Optimal. Leaf size=66 \[ -\frac {5 b x}{2 c^3}+\frac {5 x^3}{6 c^2}-\frac {x^5}{2 c \left (b+c x^2\right )}+\frac {5 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 c^{7/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 66, 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 {5 b^{3/2} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 c^{7/2}}-\frac {5 b x}{2 c^3}-\frac {x^5}{2 c \left (b+c x^2\right )}+\frac {5 x^3}{6 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^{10}}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac {x^6}{\left (b+c x^2\right )^2} \, dx\\ &=-\frac {x^5}{2 c \left (b+c x^2\right )}+\frac {5 \int \frac {x^4}{b+c x^2} \, dx}{2 c}\\ &=-\frac {x^5}{2 c \left (b+c x^2\right )}+\frac {5 \int \left (-\frac {b}{c^2}+\frac {x^2}{c}+\frac {b^2}{c^2 \left (b+c x^2\right )}\right ) \, dx}{2 c}\\ &=-\frac {5 b x}{2 c^3}+\frac {5 x^3}{6 c^2}-\frac {x^5}{2 c \left (b+c x^2\right )}+\frac {\left (5 b^2\right ) \int \frac {1}{b+c x^2} \, dx}{2 c^3}\\ &=-\frac {5 b x}{2 c^3}+\frac {5 x^3}{6 c^2}-\frac {x^5}{2 c \left (b+c x^2\right )}+\frac {5 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 c^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 60, normalized size = 0.91 \begin {gather*} \frac {x \left (-12 b+2 c x^2-\frac {3 b^2}{b+c x^2}\right )}{6 c^3}+\frac {5 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 54, normalized size = 0.82
method | result | size |
default | \(-\frac {-\frac {1}{3} c \,x^{3}+2 b x}{c^{3}}+\frac {b^{2} \left (-\frac {x}{2 \left (c \,x^{2}+b \right )}+\frac {5 \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \sqrt {b c}}\right )}{c^{3}}\) | \(54\) |
risch | \(\frac {x^{3}}{3 c^{2}}-\frac {2 b x}{c^{3}}-\frac {b^{2} x}{2 c^{3} \left (c \,x^{2}+b \right )}+\frac {5 \sqrt {-b c}\, b \ln \left (-\sqrt {-b c}\, x +b \right )}{4 c^{4}}-\frac {5 \sqrt {-b c}\, b \ln \left (\sqrt {-b c}\, x +b \right )}{4 c^{4}}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 59, normalized size = 0.89 \begin {gather*} -\frac {b^{2} x}{2 \, {\left (c^{4} x^{2} + b c^{3}\right )}} + \frac {5 \, b^{2} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \, \sqrt {b c} c^{3}} + \frac {c x^{3} - 6 \, b x}{3 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 164, normalized size = 2.48 \begin {gather*} \left [\frac {4 \, c^{2} x^{5} - 20 \, b c x^{3} - 30 \, b^{2} x + 15 \, {\left (b c x^{2} + b^{2}\right )} \sqrt {-\frac {b}{c}} \log \left (\frac {c x^{2} + 2 \, c x \sqrt {-\frac {b}{c}} - b}{c x^{2} + b}\right )}{12 \, {\left (c^{4} x^{2} + b c^{3}\right )}}, \frac {2 \, c^{2} x^{5} - 10 \, b c x^{3} - 15 \, b^{2} x + 15 \, {\left (b c x^{2} + b^{2}\right )} \sqrt {\frac {b}{c}} \arctan \left (\frac {c x \sqrt {\frac {b}{c}}}{b}\right )}{6 \, {\left (c^{4} x^{2} + b c^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 107, normalized size = 1.62 \begin {gather*} - \frac {b^{2} x}{2 b c^{3} + 2 c^{4} x^{2}} - \frac {2 b x}{c^{3}} - \frac {5 \sqrt {- \frac {b^{3}}{c^{7}}} \log {\left (x - \frac {c^{3} \sqrt {- \frac {b^{3}}{c^{7}}}}{b} \right )}}{4} + \frac {5 \sqrt {- \frac {b^{3}}{c^{7}}} \log {\left (x + \frac {c^{3} \sqrt {- \frac {b^{3}}{c^{7}}}}{b} \right )}}{4} + \frac {x^{3}}{3 c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.14, size = 61, normalized size = 0.92 \begin {gather*} \frac {5 \, b^{2} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \, \sqrt {b c} c^{3}} - \frac {b^{2} x}{2 \, {\left (c x^{2} + b\right )} c^{3}} + \frac {c^{4} x^{3} - 6 \, b c^{3} x}{3 \, c^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 56, normalized size = 0.85 \begin {gather*} \frac {x^3}{3\,c^2}+\frac {5\,b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )}{2\,c^{7/2}}-\frac {b^2\,x}{2\,\left (c^4\,x^2+b\,c^3\right )}-\frac {2\,b\,x}{c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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