Optimal. Leaf size=190 \[ -\frac {b \left (b^2-7 a c\right ) x}{c^2 \left (b^2-4 a c\right )^2}+\frac {x^4 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {x^2 \left (a \left (b^2-16 a c\right )+b \left (b^2-10 a c\right ) x\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {b \left (b^4-10 a b^2 c+30 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{5/2}}+\frac {\log \left (a+b x+c x^2\right )}{2 c^3} \]
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Rubi [A]
time = 0.18, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {1368, 752, 832,
787, 648, 632, 212, 642} \begin {gather*} \frac {b \left (30 a^2 c^2-10 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{5/2}}-\frac {b x \left (b^2-7 a c\right )}{c^2 \left (b^2-4 a c\right )^2}+\frac {x^2 \left (b x \left (b^2-10 a c\right )+a \left (b^2-16 a c\right )\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {x^4 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {\log \left (a+b x+c x^2\right )}{2 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 752
Rule 787
Rule 832
Rule 1368
Rubi steps
\begin {align*} \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x} \, dx &=\int \frac {x^5}{\left (a+b x+c x^2\right )^3} \, dx\\ &=\frac {x^4 (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^3 (8 a+b x)}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )}\\ &=\frac {x^4 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {x^2 \left (a \left (b^2-16 a c\right )+b \left (b^2-10 a c\right ) x\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\int \frac {x \left (2 a \left (b^2-16 a c\right )+2 b \left (b^2-7 a c\right ) x\right )}{a+b x+c x^2} \, dx}{2 c \left (b^2-4 a c\right )^2}\\ &=-\frac {b \left (b^2-7 a c\right ) x}{c^2 \left (b^2-4 a c\right )^2}+\frac {x^4 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {x^2 \left (a \left (b^2-16 a c\right )+b \left (b^2-10 a c\right ) x\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\int \frac {-2 a b \left (b^2-7 a c\right )+\left (2 a c \left (b^2-16 a c\right )-2 b^2 \left (b^2-7 a c\right )\right ) x}{a+b x+c x^2} \, dx}{2 c^2 \left (b^2-4 a c\right )^2}\\ &=-\frac {b \left (b^2-7 a c\right ) x}{c^2 \left (b^2-4 a c\right )^2}+\frac {x^4 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {x^2 \left (a \left (b^2-16 a c\right )+b \left (b^2-10 a c\right ) x\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^3}-\frac {\left (b \left (b^4-10 a b^2 c+30 a^2 c^2\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c^3 \left (b^2-4 a c\right )^2}\\ &=-\frac {b \left (b^2-7 a c\right ) x}{c^2 \left (b^2-4 a c\right )^2}+\frac {x^4 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {x^2 \left (a \left (b^2-16 a c\right )+b \left (b^2-10 a c\right ) x\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\log \left (a+b x+c x^2\right )}{2 c^3}+\frac {\left (b \left (b^4-10 a b^2 c+30 a^2 c^2\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^3 \left (b^2-4 a c\right )^2}\\ &=-\frac {b \left (b^2-7 a c\right ) x}{c^2 \left (b^2-4 a c\right )^2}+\frac {x^4 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {x^2 \left (a \left (b^2-16 a c\right )+b \left (b^2-10 a c\right ) x\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {b \left (b^4-10 a b^2 c+30 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{5/2}}+\frac {\log \left (a+b x+c x^2\right )}{2 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 221, normalized size = 1.16 \begin {gather*} \frac {\frac {-b^6+11 a b^4 c-39 a^2 b^2 c^2+32 a^3 c^3+4 b^5 c x-30 a b^3 c^2 x+50 a^2 b c^3 x}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac {2 a^3 c^2+b^5 x+a b^3 (b-5 c x)+a^2 b c (-4 b+5 c x)}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}-\frac {2 b c \left (b^4-10 a b^2 c+30 a^2 c^2\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}}+c \log (a+x (b+c x))}{2 c^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 357, normalized size = 1.88
method | result | size |
default | \(\frac {\frac {b \left (25 a^{2} c^{2}-15 a \,b^{2} c +2 b^{4}\right ) x^{3}}{c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {\left (32 a^{3} c^{3}+11 a^{2} b^{2} c^{2}-19 a \,b^{4} c +3 b^{6}\right ) x^{2}}{2 c^{3} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {a b \left (31 a^{2} c^{2}-22 a \,b^{2} c +3 b^{4}\right ) x}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c^{3}}+\frac {3 a^{2} \left (8 a^{2} c^{2}-7 a \,b^{2} c +b^{4}\right )}{2 c^{3} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {\frac {\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-7 a^{2} b c +a \,b^{3}-\frac {\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(357\) |
risch | \(\text {Expression too large to display}\) | \(1641\) |
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. 792 vs.
