Optimal. Leaf size=79 \[ \frac {7 b^2 x}{2 c^4}-\frac {7 b x^3}{6 c^3}+\frac {7 x^5}{10 c^2}-\frac {x^7}{2 c \left (b+c x^2\right )}-\frac {7 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 c^{9/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {1598, 294, 308,
211} \begin {gather*} -\frac {7 b^{5/2} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 c^{9/2}}+\frac {7 b^2 x}{2 c^4}-\frac {7 b x^3}{6 c^3}-\frac {x^7}{2 c \left (b+c x^2\right )}+\frac {7 x^5}{10 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 294
Rule 308
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^{12}}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac {x^8}{\left (b+c x^2\right )^2} \, dx\\ &=-\frac {x^7}{2 c \left (b+c x^2\right )}+\frac {7 \int \frac {x^6}{b+c x^2} \, dx}{2 c}\\ &=-\frac {x^7}{2 c \left (b+c x^2\right )}+\frac {7 \int \left (\frac {b^2}{c^3}-\frac {b x^2}{c^2}+\frac {x^4}{c}-\frac {b^3}{c^3 \left (b+c x^2\right )}\right ) \, dx}{2 c}\\ &=\frac {7 b^2 x}{2 c^4}-\frac {7 b x^3}{6 c^3}+\frac {7 x^5}{10 c^2}-\frac {x^7}{2 c \left (b+c x^2\right )}-\frac {\left (7 b^3\right ) \int \frac {1}{b+c x^2} \, dx}{2 c^4}\\ &=\frac {7 b^2 x}{2 c^4}-\frac {7 b x^3}{6 c^3}+\frac {7 x^5}{10 c^2}-\frac {x^7}{2 c \left (b+c x^2\right )}-\frac {7 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 c^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 71, normalized size = 0.90 \begin {gather*} \frac {x \left (90 b^2-20 b c x^2+6 c^2 x^4+\frac {15 b^3}{b+c x^2}\right )}{30 c^4}-\frac {7 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 c^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 65, normalized size = 0.82
method | result | size |
default | \(\frac {\frac {1}{5} c^{2} x^{5}-\frac {2}{3} b c \,x^{3}+3 b^{2} x}{c^{4}}-\frac {b^{3} \left (-\frac {x}{2 \left (c \,x^{2}+b \right )}+\frac {7 \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \sqrt {b c}}\right )}{c^{4}}\) | \(65\) |
risch | \(\frac {x^{5}}{5 c^{2}}-\frac {2 b \,x^{3}}{3 c^{3}}+\frac {3 b^{2} x}{c^{4}}+\frac {b^{3} x}{2 c^{4} \left (c \,x^{2}+b \right )}+\frac {7 \sqrt {-b c}\, b^{2} \ln \left (-\sqrt {-b c}\, x -b \right )}{4 c^{5}}-\frac {7 \sqrt {-b c}\, b^{2} \ln \left (\sqrt {-b c}\, x -b \right )}{4 c^{5}}\) | \(101\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 71, normalized size = 0.90 \begin {gather*} \frac {b^{3} x}{2 \, {\left (c^{5} x^{2} + b c^{4}\right )}} - \frac {7 \, b^{3} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \, \sqrt {b c} c^{4}} + \frac {3 \, c^{2} x^{5} - 10 \, b c x^{3} + 45 \, b^{2} x}{15 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 190, normalized size = 2.41 \begin {gather*} \left [\frac {12 \, c^{3} x^{7} - 28 \, b c^{2} x^{5} + 140 \, b^{2} c x^{3} + 210 \, b^{3} x + 105 \, {\left (b^{2} c x^{2} + b^{3}\right )} \sqrt {-\frac {b}{c}} \log \left (\frac {c x^{2} - 2 \, c x \sqrt {-\frac {b}{c}} - b}{c x^{2} + b}\right )}{60 \, {\left (c^{5} x^{2} + b c^{4}\right )}}, \frac {6 \, c^{3} x^{7} - 14 \, b c^{2} x^{5} + 70 \, b^{2} c x^{3} + 105 \, b^{3} x - 105 \, {\left (b^{2} c x^{2} + b^{3}\right )} \sqrt {\frac {b}{c}} \arctan \left (\frac {c x \sqrt {\frac {b}{c}}}{b}\right )}{30 \, {\left (c^{5} x^{2} + b c^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 124, normalized size = 1.57 \begin {gather*} \frac {b^{3} x}{2 b c^{4} + 2 c^{5} x^{2}} + \frac {3 b^{2} x}{c^{4}} - \frac {2 b x^{3}}{3 c^{3}} + \frac {7 \sqrt {- \frac {b^{5}}{c^{9}}} \log {\left (x - \frac {c^{4} \sqrt {- \frac {b^{5}}{c^{9}}}}{b^{2}} \right )}}{4} - \frac {7 \sqrt {- \frac {b^{5}}{c^{9}}} \log {\left (x + \frac {c^{4} \sqrt {- \frac {b^{5}}{c^{9}}}}{b^{2}} \right )}}{4} + \frac {x^{5}}{5 c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 7.83, size = 73, normalized size = 0.92 \begin {gather*} -\frac {7 \, b^{3} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \, \sqrt {b c} c^{4}} + \frac {b^{3} x}{2 \, {\left (c x^{2} + b\right )} c^{4}} + \frac {3 \, c^{8} x^{5} - 10 \, b c^{7} x^{3} + 45 \, b^{2} c^{6} x}{15 \, c^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 66, normalized size = 0.84 \begin {gather*} \frac {x^5}{5\,c^2}-\frac {2\,b\,x^3}{3\,c^3}+\frac {3\,b^2\,x}{c^4}-\frac {7\,b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )}{2\,c^{9/2}}+\frac {b^3\,x}{2\,\left (c^5\,x^2+b\,c^4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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