Optimal. Leaf size=140 \[ -\frac {\left (a+b x^2\right )^{7/2}}{17 a x^{17}}+\frac {2 b \left (a+b x^2\right )^{7/2}}{51 a^2 x^{15}}-\frac {16 b^2 \left (a+b x^2\right )^{7/2}}{663 a^3 x^{13}}+\frac {32 b^3 \left (a+b x^2\right )^{7/2}}{2431 a^4 x^{11}}-\frac {128 b^4 \left (a+b x^2\right )^{7/2}}{21879 a^5 x^9}+\frac {256 b^5 \left (a+b x^2\right )^{7/2}}{153153 a^6 x^7} \]
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Rubi [A]
time = 0.04, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {256 b^5 \left (a+b x^2\right )^{7/2}}{153153 a^6 x^7}-\frac {128 b^4 \left (a+b x^2\right )^{7/2}}{21879 a^5 x^9}+\frac {32 b^3 \left (a+b x^2\right )^{7/2}}{2431 a^4 x^{11}}-\frac {16 b^2 \left (a+b x^2\right )^{7/2}}{663 a^3 x^{13}}+\frac {2 b \left (a+b x^2\right )^{7/2}}{51 a^2 x^{15}}-\frac {\left (a+b x^2\right )^{7/2}}{17 a x^{17}} \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^{18}} \, dx &=-\frac {\left (a+b x^2\right )^{7/2}}{17 a x^{17}}-\frac {(10 b) \int \frac {\left (a+b x^2\right )^{5/2}}{x^{16}} \, dx}{17 a}\\ &=-\frac {\left (a+b x^2\right )^{7/2}}{17 a x^{17}}+\frac {2 b \left (a+b x^2\right )^{7/2}}{51 a^2 x^{15}}+\frac {\left (16 b^2\right ) \int \frac {\left (a+b x^2\right )^{5/2}}{x^{14}} \, dx}{51 a^2}\\ &=-\frac {\left (a+b x^2\right )^{7/2}}{17 a x^{17}}+\frac {2 b \left (a+b x^2\right )^{7/2}}{51 a^2 x^{15}}-\frac {16 b^2 \left (a+b x^2\right )^{7/2}}{663 a^3 x^{13}}-\frac {\left (32 b^3\right ) \int \frac {\left (a+b x^2\right )^{5/2}}{x^{12}} \, dx}{221 a^3}\\ &=-\frac {\left (a+b x^2\right )^{7/2}}{17 a x^{17}}+\frac {2 b \left (a+b x^2\right )^{7/2}}{51 a^2 x^{15}}-\frac {16 b^2 \left (a+b x^2\right )^{7/2}}{663 a^3 x^{13}}+\frac {32 b^3 \left (a+b x^2\right )^{7/2}}{2431 a^4 x^{11}}+\frac {\left (128 b^4\right ) \int \frac {\left (a+b x^2\right )^{5/2}}{x^{10}} \, dx}{2431 a^4}\\ &=-\frac {\left (a+b x^2\right )^{7/2}}{17 a x^{17}}+\frac {2 b \left (a+b x^2\right )^{7/2}}{51 a^2 x^{15}}-\frac {16 b^2 \left (a+b x^2\right )^{7/2}}{663 a^3 x^{13}}+\frac {32 b^3 \left (a+b x^2\right )^{7/2}}{2431 a^4 x^{11}}-\frac {128 b^4 \left (a+b x^2\right )^{7/2}}{21879 a^5 x^9}-\frac {\left (256 b^5\right ) \int \frac {\left (a+b x^2\right )^{5/2}}{x^8} \, dx}{21879 a^5}\\ &=-\frac {\left (a+b x^2\right )^{7/2}}{17 a x^{17}}+\frac {2 b \left (a+b x^2\right )^{7/2}}{51 a^2 x^{15}}-\frac {16 b^2 \left (a+b x^2\right )^{7/2}}{663 a^3 x^{13}}+\frac {32 b^3 \left (a+b x^2\right )^{7/2}}{2431 a^4 x^{11}}-\frac {128 b^4 \left (a+b x^2\right )^{7/2}}{21879 a^5 x^9}+\frac {256 b^5 \left (a+b x^2\right )^{7/2}}{153153 a^6 x^7}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 75, normalized size = 0.54 \begin {gather*} \frac {\left (a+b x^2\right )^{7/2} \left (-9009 a^5+6006 a^4 b x^2-3696 a^3 b^2 x^4+2016 a^2 b^3 x^6-896 a b^4 x^8+256 b^5 x^{10}\right )}{153153 a^6 x^{17}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.99, size = 133, normalized size = 0.95
method | result | size |
gosper | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (-256 b^{5} x^{10}+896 a \,b^{4} x^{8}-2016 a^{2} b^{3} x^{6}+3696 a^{3} b^{2} x^{4}-6006 a^{4} b \,x^{2}+9009 a^{5}\right )}{153153 x^{17} a^{6}}\) | \(72\) |
trager | \(-\frac {\left (-256 b^{8} x^{16}+128 a \,b^{7} x^{14}-96 a^{2} b^{6} x^{12}+80 a^{3} b^{5} x^{10}-70 a^{4} b^{4} x^{8}+63 a^{5} b^{3} x^{6}+12705 a^{6} b^{2} x^{4}+21021 a^{7} b \,x^{2}+9009 a^{8}\right ) \sqrt {b \,x^{2}+a}}{153153 x^{17} a^{6}}\) | \(105\) |
risch | \(-\frac {\left (-256 b^{8} x^{16}+128 a \,b^{7} x^{14}-96 a^{2} b^{6} x^{12}+80 a^{3} b^{5} x^{10}-70 a^{4} b^{4} x^{8}+63 a^{5} b^{3} x^{6}+12705 a^{6} b^{2} x^{4}+21021 a^{7} b \,x^{2}+9009 a^{8}\right ) \sqrt {b \,x^{2}+a}}{153153 x^{17} a^{6}}\) | \(105\) |
default | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{17 a \,x^{17}}-\frac {10 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{15 a \,x^{15}}-\frac {8 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{13 a \,x^{13}}-\frac {6 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{11 a \,x^{11}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{9 a \,x^{9}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{63 a^{2} x^{7}}\right )}{11 a}\right )}{13 a}\right )}{15 a}\right )}{17 a}\) | \(133\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 116, normalized size = 0.83 \begin {gather*} \frac {256 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{5}}{153153 \, a^{6} x^{7}} - \frac {128 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}}{21879 \, a^{5} x^{9}} + \frac {32 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}}{2431 \, a^{4} x^{11}} - \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}}{663 \, a^{3} x^{13}} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b}{51 \, a^{2} x^{15}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}}}{17 \, a x^{17}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.15, size = 104, normalized size = 0.74 \begin {gather*} \frac {{\left (256 \, b^{8} x^{16} - 128 \, a b^{7} x^{14} + 96 \, a^{2} b^{6} x^{12} - 80 \, a^{3} b^{5} x^{10} + 70 \, a^{4} b^{4} x^{8} - 63 \, a^{5} b^{3} x^{6} - 12705 \, a^{6} b^{2} x^{4} - 21021 \, a^{7} b x^{2} - 9009 \, a^{8}\right )} \sqrt {b x^{2} + a}}{153153 \, a^{6} x^{17}} \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. 1346 vs.
\(2 (133) = 266\).
time = 2.36, size = 1346, normalized size = 9.61 \begin {gather*} - \frac {9009 a^{13} b^{\frac {51}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{153153 a^{11} b^{25} x^{16} + 765765 a^{10} b^{26} x^{18} + 1531530 a^{9} b^{27} x^{20} + 1531530 a^{8} b^{28} x^{22} + 765765 a^{7} b^{29} x^{24} + 153153 a^{6} b^{30} x^{26}} - \frac {66066 a^{12} b^{\frac {53}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{153153 a^{11} b^{25} x^{16} + 765765 a^{10} b^{26} x^{18} + 1531530 a^{9} b^{27} x^{20} + 1531530 a^{8} b^{28} x^{22} + 765765 a^{7} b^{29} x^{24} + 153153 a^{6} b^{30} x^{26}} - \frac {207900 a^{11} b^{\frac {55}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{153153 a^{11} b^{25} x^{16} + 765765 a^{10} b^{26} x^{18} + 1531530 a^{9} b^{27} x^{20} + 1531530 a^{8} b^{28} x^{22} + 765765 a^{7} b^{29} x^{24} + 153153 a^{6} b^{30} x^{26}} - \frac {363888 a^{10} b^{\frac {57}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{153153 a^{11} b^{25} x^{16} + 765765 a^{10} b^{26} x^{18} + 1531530 a^{9} b^{27} x^{20} + 1531530 a^{8} b^{28} x^{22} + 765765 a^{7} b^{29} x^{24} + 153153 a^{6} b^{30} x^{26}} - \frac {382550 a^{9} b^{\frac {59}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{153153 a^{11} b^{25} x^{16} + 765765 a^{10} b^{26} x^{18} + 1531530 a^{9} b^{27} x^{20} + 1531530 a^{8} b^{28} x^{22} + 765765 a^{7} b^{29} x^{24} + 153153 a^{6} b^{30} x^{26}} - \frac {241524 a^{8} b^{\frac {61}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{153153 a^{11} b^{25} x^{16} + 765765 a^{10} b^{26} x^{18} + 1531530 a^{9} b^{27} x^{20} + 1531530 a^{8} b^{28} x^{22} + 765765 a^{7} b^{29} x^{24} + 153153 a^{6} b^{30} x^{26}} - \frac {84780 a^{7} b^{\frac {63}{2}} x^{12} \sqrt {\frac {a}{b x^{2}} + 1}}{153153 a^{11} b^{25} x^{16} + 765765 a^{10} b^{26} x^{18} + 1531530 a^{9} b^{27} x^{20} + 1531530 a^{8} b^{28} x^{22} + 765765 a^{7} b^{29} x^{24} + 153153 