Optimal. Leaf size=68 \[ -\frac {5}{6 b^2 x^3}+\frac {5 c}{2 b^3 x}+\frac {1}{2 b x^3 \left (b+c x^2\right )}+\frac {5 c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 b^{7/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1607, 296, 331,
211} \begin {gather*} \frac {5 c^{3/2} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 b^{7/2}}+\frac {5 c}{2 b^3 x}-\frac {5}{6 b^2 x^3}+\frac {1}{2 b x^3 \left (b+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 296
Rule 331
Rule 1607
Rubi steps
\begin {align*} \int \frac {1}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac {1}{x^4 \left (b+c x^2\right )^2} \, dx\\ &=\frac {1}{2 b x^3 \left (b+c x^2\right )}+\frac {5 \int \frac {1}{x^4 \left (b+c x^2\right )} \, dx}{2 b}\\ &=-\frac {5}{6 b^2 x^3}+\frac {1}{2 b x^3 \left (b+c x^2\right )}-\frac {(5 c) \int \frac {1}{x^2 \left (b+c x^2\right )} \, dx}{2 b^2}\\ &=-\frac {5}{6 b^2 x^3}+\frac {5 c}{2 b^3 x}+\frac {1}{2 b x^3 \left (b+c x^2\right )}+\frac {\left (5 c^2\right ) \int \frac {1}{b+c x^2} \, dx}{2 b^3}\\ &=-\frac {5}{6 b^2 x^3}+\frac {5 c}{2 b^3 x}+\frac {1}{2 b x^3 \left (b+c x^2\right )}+\frac {5 c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 b^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 67, normalized size = 0.99 \begin {gather*} -\frac {1}{3 b^2 x^3}+\frac {2 c}{b^3 x}+\frac {c^2 x}{2 b^3 \left (b+c x^2\right )}+\frac {5 c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 55, normalized size = 0.81
method | result | size |
default | \(\frac {c^{2} \left (\frac {x}{2 c \,x^{2}+2 b}+\frac {5 \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \sqrt {b c}}\right )}{b^{3}}-\frac {1}{3 b^{2} x^{3}}+\frac {2 c}{b^{3} x}\) | \(55\) |
risch | \(\frac {\frac {5 c^{2} x^{4}}{2 b^{3}}+\frac {5 c \,x^{2}}{3 b^{2}}-\frac {1}{3 b}}{x^{3} \left (c \,x^{2}+b \right )}+\frac {5 \left (\munderset {\textit {\_R} =\RootOf \left (b^{7} \textit {\_Z}^{2}+c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} b^{7}+2 c^{3}\right ) x -b^{4} c \textit {\_R} \right )\right )}{4}\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 64, normalized size = 0.94 \begin {gather*} \frac {15 \, c^{2} x^{4} + 10 \, b c x^{2} - 2 \, b^{2}}{6 \, {\left (b^{3} c x^{5} + b^{4} x^{3}\right )}} + \frac {5 \, c^{2} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \, \sqrt {b c} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 172, normalized size = 2.53 \begin {gather*} \left [\frac {30 \, c^{2} x^{4} + 20 \, b c x^{2} + 15 \, {\left (c^{2} x^{5} + b c x^{3}\right )} \sqrt {-\frac {c}{b}} \log \left (\frac {c x^{2} + 2 \, b x \sqrt {-\frac {c}{b}} - b}{c x^{2} + b}\right ) - 4 \, b^{2}}{12 \, {\left (b^{3} c x^{5} + b^{4} x^{3}\right )}}, \frac {15 \, c^{2} x^{4} + 10 \, b c x^{2} + 15 \, {\left (c^{2} x^{5} + b c x^{3}\right )} \sqrt {\frac {c}{b}} \arctan \left (x \sqrt {\frac {c}{b}}\right ) - 2 \, b^{2}}{6 \, {\left (b^{3} c x^{5} + b^{4} x^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.18, size = 114, normalized size = 1.68 \begin {gather*} - \frac {5 \sqrt {- \frac {c^{3}}{b^{7}}} \log {\left (- \frac {b^{4} \sqrt {- \frac {c^{3}}{b^{7}}}}{c^{2}} + x \right )}}{4} + \frac {5 \sqrt {- \frac {c^{3}}{b^{7}}} \log {\left (\frac {b^{4} \sqrt {- \frac {c^{3}}{b^{7}}}}{c^{2}} + x \right )}}{4} + \frac {- 2 b^{2} + 10 b c x^{2} + 15 c^{2} x^{4}}{6 b^{4} x^{3} + 6 b^{3} c x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.56, size = 59, normalized size = 0.87 \begin {gather*} \frac {5 \, c^{2} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \, \sqrt {b c} b^{3}} + \frac {c^{2} x}{2 \, {\left (c x^{2} + b\right )} b^{3}} + \frac {6 \, c x^{2} - b}{3 \, b^{3} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.17, size = 58, normalized size = 0.85 \begin {gather*} \frac {\frac {5\,c\,x^2}{3\,b^2}-\frac {1}{3\,b}+\frac {5\,c^2\,x^4}{2\,b^3}}{c\,x^5+b\,x^3}+\frac {5\,c^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )}{2\,b^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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