a^{2} b^{6} c^{2} + 20 a b^{8} c - b^{10}\right )}\right ) - 3 a b^{5} - 12 a b^{4} c^{4} \left (- \frac {3 b}{2 c^{4}} + \frac {3 \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (20 a^{3} c^{3} - 30 a^{2} b^{2} c^{2} + 10 a b^{4} c - b^{6}\right )}{2 c^{4} \cdot \left (1024 a^{5} c^{5} - 1280 a^{4} b^{2} c^{4} + 640 a^{3} b^{4} c^{3} - 160 a^{2} b^{6} c^{2} + 20 a b^{8} c - b^{10}\right )}\right ) + b^{6} c^{3} \left (- \frac {3 b}{2 c^{4}} + \frac {3 \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (20 a^{3} c^{3} - 30 a^{2} b^{2} c^{2} + 10 a b^{4} c - b^{6}\right )}{2 c^{4} \cdot \left (1024 a^{5} c^{5} - 1280 a^{4} b^{2} c^{4} + 640 a^{3} b^{4} c^{3} - 160 a^{2} b^{6} c^{2} + 20 a b^{8} c - b^{10}\right )}\right )}{60 a^{3} c^{3} - 90 a^{2} b^{2} c^{2} + 30 a b^{4} c - 3 b^{6}} \right )} + \frac {- 58 a^{4} b c^{2} + 36 a^{3} b^{3} c - 5 a^{2} b^{5} + x^{3} \cdot \left (36 a^{3} c^{4} - 102 a^{2} b^{2} c^{3} + 48 a b^{4} c^{2} - 6 b^{6} c\right ) + x^{2} \left (- 42 a^{3} b c^{3} - 41 a^{2} b^{3} c^{2} + 34 a b^{5} c - 5 b^{7}\right ) + x \left (28 a^{4} c^{3} - 142 a^{3} b^{2} c^{2} + 76 a^{2} b^{4} c - 10 a b^{6}\right )}{32 a^{4} c^{6} - 16 a^{3} b^{2} c^{5} + 2 a^{2} b^{4} c^{4} + x^{4} \cdot \left (32 a^{2} c^{8} - 16 a b^{2} c^{7} + 2 b^{4} c^{6}\right ) + x^{3} \cdot \left (64 a^{2} b c^{7} - 32 a b^{3} c^{6} + 4 b^{5} c^{5}\right ) + x^{2} \cdot \left (64 a^{3} c^{7} - 12 a b^{4} c^{5} + 2 b^{6} c^{4}\right ) + x \left (64 a^{3} b c^{6} - 32 a^{2} b^{3} c^{5} + 4 a b^{5} c^{4}\right )} + \frac {x}{c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.91, size = 282, normalized size = 1.18 \begin {gather*} \frac {3 \, {\left (b^{6} - 10 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2} - 20 \, a^{3} c^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {x}{c^{3}} - \frac {3 \, b \log \left (c x^{2} + b x + a\right )}{2 \, c^{4}} - \frac {5 \, a^{2} b^{5} - 36 \, a^{3} b^{3} c + 58 \, a^{4} b c^{2} + 6 \, {\left (b^{6} c - 8 \, a b^{4} c^{2} + 17 \, a^{2} b^{2} c^{3} - 6 \, a^{3} c^{4}\right )} x^{3} + {\left (5 \, b^{7} - 34 \, a b^{5} c + 41 \, a^{2} b^{3} c^{2} + 42 \, a^{3} b c^{3}\right )} x^{2} + 2 \, {\left (5 \, a b^{6} - 38 \, a^{2} b^{4} c + 71 \, a^{3} b^{2} c^{2} - 14 \, a^{4} c^{3}\right )} x}{2 \, {\left (c x^{2} + b x + a\right )}^{2} {\left (b^{2} - 4 \, a c\right )}^{2} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.00, size = 705, normalized size = 2.96 \begin {gather*} \frac {x}{c^3}-\frac {\frac {3\,x^3\,\left (-6\,a^3\,c^3+17\,a^2\,b^2\,c^2-8\,a\,b^4\,c+b^6\right )}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {x^2\,\left (42\,a^3\,b\,c^3+41\,a^2\,b^3\,c^2-34\,a\,b^5\,c+5\,b^7\right )}{2\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {a^2\,\left (58\,a^2\,b\,c^2-36\,a\,b^3\,c+5\,b^5\right )}{2\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {a\,x\,\left (-14\,a^3\,c^3+71\,a^2\,b^2\,c^2-38\,a\,b^4\,c+5\,b^6\right )}{c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{a^2\,c^3+c^5\,x^4+x^2\,\left (b^2\,c^3+2\,a\,c^4\right )+2\,b\,c^4\,x^3+2\,a\,b\,c^3\,x}+\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (-3072\,a^5\,b\,c^5+3840\,a^4\,b^3\,c^4-1920\,a^3\,b^5\,c^3+480\,a^2\,b^7\,c^2-60\,a\,b^9\,c+3\,b^{11}\right )}{2\,\left (1024\,a^5\,c^9-1280\,a^4\,b^2\,c^8+640\,a^3\,b^4\,c^7-160\,a^2\,b^6\,c^6+20\,a\,b^8\,c^5-b^{10}\,c^4\right )}+\frac {3\,\mathrm {atan}\left (\frac {\left (\frac {3\,x\,\left (-20\,a^3\,c^3+30\,a^2\,b^2\,c^2-10\,a\,b^4\,c+b^6\right )}{c^3\,{\left (4\,a\,c-b^2\right )}^5}+\frac {3\,\left (16\,a^2\,b\,c^5-8\,a\,b^3\,c^4+b^5\,c^3\right )\,\left (-20\,a^3\,c^3+30\,a^2\,b^2\,c^2-10\,a\,b^4\,c+b^6\right )}{2\,c^7\,{\left (4\,a\,c-b^2\right )}^5\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )\,\left (32\,a^2\,c^6\,{\left (4\,a\,c-b^2\right )}^{5/2}+2\,b^4\,c^4\,{\left (4\,a\,c-b^2\right )}^{5/2}-16\,a\,b^2\,c^5\,{\left (4\,a\,c-b^2\right )}^{5/2}\right )}{-60\,a^3\,c^3+90\,a^2\,b^2\,c^2-30\,a\,b^4\,c+3\,b^6}\right )\,\left (-20\,a^3\,c^3+30\,a^2\,b^2\,c^2-10\,a\,b^4\,c+b^6\right )}{c^4\,{\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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