a^{6} b^{30} x^{26}} - \frac {12768 a^{6} b^{\frac {65}{2}} x^{14} \sqrt {\frac {a}{b x^{2}} + 1}}{153153 a^{11} b^{25} x^{16} + 765765 a^{10} b^{26} x^{18} + 1531530 a^{9} b^{27} x^{20} + 1531530 a^{8} b^{28} x^{22} + 765765 a^{7} b^{29} x^{24} + 153153 a^{6} b^{30} x^{26}} + \frac {63 a^{5} b^{\frac {67}{2}} x^{16} \sqrt {\frac {a}{b x^{2}} + 1}}{153153 a^{11} b^{25} x^{16} + 765765 a^{10} b^{26} x^{18} + 1531530 a^{9} b^{27} x^{20} + 1531530 a^{8} b^{28} x^{22} + 765765 a^{7} b^{29} x^{24} + 153153 a^{6} b^{30} x^{26}} + \frac {630 a^{4} b^{\frac {69}{2}} x^{18} \sqrt {\frac {a}{b x^{2}} + 1}}{153153 a^{11} b^{25} x^{16} + 765765 a^{10} b^{26} x^{18} + 1531530 a^{9} b^{27} x^{20} + 1531530 a^{8} b^{28} x^{22} + 765765 a^{7} b^{29} x^{24} + 153153 a^{6} b^{30} x^{26}} + \frac {1680 a^{3} b^{\frac {71}{2}} x^{20} \sqrt {\frac {a}{b x^{2}} + 1}}{153153 a^{11} b^{25} x^{16} + 765765 a^{10} b^{26} x^{18} + 1531530 a^{9} b^{27} x^{20} + 1531530 a^{8} b^{28} x^{22} + 765765 a^{7} b^{29} x^{24} + 153153 a^{6} b^{30} x^{26}} + \frac {2016 a^{2} b^{\frac {73}{2}} x^{22} \sqrt {\frac {a}{b x^{2}} + 1}}{153153 a^{11} b^{25} x^{16} + 765765 a^{10} b^{26} x^{18} + 1531530 a^{9} b^{27} x^{20} + 1531530 a^{8} b^{28} x^{22} + 765765 a^{7} b^{29} x^{24} + 153153 a^{6} b^{30} x^{26}} + \frac {1152 a b^{\frac {75}{2}} x^{24} \sqrt {\frac {a}{b x^{2}} + 1}}{153153 a^{11} b^{25} x^{16} + 765765 a^{10} b^{26} x^{18} + 1531530 a^{9} b^{27} x^{20} + 1531530 a^{8} b^{28} x^{22} + 765765 a^{7} b^{29} x^{24} + 153153 a^{6} b^{30} x^{26}} + \frac {256 b^{\frac {77}{2}} x^{26} \sqrt {\frac {a}{b x^{2}} + 1}}{153153 a^{11} b^{25} x^{16} + 765765 a^{10} b^{26} x^{18} + 1531530 a^{9} b^{27} x^{20} + 1531530 a^{8} b^{28} x^{22} + 765765 a^{7} b^{29} x^{24} + 153153 a^{6} b^{30} x^{26}} \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.
\(2 (116) = 232\).
time = 1.54, size = 328, normalized size = 2.34 \begin {gather*} \frac {512 \, {\left (102102 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{22} b^{\frac {17}{2}} + 364650 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{20} a b^{\frac {17}{2}} + 692835 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{18} a^{2} b^{\frac {17}{2}} + 668525 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} a^{3} b^{\frac {17}{2}} + 384098 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a^{4} b^{\frac {17}{2}} + 89726 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{5} b^{\frac {17}{2}} + 6188 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{6} b^{\frac {17}{2}} - 2380 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{7} b^{\frac {17}{2}} + 680 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{8} b^{\frac {17}{2}} - 136 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{9} b^{\frac {17}{2}} + 17 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{10} b^{\frac {17}{2}} - a^{11} b^{\frac {17}{2}}\right )}}{153153 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{17}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.87, size = 171, normalized size = 1.22 \begin {gather*} \frac {10\,b^4\,\sqrt {b\,x^2+a}}{21879\,a^2\,x^9}-\frac {55\,b^2\,\sqrt {b\,x^2+a}}{663\,x^{13}}-\frac {b^3\,\sqrt {b\,x^2+a}}{2431\,a\,x^{11}}-\frac {a^2\,\sqrt {b\,x^2+a}}{17\,x^{17}}-\frac {80\,b^5\,\sqrt {b\,x^2+a}}{153153\,a^3\,x^7}+\frac {32\,b^6\,\sqrt {b\,x^2+a}}{51051\,a^4\,x^5}-\frac {128\,b^7\,\sqrt {b\,x^2+a}}{153153\,a^5\,x^3}+\frac {256\,b^8\,\sqrt {b\,x^2+a}}{153153\,a^6\,x}-\frac {7\,a\,b\,\sqrt {b\,x^2+a}}{51\,x^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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