Optimal. Leaf size=252 \[ \frac {b \left (2 b^2-3 a c\right ) \log \left (a+b x+c x^2\right )}{a^5}-\frac {2 b \log (x) \left (2 b^2-3 a c\right )}{a^5}+\frac {b \left (2 b^2-7 a c\right )}{a^3 x^2 \left (b^2-4 a c\right )}-\frac {2 \left (2 b^2-5 a c\right )}{3 a^2 x^3 \left (b^2-4 a c\right )}-\frac {2 \left (5 a^2 c^2-9 a b^2 c+2 b^4\right )}{a^4 x \left (b^2-4 a c\right )}-\frac {2 \left (-10 a^3 c^3+30 a^2 b^2 c^2-15 a b^4 c+2 b^6\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^5 \left (b^2-4 a c\right )^{3/2}}+\frac {-2 a c+b^2+b c x}{a x^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]
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Rubi [A] time = 0.32, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {1594, 740, 800, 634, 618, 206, 628} \[ -\frac {2 \left (5 a^2 c^2-9 a b^2 c+2 b^4\right )}{a^4 x \left (b^2-4 a c\right )}-\frac {2 \left (30 a^2 b^2 c^2-10 a^3 c^3-15 a b^4 c+2 b^6\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^5 \left (b^2-4 a c\right )^{3/2}}+\frac {b \left (2 b^2-7 a c\right )}{a^3 x^2 \left (b^2-4 a c\right )}-\frac {2 \left (2 b^2-5 a c\right )}{3 a^2 x^3 \left (b^2-4 a c\right )}+\frac {b \left (2 b^2-3 a c\right ) \log \left (a+b x+c x^2\right )}{a^5}-\frac {2 b \log (x) \left (2 b^2-3 a c\right )}{a^5}+\frac {-2 a c+b^2+b c x}{a x^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 740
Rule 800
Rule 1594
Rubi steps
\begin {align*} \int \frac {1}{\left (a x^2+b x^3+c x^4\right )^2} \, dx &=\int \frac {1}{x^4 \left (a+b x+c x^2\right )^2} \, dx\\ &=\frac {b^2-2 a c+b c x}{a \left (b^2-4 a c\right ) x^3 \left (a+b x+c x^2\right )}-\frac {\int \frac {-2 \left (2 b^2-5 a c\right )-4 b c x}{x^4 \left (a+b x+c x^2\right )} \, dx}{a \left (b^2-4 a c\right )}\\ &=\frac {b^2-2 a c+b c x}{a \left (b^2-4 a c\right ) x^3 \left (a+b x+c x^2\right )}-\frac {\int \left (\frac {2 \left (-2 b^2+5 a c\right )}{a x^4}-\frac {2 \left (-2 b^3+7 a b c\right )}{a^2 x^3}-\frac {2 \left (2 b^4-9 a b^2 c+5 a^2 c^2\right )}{a^3 x^2}+\frac {2 b \left (b^2-4 a c\right ) \left (2 b^2-3 a c\right )}{a^4 x}+\frac {2 \left (-2 b^6+13 a b^4 c-21 a^2 b^2 c^2+5 a^3 c^3-b c \left (b^2-4 a c\right ) \left (2 b^2-3 a c\right ) x\right )}{a^4 \left (a+b x+c x^2\right )}\right ) \, dx}{a \left (b^2-4 a c\right )}\\ &=-\frac {2 \left (2 b^2-5 a c\right )}{3 a^2 \left (b^2-4 a c\right ) x^3}+\frac {b \left (2 b^2-7 a c\right )}{a^3 \left (b^2-4 a c\right ) x^2}-\frac {2 \left (2 b^4-9 a b^2 c+5 a^2 c^2\right )}{a^4 \left (b^2-4 a c\right ) x}+\frac {b^2-2 a c+b c x}{a \left (b^2-4 a c\right ) x^3 \left (a+b x+c x^2\right )}-\frac {2 b \left (2 b^2-3 a c\right ) \log (x)}{a^5}-\frac {2 \int \frac {-2 b^6+13 a b^4 c-21 a^2 b^2 c^2+5 a^3 c^3-b c \left (b^2-4 a c\right ) \left (2 b^2-3 a c\right ) x}{a+b x+c x^2} \, dx}{a^5 \left (b^2-4 a c\right )}\\ &=-\frac {2 \left (2 b^2-5 a c\right )}{3 a^2 \left (b^2-4 a c\right ) x^3}+\frac {b \left (2 b^2-7 a c\right )}{a^3 \left (b^2-4 a c\right ) x^2}-\frac {2 \left (2 b^4-9 a b^2 c+5 a^2 c^2\right )}{a^4 \left (b^2-4 a c\right ) x}+\frac {b^2-2 a c+b c x}{a \left (b^2-4 a c\right ) x^3 \left (a+b x+c x^2\right )}-\frac {2 b \left (2 b^2-3 a c\right ) \log (x)}{a^5}+\frac {\left (b \left (2 b^2-3 a c\right )\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{a^5}+\frac {\left (2 b^6-15 a b^4 c+30 a^2 b^2 c^2-10 a^3 c^3\right ) \int \frac {1}{a+b x+c x^2} \, dx}{a^5 \left (b^2-4 a c\right )}\\ &=-\frac {2 \left (2 