Optimal. Leaf size=294 \[ -\frac {3 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) x}{c^4 \left (b^2-4 a c\right )^2}+\frac {3 \left (2 b^4-13 a b^2 c+16 a^2 c^2\right ) x^2}{2 c^3 \left (b^2-4 a c\right )^2}-\frac {b \left (2 b^2-11 a c\right ) x^3}{c^2 \left (b^2-4 a c\right )^2}+\frac {x^6 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 x^4 \left (a \left (b^2-8 a c\right )+b \left (b^2-6 a c\right ) x\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {3 b \left (2 b^6-21 a b^4 c+70 a^2 b^2 c^2-70 a^3 c^3\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^5 \left (b^2-4 a c\right )^{5/2}}+\frac {3 \left (2 b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 c^5} \]
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Rubi [A]
time = 0.29, antiderivative size = 294, 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 = {752, 832, 814,
648, 632, 212, 642} \begin {gather*} \frac {3 x^2 \left (16 a^2 c^2-13 a b^2 c+2 b^4\right )}{2 c^3 \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 )}{c^5 \left (b^2-4 a c\right )^{5/2}}+\frac {3 \left (2 b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 c^5}-\frac {3 b x \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{c^4 \left (b^2-4 a c\right )^2}-\frac {b x^3 \left (2 b^2-11 a c\right )}{c^2 \left (b^2-4 a c\right )^2}+\frac {x^6 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 x^4 \left (b x \left (b^2-6 a c\right )+a \left (b^2-8 a c\right )\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 752
Rule 814
Rule 832
Rubi steps
\begin {align*} \int \frac {x^7}{\left (a+b x+c x^2\right )^3} \, dx &=\frac {x^6 (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^5 (12 a+3 b x)}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )}\\ &=\frac {x^6 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 x^4 \left (a \left (b^2-8 a c\right )+b \left (b^2-6 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^3 \left (12 a \left (b^2-8 a c\right )+6 b \left (2 b^2-11 a c\right ) x\right )}{a+b x+c x^2} \, dx}{2 c \left (b^2-4 a c\right )^2}\\ &=\frac {x^6 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 x^4 \left (a \left (b^2-8 a c\right )+b \left (b^2-6 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 \left (\frac {6 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{c^3}-\frac {6 \left (2 b^4-13 a b^2 c+16 a^2 c^2\right ) x}{c^2}+\frac {6 b \left (2 b^2-11 a c\right ) x^2}{c}-\frac {6 \left (a b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )+\left (b^2-4 a c\right )^2 \left (2 b^2-a c\right ) x\right )}{c^3 \left (a+b x+c x^2\right )}\right ) \, dx}{2 c \left (b^2-4 a c\right )^2}\\ &=-\frac {3 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) x}{c^4 \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 ) x^2}{2 c^3 \left (b^2-4 a c\right )^2}-\frac {b \left (2 b^2-11 a c\right ) x^3}{c^2 \left (b^2-4 a c\right )^2}+\frac {x^6 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 x^4 \left (a \left (b^2-8 a c\right )+b \left (b^2-6 a c\right ) x\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {3 \int \frac {a b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )+\left (b^2-4 a c\right )^2 \left (2 b^2-a c\right ) x}{a+b x+c x^2} \, dx}{c^4 \left (b^2-4 a c\right )^2}\\ &=-\frac {3 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) x}{c^4 \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 ) x^2}{2 c^3 \left (b^2-4 a c\right )^2}-\frac {b \left (2 b^2-11 a c\right ) x^3}{c^2 \left (b^2-4 a c\right )^2}+\frac {x^6 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 x^4 \left (a \left (b^2-8 a c\right )+b \left (b^2-6 a c\right ) x\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\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 c^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 c^5 \left (b^2-4 a c\right )^2}\\ &=-\frac {3 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) x}{c^4 \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 ) x^2}{2 c^3 \left (b^2-4 a c\right )^2}-\frac {b \left (2 b^2-11 a c\right ) x^3}{c^2 \left (b^2-4 a c\right )^2}+\frac {x^6 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 x^4 \left (a \left (b^2-8 a c\right )+b \left (b^2-6 a c\right ) x\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {3 \left (2 b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 c^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 )}{c^5 \left (b^2-4 a c\right )^2}\\ &=-\frac {3 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) x}{c^4 \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 ) x^2}{2 c^3 \left (b^2-4 a c\right )^2}-\frac {b \left (2 b^2-11 a c\right ) x^3}{c^2 \left (b^2-4 a c\right )^2}+\frac {x^6 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 x^4 \left (a \left (b^2-8 a c\right )+b \left (b^2-6 a c\right ) x\right )}{2 c \left (b^2-4 a c\right )^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 )}{c^5 \left (b^2-4 a c\right )^{5/2}}+\frac {3 \left (2 b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 c^5}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 299, normalized size = 1.02 \begin {gather*} \frac {-6 b c^2 x+c^3 x^2-\frac {b^8-17 a b^6 c+88 a^2 b^4 c^2-153 a^3 b^2 c^3+48 a^4 c^4-8 b^7 c x+70 a b^5 c^2 x-182 a^2 b^3 c^3 x+126 a^3 b c^4 x}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac {-2 a^4 c^3+b^7 x+a b^5 (b-7 c x)+a^3 b c^2 (9 b-7 c x)+2 a^2 b^3 c (-3 b+7 c x)}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}+\frac {6 b c \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}}-3 c \left (-2 b^2+a c\right ) \log (a+x (b+c x))}{2 c^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.82, size = 459, normalized size = 1.56
method | result | size |
default | \(-\frac {-\frac {1}{2} c \,x^{2}+3 b x}{c^{4}}+\frac {\frac {-\frac {b \left (63 a^{3} c^{3}-91 a^{2} b^{2} c^{2}+35 a \,b^{4} c -4 b^{6}\right ) x^{3}}{16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}}-\frac {\left (48 a^{4} c^{4}-27 a^{3} b^{2} c^{3}-94 a^{2} b^{4} c^{2}+53 a \,b^{6} c -7 b^{8}\right ) x^{2}}{2 c \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}-\frac {a b \left (73 a^{3} c^{3}-136 a^{2} b^{2} c^{2}+58 a \,b^{4} c -7 b^{6}\right ) x}{c \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}-\frac {a^{2} \left (40 a^{3} c^{3}-115 a^{2} b^{2} c^{2}+55 a \,b^{4} c -7 b^{6}\right )}{2 c \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {\frac {3 \left (-16 a^{3} c^{3}+40 a^{2} b^{2} c^{2}-17 a \,b^{4} c +2 b^{6}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {6 \left (27 c^{2} a^{3} b -15 a^{2} c \,b^{3}+2 a \,b^{5}-\frac {\left (-16 a^{3} c^{3}+40 a^{2} b^{2} c^{2}-17 a \,b^{4} c +2 b^{6}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}}}{c^{4}}\) | \(459\) |
risch | \(\text {Expression too large to display}\) | \(2479\) |
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. 1093 vs.
\(2 (282) = 564\).
time = 2.60, size = 2207, normalized size = 7.51 \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. 1875 vs.
\(2 (289) = 578\).