\(2 (180) = 360\).
time = 0.39, size = 1603, normalized size = 8.44 \begin {gather*} \left [\frac {3 \, a^{2} b^{6} - 33 \, a^{3} b^{4} c + 108 \, a^{4} b^{2} c^{2} - 96 \, a^{5} c^{3} + 2 \, {\left (2 \, b^{7} c - 23 \, a b^{5} c^{2} + 85 \, a^{2} b^{3} c^{3} - 100 \, a^{3} b c^{4}\right )} x^{3} + {\left (3 \, b^{8} - 31 \, a b^{6} c + 87 \, a^{2} b^{4} c^{2} - 12 \, a^{3} b^{2} c^{3} - 128 \, a^{4} c^{4}\right )} x^{2} + {\left (a^{2} b^{5} - 10 \, a^{3} b^{3} c + 30 \, a^{4} b c^{2} + {\left (b^{5} c^{2} - 10 \, a b^{3} c^{3} + 30 \, a^{2} b c^{4}\right )} x^{4} + 2 \, {\left (b^{6} c - 10 \, a b^{4} c^{2} + 30 \, a^{2} b^{2} c^{3}\right )} x^{3} + {\left (b^{7} - 8 \, a b^{5} c + 10 \, a^{2} b^{3} c^{2} + 60 \, a^{3} b c^{3}\right )} x^{2} + 2 \, {\left (a b^{6} - 10 \, a^{2} b^{4} c + 30 \, a^{3} b^{2} c^{2}\right )} x\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 ) + 2 \, {\left (3 \, a b^{7} - 34 \, a^{2} b^{5} c + 119 \, a^{3} b^{3} c^{2} - 124 \, a^{4} b c^{3}\right )} x + {\left (a^{2} b^{6} - 12 \, a^{3} b^{4} c + 48 \, a^{4} b^{2} c^{2} - 64 \, a^{5} c^{3} + {\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} x^{4} + 2 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} x^{3} + {\left (b^{8} - 10 \, a b^{6} c + 24 \, a^{2} b^{4} c^{2} + 32 \, a^{3} b^{2} c^{3} - 128 \, a^{4} c^{4}\right )} x^{2} + 2 \, {\left (a b^{7} - 12 \, a^{2} b^{5} c + 48 \, a^{3} b^{3} c^{2} - 64 \, a^{4} b c^{3}\right )} x\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (a^{2} b^{6} c^{3} - 12 \, a^{3} b^{4} c^{4} + 48 \, a^{4} b^{2} c^{5} - 64 \, a^{5} c^{6} + {\left (b^{6} c^{5} - 12 \, a b^{4} c^{6} + 48 \, a^{2} b^{2} c^{7} - 64 \, a^{3} c^{8}\right )} x^{4} + 2 \, {\left (b^{7} c^{4} - 12 \, a b^{5} c^{5} + 48 \, a^{2} b^{3} c^{6} - 64 \, a^{3} b c^{7}\right )} x^{3} + {\left (b^{8} c^{3} - 10 \, a b^{6} c^{4} + 24 \, a^{2} b^{4} c^{5} + 32 \, a^{3} b^{2} c^{6} - 128 \, a^{4} c^{7}\right )} x^{2} + 2 \, {\left (a b^{7} c^{3} - 12 \, a^{2} b^{5} c^{4} + 48 \, a^{3} b^{3} c^{5} - 64 \, a^{4} b c^{6}\right )} x\right )}}, \frac {3 \, a^{2} b^{6} - 33 \, a^{3} b^{4} c + 108 \, a^{4} b^{2} c^{2} - 96 \, a^{5} c^{3} + 2 \, {\left (2 \, b^{7} c - 23 \, a b^{5} c^{2} + 85 \, a^{2} b^{3} c^{3} - 100 \, a^{3} b c^{4}\right )} x^{3} + {\left (3 \, b^{8} - 31 \, a b^{6} c + 87 \, a^{2} b^{4} c^{2} - 12 \, a^{3} b^{2} c^{3} - 128 \, a^{4} c^{4}\right )} x^{2} + 2 \, {\left (a^{2} b^{5} - 10 \, a^{3} b^{3} c + 30 \, a^{4} b c^{2} + {\left (b^{5} c^{2} - 10 \, a b^{3} c^{3} + 30 \, a^{2} b c^{4}\right )} x^{4} + 2 \, {\left (b^{6} c - 10 \, a b^{4} c^{2} + 30 \, a^{2} b^{2} c^{3}\right )} x^{3} + {\left (b^{7} - 8 \, a b^{5} c + 10 \, a^{2} b^{3} c^{2} + 60 \, a^{3} b c^{3}\right )} x^{2} + 2 \, {\left (a b^{6} - 10 \, a^{2} b^{4} c + 30 \, a^{3} b^{2} c^{2}\right )} x\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 ) + 2 \, {\left (3 \, a b^{7} - 34 \, a^{2} b^{5} c + 119 \, a^{3} b^{3} c^{2} - 124 \, a^{4} b c^{3}\right )} x + {\left (a^{2} b^{6} - 12 \, a^{3} b^{4} c + 48 \, a^{4} b^{2} c^{2} - 64 \, a^{5} c^{3} + {\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} x^{4} + 2 