Optimal. Leaf size=196 \[ \frac {b \left (30 a^2 c^2-20 a b^2 c+3 b^4\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{3/2}}+\frac {\left (3 b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 c^4}-\frac {b x \left (3 b^2-11 a c\right )}{c^3 \left (b^2-4 a c\right )}+\frac {x^2 \left (3 b^2-8 a c\right )}{2 c^2 \left (b^2-4 a c\right )}-\frac {b x^3}{c \left (b^2-4 a c\right )}+\frac {x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]
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Rubi [A] time = 0.20, antiderivative size = 196, 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 = {1354, 738, 800, 634, 618, 206, 628} \[ \frac {b \left (30 a^2 c^2-20 a b^2 c+3 b^4\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{3/2}}+\frac {x^2 \left (3 b^2-8 a c\right )}{2 c^2 \left (b^2-4 a c\right )}+\frac {\left (3 b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 c^4}-\frac {b x \left (3 b^2-11 a c\right )}{c^3 \left (b^2-4 a c\right )}+\frac {x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {b x^3}{c \left (b^2-4 a c\right )} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 738
Rule 800
Rule 1354
Rubi steps
\begin {align*} \int \frac {x}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^2} \, dx &=\int \frac {x^5}{\left (a+b x+c x^2\right )^2} \, dx\\ &=\frac {x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \frac {x^3 (8 a+3 b x)}{a+b x+c x^2} \, dx}{-b^2+4 a c}\\ &=\frac {x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \left (\frac {b \left (3 b^2-11 a c\right )}{c^3}-\frac {\left (3 b^2-8 a c\right ) x}{c^2}+\frac {3 b x^2}{c}-\frac {a b \left (3 b^2-11 a c\right )+\left (b^2-4 a c\right ) \left (3 b^2-2 a c\right ) x}{c^3 \left (a+b x+c x^2\right )}\right ) \, dx}{-b^2+4 a c}\\ &=-\frac {b \left (3 b^2-11 a c\right ) x}{c^3 \left (b^2-4 a c\right )}+\frac {\left (3 b^2-8 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}-\frac {b x^3}{c \left (b^2-4 a c\right )}+\frac {x^4 (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 (3 b^2-11 a c\right )+\left (b^2-4 a c\right ) \left (3 b^2-2 a c\right ) x}{a+b x+c x^2} \, dx}{c^3 \left (b^2-4 a c\right )}\\ &=-\frac {b \left (3 b^2-11 a c\right ) x}{c^3 \left (b^2-4 a c\right )}+\frac {\left (3 b^2-8 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}-\frac {b x^3}{c \left (b^2-4 a c\right )}+\frac {x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (3 b^2-2 a c\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^4}-\frac {\left (b \left (3 b^4-20 a b^2 c+30 a^2 c^2\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c^4 \left (b^2-4 a c\right )}\\ &=-\frac {b \left (3 b^2-11 a c\right ) x}{c^3 \left (b^2-4 a c\right )}+\frac {\left (3 b^2-8 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}-\frac {b x^3}{c \left (b^2-4 a c\right )}+\frac {x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (3 b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac {\left (b \left (3 b^4-20 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 )}{c^4 \left (b^2-4 a c\right )}\\ &=-\frac {b \left (3 b^2-11 a c\right ) x}{c^3 \left (b^2-4 a c\right )}+\frac {\left (3 b^2-8 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}-\frac {b x^3}{c \left (b^2-4 a c\right )}+\frac {x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {b \left (3 b^4-20 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 )}{c^4 \left (b^2-4 a c\right )^{3/2}}+\frac {\left (3 b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 c^4}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 163, normalized size = 0.83 \[ \frac {\frac {2 b \left (30 a^2 c^2-20 a b^2 c+3 b^4\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+\frac {2 \left (2 a^3 c^2+a^2 b c (5 c x-4 b)+a b^3 (b-5 c x)+b^5 x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\left (3 b^2-2 a c\right ) \log (a+x (b+c x))-4 b c x+c^2 x^2}{2 c^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.92, size = 1029, normalized size = 5.25 \[ \left [\frac {2 \, a b^{6} - 16 \, a^{2} b^{4} c + 36 \, a^{3} b^{2} c^{2} - 16 \, a^{4} c^{3} + {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} x^{4} - 3 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} x^{3} - {\left (4 \, b^{6} c - 33 \, a b^{4} c^{2} + 72 \, a^{2} b^{2} c^{3} - 16 \, a^{3} c^{4}\right )} x^{2} - {\left (3 \, a b^{5} - 20 \, a^{2} b^{3} c + 30 \, a^{3} b c^{2} + {\left (3 \, b^{5} c - 20 \, a b^{3} c^{2} + 30 \, a^{2} b c^{3}\right )} x^{2} + {\left (3 \, b^{6} - 20 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2}\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^{7} - 11 \, a b^{5} c + 41 \, a^{2} b^{3} c^{2} - 52 \, a^{3} b c^{3}\right )} x + {\left (3 \, a b^{6} - 26 \, a^{2} b^{4} c + 64 \, a^{3} b^{2} c^{2} - 32 \, a^{4} c^{3} + {\left (3 \, b^{6} c - 26 \, a b^{4} c^{2} + 64 \, a^{2} b^{2} c^{3} - 32 \, a^{3} c^{4}\right )} x^{2} + {\left (3 \, b^{7} - 26 \, a b^{5} c + 64 \, a^{2} b^{3} c^{2} - 32 \, a^{3} b c^{3}\right )} x\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (a b^{4} c^{4} - 8 \, a^{2} b^{2} c^{5} + 16 \, a^{3} c^{6} + {\left (b^{4} c^{5} - 8 \, a b^{2} c^{6} + 16 \, a^{2} c^{7}\right )} x^{2} + {\left (b^{5} c^{4} - 8 \, a b^{3} c^{5} + 16 \, a^{2} b c^{6}\right )} x\right )}}, \frac {2 \, a b^{6} - 16 \, a^{2} b^{4} c + 36 \, a^{3} b^{2} c^{2} - 16 \, a^{4} c^{3} + {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} x^{4} - 3 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} x^{3} - {\left (4 \, b^{6} c - 33 \, a b^{4} c^{2} + 72 \, a^{2} b^{2} c^{3} - 16 \, a^{3} c^{4}\right )} x^{2} + 2 \, {\left (3 \, a b^{5} - 20 \, a^{2} b^{3} c + 30 \, a^{3} b c^{2} + {\left (3 \, b^{5} c - 20 \, a b^{3} c^{2} + 30 \, a^{2} b c^{3}\right )} x^{2} + {\left (3 \, b^{6} - 20 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2}\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^{7} - 11 \, a b^{5} c + 41 \, a^{2} b^{3} c^{2} - 52 \, a^{3} b c^{3}\right )} x + {\left (3 \, a b^{6} - 26 \, a^{2} b^{4} c + 64 \, a^{3} b^{2} c^{2} - 32 \, a^{4} c^{3} + {\left (3 \, b^{6} c - 26 \, a b^{4} c^{2} + 64 \, a^{2} b^{2} c^{3} - 32 \, a^{3} c^{4}\right )} x^{2} + {\left (3 \, b^{7} - 26 \, a b^{5} c + 64 \, a^{2} b^{3} c^{2} - 32 \, a^{3} b c^{3}\right )} x\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (a b^{4} c^{4} - 8 \, a^{2} b^{2} c^{5} + 16 \, a^{3} c^{6} + {\left (b^{4} c^{5} - 8 \, a b^{2} c^{6} + 16 \, a^{2} c^{7}\right )} x^{2} + {\left (b^{5} c^{4} - 8 \, a b^{3} c^{5} + 16 \, a^{2} b c^{6}\right )} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 188, normalized size = 0.96 \[ -\frac {{\left (3 \, b^{5} - 20 \, a b^{3} c + 30 \, a^{2} b c^{2}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {{\left (3 \, b^{2} - 2 \, a c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{4}} + \frac {c^{2} x^{2} - 4 \, b c x}{2 \, c^{4}} + \frac {a b^{4} - 4 \, a^{2} b^{2} c + 2 \, a^{3} c^{2} + {\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b c^{2}\right )} x}{{\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 434, normalized size = 2.21 \[ -\frac {5 a^{2} b x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{2}}+\frac {30 a^{2} b \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} c^{2}}+\frac {5 a \,b^{3} x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{3}}-\frac {20 a \,b^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} c^{3}}-\frac {b^{5} x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{4}}+\frac {3 b^{5} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} c^{4}}-\frac {2 a^{3}}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{2}}+\frac {4 a^{2} b^{2}}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{3}}-\frac {4 a^{2} \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) c^{2}}-\frac {a \,b^{4}}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{4}}+\frac {7 a \,b^{2} \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) c^{3}}-\frac {3 b^{4} \ln \left (c \,x^{2}+b x +a \right )}{2 \left (4 a c -b^{2}\right ) c^{4}}+\frac {x^{2}}{2 c^{2}}-\frac {2 b x}{c^{3}} \]
