Optimal. Leaf size=148 \[ -\frac {2 \left (b^2-3 a c\right )}{a^2 \left (b^2-4 a c\right ) x}+\frac {b^2-2 a c+b c x}{a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )}-\frac {2 \left (b^4-6 a b^2 c+6 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^3 \left (b^2-4 a c\right )^{3/2}}-\frac {2 b \log (x)}{a^3}+\frac {b \log \left (a+b x+c x^2\right )}{a^3} \]
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Rubi [A]
time = 0.13, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1599, 754, 814,
648, 632, 212, 642} \begin {gather*} \frac {b \log \left (a+b x+c x^2\right )}{a^3}-\frac {2 b \log (x)}{a^3}-\frac {2 \left (b^2-3 a c\right )}{a^2 x \left (b^2-4 a c\right )}-\frac {2 \left (6 a^2 c^2-6 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 )^{3/2}}+\frac {-2 a c+b^2+b c x}{a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 754
Rule 814
Rule 1599
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a x^2+b x^3+c x^4\right )^2} \, dx &=\int \frac {1}{x^2 \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 \left (a+b x+c x^2\right )}-\frac {\int \frac {-2 \left (b^2-3 a c\right )-2 b c x}{x^2 \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 \left (a+b x+c x^2\right )}-\frac {\int \left (\frac {2 \left (-b^2+3 a c\right )}{a x^2}-\frac {2 b \left (-b^2+4 a c\right )}{a^2 x}+\frac {2 \left (-b^4+5 a b^2 c-3 a^2 c^2-b c \left (b^2-4 a c\right ) x\right )}{a^2 \left (a+b x+c x^2\right )}\right ) \, dx}{a \left (b^2-4 a c\right )}\\ &=-\frac {2 \left (b^2-3 a c\right )}{a^2 \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 \left (a+b x+c x^2\right )}-\frac {2 b \log (x)}{a^3}-\frac {2 \int \frac {-b^4+5 a b^2 c-3 a^2 c^2-b c \left (b^2-4 a c\right ) x}{a+b x+c x^2} \, dx}{a^3 \left (b^2-4 a c\right )}\\ &=-\frac {2 \left (b^2-3 a c\right )}{a^2 \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 \left (a+b x+c x^2\right )}-\frac {2 b \log (x)}{a^3}+\frac {b \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{a^3}+\frac {\left (b^4-6 a b^2 c+6 a^2 c^2\right ) \int \frac {1}{a+b x+c x^2} \, dx}{a^3 \left (b^2-4 a c\right )}\\ &=-\frac {2 \left (b^2-3 a c\right )}{a^2 \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 \left (a+b x+c x^2\right )}-\frac {2 b \log (x)}{a^3}+\frac {b \log \left (a+b x+c x^2\right )}{a^3}-\frac {\left (2 \left (b^4-6 a b^2 c+6 a^2 c^2\right )\right ) \text {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 )}\\ &=-\frac {2 \left (b^2-3 a c\right )}{a^2 \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 \left (a+b x+c x^2\right )}-\frac {2 \left (b^4-6 a b^2 c+6 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 )^{3/2}}-\frac {2 b \log (x)}{a^3}+\frac {b \log \left (a+b x+c x^2\right )}{a^3}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 131, normalized size = 0.89 \begin {gather*} -\frac {\frac {a}{x}+\frac {a \left (b^3-3 a b c+b^2 c x-2 a c^2 x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac {2 \left (b^4-6 a b^2 c+6 a^2 c^2\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}}+2 b \log (x)-b \log (a+x (b+c x))}{a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 205, normalized size = 1.39
method | result | size |
default | \(-\frac {\frac {\frac {a c \left (2 a c -b^{2}\right ) x}{4 a c -b^{2}}+\frac {a b \left (3 a c -b^{2}\right )}{4 a c -b^{2}}}{c \,x^{2}+b x +a}+\frac {\frac {\left (-4 a b \,c^{2}+b^{3} c \right ) \ln \left (c \,x^{2}+b x +a \right )}{c}+\frac {4 \left (3 a^{2} c^{2}-5 a \,b^{2} c +b^{4}-\frac {\left (-4 a b \,c^{2}+b^{3} c \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{a^{3}}-\frac {1}{a^{2} x}-\frac {2 b \ln \left (x \right )}{a^{3}}\) | \(205\) |
risch | \(\text {Expression too large to display}\) | \(3181\) |
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. 478 vs.
