Optimal. Leaf size=92 \[ -\frac {\left (a+b x^2\right )^{7/2}}{13 a x^{13}}+\frac {6 b \left (a+b x^2\right )^{7/2}}{143 a^2 x^{11}}-\frac {8 b^2 \left (a+b x^2\right )^{7/2}}{429 a^3 x^9}+\frac {16 b^3 \left (a+b x^2\right )^{7/2}}{3003 a^4 x^7} \]
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Rubi [A]
time = 0.02, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 4, 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 {16 b^3 \left (a+b x^2\right )^{7/2}}{3003 a^4 x^7}-\frac {8 b^2 \left (a+b x^2\right )^{7/2}}{429 a^3 x^9}+\frac {6 b \left (a+b x^2\right )^{7/2}}{143 a^2 x^{11}}-\frac {\left (a+b x^2\right )^{7/2}}{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 )^{5/2}}{x^{14}} \, dx &=-\frac {\left (a+b x^2\right )^{7/2}}{13 a x^{13}}-\frac {(6 b) \int \frac {\left (a+b x^2\right )^{5/2}}{x^{12}} \, dx}{13 a}\\ &=-\frac {\left (a+b x^2\right )^{7/2}}{13 a x^{13}}+\frac {6 b \left (a+b x^2\right )^{7/2}}{143 a^2 x^{11}}+\frac {\left (24 b^2\right ) \int \frac {\left (a+b x^2\right )^{5/2}}{x^{10}} \, dx}{143 a^2}\\ &=-\frac {\left (a+b x^2\right )^{7/2}}{13 a x^{13}}+\frac {6 b \left (a+b x^2\right )^{7/2}}{143 a^2 x^{11}}-\frac {8 b^2 \left (a+b x^2\right )^{7/2}}{429 a^3 x^9}-\frac {\left (16 b^3\right ) \int \frac {\left (a+b x^2\right )^{5/2}}{x^8} \, dx}{429 a^3}\\ &=-\frac {\left (a+b x^2\right )^{7/2}}{13 a x^{13}}+\frac {6 b \left (a+b x^2\right )^{7/2}}{143 a^2 x^{11}}-\frac {8 b^2 \left (a+b x^2\right )^{7/2}}{429 a^3 x^9}+\frac {16 b^3 \left (a+b x^2\right )^{7/2}}{3003 a^4 x^7}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 53, normalized size = 0.58 \begin {gather*} \frac {\left (a+b x^2\right )^{7/2} \left (-231 a^3+126 a^2 b x^2-56 a b^2 x^4+16 b^3 x^6\right )}{3003 a^4 x^{13}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 85, normalized size = 0.92
method | result | size |
gosper | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (-16 b^{3} x^{6}+56 a \,b^{2} x^{4}-126 a^{2} b \,x^{2}+231 a^{3}\right )}{3003 x^{13} a^{4}}\) | \(50\) |
trager | \(-\frac {\left (-16 b^{6} x^{12}+8 a \,b^{5} x^{10}-6 a^{2} b^{4} x^{8}+5 a^{3} x^{6} b^{3}+371 a^{4} b^{2} x^{4}+567 a^{5} b \,x^{2}+231 a^{6}\right ) \sqrt {b \,x^{2}+a}}{3003 x^{13} a^{4}}\) | \(83\) |
risch | \(-\frac {\left (-16 b^{6} x^{12}+8 a \,b^{5} x^{10}-6 a^{2} b^{4} x^{8}+5 a^{3} x^{6} b^{3}+371 a^{4} b^{2} x^{4}+567 a^{5} b \,x^{2}+231 a^{6}\right ) \sqrt {b \,x^{2}+a}}{3003 x^{13} a^{4}}\) | \(83\) |
default | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{13 a \,x^{13}}-\frac {6 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{11 a \,x^{11}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{9 a \,x^{9}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{63 a^{2} x^{7}}\right )}{11 a}\right )}{13 a}\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 76, normalized size = 0.83 \begin {gather*} \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}}{3003 \, a^{4} x^{7}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}}{429 \, a^{3} x^{9}} + \frac {6 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b}{143 \, a^{2} x^{11}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}}}{13 \, a x^{13}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.17, size = 82, normalized size = 0.89 \begin {gather*} \frac {{\left (16 \, b^{6} x^{12} - 8 \, a b^{5} x^{10} + 6 \, a^{2} b^{4} x^{8} - 5 \, a^{3} b^{3} x^{6} - 371 \, a^{4} b^{2} x^{4} - 567 \, a^{5} b x^{2} - 231 \, a^{6}\right )} \sqrt {b x^{2} + a}}{3003 \, a^{4} 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. 721 vs.
\(2 (85) = 170\).
time = 1.42, size = 721, normalized size = 7.84 \begin {gather*} - \frac {231 a^{9} b^{\frac {19}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} - \frac {1260 a^{8} b^{\frac {21}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} - \frac {2765 a^{7} b^{\frac {23}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} - \frac {3050 a^{6} b^{\frac {25}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} - \frac {1689 a^{5} b^{\frac {27}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} - \frac {376 a^{4} b^{\frac {29}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} + \frac {5 a^{3} b^{\frac {31}{2}} x^{12} \sqrt {\frac {a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} + \frac {30 a^{2} b^{\frac {33}{2}} x^{14} \sqrt {\frac {a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} + \frac {40 a b^{\frac {35}{2}} x^{16} \sqrt {\frac {a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} + \frac {16 b^{\frac {37}{2}} x^{18} \sqrt {\frac {a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} \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. 274 vs.
\(2 (76) = 152\).
time = 0.73, size = 274, normalized size = 2.98 \begin {gather*} \frac {32 \, {\left (3003 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{18} b^{\frac {13}{2}} + 9009 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} a b^{\frac {13}{2}} + 18018 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a^{2} b^{\frac {13}{2}} + 16302 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{3} b^{\frac {13}{2}} + 10296 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{4} b^{\frac {13}{2}} + 2288 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{5} b^{\frac {13}{2}} + 286 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{6} b^{\frac {13}{2}} - 78 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{7} b^{\frac {13}{2}} + 13 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{8} b^{\frac {13}{2}} - a^{9} b^{\frac {13}{2}}\right )}}{3003 \, {\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 = 5.98, size = 131, normalized size = 1.42 \begin {gather*} \frac {2\,b^4\,\sqrt {b\,x^2+a}}{1001\,a^2\,x^5}-\frac {53\,b^2\,\sqrt {b\,x^2+a}}{429\,x^9}-\frac {5\,b^3\,\sqrt {b\,x^2+a}}{3003\,a\,x^7}-\frac {a^2\,\sqrt {b\,x^2+a}}{13\,x^{13}}-\frac {8\,b^5\,\sqrt {b\,x^2+a}}{3003\,a^3\,x^3}+\frac {16\,b^6\,\sqrt {b\,x^2+a}}{3003\,a^4\,x}-\frac {27\,a\,b\,\sqrt {b\,x^2+a}}{143\,x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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