Optimal. Leaf size=52 \[ -\frac {a^2}{8 c^3 \left (a+c x^4\right )^2}+\frac {a}{2 c^3 \left (a+c x^4\right )}+\frac {\log \left (a+c x^4\right )}{4 c^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45}
\begin {gather*} -\frac {a^2}{8 c^3 \left (a+c x^4\right )^2}+\frac {a}{2 c^3 \left (a+c x^4\right )}+\frac {\log \left (a+c x^4\right )}{4 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^{11}}{\left (a+c x^4\right )^3} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {x^2}{(a+c x)^3} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (\frac {a^2}{c^2 (a+c x)^3}-\frac {2 a}{c^2 (a+c x)^2}+\frac {1}{c^2 (a+c x)}\right ) \, dx,x,x^4\right )\\ &=-\frac {a^2}{8 c^3 \left (a+c x^4\right )^2}+\frac {a}{2 c^3 \left (a+c x^4\right )}+\frac {\log \left (a+c x^4\right )}{4 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 39, normalized size = 0.75 \begin {gather*} \frac {\frac {a \left (3 a+4 c x^4\right )}{\left (a+c x^4\right )^2}+2 \log \left (a+c x^4\right )}{8 c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 47, normalized size = 0.90
method | result | size |
norman | \(\frac {\frac {3 a^{2}}{8 c^{3}}+\frac {a \,x^{4}}{2 c^{2}}}{\left (x^{4} c +a \right )^{2}}+\frac {\ln \left (x^{4} c +a \right )}{4 c^{3}}\) | \(43\) |
risch | \(\frac {\frac {3 a^{2}}{8 c^{3}}+\frac {a \,x^{4}}{2 c^{2}}}{\left (x^{4} c +a \right )^{2}}+\frac {\ln \left (x^{4} c +a \right )}{4 c^{3}}\) | \(43\) |
default | \(-\frac {a^{2}}{8 c^{3} \left (x^{4} c +a \right )^{2}}+\frac {a}{2 c^{3} \left (x^{4} c +a \right )}+\frac {\ln \left (x^{4} c +a \right )}{4 c^{3}}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 55, normalized size = 1.06 \begin {gather*} \frac {4 \, a c x^{4} + 3 \, a^{2}}{8 \, {\left (c^{5} x^{8} + 2 \, a c^{4} x^{4} + a^{2} c^{3}\right )}} + \frac {\log \left (c x^{4} + a\right )}{4 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 69, normalized size = 1.33 \begin {gather*} \frac {4 \, a c x^{4} + 3 \, a^{2} + 2 \, {\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \log \left (c x^{4} + a\right )}{8 \, {\left (c^{5} x^{8} + 2 \, a c^{4} x^{4} + a^{2} c^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.26, size = 53, normalized size = 1.02 \begin {gather*} \frac {3 a^{2} + 4 a c x^{4}}{8 a^{2} c^{3} + 16 a c^{4} x^{4} + 8 c^{5} x^{8}} + \frac {\log {\left (a + c x^{4} \right )}}{4 c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.87, size = 42, normalized size = 0.81 \begin {gather*} \frac {\log \left ({\left | c x^{4} + a \right |}\right )}{4 \, c^{3}} - \frac {3 \, c x^{8} + 2 \, a x^{4}}{8 \, {\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 = 0.07, size = 53, normalized size = 1.02 \begin {gather*} \frac {\frac {3\,a^2}{8\,c^3}+\frac {a\,x^4}{2\,c^2}}{a^2+2\,a\,c\,x^4+c^2\,x^8}+\frac {\ln \left (c\,x^4+a\right )}{4\,c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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