\(2 (144) = 288\).
time = 0.45, size = 975, normalized size = 6.59 \begin {gather*} \left [-\frac {a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + 2 \, {\left (a b^{4} c - 7 \, a^{2} b^{2} c^{2} + 12 \, a^{3} c^{3}\right )} x^{2} + {\left ({\left (b^{4} c - 6 \, a b^{2} c^{2} + 6 \, a^{2} c^{3}\right )} x^{3} + {\left (b^{5} - 6 \, a b^{3} c + 6 \, a^{2} b c^{2}\right )} x^{2} + {\left (a b^{4} - 6 \, a^{2} b^{2} c + 6 \, a^{3} 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 ) + {\left (2 \, a b^{5} - 15 \, a^{2} b^{3} c + 28 \, a^{3} b c^{2}\right )} x - {\left ({\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2}\right )} x^{2} + {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )} \log \left (c x^{2} + b x + a\right ) + 2 \, {\left ({\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2}\right )} x^{2} + {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )} \log \left (x\right )}{{\left (a^{3} b^{4} c - 8 \, a^{4} b^{2} c^{2} + 16 \, a^{5} c^{3}\right )} x^{3} + {\left (a^{3} b^{5} - 8 \, a^{4} b^{3} c + 16 \, a^{5} b c^{2}\right )} x^{2} + {\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )} x}, -\frac {a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + 2 \, {\left (a b^{4} c - 7 \, a^{2} b^{2} c^{2} + 12 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left ({\left (b^{4} c - 6 \, a b^{2} c^{2} + 6 \, a^{2} c^{3}\right )} x^{3} + {\left (b^{5} - 6 \, a b^{3} c + 6 \, a^{2} b c^{2}\right )} x^{2} + {\left (a b^{4} - 6 \, a^{2} b^{2} c + 6 \, a^{3} 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 ) + {\left (2 \, a b^{5} - 15 \, a^{2} b^{3} c + 28 \, a^{3} b c^{2}\right )} x - {\left ({\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2}\right )} x^{2} + {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )} \log \left (c x^{2} + b x + a\right ) + 2 \, {\left ({\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2}\right )} x^{2} + {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )} \log \left (x\right )}{{\left (a^{3} b^{4} c - 8 \, a^{4} b^{2} c^{2} + 16 \, a^{5} c^{3}\right )} x^{3} + {\left (a^{3} b^{5} - 8 \, a^{4} b^{3} c + 16 \, a^{5} b c^{2}\right )} x^{2} + {\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.59, size = 171, normalized size = 1.16 \begin {gather*} \frac {2 \, {\left (b^{4} - 6 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, b^{2} c x^{2} - 6 \, a c^{2} x^{2} + 2 \, b^{3} x - 7 \, a b c x + a b^{2} - 4 \, a^{2} c}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} {\left (c x^{3} + b x^{2} + a x\right )}} + \frac {b \log \left (c x^{2} + b x + a\right )}{a^{3}} - \frac {2 \, b \log \left ({\left | x \right |}\right )}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.83, size = 775, normalized size = 5.24 \begin {gather*} \ln \left (2\,a\,b^7+2\,b^8\,x+2\,a\,b^4\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-23\,a^2\,b^5\,c-108\,a^4\,b\,c^3+24\,a^4\,c^4\,x+2\,b^5\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+87\,a^3\,b^3\,c^2+3\,a^3\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-9\,a^2\,b^2\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+97\,a^2\,b^4\,c^2\,x-138\,a^3\,b^2\,c^3\,x-24\,a\,b^6\,c\,x-12\,a\,b^3\,c\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+15\,a^2\,b\,c^2\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )\,\left (\frac {b^4\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+6\,a^2\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-6\,a\,b^2\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}}{-64\,a^6\,c^3+48\,a^5\,b^2\,c^2-12\,a^4\,b^4\,c+a^3\,b^6}+\frac {b}{a^3}\right )-\frac {\frac {1}{a}-\frac {x\,\left (2\,b^3-7\,a\,b\,c\right )}{a^2\,\left (4\,a\,c-b^2\right )}+\frac {2\,c\,x^2\,\left (3\,a\,c-b^2\right )}{a^2\,\left (4\,a\,c-b^2\right )}}{c\,x^3+b\,x^2+a\,x}-\ln \left (2\,a\,b^4\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-2\,b^8\,x-2\,a\,b^7+23\,a^2\,b^5\,c+108\,a^4\,b\,c^3-24\,a^4\,c^4\,x+2\,b^5\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-87\,a^3\,b^3\,c^2+3\,a^3\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-9\,a^2\,b^2\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-97\,a^2\,b^4\,c^2\,x+138\,a^3\,b^2\,c^3\,x+24\,a\,b^6\,c\,x-12\,a\,b^3\,c\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+15\,a^2\,b\,c^2\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )\,\left (\frac {b^4\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+6\,a^2\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-6\,a\,b^2\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}}{-64\,a^6\,c^3+48\,a^5\,b^2\,c^2-12\,a^4\,b^4\,c+a^3\,b^6}-\frac {b}{a^3}\right )-\frac {2\,b\,\ln \left (x\right )}{a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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