Optimal. Leaf size=71 \[ -\frac {x^2}{8 c \left (a+c x^4\right )^2}+\frac {x^2}{16 a c \left (a+c x^4\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{3/2} c^{3/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {281, 294, 205,
211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{3/2} c^{3/2}}+\frac {x^2}{16 a c \left (a+c x^4\right )}-\frac {x^2}{8 c \left (a+c x^4\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 211
Rule 281
Rule 294
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a+c x^4\right )^3} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{\left (a+c x^2\right )^3} \, dx,x,x^2\right )\\ &=-\frac {x^2}{8 c \left (a+c x^4\right )^2}+\frac {\text {Subst}\left (\int \frac {1}{\left (a+c x^2\right )^2} \, dx,x,x^2\right )}{8 c}\\ &=-\frac {x^2}{8 c \left (a+c x^4\right )^2}+\frac {x^2}{16 a c \left (a+c x^4\right )}+\frac {\text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{16 a c}\\ &=-\frac {x^2}{8 c \left (a+c x^4\right )^2}+\frac {x^2}{16 a c \left (a+c x^4\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{3/2} c^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 62, normalized size = 0.87 \begin {gather*} \frac {\frac {\sqrt {a} \sqrt {c} x^2 \left (-a+c x^4\right )}{\left (a+c x^4\right )^2}+\tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{3/2} c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 54, normalized size = 0.76
method | result | size |
default | \(\frac {\frac {x^{6}}{8 a}-\frac {x^{2}}{8 c}}{2 \left (x^{4} c +a \right )^{2}}+\frac {\arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{16 a c \sqrt {a c}}\) | \(54\) |
risch | \(\frac {\frac {x^{6}}{16 a}-\frac {x^{2}}{16 c}}{\left (x^{4} c +a \right )^{2}}-\frac {\ln \left (x^{2} \sqrt {-a c}-a \right )}{32 \sqrt {-a c}\, c a}+\frac {\ln \left (x^{2} \sqrt {-a c}+a \right )}{32 \sqrt {-a c}\, c a}\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 66, normalized size = 0.93 \begin {gather*} \frac {c x^{6} - a x^{2}}{16 \, {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )}} + \frac {\arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 199, normalized size = 2.80 \begin {gather*} \left [\frac {2 \, a c^{2} x^{6} - 2 \, a^{2} c x^{2} - {\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^{2} c^{4} x^{8} + 2 \, a^{3} c^{3} x^{4} + a^{4} c^{2}\right )}}, \frac {a c^{2} x^{6} - a^{2} c x^{2} - {\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^{2} c^{4} x^{8} + 2 \, a^{3} c^{3} x^{4} + a^{4} c^{2}\right )}}\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. 116 vs.
\(2 (56) = 112\).
time = 0.22, size = 116, normalized size = 1.63 \begin {gather*} - \frac {\sqrt {- \frac {1}{a^{3} c^{3}}} \log {\left (- a^{2} c \sqrt {- \frac {1}{a^{3} c^{3}}} + x^{2} \right )}}{32} + \frac {\sqrt {- \frac {1}{a^{3} c^{3}}} \log {\left (a^{2} c \sqrt {- \frac {1}{a^{3} c^{3}}} + x^{2} \right )}}{32} + \frac {- a x^{2} + c x^{6}}{16 a^{3} c + 32 a^{2} c^{2} x^{4} + 16 a c^{3} x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.55, size = 54, normalized size = 0.76 \begin {gather*} \frac {\arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a c} + \frac {c x^{6} - a x^{2}}{16 \, {\left (c x^{4} + a\right )}^{2} a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.04, size = 58, normalized size = 0.82 \begin {gather*} \frac {\frac {x^6}{16\,a}-\frac {x^2}{16\,c}}{a^2+2\,a\,c\,x^4+c^2\,x^8}+\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x^2}{\sqrt {a}}\right )}{16\,a^{3/2}\,c^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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