Optimal. Leaf size=68 \[ -\frac {\left (a+c x^4\right )^{3/2}}{14 a x^{14}}+\frac {2 c \left (a+c x^4\right )^{3/2}}{35 a^2 x^{10}}-\frac {4 c^2 \left (a+c x^4\right )^{3/2}}{105 a^3 x^6} \]
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Rubi [A]
time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {277, 270}
\begin {gather*} -\frac {4 c^2 \left (a+c x^4\right )^{3/2}}{105 a^3 x^6}+\frac {2 c \left (a+c x^4\right )^{3/2}}{35 a^2 x^{10}}-\frac {\left (a+c x^4\right )^{3/2}}{14 a x^{14}} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 277
Rubi steps
\begin {align*} \int \frac {\sqrt {a+c x^4}}{x^{15}} \, dx &=-\frac {\left (a+c x^4\right )^{3/2}}{14 a x^{14}}-\frac {(4 c) \int \frac {\sqrt {a+c x^4}}{x^{11}} \, dx}{7 a}\\ &=-\frac {\left (a+c x^4\right )^{3/2}}{14 a x^{14}}+\frac {2 c \left (a+c x^4\right )^{3/2}}{35 a^2 x^{10}}+\frac {\left (8 c^2\right ) \int \frac {\sqrt {a+c x^4}}{x^7} \, dx}{35 a^2}\\ &=-\frac {\left (a+c x^4\right )^{3/2}}{14 a x^{14}}+\frac {2 c \left (a+c x^4\right )^{3/2}}{35 a^2 x^{10}}-\frac {4 c^2 \left (a+c x^4\right )^{3/2}}{105 a^3 x^6}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 53, normalized size = 0.78 \begin {gather*} \frac {\sqrt {a+c x^4} \left (-15 a^3-3 a^2 c x^4+4 a c^2 x^8-8 c^3 x^{12}\right )}{210 a^3 x^{14}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 39, normalized size = 0.57
method | result | size |
gosper | \(-\frac {\left (x^{4} c +a \right )^{\frac {3}{2}} \left (8 c^{2} x^{8}-12 a c \,x^{4}+15 a^{2}\right )}{210 x^{14} a^{3}}\) | \(39\) |
default | \(-\frac {\left (x^{4} c +a \right )^{\frac {3}{2}} \left (8 c^{2} x^{8}-12 a c \,x^{4}+15 a^{2}\right )}{210 x^{14} a^{3}}\) | \(39\) |
elliptic | \(-\frac {\left (x^{4} c +a \right )^{\frac {3}{2}} \left (8 c^{2} x^{8}-12 a c \,x^{4}+15 a^{2}\right )}{210 x^{14} a^{3}}\) | \(39\) |
trager | \(-\frac {\left (8 c^{3} x^{12}-4 a \,c^{2} x^{8}+3 a^{2} c \,x^{4}+15 a^{3}\right ) \sqrt {x^{4} c +a}}{210 x^{14} a^{3}}\) | \(50\) |
risch | \(-\frac {\left (8 c^{3} x^{12}-4 a \,c^{2} x^{8}+3 a^{2} c \,x^{4}+15 a^{3}\right ) \sqrt {x^{4} c +a}}{210 x^{14} a^{3}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 52, normalized size = 0.76 \begin {gather*} -\frac {\frac {35 \, {\left (c x^{4} + a\right )}^{\frac {3}{2}} c^{2}}{x^{6}} - \frac {42 \, {\left (c x^{4} + a\right )}^{\frac {5}{2}} c}{x^{10}} + \frac {15 \, {\left (c x^{4} + a\right )}^{\frac {7}{2}}}{x^{14}}}{210 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 49, normalized size = 0.72 \begin {gather*} -\frac {{\left (8 \, c^{3} x^{12} - 4 \, a c^{2} x^{8} + 3 \, a^{2} c x^{4} + 15 \, a^{3}\right )} \sqrt {c x^{4} + a}}{210 \, a^{3} x^{14}} \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. 359 vs.
\(2 (61) = 122\).
time = 0.97, size = 359, normalized size = 5.28 \begin {gather*} - \frac {15 a^{5} c^{\frac {9}{2}} \sqrt {\frac {a}{c x^{4}} + 1}}{210 a^{5} c^{4} x^{12} + 420 a^{4} c^{5} x^{16} + 210 a^{3} c^{6} x^{20}} - \frac {33 a^{4} c^{\frac {11}{2}} x^{4} \sqrt {\frac {a}{c x^{4}} + 1}}{210 a^{5} c^{4} x^{12} + 420 a^{4} c^{5} x^{16} + 210 a^{3} c^{6} x^{20}} - \frac {17 a^{3} c^{\frac {13}{2}} x^{8} \sqrt {\frac {a}{c x^{4}} + 1}}{210 a^{5} c^{4} x^{12} + 420 a^{4} c^{5} x^{16} + 210 a^{3} c^{6} x^{20}} - \frac {3 a^{2} c^{\frac {15}{2}} x^{12} \sqrt {\frac {a}{c x^{4}} + 1}}{210 a^{5} c^{4} x^{12} + 420 a^{4} c^{5} x^{16} + 210 a^{3} c^{6} x^{20}} - \frac {12 a c^{\frac {17}{2}} x^{16} \sqrt {\frac {a}{c x^{4}} + 1}}{210 a^{5} c^{4} x^{12} + 420 a^{4} c^{5} x^{16} + 210 a^{3} c^{6} x^{20}} - \frac {8 c^{\frac {19}{2}} x^{20} \sqrt {\frac {a}{c x^{4}} + 1}}{210 a^{5} c^{4} x^{12} + 420 a^{4} c^{5} x^{16} + 210 a^{3} c^{6} x^{20}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 148 vs.
\(2 (56) = 112\).
time = 0.53, size = 148, normalized size = 2.18 \begin {gather*} \frac {8 \, {\left (70 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{8} c^{\frac {7}{2}} + 35 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{6} a c^{\frac {7}{2}} + 21 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{4} a^{2} c^{\frac {7}{2}} - 7 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} a^{3} c^{\frac {7}{2}} + a^{4} c^{\frac {7}{2}}\right )}}{105 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} - a\right )}^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.51, size = 73, normalized size = 1.07 \begin {gather*} \frac {2\,c^2\,\sqrt {c\,x^4+a}}{105\,a^2\,x^6}-\frac {c\,\sqrt {c\,x^4+a}}{70\,a\,x^{10}}-\frac {4\,c^3\,\sqrt {c\,x^4+a}}{105\,a^3\,x^2}-\frac {\sqrt {c\,x^4+a}}{14\,x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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