b^2-5 a c\right )}{3 a^2 \left (b^2-4 a c\right ) x^3}+\frac {b \left (2 b^2-7 a c\right )}{a^3 \left (b^2-4 a c\right ) x^2}-\frac {2 \left (2 b^4-9 a b^2 c+5 a^2 c^2\right )}{a^4 \left (b^2-4 a c\right ) x}+\frac {b^2-2 a c+b c x}{a \left (b^2-4 a c\right ) x^3 \left (a+b x+c x^2\right )}-\frac {2 b \left (2 b^2-3 a c\right ) \log (x)}{a^5}+\frac {b \left (2 b^2-3 a c\right ) \log \left (a+b x+c x^2\right )}{a^5}-\frac {\left (2 \left (2 b^6-15 a b^4 c+30 a^2 b^2 c^2-10 a^3 c^3\right )\right ) \operatorname {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 )}\\ &=-\frac {2 \left (2 b^2-5 a c\right )}{3 a^2 \left (b^2-4 a c\right ) x^3}+\frac {b \left (2 b^2-7 a c\right )}{a^3 \left (b^2-4 a c\right ) x^2}-\frac {2 \left (2 b^4-9 a b^2 c+5 a^2 c^2\right )}{a^4 \left (b^2-4 a c\right ) x}+\frac {b^2-2 a c+b c x}{a \left (b^2-4 a c\right ) x^3 \left (a+b x+c x^2\right )}-\frac {2 \left (2 b^6-15 a b^4 c+30 a^2 b^2 c^2-10 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 )^{3/2}}-\frac {2 b \left (2 b^2-3 a c\right ) \log (x)}{a^5}+\frac {b \left (2 b^2-3 a c\right ) \log \left (a+b x+c x^2\right )}{a^5}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 218, normalized size = 0.87 \[ \frac {-\frac {a^3}{x^3}-\frac {3 a \left (5 a^2 b c^2+2 a^2 c^3 x-5 a b^3 c-4 a b^2 c^2 x+b^5+b^4 c x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac {3 a^2 b}{x^2}-\frac {6 \left (-10 a^3 c^3+30 a^2 b^2 c^2-15 a b^4 c+2 b^6\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+6 \log (x) \left (3 a b c-2 b^3\right )+3 \left (2 b^3-3 a b c\right ) \log (a+x (b+c x))+\frac {3 a \left (2 a c-3 b^2\right )}{x}}{3 a^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.32, size = 1407, normalized size = 5.58 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.55, size = 282, normalized size = 1.12 \[ \frac {2 \, {\left (2 \, b^{6} - 15 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2} - 10 \, a^{3} c^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{5} b^{2} - 4 \, a^{6} c\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {{\left (2 \, b^{3} - 3 \, a b c\right )} \log \left (c x^{2} + b x + a\right )}{a^{5}} - \frac {2 \, {\left (2 \, b^{3} - 3 \, a b c\right )} \log \left ({\left | x \right |}\right )}{a^{5}} - \frac {a^{4} b^{2} - 4 \, a^{5} c + 6 \, {\left (2 \, a b^{4} c - 9 \, a^{2} b^{2} c^{2} + 5 \, a^{3} c^{3}\right )} x^{4} + 3 \, {\left (4 \, a b^{5} - 20 \, a^{2} b^{3} c + 17 \, a^{3} b c^{2}\right )} x^{3} + {\left (6 \, a^{2} b^{4} - 29 \, a^{3} b^{2} c + 20 \, a^{4} c^{2}\right )} x^{2} - 2 \, {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x}{3 \, {\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} a^{5} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 515, normalized size = 2.04 \[ \frac {2 c^{3} x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a^{2}}+\frac {20 c^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} a^{2}}-\frac {4 b^{2} c^{2} x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a^{3}}-\frac {60 b^{2} c^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} a^{3}}+\frac {b^{4} c x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a^{4}}+\frac {30 b^{4} c \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} a^{4}}-\frac {4 b^{6} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} a^{5}}+\frac {5 b \,c^{2}}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a^{2}}-\frac {5 