Optimal. Leaf size=68 \[ -\frac {3 c \sqrt {a+c x^4}}{16 x^4}-\frac {\left (a+c x^4\right )^{3/2}}{8 x^8}-\frac {3 c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}} \]
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Rubi [A]
time = 0.03, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 43, 65,
214} \begin {gather*} -\frac {3 c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}}-\frac {3 c \sqrt {a+c x^4}}{16 x^4}-\frac {\left (a+c x^4\right )^{3/2}}{8 x^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 214
Rule 272
Rubi steps
\begin {align*} \int \frac {\left (a+c x^4\right )^{3/2}}{x^9} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {(a+c x)^{3/2}}{x^3} \, dx,x,x^4\right )\\ &=-\frac {\left (a+c x^4\right )^{3/2}}{8 x^8}+\frac {1}{16} (3 c) \text {Subst}\left (\int \frac {\sqrt {a+c x}}{x^2} \, dx,x,x^4\right )\\ &=-\frac {3 c \sqrt {a+c x^4}}{16 x^4}-\frac {\left (a+c x^4\right )^{3/2}}{8 x^8}+\frac {1}{32} \left (3 c^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^4\right )\\ &=-\frac {3 c \sqrt {a+c x^4}}{16 x^4}-\frac {\left (a+c x^4\right )^{3/2}}{8 x^8}+\frac {1}{16} (3 c) \text {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^4}\right )\\ &=-\frac {3 c \sqrt {a+c x^4}}{16 x^4}-\frac {\left (a+c x^4\right )^{3/2}}{8 x^8}-\frac {3 c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 59, normalized size = 0.87 \begin {gather*} \frac {\left (-2 a-5 c x^4\right ) \sqrt {a+c x^4}}{16 x^8}-\frac {3 c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 63, normalized size = 0.93
method | result | size |
risch | \(-\frac {\sqrt {x^{4} c +a}\, \left (5 x^{4} c +2 a \right )}{16 x^{8}}-\frac {3 c^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {x^{4} c +a}}{x^{2}}\right )}{16 \sqrt {a}}\) | \(57\) |
default | \(-\frac {3 c^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {x^{4} c +a}}{x^{2}}\right )}{16 \sqrt {a}}-\frac {a \sqrt {x^{4} c +a}}{8 x^{8}}-\frac {5 c \sqrt {x^{4} c +a}}{16 x^{4}}\) | \(63\) |
elliptic | \(-\frac {3 c^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {x^{4} c +a}}{x^{2}}\right )}{16 \sqrt {a}}-\frac {a \sqrt {x^{4} c +a}}{8 x^{8}}-\frac {5 c \sqrt {x^{4} c +a}}{16 x^{4}}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 98, normalized size = 1.44 \begin {gather*} \frac {3 \, c^{2} \log \left (\frac {\sqrt {c x^{4} + a} - \sqrt {a}}{\sqrt {c x^{4} + a} + \sqrt {a}}\right )}{32 \, \sqrt {a}} - \frac {5 \, {\left (c x^{4} + a\right )}^{\frac {3}{2}} c^{2} - 3 \, \sqrt {c x^{4} + a} a c^{2}}{16 \, {\left ({\left (c x^{4} + a\right )}^{2} - 2 \, {\left (c x^{4} + a\right )} a + a^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 138, normalized size = 2.03 \begin {gather*} \left [\frac {3 \, \sqrt {a} c^{2} x^{8} \log \left (\frac {c x^{4} - 2 \, \sqrt {c x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) - 2 \, {\left (5 \, a c x^{4} + 2 \, a^{2}\right )} \sqrt {c x^{4} + a}}{32 \, a x^{8}}, \frac {3 \, \sqrt {-a} c^{2} x^{8} \arctan \left (\frac {\sqrt {c x^{4} + a} \sqrt {-a}}{a}\right ) - {\left (5 \, a c x^{4} + 2 \, a^{2}\right )} \sqrt {c x^{4} + a}}{16 \, a x^{8}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.73, size = 75, normalized size = 1.10 \begin {gather*} - \frac {a \sqrt {c} \sqrt {\frac {a}{c x^{4}} + 1}}{8 x^{6}} - \frac {5 c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{4}} + 1}}{16 x^{2}} - \frac {3 c^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x^{2}} \right )}}{16 \sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.72, size = 70, normalized size = 1.03 \begin {gather*} \frac {\frac {3 \, c^{3} \arctan \left (\frac {\sqrt {c x^{4} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {5 \, {\left (c x^{4} + a\right )}^{\frac {3}{2}} c^{3} - 3 \, \sqrt {c x^{4} + a} a c^{3}}{c^{2} x^{8}}}{16 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.49, size = 52, normalized size = 0.76 \begin {gather*} \frac {3\,a\,\sqrt {c\,x^4+a}}{16\,x^8}-\frac {3\,c^2\,\mathrm {atanh}\left (\frac {\sqrt {c\,x^4+a}}{\sqrt {a}}\right )}{16\,\sqrt {a}}-\frac {5\,{\left (c\,x^4+a\right )}^{3/2}}{16\,x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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