Optimal. Leaf size=68 \[ -\frac {x^6}{8 c \left (a+c x^4\right )^2}-\frac {3 x^2}{16 c^2 \left (a+c x^4\right )}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 \sqrt {a} c^{5/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {281, 294, 211}
\begin {gather*} \frac {3 \text {ArcTan}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 \sqrt {a} c^{5/2}}-\frac {3 x^2}{16 c^2 \left (a+c x^4\right )}-\frac {x^6}{8 c \left (a+c x^4\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 281
Rule 294
Rubi steps
\begin {align*} \int \frac {x^9}{\left (a+c x^4\right )^3} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^4}{\left (a+c x^2\right )^3} \, dx,x,x^2\right )\\ &=-\frac {x^6}{8 c \left (a+c x^4\right )^2}+\frac {3 \text {Subst}\left (\int \frac {x^2}{\left (a+c x^2\right )^2} \, dx,x,x^2\right )}{8 c}\\ &=-\frac {x^6}{8 c \left (a+c x^4\right )^2}-\frac {3 x^2}{16 c^2 \left (a+c x^4\right )}+\frac {3 \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{16 c^2}\\ &=-\frac {x^6}{8 c \left (a+c x^4\right )^2}-\frac {3 x^2}{16 c^2 \left (a+c x^4\right )}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 \sqrt {a} c^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 58, normalized size = 0.85 \begin {gather*} \frac {1}{16} \left (\frac {-3 a x^2-5 c x^6}{c^2 \left (a+c x^4\right )^2}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{\sqrt {a} c^{5/2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 52, normalized size = 0.76
method | result | size |
default | \(\frac {-\frac {5 x^{6}}{8 c}-\frac {3 a \,x^{2}}{8 c^{2}}}{2 \left (x^{4} c +a \right )^{2}}+\frac {3 \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{16 c^{2} \sqrt {a c}}\) | \(52\) |
risch | \(\frac {-\frac {5 x^{6}}{16 c}-\frac {3 a \,x^{2}}{16 c^{2}}}{\left (x^{4} c +a \right )^{2}}-\frac {3 \ln \left (x^{2} \sqrt {-a c}-a \right )}{32 \sqrt {-a c}\, c^{2}}+\frac {3 \ln \left (x^{2} \sqrt {-a c}+a \right )}{32 \sqrt {-a c}\, c^{2}}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 63, normalized size = 0.93 \begin {gather*} -\frac {5 \, c x^{6} + 3 \, a x^{2}}{16 \, {\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )}} + \frac {3 \, \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 196, normalized size = 2.88 \begin {gather*} \left [-\frac {10 \, a c^{2} x^{6} + 6 \, a^{2} c x^{2} + 3 \, {\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{4} - 2 \, \sqrt {-a c} x^{2} - a}{c x^{4} + a}\right )}{32 \, {\left (a c^{5} x^{8} + 2 \, a^{2} c^{4} x^{4} + a^{3} c^{3}\right )}}, -\frac {5 \, a c^{2} x^{6} + 3 \, a^{2} c x^{2} + 3 \, {\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c}}{c x^{2}}\right )}{16 \, {\left (a c^{5} x^{8} + 2 \, a^{2} c^{4} x^{4} + a^{3} c^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.28, size = 116, normalized size = 1.71 \begin {gather*} - \frac {3 \sqrt {- \frac {1}{a c^{5}}} \log {\left (- a c^{2} \sqrt {- \frac {1}{a c^{5}}} + x^{2} \right )}}{32} + \frac {3 \sqrt {- \frac {1}{a c^{5}}} \log {\left (a c^{2} \sqrt {- \frac {1}{a c^{5}}} + x^{2} \right )}}{32} + \frac {- 3 a x^{2} - 5 c x^{6}}{16 a^{2} c^{2} + 32 a c^{3} x^{4} + 16 c^{4} x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.76, size = 49, normalized size = 0.72 \begin {gather*} \frac {3 \, \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} c^{2}} - \frac {5 \, c x^{6} + 3 \, a x^{2}}{16 \, {\left (c x^{4} + a\right )}^{2} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.04, size = 60, normalized size = 0.88 \begin {gather*} \frac {3\,\mathrm {atan}\left (\frac {\sqrt {c}\,x^2}{\sqrt {a}}\right )}{16\,\sqrt {a}\,c^{5/2}}-\frac {\frac {5\,x^6}{16\,c}+\frac {3\,a\,x^2}{16\,c^2}}{a^2+2\,a\,c\,x^4+c^2\,x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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