Optimal. Leaf size=114 \[ -\frac {b x}{c \left (b^2-4 a c\right )}+\frac {x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac {\log \left (a+b x+c x^2\right )}{2 c^2} \]
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Rubi [A]
time = 0.07, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {1368, 752, 787,
648, 632, 212, 642} \begin {gather*} \frac {b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac {x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {b x}{c \left (b^2-4 a c\right )}+\frac {\log \left (a+b x+c x^2\right )}{2 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 752
Rule 787
Rule 1368
Rubi steps
\begin {align*} \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^2 x} \, dx &=\int \frac {x^3}{\left (a+b x+c x^2\right )^2} \, dx\\ &=\frac {x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \frac {x (4 a+b x)}{a+b x+c x^2} \, dx}{-b^2+4 a c}\\ &=-\frac {b x}{c \left (b^2-4 a c\right )}+\frac {x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {-a b+\left (-b^2+4 a c\right ) x}{a+b x+c x^2} \, dx}{c \left (b^2-4 a c\right )}\\ &=-\frac {b x}{c \left (b^2-4 a c\right )}+\frac {x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^2}-\frac {\left (b \left (b^2-6 a c\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c^2 \left (b^2-4 a c\right )}\\ &=-\frac {b x}{c \left (b^2-4 a c\right )}+\frac {x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\log \left (a+b x+c x^2\right )}{2 c^2}+\frac {\left (b \left (b^2-6 a c\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^2 \left (b^2-4 a c\right )}\\ &=-\frac {b x}{c \left (b^2-4 a c\right )}+\frac {x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac {\log \left (a+b x+c x^2\right )}{2 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 109, normalized size = 0.96 \begin {gather*} \frac {\frac {2 \left (-2 a^2 c+b^3 x+a b (b-3 c x)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac {2 b \left (b^2-6 a c\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+\log (a+x (b+c x))}{2 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 169, normalized size = 1.48
method | result | size |
default | \(\frac {\frac {b \left (3 a c -b^{2}\right ) x}{c^{2} \left (4 a c -b^{2}\right )}+\frac {a \left (2 a c -b^{2}\right )}{\left (4 a c -b^{2}\right ) c^{2}}}{c \,x^{2}+b x +a}+\frac {\frac {\left (4 a c -b^{2}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-a b -\frac {\left (4 a c -b^{2}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{c \left (4 a c -b^{2}\right )}\) | \(169\) |
risch | \(\frac {\frac {b \left (3 a c -b^{2}\right ) x}{c^{2} \left (4 a c -b^{2}\right )}+\frac {a \left (2 a c -b^{2}\right )}{\left (4 a c -b^{2}\right ) c^{2}}}{c \,x^{2}+b x +a}+\frac {8 \ln \left (-24 a^{2} b \,c^{2}+10 a \,b^{3} c -b^{5}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, c x -\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, b \right ) a^{2}}{\left (4 a c -b^{2}\right )^{2}}-\frac {4 \ln \left (-24 a^{2} b \,c^{2}+10 a \,b^{3} c -b^{5}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, c x -\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, b \right ) a \,b^{2}}{\left (4 a c -b^{2}\right )^{2} c}+\frac {\ln \left (-24 a^{2} b \,c^{2}+10 a \,b^{3} c -b^{5}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, c x -\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, b \right ) b^{4}}{2 \left (4 a c -b^{2}\right )^{2} c^{2}}+\frac {\ln \left (-24 a^{2} b \,c^{2}+10 a \,b^{3} c -b^{5}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, c x -\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, b \right ) \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}}{2 \left (4 a c -b^{2}\right )^{2} c^{2}}+\frac {8 \ln \left (-24 a^{2} b \,c^{2}+10 a \,b^{3} c -b^{5}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, c x +\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, b \right ) a^{2}}{\left (4 a c -b^{2}\right )^{2}}-\frac {4 \ln \left (-24 a^{2} b \,c^{2}+10 a \,b^{3} c -b^{5}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, c x +\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, b \right ) a \,b^{2}}{\left (4 a c -b^{2}\right )^{2} c}+\frac {\ln \left (-24 a^{2} b \,c^{2}+10 a \,b^{3} c -b^{5}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, c x +\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, b \right ) b^{4}}{2 \left (4 a c -b^{2}\right )^{2} c^{2}}-\frac {\ln \left (-24 a^{2} b \,c^{2}+10 a \,b^{3} c -b^{5}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, c x +\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, b \right ) \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}}{2 \left (4 a c -b^{2}\right )^{2} c^{2}}\) | \(979\) |
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. 308 vs.
