Optimal. Leaf size=21 \[ -\frac {\left (a+b x^2\right )^{7/2}}{7 a x^7} \]
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Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {270}
\begin {gather*} -\frac {\left (a+b x^2\right )^{7/2}}{7 a x^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{5/2}}{x^8} \, dx &=-\frac {\left (a+b x^2\right )^{7/2}}{7 a x^7}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 21, normalized size = 1.00 \begin {gather*} -\frac {\left (a+b x^2\right )^{7/2}}{7 a x^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 18, normalized size = 0.86
method | result | size |
gosper | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{7 a \,x^{7}}\) | \(18\) |
default | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{7 a \,x^{7}}\) | \(18\) |
trager | \(-\frac {\left (b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}\right ) \sqrt {b \,x^{2}+a}}{7 a \,x^{7}}\) | \(47\) |
risch | \(-\frac {\left (b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}\right ) \sqrt {b \,x^{2}+a}}{7 a \,x^{7}}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 17, normalized size = 0.81 \begin {gather*} -\frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}}}{7 \, a x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs.
\(2 (17) = 34\).
time = 0.97, size = 46, normalized size = 2.19 \begin {gather*} -\frac {{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \sqrt {b x^{2} + a}}{7 \, a x^{7}} \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. 95 vs.
\(2 (17) = 34\).
time = 0.62, size = 95, normalized size = 4.52 \begin {gather*} - \frac {a^{2} \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{7 x^{6}} - \frac {3 a b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{7 x^{4}} - \frac {3 b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{7 x^{2}} - \frac {b^{\frac {7}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{7 a} \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. 113 vs.
\(2 (17) = 34\).
time = 1.38, size = 113, normalized size = 5.38 \begin {gather*} \frac {2 \, {\left (7 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} b^{\frac {7}{2}} + 35 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{2} b^{\frac {7}{2}} + 21 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{4} b^{\frac {7}{2}} + a^{6} b^{\frac {7}{2}}\right )}}{7 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + 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 = 5.07, size = 71, normalized size = 3.38 \begin {gather*} -\frac {a^2\,\sqrt {b\,x^2+a}}{7\,x^7}-\frac {3\,b^2\,\sqrt {b\,x^2+a}}{7\,x^3}-\frac {b^3\,\sqrt {b\,x^2+a}}{7\,a\,x}-\frac {3\,a\,b\,\sqrt {b\,x^2+a}}{7\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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