Optimal. Leaf size=44 \[ -\frac {\left (a+b x^2\right )^{11/2}}{13 a x^{13}}+\frac {2 b \left (a+b x^2\right )^{11/2}}{143 a^2 x^{11}} \]
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Rubi [A]
time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 2, 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 {2 b \left (a+b x^2\right )^{11/2}}{143 a^2 x^{11}}-\frac {\left (a+b x^2\right )^{11/2}}{13 a x^{13}} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 277
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{9/2}}{x^{14}} \, dx &=-\frac {\left (a+b x^2\right )^{11/2}}{13 a x^{13}}-\frac {(2 b) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{12}} \, dx}{13 a}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{13 a x^{13}}+\frac {2 b \left (a+b x^2\right )^{11/2}}{143 a^2 x^{11}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 31, normalized size = 0.70 \begin {gather*} \frac {\left (a+b x^2\right )^{11/2} \left (-11 a+2 b x^2\right )}{143 a^2 x^{13}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 37, normalized size = 0.84
method | result | size |
gosper | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (-2 b \,x^{2}+11 a \right )}{143 x^{13} a^{2}}\) | \(28\) |
default | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{13 a \,x^{13}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {11}{2}}}{143 a^{2} x^{11}}\) | \(37\) |
trager | \(-\frac {\left (-2 b^{6} x^{12}+a \,b^{5} x^{10}+35 a^{2} b^{4} x^{8}+90 a^{3} x^{6} b^{3}+100 a^{4} b^{2} x^{4}+53 a^{5} b \,x^{2}+11 a^{6}\right ) \sqrt {b \,x^{2}+a}}{143 x^{13} a^{2}}\) | \(82\) |
risch | \(-\frac {\left (-2 b^{6} x^{12}+a \,b^{5} x^{10}+35 a^{2} b^{4} x^{8}+90 a^{3} x^{6} b^{3}+100 a^{4} b^{2} x^{4}+53 a^{5} b \,x^{2}+11 a^{6}\right ) \sqrt {b \,x^{2}+a}}{143 x^{13} a^{2}}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 36, normalized size = 0.82 \begin {gather*} \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b}{143 \, a^{2} x^{11}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{13 \, a x^{13}} \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. 82 vs.
\(2 (36) = 72\).
time = 1.53, size = 82, normalized size = 1.86 \begin {gather*} \frac {{\left (2 \, b^{6} x^{12} - a b^{5} x^{10} - 35 \, a^{2} b^{4} x^{8} - 90 \, a^{3} b^{3} x^{6} - 100 \, a^{4} b^{2} x^{4} - 53 \, a^{5} b x^{2} - 11 \, a^{6}\right )} \sqrt {b x^{2} + a}}{143 \, a^{2} x^{13}} \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. 175 vs.
\(2 (37) = 74\).
time = 1.65, size = 175, normalized size = 3.98 \begin {gather*} - \frac {a^{4} \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{13 x^{12}} - \frac {53 a^{3} b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{143 x^{10}} - \frac {100 a^{2} b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{143 x^{8}} - \frac {90 a b^{\frac {7}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{143 x^{6}} - \frac {35 b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{143 x^{4}} - \frac {b^{\frac {11}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{143 a x^{2}} + \frac {2 b^{\frac {13}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{143 a^{2}} \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. 328 vs.
\(2 (36) = 72\).
time = 0.71, size = 328, normalized size = 7.45 \begin {gather*} \frac {4 \, {\left (143 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{22} b^{\frac {13}{2}} + 429 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{20} a b^{\frac {13}{2}} + 2145 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{18} a^{2} b^{\frac {13}{2}} + 3003 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} a^{3} b^{\frac {13}{2}} + 6006 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a^{4} b^{\frac {13}{2}} + 4290 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{5} b^{\frac {13}{2}} + 4290 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{6} b^{\frac {13}{2}} + 1430 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{7} b^{\frac {13}{2}} + 715 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{8} b^{\frac {13}{2}} + 65 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{9} b^{\frac {13}{2}} + 13 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{10} b^{\frac {13}{2}} - a^{11} b^{\frac {13}{2}}\right )}}{143 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{13}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.85, size = 131, normalized size = 2.98 \begin {gather*} \frac {2\,b^6\,\sqrt {b\,x^2+a}}{143\,a^2\,x}-\frac {35\,b^4\,\sqrt {b\,x^2+a}}{143\,x^5}-\frac {90\,a\,b^3\,\sqrt {b\,x^2+a}}{143\,x^7}-\frac {53\,a^3\,b\,\sqrt {b\,x^2+a}}{143\,x^{11}}-\frac {b^5\,\sqrt {b\,x^2+a}}{143\,a\,x^3}-\frac {a^4\,\sqrt {b\,x^2+a}}{13\,x^{13}}-\frac {100\,a^2\,b^2\,\sqrt {b\,x^2+a}}{143\,x^9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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