\(2 (108) = 216\).
time = 0.34, size = 635, normalized size = 5.57 \begin {gather*} \left [\frac {2 \, a b^{4} - 12 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (a b^{3} - 6 \, a^{2} b c + {\left (b^{3} c - 6 \, a b c^{2}\right )} x^{2} + {\left (b^{4} - 6 \, a b^{2} c\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 (b^{5} - 7 \, a b^{3} c + 12 \, a^{2} b c^{2}\right )} x + {\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{2} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4} + {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} x^{2} + {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} x\right )}}, \frac {2 \, a b^{4} - 12 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + 2 \, {\left (a b^{3} - 6 \, a^{2} b c + {\left (b^{3} c - 6 \, a b c^{2}\right )} x^{2} + {\left (b^{4} - 6 \, a b^{2} c\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 (b^{5} - 7 \, a b^{3} c + 12 \, a^{2} b c^{2}\right )} x + {\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{2} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4} + {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} x^{2} + {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\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. 729 vs.
\(2 (104) = 208\).
time = 0.76, size = 729, normalized size = 6.39 \begin {gather*} \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a c - b^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{2 c^{2}}\right ) \log {\left (x + \frac {- 16 a^{2} c^{3} \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a c - b^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{2 c^{2}}\right ) + 8 a^{2} c + 8 a b^{2} c^{2} \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a c - b^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{2 c^{2}}\right ) - a b^{2} - b^{4} c \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a c - b^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{2 c^{2}}\right )}{6 a b c - b^{3}} \right )} + \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a c - b^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{2 c^{2}}\right ) \log {\left (x + \frac {- 16 a^{2} c^{3} \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a c - b^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{2 c^{2}}\right ) + 8 a^{2} c + 8 a b^{2} c^{2} \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a c - b^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{2 c^{2}}\right ) - a b^{2} - b^{4} c \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a c - b^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{2 c^{2}}\right )}{6 a b c - b^{3}} \right )} + \frac {2 a^{2} c - a b^{2} + x \left (3 a b c - b^{3}\right )}{4 a^{2} c^{3} - a b^{2} c^{2} + x^{2} \cdot \left (4 a c^{4} - b^{2} c^{3}\right ) + x \left (4 a b c^{3} - b^{3} c^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.78, size = 125, normalized size = 1.10 \begin {gather*} -\frac {{\left (b^{3} - 6 \, a b c\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {\log \left (c x^{2} + b x + a\right )}{2 \, c^{2}} + \frac {a b^{2} - 2 \, a^{2} c + {\left (b^{3} - 3 \, a b c\right )} x}{{\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.86, size = 279, normalized size = 2.45 \begin {gather*} \frac {\frac {a\,\left (2\,a\,c-b^2\right )}{c^2\,\left (4\,a\,c-b^2\right )}+\frac {b\,x\,\left (3\,a\,c-b^2\right )}{c^2\,\left (4\,a\,c-b^2\right )}}{c\,x^2+b\,x+a}-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}{2\,\left (64\,a^3\,c^5-48\,a^2\,b^2\,c^4+12\,a\,b^4\,c^3-b^6\,c^2\right )}+\frac {b\,\mathrm {atan}\left (\frac {c^2\,{\left (4\,a\,c-b^2\right )}^{5/2}\,\left (\frac {2\,b\,x\,\left (6\,a\,c-b^2\right )}{c\,{\left (4\,a\,c-b^2\right )}^3}+\frac {b^2\,\left (4\,a\,c^2-b^2\,c\right )\,\left (6\,a\,c-b^2\right )}{c^3\,{\left (4\,a\,c-b^2\right )}^4}\right )}{b^3-6\,a\,b\,c}\right )\,\left (6\,a\,c-b^2\right )}{c^2\,{\left (4\,a\,c-b^2\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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