b^{3} c}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a^{3}}-\frac {12 b \,c^{2} \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) a^{3}}+\frac {b^{5}}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a^{4}}+\frac {11 b^{3} c \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) a^{4}}-\frac {2 b^{5} \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) a^{5}}+\frac {6 b c \ln \relax (x )}{a^{4}}-\frac {4 b^{3} \ln \relax (x )}{a^{5}}+\frac {2 c}{a^{3} x}-\frac {3 b^{2}}{a^{4} x}+\frac {b}{a^{3} x^{2}}-\frac {1}{3 a^{2} x^{3}} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.06, size = 1120, normalized size = 4.44 \[ \frac {\frac {x^2\,\left (5\,a\,c-6\,b^2\right )}{3\,a^3}-\frac {1}{3\,a}+\frac {2\,b\,x}{3\,a^2}+\frac {x^3\,\left (17\,a^2\,b\,c^2-20\,a\,b^3\,c+4\,b^5\right )}{a^4\,\left (4\,a\,c-b^2\right )}+\frac {2\,c\,x^4\,\left (5\,a^2\,c^2-9\,a\,b^2\,c+2\,b^4\right )}{a^4\,\left (4\,a\,c-b^2\right )}}{c\,x^5+b\,x^4+a\,x^3}+\frac {\ln \left (4\,a\,b^9+4\,b^{10}\,x-4\,a\,b^6\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-52\,a^2\,b^7\,c+308\,a^5\,b\,c^4-40\,a^5\,c^5\,x-4\,b^7\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+243\,a^3\,b^5\,c^2-473\,a^4\,b^3\,c^3+5\,a^4\,c^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+24\,a^2\,b^4\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+266\,a^2\,b^6\,c^2\,x-563\,a^3\,b^4\,c^3\,x+438\,a^4\,b^2\,c^4\,x-54\,a\,b^8\,c\,x-33\,a^3\,b^2\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+30\,a\,b^5\,c\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+41\,a^3\,b\,c^3\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-66\,a^2\,b^3\,c^2\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )\,\left (a^2\,\left (132\,b^5\,c^2-30\,b^2\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )-a^3\,\left (272\,b^3\,c^3-10\,c^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )+2\,b^9-2\,b^6\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-a\,\left (27\,b^7\,c-15\,b^4\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )+192\,a^4\,b\,c^4\right )}{-64\,a^8\,c^3+48\,a^7\,b^2\,c^2-12\,a^6\,b^4\,c+a^5\,b^6}+\frac {\ln \left (4\,a\,b^9+4\,b^{10}\,x+4\,a\,b^6\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-52\,a^2\,b^7\,c+308\,a^5\,b\,c^4-40\,a^5\,c^5\,x+4\,b^7\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+243\,a^3\,b^5\,c^2-473\,a^4\,b^3\,c^3-5\,a^4\,c^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-24\,a^2\,b^4\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+266\,a^2\,b^6\,c^2\,x-563\,a^3\,b^4\,c^3\,x+438\,a^4\,b^2\,c^4\,x-54\,a\,b^8\,c\,x+33\,a^3\,b^2\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-30\,a\,b^5\,c\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-41\,a^3\,b\,c^3\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+66\,a^2\,b^3\,c^2\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )\,\left (a^2\,\left (132\,b^5\,c^2+30\,b^2\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )-a^3\,\left (272\,b^3\,c^3+10\,c^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )+2\,b^9+2\,b^6\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-a\,\left (27\,b^7\,c+15\,b^4\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )+192\,a^4\,b\,c^4\right )}{-64\,a^8\,c^3+48\,a^7\,b^2\,c^2-12\,a^6\,b^4\,c+a^5\,b^6}+\frac {2\,b\,\ln \relax (x)\,\left (3\,a\,c-2\,b^2\right )}{a^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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