Optimal. Leaf size=185 \[ -\frac {\log \left (a+b x+c x^2\right )}{2 a^3}+\frac {\log (x)}{a^3}+\frac {16 a^2 c^2+2 b c x \left (b^2-7 a c\right )-15 a b^2 c+2 b^4}{2 a^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {b \left (30 a^2 c^2-10 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^3 \left (b^2-4 a c\right )^{5/2}}+\frac {-2 a c+b^2+b c x}{2 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]
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Rubi [A] time = 0.22, antiderivative size = 185, 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} \[ \frac {16 a^2 c^2+2 b c x \left (b^2-7 a c\right )-15 a b^2 c+2 b^4}{2 a^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {b \left (30 a^2 c^2-10 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^3 \left (b^2-4 a c\right )^{5/2}}-\frac {\log \left (a+b x+c x^2\right )}{2 a^3}+\frac {\log (x)}{a^3}+\frac {-2 a c+b^2+b c x}{2 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]
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^7} \, dx &=\int \frac {1}{x \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 ) \left (a+b x+c x^2\right )^2}-\frac {\int \frac {-2 \left (b^2-4 a c\right )-3 b c x}{x \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 ) \left (a+b x+c x^2\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\int \frac {2 \left (b^2-4 a c\right )^2+2 b c \left (b^2-7 a c\right ) x}{x \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 ) \left (a+b x+c x^2\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\int \left (\frac {2 \left (-b^2+4 a c\right )^2}{a x}+\frac {2 \left (-b \left (b^4-9 a b^2 c+23 a^2 c^2\right )-c \left (b^2-4 a c\right )^2 x\right )}{a \left (a+b x+c x^2\right )}\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 ) \left (a+b x+c x^2\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\log (x)}{a^3}+\frac {\int \frac {-b \left (b^4-9 a b^2 c+23 a^2 c^2\right )-c \left (b^2-4 a c\right )^2 x}{a+b x+c x^2} \, dx}{a^3 \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 ) \left (a+b x+c x^2\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\log (x)}{a^3}-\frac {\int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 a^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 a^3 \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 ) \left (a+b x+c x^2\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\log (x)}{a^3}-\frac {\log \left (a+b x+c x^2\right )}{2 a^3}+\frac {\left (b \left (b^4-10 a b^2 c+30 a^2 c^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{a^3 \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 ) \left (a+b x+c x^2\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x}{2 a^2 \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 )}{a^3 \left (b^2-4 a c\right )^{5/2}}+\frac {\log (x)}{a^3}-\frac {\log \left (a+b x+c x^2\right )}{2 a^3}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 178, normalized size = 0.96 \[ \frac {\frac {a^2 \left (-2 a c+b^2+b c x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}-\frac {2 b \left (30 a^2 c^2-10 a b^2 c+b^4\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}+\frac {a \left (16 a^2 c^2-15 a b^2 c-14 a b c^2 x+2 b^4+2 b^3 c x\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}-\log (a+x (b+c x))+2 \log (x)}{2 a^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.79, size = 1985, normalized size = 10.73 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 239, normalized size = 1.29 \[ -\frac {{\left (b^{5} - 10 \, a b^{3} c + 30 \, a^{2} b c^{2}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{3} b^{4} - 8 \, a^{4} b^{2} c + 16 \, a^{5} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {\log \left (c x^{2} + b x + a\right )}{2 \, a^{3}} + \frac {\log \left ({\left | x \right |}\right )}{a^{3}} + \frac {3 \, a^{2} b^{4} - 21 \, a^{3} b^{2} c + 24 \, a^{4} c^{2} + 2 \, {\left (a b^{3} c^{2} - 7 \, a^{2} b c^{3}\right )} x^{3} + {\left (4 \, a b^{4} c - 29 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (a b^{5} - 6 \, a^{2} b^{3} c - a^{3} b c^{2}\right )} x}{2 \, {\left (c x^{2} + b x + a\right )}^{2} {\left (b^{2} - 4 \, a c\right )}^{2} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 781, normalized size = 4.22 \[ -\frac {7 b \,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}+\frac {b^{3} 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^{2}}-\frac {29 b^{2} 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}+\frac {2 b^{4} 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^{2}}+\frac {8 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 )}-\frac {6 b^{3} 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}+\frac {b^{5} 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 {b \,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 )}+\frac {12 a \,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 {3 b^{4}}{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 {30 b \,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}+\frac {10 b^{3} 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^{2}}-\frac {b^{5} \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 {21 b^{2} 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 {8 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}+\frac {4 b^{2} 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^{2}}-\frac {b^{4} \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^{3}}+\frac {\ln \relax (x )}{a^{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 = 2.46, size = 1089, normalized size = 5.89 \[ \frac {\ln \relax (x)}{a^3}+\frac {\frac {3\,\left (8\,a^2\,c^2-7\,a\,b^2\,c+b^4\right )}{2\,a\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x^2\,\left (16\,a^2\,c^3-29\,a\,b^2\,c^2+4\,b^4\,c\right )}{2\,a^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}-\frac {b\,x\,\left (a^2\,c^2+6\,a\,b^2\,c-b^4\right )}{a^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}-\frac {b\,c^2\,x^3\,\left (7\,a\,c-b^2\right )}{a^2\,\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 (1536\,a^6\,c^5-2\,b^{11}\,x-2\,a\,b^{10}+2\,a\,b^5\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+39\,a^2\,b^8\,c+2\,b^6\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-303\,a^3\,b^6\,c^2+1160\,a^4\,b^4\,c^3-2160\,a^5\,b^2\,c^4-17\,a^2\,b^3\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+39\,a^3\,b\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-321\,a^2\,b^7\,c^2\,x+1286\,a^3\,b^5\,c^3\,x-2560\,a^4\,b^3\,c^4\,x-48\,a^3\,c^3\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+40\,a\,b^9\,c\,x+2016\,a^5\,b\,c^5\,x-20\,a\,b^4\,c\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+63\,a^2\,b^2\,c^2\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}\right )\,\left (1024\,a^5\,c^5-b^{10}+b^5\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-160\,a^2\,b^6\,c^2+640\,a^3\,b^4\,c^3-1280\,a^4\,b^2\,c^4+20\,a\,b^8\,c+30\,a^2\,b\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-10\,a\,b^3\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}\right )}{2\,a^3\,{\left (4\,a\,c-b^2\right )}^5}+\frac {\ln \left (2\,a\,b^{10}+2\,b^{11}\,x-1536\,a^6\,c^5+2\,a\,b^5\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-39\,a^2\,b^8\,c+2\,b^6\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+303\,a^3\,b^6\,c^2-1160\,a^4\,b^4\,c^3+2160\,a^5\,b^2\,c^4-17\,a^2\,b^3\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+39\,a^3\,b\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+321\,a^2\,b^7\,c^2\,x-1286\,a^3\,b^5\,c^3\,x+2560\,a^4\,b^3\,c^4\,x-48\,a^3\,c^3\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-40\,a\,b^9\,c\,x-2016\,a^5\,b\,c^5\,x-20\,a\,b^4\,c\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+63\,a^2\,b^2\,c^2\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}\right )\,\left (b^{10}-1024\,a^5\,c^5+b^5\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+160\,a^2\,b^6\,c^2-640\,a^3\,b^4\,c^3+1280\,a^4\,b^2\,c^4-20\,a\,b^8\,c+30\,a^2\,b\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-10\,a\,b^3\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}\right )}{2\,a^3\,{\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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