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} x^{3} + {\left (b^{8} - 10 \, a b^{6} c + 24 \, a^{2} b^{4} c^{2} + 32 \, a^{3} b^{2} c^{3} - 128 \, a^{4} c^{4}\right )} x^{2} + 2 \, {\left (a b^{7} - 12 \, a^{2} b^{5} c + 48 \, a^{3} b^{3} c^{2} - 64 \, a^{4} b c^{3}\right )} x\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (a^{2} b^{6} c^{3} - 12 \, a^{3} b^{4} c^{4} + 48 \, a^{4} b^{2} c^{5} - 64 \, a^{5} c^{6} + {\left (b^{6} c^{5} - 12 \, a b^{4} c^{6} + 48 \, a^{2} b^{2} c^{7} - 64 \, a^{3} c^{8}\right )} x^{4} + 2 \, {\left (b^{7} c^{4} - 12 \, a b^{5} c^{5} + 48 \, a^{2} b^{3} c^{6} - 64 \, a^{3} b c^{7}\right )} x^{3} + {\left (b^{8} c^{3} - 10 \, a b^{6} c^{4} + 24 \, a^{2} b^{4} c^{5} + 32 \, a^{3} b^{2} c^{6} - 128 \, a^{4} c^{7}\right )} x^{2} + 2 \, {\left (a b^{7} c^{3} - 12 \, a^{2} b^{5} c^{4} + 48 \, a^{3} b^{3} c^{5} - 64 \, a^{4} b c^{6}\right )} x\right )}}\right ] \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. 1510 vs.
\(2 (180) = 360\).
time = 2.04, size = 1510, normalized size = 7.95 \begin {gather*} \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (30 a^{2} c^{2} - 10 a b^{2} c + b^{4}\right )}{2 c^{3} \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 )} + \frac {1}{2 c^{3}}\right ) \log {\left (x + \frac {- 64 a^{3} c^{5} \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (30 a^{2} c^{2} - 10 a b^{2} c + b^{4}\right )}{2 c^{3} \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 )} + \frac {1}{2 c^{3}}\right ) + 32 a^{3} c^{2} + 48 a^{2} b^{2} c^{4} \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (30 a^{2} c^{2} - 10 a b^{2} c + b^{4}\right )}{2 c^{3} \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 )} + \frac {1}{2 c^{3}}\right ) - 9 a^{2} b^{2} c - 12 a b^{4} c^{3} \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (30 a^{2} c^{2} - 10 a b^{2} c + b^{4}\right )}{2 c^{3} \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 )} + \frac {1}{2 c^{3}}\right ) + a b^{4} + b^{6} c^{2} \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (30 a^{2} c^{2} - 10 a b^{2} c + b^{4}\right )}{2 c^{3} \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 )} + \frac {1}{2 c^{3}}\right )}{30 a^{2} b c^{2} - 10 a b^{3} c + b^{5}} \right )} + \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (30 a^{2} c^{2} - 10 a b^{2} c + b^{4}\right )}{2 c^{3} \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 )} + \frac {1}{2 c^{3}}\right ) \log {\left (x + \frac {- 64 a^{3} c^{5} \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (30 a^{2} c^{2} - 10 a b^{2} c + b^{4}\right )}{2 c^{3} \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 )} + \frac {1}{2 c^{3}}\right ) + 32 a^{3} c^{2} + 48 a^{2} b^{2} c^{4} \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (30 a^{2} c^{2} - 10 a b^{2} c + b^{4}\right )}{2 c^{3} \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 )} + \frac {1}{2 c^{3}}\right ) - 9 a^{2} b^{2} c - 12 a b^{4} c^{3} \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (30 a^{2} c^{2} - 10 a b^{2} c + b^{4}\right )}{2 c^{3} \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 )} + \frac {1}{2 c^{3}}\right ) + a