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 = 1.82, size = 382, normalized size = 1.95 \[ \frac {x^2}{2\,c^2}-\frac {\frac {a\,\left (2\,a^2\,c^2-4\,a\,b^2\,c+b^4\right )}{c\,\left (4\,a\,c-b^2\right )}+\frac {b\,x\,\left (5\,a^2\,c^2-5\,a\,b^2\,c+b^4\right )}{c\,\left (4\,a\,c-b^2\right )}}{c^4\,x^2+b\,c^3\,x+a\,c^3}-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (128\,a^4\,c^4-288\,a^3\,b^2\,c^3+168\,a^2\,b^4\,c^2-38\,a\,b^6\,c+3\,b^8\right )}{2\,\left (64\,a^3\,c^7-48\,a^2\,b^2\,c^6+12\,a\,b^4\,c^5-b^6\,c^4\right )}-\frac {2\,b\,x}{c^3}+\frac {b\,\mathrm {atan}\left (\frac {c^4\,\left (\frac {2\,b\,x\,\left (30\,a^2\,c^2-20\,a\,b^2\,c+3\,b^4\right )}{c^3\,{\left (4\,a\,c-b^2\right )}^3}-\frac {b\,\left (b^3\,c^3-4\,a\,b\,c^4\right )\,\left (30\,a^2\,c^2-20\,a\,b^2\,c+3\,b^4\right )}{c^7\,{\left (4\,a\,c-b^2\right )}^4}\right )\,{\left (4\,a\,c-b^2\right )}^{5/2}}{30\,a^2\,b\,c^2-20\,a\,b^3\,c+3\,b^5}\right )\,\left (30\,a^2\,c^2-20\,a\,b^2\,c+3\,b^4\right )}{c^4\,{\left (4\,a\,c-b^2\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.37, size = 1012, normalized size = 5.16 \[ - \frac {2 b x}{c^{3}} + \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (30 a^{2} c^{2} - 20 a b^{2} c + 3 b^{4}\right )}{2 c^{4} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 a c - 3 b^{2}}{2 c^{4}}\right ) \log {\left (x + \frac {16 a^{3} c^{2} - 17 a^{2} b^{2} c + 16 a^{2} c^{5} \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (30 a^{2} c^{2} - 20 a b^{2} c + 3 b^{4}\right )}{2 c^{4} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 a c - 3 b^{2}}{2 c^{4}}\right ) + 3 a b^{4} - 8 a b^{2} c^{4} \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (30 a^{2} c^{2} - 20 a b^{2} c + 3 b^{4}\right )}{2 c^{4} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 a c - 3 b^{2}}{2 c^{4}}\right ) + b^{4} c^{3} \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (30 a^{2} c^{2} - 20 a b^{2} c + 3 b^{4}\right )}{2 c^{4} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 a c - 3 b^{2}}{2 c^{4}}\right )}{30 a^{2} b c^{2} - 20 a b^{3} c + 3 b^{5}} \right )} + \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (30 a^{2} c^{2} - 20 a b^{2} c + 3 b^{4}\right )}{2 c^{4} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 a c - 3 b^{2}}{2 c^{4}}\right ) \log {\left (x + \frac {16 a^{3} c^{2} - 17 a^{2} b^{2} c + 16 a^{2} c^{5} \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (30 a^{2} c^{2} - 20 a b^{2} c + 3 b^{4}\right )}{2 c^{4} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 a c - 3 b^{2}}{2 c^{4}}\right ) + 3 a b^{4} - 8 a b^{2} c^{4} \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (30 a^{2} c^{2} - 20 a b^{2} c + 3 b^{4}\right )}{2 c^{4} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 a c - 3 b^{2}}{2 c^{4}}\right ) + b^{4} c^{3} \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (30 a^{2} c^{2} - 20 a b^{2} c + 3 b^{4}\right )}{2 c^{4} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 a c - 3 b^{2}}{2 c^{4}}\right )}{30 a^{2} b c^{2} - 20 a b^{3} c + 3 b^{5}} \right )} + \frac {- 2 a^{3} c^{2} + 4 a^{2} b^{2} c - a b^{4} + x \left (- 5 a^{2} b c^{2} + 5 a b^{3} c - b^{5}\right )}{4 a^{2} c^{5} - a b^{2} c^{4} + x^{2} \left (4 a c^{6} - b^{2} c^{5}\right ) + x \left (4 a b c^{5} - b^{3} c^{4}\right )} + \frac {x^{2}}{2 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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