time = 4.07, size = 1875, normalized size = 6.38 \begin {gather*} - \frac {3 b x}{c^{4}} + \left (- \frac {3 b \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (70 a^{3} c^{3} - 70 a^{2} b^{2} c^{2} + 21 a b^{4} c - 2 b^{6}\right )}{2 c^{5} \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 {3 \left (a c - 2 b^{2}\right )}{2 c^{5}}\right ) \log {\left (x + \frac {96 a^{4} c^{3} - 159 a^{3} b^{2} c^{2} + 64 a^{3} c^{7} \left (- \frac {3 b \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (70 a^{3} c^{3} - 70 a^{2} b^{2} c^{2} + 21 a b^{4} c - 2 b^{6}\right )}{2 c^{5} \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 {3 \left (a c - 2 b^{2}\right )}{2 c^{5}}\right ) + 57 a^{2} b^{4} c - 48 a^{2} b^{2} c^{6} \left (- \frac {3 b \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (70 a^{3} c^{3} - 70 a^{2} b^{2} c^{2} + 21 a b^{4} c - 2 b^{6}\right )}{2 c^{5} \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 {3 \left (a c - 2 b^{2}\right )}{2 c^{5}}\right ) - 6 a b^{6} + 12 a b^{4} c^{5} \left (- \frac {3 b \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (70 a^{3} c^{3} - 70 a^{2} b^{2} c^{2} + 21 a b^{4} c - 2 b^{6}\right )}{2 c^{5} \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 {3 \left (a c - 2 b^{2}\right )}{2 c^{5}}\right ) - b^{6} c^{4} \left (- \frac {3 b \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (70 a^{3} c^{3} - 70 a^{2} b^{2} c^{2} + 21 a b^{4} c - 2 b^{6}\right )}{2 c^{5} \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 {3 \left (a c - 2 b^{2}\right )}{2 c^{5}}\right )}{210 a^{3} b c^{3} - 210 a^{2} b^{3} c^{2} + 63 a b^{5} c - 6 b^{7}} \right )} + \left (\frac {3 b \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (70 a^{3} c^{3} - 70 a^{2} b^{2} c^{2} + 21 a b^{4} c - 2 b^{6}\right )}{2 c^{5} \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 {3 \left (a c - 2 b^{2}\right )}{2 c^{5}}\right ) \log {\left (x + \frac {96 a^{4} c^{3} - 159 a^{3} b^{2} c^{2} + 64 a^{3} c^{7} \cdot \left (\frac {3 b \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (70 a^{3} c^{3} - 70 a^{2} b^{2} c^{2} + 21 a b^{4} c - 2 b^{6}\right )}{2 c^{5} \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 {3 \left (a c - 2 b^{2}\right )}{2 c^{5}}\right ) + 57 a^{2} b^{4} c - 48 a^{2} b^{2} c^{6} \cdot \left (\frac {3 b \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (70 a^{3} c^{3} - 70 a^{2} b^{2} c^{2} + 21 a b^{4} c - 2 b^{6}\right )}{2 c^{5} \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 {3 \left (a c - 2 b^{2}\right )}{2 c^{5}}\right ) - 6 a b^{6} + 12 a b^{4} c^{5} \cdot \left (\frac {3 b \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (70 a^{3} c^{3} - 70 a^{2} b^{2} c^{2} + 21 a b^{4} c - 2 b^{6}\right )}{2 c^{5} \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 {3 \left (a c - 2 b^{2}\right )}{2 c^{5}}\right ) - b^{6} c^{4} \cdot \left (\frac {3 b \sqrt {- \left (4 a c - b^{2}\right )^{5}} \cdot \left (70 a^{3} c^{3} - 70 a^{2} b^{2} c^{2} + 21 a b^{4} c - 2 b^{6}\right )}{2 c^{5} \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 {3 \left (a c - 2 b^{2}\right )}{2 c^{5}}\right )}{210 a^{3} b c^{3} - 210 a^{2} b^{3} c^{2} + 63 a b^{5} c - 6 b^{7}} \right )} + \frac {- 40 a^{5} c^{3} + 115 a^{4} b^{2} c^{2} - 55 a^{3} b^{4} c + 7 a^{2} b^{6} + x^{3} \left (- 126 a^{3} b c^{4} + 182 a^{2} b^{3} c^{3} - 70 a b^{5} c^{2} + 8 b^{7} c\right ) + x^{2} \left (- 48 a^{4} c^{4} + 27 a^{3} b^{2} c^{3} + 94 a^{2} b^{4} c^{2} - 53 a b^{6} c + 7 b^{8}\right ) + x \left (- 146 a^{4} b c^{3} + 272 a^{3} b^{3} c^{2} - 116 a^{2} b^{5} c + 14 a b^{7}\right )}{32 a^{4} c^{7} - 16 a^{3} b^{2} c^{6} + 2 a^{2} b^{4} c^{5} + x^{4} \cdot \left (32 a^{2} c^{9} - 16 a b^{2} c^{8} + 2 b^{4} c^{7}\right ) + x^{3} \cdot \left (64 a^{2} b c^{8} - 32 a b^{3} c^{7} + 4 b^{5} c^{6}\right ) + x^{2} \cdot \left (64 a^{3} c^{8} - 12 a b^{4} c^{6} + 2 b^{6} c^{5}\right ) + x \left (64 a^{3} b c^{7} - 32 a^{2} b^{3} c^{6} + 4 a b^{5} c^{5}\right )} + \frac {x^{2}}{2 c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.62, size = 332, normalized size = 1.13 \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 (b^{4} c^{5} - 8 \, a b^{2} c^{6} + 16 \, a^{2} c^{7}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {3 \, {\left (2 \, b^{2} - a c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{5}} + \frac {c^{3} x^{2} - 6 \, b c^{2} x}{2 \, c^{6}} + \frac {7 \, a^{2} b^{6} - 55 \, a^{3} b^{4} c + 115 \, a^{4} b^{2} c^{2} - 40 \, a^{5} c^{3} + 2 \, {\left (4 \, b^{7} c - 35 \, a b^{5} c^{2} + 91 \, a^{2} b^{3} c^{3} - 63 \, a^{3} b c^{4}\right )} x^{3} + {\left (7 \, b^{8} - 53 \, a b^{6} c + 94 \, a^{2} b^{4} c^{2} + 27 \, a^{3} b^{2} c^{3} - 48 \, a^{4} c^{4}\right )} x^{2} + 2 \, {\left (7 \, a b^{7} - 58 \, a^{2} b^{5} c + 136 \, a^{3} b^{3} c^{2} - 73 \, a^{4} b c^{3}\right )} x}{2 \, {\left (c x^{2} + b x + a\right )}^{2} {\left (b^{2} - 4 \, a c\right )}^{2} c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.47, size = 762, normalized size = 2.59 \begin {gather*} \frac {\frac {a\,\left (-40\,a^4\,c^3+115\,a^3\,b^2\,c^2-55\,a^2\,b^4\,c+7\,a\,b^6\right )}{2\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {b\,x^3\,\left (-63\,a^3\,c^3+91\,a^2\,b^2\,c^2-35\,a\,b^4\,c+4\,b^6\right )}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {x^2\,\left (-48\,a^4\,c^4+27\,a^3\,b^2\,c^3+94\,a^2\,b^4\,c^2-53\,a\,b^6\,c+7\,b^8\right )}{2\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {b\,x\,\left (-73\,a^4\,c^3+136\,a^3\,b^2\,c^2-58\,a^2\,b^4\,c+7\,a\,b^6\right )}{c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{a^2\,c^4+c^6\,x^4+x^2\,\left (b^2\,c^4+2\,a\,c^5\right )+2\,b\,c^5\,x^3+2\,a\,b\,c^4\,x}+\frac {x^2}{2\,c^3}-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (3072\,a^6\,c^6-9984\,a^5\,b^2\,c^5+9600\,a^4\,b^4\,c^4-4320\,a^3\,b^6\,c^3+1020\,a^2\,b^8\,c^2-123\,a\,b^{10}\,c+6\,b^{12}\right )}{2\,\left (1024\,a^5\,c^{10}-1280\,a^4\,b^2\,c^9+640\,a^3\,b^4\,c^8-160\,a^2\,b^6\,c^7+20\,a\,b^8\,c^6-b^{10}\,c^5\right )}-\frac {3\,b\,x}{c^4}-\frac {3\,b\,\mathrm {atan}\left (\frac {\left (\frac {3\,b\,x\,\left (-70\,a^3\,c^3+70\,a^2\,b^2\,c^2-21\,a\,b^4\,c+2\,b^6\right )}{c^4\,{\left (4\,a\,c-b^2\right )}^5}+\frac {3\,b^2\,\left (16\,a^2\,c^6-8\,a\,b^2\,c^5+b^4\,c^4\right )\,\left (-70\,a^3\,c^3+70\,a^2\,b^2\,c^2-21\,a\,b^4\,c+2\,b^6\right )}{2\,c^9\,{\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^7\,{\left (4\,a\,c-b^2\right )}^{5/2}+2\,b^4\,c^5\,{\left (4\,a\,c-b^2\right )}^{5/2}-16\,a\,b^2\,c^6\,{\left (4\,a\,c-b^2\right )}^{5/2}\right )}{-210\,a^3\,b\,c^3+210\,a^2\,b^3\,c^2-63\,a\,b^5\,c+6\,b^7}\right )\,\left (-70\,a^3\,c^3+70\,a^2\,b^2\,c^2-21\,a\,b^4\,c+2\,b^6\right )}{c^5\,{\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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