b^{4} + b^{6} c^{2} \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (30 a^{2} c^{2} - 10 a b^{2} c + b^{4}\right )}{2 c^{3} \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 )} + \frac {1}{2 c^{3}}\right )}{30 a^{2} b c^{2} - 10 a b^{3} c + b^{5}} \right )} + \frac {24 a^{4} c^{2} - 21 a^{3} b^{2} c + 3 a^{2} b^{4} + x^{3} \cdot \left (50 a^{2} b c^{3} - 30 a b^{3} c^{2} + 4 b^{5} c\right ) + x^{2} \cdot \left (32 a^{3} c^{3} + 11 a^{2} b^{2} c^{2} - 19 a b^{4} c + 3 b^{6}\right ) + x \left (62 a^{3} b c^{2} - 44 a^{2} b^{3} c + 6 a b^{5}\right )}{32 a^{4} c^{5} - 16 a^{3} b^{2} c^{4} + 2 a^{2} b^{4} c^{3} + x^{4} \cdot \left (32 a^{2} c^{7} - 16 a b^{2} c^{6} + 2 b^{4} c^{5}\right ) + x^{3} \cdot \left (64 a^{2} b c^{6} - 32 a b^{3} c^{5} + 4 b^{5} c^{4}\right ) + x^{2} \cdot \left (64 a^{3} c^{6} - 12 a b^{4} c^{4} + 2 b^{6} c^{3}\right ) + x \left (64 a^{3} b c^{5} - 32 a^{2} b^{3} c^{4} + 4 a b^{5} c^{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.39, size = 245, normalized size = 1.29 \begin {gather*} -\frac {{\left (b^{5} - 10 \, a b^{3} c + 30 \, a^{2} b c^{2}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {\log \left (c x^{2} + b x + a\right )}{2 \, c^{3}} + \frac {3 \, a^{2} b^{4} - 21 \, a^{3} b^{2} c + 24 \, a^{4} c^{2} + 2 \, {\left (2 \, b^{5} c - 15 \, a b^{3} c^{2} + 25 \, a^{2} b c^{3}\right )} x^{3} + {\left (3 \, b^{6} - 19 \, a b^{4} c + 11 \, a^{2} b^{2} c^{2} + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (3 \, a b^{5} - 22 \, a^{2} b^{3} c + 31 \, a^{3} b c^{2}\right )} x}{2 \, {\left (c x^{2} + b x + a\right )}^{2} {\left (b^{2} - 4 \, a c\right )}^{2} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.20, size = 620, normalized size = 3.26 \begin {gather*} \frac {\frac {3\,a^2\,\left (8\,a^2\,c^2-7\,a\,b^2\,c+b^4\right )}{2\,c^3\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x^2\,\left (32\,a^3\,c^3+11\,a^2\,b^2\,c^2-19\,a\,b^4\,c+3\,b^6\right )}{2\,c^3\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {b\,x^3\,\left (25\,a^2\,c^2-15\,a\,b^2\,c+2\,b^4\right )}{c^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {a\,b\,x\,\left (31\,a^2\,c^2-22\,a\,b^2\,c+3\,b^4\right )}{c^3\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{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 {\ln \left (c\,x^2+b\,x+a\right )\,\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 )}{2\,\left (1024\,a^5\,c^8-1280\,a^4\,b^2\,c^7+640\,a^3\,b^4\,c^6-160\,a^2\,b^6\,c^5+20\,a\,b^8\,c^4-b^{10}\,c^3\right )}-\frac {b\,\mathrm {atan}\left (\frac {\left (\frac {b\,x\,\left (30\,a^2\,c^2-10\,a\,b^2\,c+b^4\right )}{c^2\,{\left (4\,a\,c-b^2\right )}^5}+\frac {b^2\,\left (16\,a^2\,c^4-8\,a\,b^2\,c^3+b^4\,c^2\right )\,\left (30\,a^2\,c^2-10\,a\,b^2\,c+b^4\right )}{2\,c^5\,{\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^5\,{\left (4\,a\,c-b^2\right )}^{5/2}+2\,b^4\,c^3\,{\left (4\,a\,c-b^2\right )}^{5/2}-16\,a\,b^2\,c^4\,{\left (4\,a\,c-b^2\right )}^{5/2}\right )}{30\,a^2\,b\,c^2-10\,a\,b^3\,c+b^5}\right )\,\left (30\,a^2\,c^2-10\,a\,b^2\,c+b^4\right )}{c^3\,{\left (4\,a\,c-b^2\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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