Optimal. Leaf size=239 \[ \frac {3 b \log \left (a+b x+c x^2\right )}{2 a^4}-\frac {3 b \log (x)}{a^4}-\frac {3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{a^3 x \left (b^2-4 a c\right )^2}+\frac {20 a^2 c^2+3 b c x \left (b^2-6 a c\right )-20 a b^2 c+3 b^4}{2 a^2 x \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {3 \left (-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^4 \left (b^2-4 a c\right )^{5/2}}+\frac {-2 a c+b^2+b c x}{2 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]
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Rubi [A] time = 0.28, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {1354, 740, 822, 800, 634, 618, 206, 628} \[ \frac {20 a^2 c^2+3 b c x \left (b^2-6 a c\right )-20 a b^2 c+3 b^4}{2 a^2 x \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {3 \left (30 a^2 b^2 c^2-20 a^3 c^3-10 a b^4 c+b^6\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^4 \left (b^2-4 a c\right )^{5/2}}-\frac {3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{a^3 x \left (b^2-4 a c\right )^2}+\frac {3 b \log \left (a+b x+c x^2\right )}{2 a^4}-\frac {3 b \log (x)}{a^4}+\frac {-2 a c+b^2+b c x}{2 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 740
Rule 800
Rule 822
Rule 1354
Rubi steps
\begin {align*} \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^8} \, dx &=\int \frac {1}{x^2 \left (a+b x+c x^2\right )^3} \, dx\\ &=\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )^2}-\frac {\int \frac {-3 b^2+10 a c-4 b c x}{x^2 \left (a+b x+c x^2\right )^2} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )^2}+\frac {3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x+c x^2\right )}+\frac {\int \frac {6 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )+6 b c \left (b^2-6 a c\right ) x}{x^2 \left (a+b x+c x^2\right )} \, dx}{2 a^2 \left (b^2-4 a c\right )^2}\\ &=\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )^2}+\frac {3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x+c x^2\right )}+\frac {\int \left (\frac {6 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{a x^2}-\frac {6 b \left (-b^2+4 a c\right )^2}{a^2 x}+\frac {6 \left (b^6-9 a b^4 c+23 a^2 b^2 c^2-10 a^3 c^3+b c \left (b^2-4 a c\right )^2 x\right )}{a^2 \left (a+b x+c x^2\right )}\right ) \, dx}{2 a^2 \left (b^2-4 a c\right )^2}\\ &=-\frac {3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{a^3 \left (b^2-4 a c\right )^2 x}+\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )^2}+\frac {3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x+c x^2\right )}-\frac {3 b \log (x)}{a^4}+\frac {3 \int \frac {b^6-9 a b^4 c+23 a^2 b^2 c^2-10 a^3 c^3+b c \left (b^2-4 a c\right )^2 x}{a+b x+c x^2} \, dx}{a^4 \left (b^2-4 a c\right )^2}\\ &=-\frac {3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{a^3 \left (b^2-4 a c\right )^2 x}+\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )^2}+\frac {3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x+c x^2\right )}-\frac {3 b \log (x)}{a^4}+\frac {(3 b) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 a^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 a^4 \left (b^2-4 a c\right )^2}\\ &=-\frac {3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{a^3 \left (b^2-4 a c\right )^2 x}+\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )^2}+\frac {3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x+c x^2\right )}-\frac {3 b \log (x)}{a^4}+\frac {3 b \log \left (a+b x+c x^2\right )}{2 a^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 ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{a^4 \left (b^2-4 a c\right )^2}\\ &=-\frac {3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{a^3 \left (b^2-4 a c\right )^2 x}+\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )^2}+\frac {3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 x \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 )}{a^4 \left (b^2-4 a c\right )^{5/2}}-\frac {3 b \log (x)}{a^4}+\frac {3 b \log \left (a+b x+c x^2\right )}{2 a^4}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 221, normalized size = 0.92 \[ \frac {\frac {a^2 \left (-3 a b c-2 a c^2 x+b^3+b^2 c x\right )}{\left (4 a c-b^2\right ) (a+x (b+c x))^2}-\frac {a \left (46 a^2 b c^2+28 a^2 c^3 x-29 a b^3 c-26 a b^2 c^2 x+4 b^5+4 b^4 c x\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac {6 \left (-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}+3 b \log (a+x (b+c x))-\frac {2 a}{x}-6 b \log (x)}{2 a^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 2.17, size = 2280, normalized size = 9.54 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 309, normalized size = 1.29 \[ \frac {3 \, {\left (b^{6} - 10 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2} - 20 \, a^{3} c^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {3 \, b \log \left (c x^{2} + b x + a\right )}{2 \, a^{4}} - \frac {3 \, b \log \left ({\left | x \right |}\right )}{a^{4}} - \frac {2 \, a^{3} b^{4} - 16 \, a^{4} b^{2} c + 32 \, a^{5} c^{2} + 6 \, {\left (a b^{4} c^{2} - 7 \, a^{2} b^{2} c^{3} + 10 \, a^{3} c^{4}\right )} x^{4} + 3 \, {\left (4 \, a b^{5} c - 29 \, a^{2} b^{3} c^{2} + 46 \, a^{3} b c^{3}\right )} x^{3} + 2 \, {\left (3 \, a b^{6} - 18 \, a^{2} b^{4} c + 7 \, a^{3} b^{2} c^{2} + 50 \, a^{4} c^{3}\right )} x^{2} + {\left (9 \, a^{2} b^{5} - 68 \, a^{3} b^{3} c + 122 \, a^{4} b c^{2}\right )} x}{2 \, {\left (c x^{2} + b x + a\right )}^{2} {\left (b^{2} - 4 \, a c\right )}^{2} a^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 954, normalized size = 3.99 \[ -\frac {14 c^{4} x^{3}}{\left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {13 b^{2} c^{3} x^{3}}{\left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}-\frac {2 b^{4} c^{2} x^{3}}{\left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{3}}-\frac {37 b \,c^{3} x^{2}}{\left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {55 b^{3} c^{2} x^{2}}{2 \left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}-\frac {4 b^{5} c \,x^{2}}{\left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{3}}-\frac {7 b^{2} c^{2} x}{\left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {12 b^{4} c x}{\left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}-\frac {2 b^{6} x}{\left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{3}}-\frac {18 c^{3} x}{\left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {18 b^{3} c}{\left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}-\frac {60 c^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}\, a}-\frac {5 b^{5}}{2 \left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}+\frac {90 b^{2} c^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}\, a^{2}}-\frac {30 b^{4} c \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}\, a^{3}}+\frac {3 b^{6} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}\, a^{4}}-\frac {29 b \,c^{2}}{\left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {24 b \,c^{2} \ln \left (c \,x^{2}+b x +a \right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}-\frac {12 b^{3} c \ln \left (c \,x^{2}+b x +a \right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{3}}+\frac {3 b^{5} \ln \left (c \,x^{2}+b x +a \right )}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{4}}-\frac {3 b \ln \relax (x )}{a^{4}}-\frac {1}{a^{3} x} \]
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 = 2.55, size = 1255, normalized size = 5.25 \[ -\frac {\frac {1}{a}+\frac {x^2\,\left (50\,a^3\,c^3+7\,a^2\,b^2\,c^2-18\,a\,b^4\,c+3\,b^6\right )}{a^3\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x\,\left (122\,a^2\,b\,c^2-68\,a\,b^3\,c+9\,b^5\right )}{2\,a^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {3\,x^3\,\left (46\,a^2\,b\,c^3-29\,a\,b^3\,c^2+4\,b^5\,c\right )}{2\,a^3\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {3\,c^2\,x^4\,\left (10\,a^2\,c^2-7\,a\,b^2\,c+b^4\right )}{a^3\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{x^3\,\left (b^2+2\,a\,c\right )+a^2\,x+c^2\,x^5+2\,a\,b\,x^2+2\,b\,c\,x^4}-\frac {3\,b\,\ln \relax (x)}{a^4}-\frac {3\,\ln \left (2\,a\,b^{11}+2\,b^{12}\,x+2\,a\,b^6\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-39\,a^2\,b^9\,c-1696\,a^6\,b\,c^5+320\,a^6\,c^6\,x+2\,b^7\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+303\,a^3\,b^7\,c^2-1170\,a^4\,b^5\,c^3+2240\,a^5\,b^3\,c^4-10\,a^4\,c^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-17\,a^2\,b^4\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+321\,a^2\,b^8\,c^2\,x-1296\,a^3\,b^6\,c^3\,x+2660\,a^4\,b^4\,c^4\,x-2336\,a^5\,b^2\,c^5\,x-40\,a\,b^{10}\,c\,x+39\,a^3\,b^2\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-20\,a\,b^5\,c\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-58\,a^3\,b\,c^3\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+63\,a^2\,b^3\,c^2\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}\right )\,\left (b^{11}+b^6\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-1024\,a^5\,b\,c^5+160\,a^2\,b^7\,c^2-640\,a^3\,b^5\,c^3+1280\,a^4\,b^3\,c^4-20\,a^3\,c^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-20\,a\,b^9\,c+30\,a^2\,b^2\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-10\,a\,b^4\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}\right )}{2\,a^4\,{\left (4\,a\,c-b^2\right )}^5}-\frac {3\,\ln \left (2\,a\,b^{11}+2\,b^{12}\,x-2\,a\,b^6\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-39\,a^2\,b^9\,c-1696\,a^6\,b\,c^5+320\,a^6\,c^6\,x-2\,b^7\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+303\,a^3\,b^7\,c^2-1170\,a^4\,b^5\,c^3+2240\,a^5\,b^3\,c^4+10\,a^4\,c^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+17\,a^2\,b^4\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+321\,a^2\,b^8\,c^2\,x-1296\,a^3\,b^6\,c^3\,x+2660\,a^4\,b^4\,c^4\,x-2336\,a^5\,b^2\,c^5\,x-40\,a\,b^{10}\,c\,x-39\,a^3\,b^2\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+20\,a\,b^5\,c\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+58\,a^3\,b\,c^3\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-63\,a^2\,b^3\,c^2\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}\right )\,\left (b^{11}-b^6\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-1024\,a^5\,b\,c^5+160\,a^2\,b^7\,c^2-640\,a^3\,b^5\,c^3+1280\,a^4\,b^3\,c^4+20\,a^3\,c^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-20\,a\,b^9\,c-30\,a^2\,b^2\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+10\,a\,b^4\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}\right )}{2\,a^4\,{\left (4\,a\,c-b^2\right )}^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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