Optimal. Leaf size=66 \[ -\frac {b+2 c x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {4 c \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1368, 628, 632,
212} \begin {gather*} \frac {4 c \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c 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 628
Rule 632
Rule 1368
Rubi steps
\begin {align*} \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^2 x^4} \, dx &=\int \frac {1}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac {b+2 c x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {(2 c) \int \frac {1}{a+b x+c x^2} \, dx}{b^2-4 a c}\\ &=-\frac {b+2 c x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {(4 c) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{b^2-4 a c}\\ &=-\frac {b+2 c x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {4 c \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 70, normalized size = 1.06 \begin {gather*} -\frac {\frac {b+2 c x}{a+x (b+c x)}+\frac {4 c \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}}{b^2-4 a c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 68, normalized size = 1.03
method | result | size |
default | \(\frac {2 c x +b}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )}+\frac {4 c \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}\) | \(68\) |
risch | \(\frac {\frac {2 c x}{4 a c -b^{2}}+\frac {b}{4 a c -b^{2}}}{c \,x^{2}+b x +a}+\frac {2 c \ln \left (\left (-8 c^{2} a +2 b^{2} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}}-4 a b c +b^{3}\right )}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}-\frac {2 c \ln \left (\left (8 c^{2} a -2 b^{2} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}}+4 a b c -b^{3}\right )}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}\) | \(144\) |
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. 160 vs.
\(2 (62) = 124\).
time = 0.37, size = 341, normalized size = 5.17 \begin {gather*} \left [-\frac {b^{3} - 4 \, a b c + 2 \, {\left (c^{2} x^{2} + b c x + a c\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^{2} c - 4 \, a c^{2}\right )} x}{a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{2} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x}, -\frac {b^{3} - 4 \, a b c - 4 \, {\left (c^{2} x^{2} + b c x + a c\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^{2} c - 4 \, a c^{2}\right )} x}{a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{2} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 265 vs.
\(2 (61) = 122\).
time = 0.31, size = 265, normalized size = 4.02 \begin {gather*} - 2 c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \log {\left (x + \frac {- 32 a^{2} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + 16 a b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} - 2 b^{4} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + 2 b c}{4 c^{2}} \right )} + 2 c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \log {\left (x + \frac {32 a^{2} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} - 16 a b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + 2 b^{4} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + 2 b c}{4 c^{2}} \right )} + \frac {b + 2 c x}{4 a^{2} c - a b^{2} + x^{2} \cdot \left (4 a c^{2} - b^{2} c\right ) + x \left (4 a b c - b^{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.50, size = 76, normalized size = 1.15 \begin {gather*} -\frac {4 \, c \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, c x + b}{{\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 119, normalized size = 1.80 \begin {gather*} \frac {\frac {b}{4\,a\,c-b^2}+\frac {2\,c\,x}{4\,a\,c-b^2}}{c\,x^2+b\,x+a}-\frac {4\,c\,\mathrm {atan}\left (\frac {\left (\frac {2\,c\,\left (b^3-4\,a\,b\,c\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {4\,c^2\,x}{{\left (4\,a\,c-b^2\right )}^{3/2}}\right )\,\left (4\,a\,c-b^2\right )}{2\,c}\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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