Optimal. Leaf size=132 \[ -\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}}+\frac {10 b}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac {80 b^2 x}{21 a^3 \left (a+b x^2\right )^{7/2}}+\frac {32 b^2 x}{7 a^4 \left (a+b x^2\right )^{5/2}}+\frac {128 b^2 x}{21 a^5 \left (a+b x^2\right )^{3/2}}+\frac {256 b^2 x}{21 a^6 \sqrt {a+b x^2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {277, 198, 197}
\begin {gather*} \frac {256 b^2 x}{21 a^6 \sqrt {a+b x^2}}+\frac {128 b^2 x}{21 a^5 \left (a+b x^2\right )^{3/2}}+\frac {32 b^2 x}{7 a^4 \left (a+b x^2\right )^{5/2}}+\frac {80 b^2 x}{21 a^3 \left (a+b x^2\right )^{7/2}}+\frac {10 b}{3 a^2 x \left (a+b x^2\right )^{7/2}}-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 198
Rule 277
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+b x^2\right )^{9/2}} \, dx &=-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}}-\frac {(10 b) \int \frac {1}{x^2 \left (a+b x^2\right )^{9/2}} \, dx}{3 a}\\ &=-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}}+\frac {10 b}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac {\left (80 b^2\right ) \int \frac {1}{\left (a+b x^2\right )^{9/2}} \, dx}{3 a^2}\\ &=-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}}+\frac {10 b}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac {80 b^2 x}{21 a^3 \left (a+b x^2\right )^{7/2}}+\frac {\left (160 b^2\right ) \int \frac {1}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a^3}\\ &=-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}}+\frac {10 b}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac {80 b^2 x}{21 a^3 \left (a+b x^2\right )^{7/2}}+\frac {32 b^2 x}{7 a^4 \left (a+b x^2\right )^{5/2}}+\frac {\left (128 b^2\right ) \int \frac {1}{\left (a+b x^2\right )^{5/2}} \, dx}{7 a^4}\\ &=-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}}+\frac {10 b}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac {80 b^2 x}{21 a^3 \left (a+b x^2\right )^{7/2}}+\frac {32 b^2 x}{7 a^4 \left (a+b x^2\right )^{5/2}}+\frac {128 b^2 x}{21 a^5 \left (a+b x^2\right )^{3/2}}+\frac {\left (256 b^2\right ) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{21 a^5}\\ &=-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}}+\frac {10 b}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac {80 b^2 x}{21 a^3 \left (a+b x^2\right )^{7/2}}+\frac {32 b^2 x}{7 a^4 \left (a+b x^2\right )^{5/2}}+\frac {128 b^2 x}{21 a^5 \left (a+b x^2\right )^{3/2}}+\frac {256 b^2 x}{21 a^6 \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 75, normalized size = 0.57 \begin {gather*} \frac {-7 a^5+70 a^4 b x^2+560 a^3 b^2 x^4+1120 a^2 b^3 x^6+896 a b^4 x^8+256 b^5 x^{10}}{21 a^6 x^3 \left (a+b x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 122, normalized size = 0.92
method | result | size |
gosper | \(-\frac {-256 b^{5} x^{10}-896 a \,b^{4} x^{8}-1120 a^{2} b^{3} x^{6}-560 a^{3} b^{2} x^{4}-70 a^{4} b \,x^{2}+7 a^{5}}{21 x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{6}}\) | \(72\) |
trager | \(-\frac {-256 b^{5} x^{10}-896 a \,b^{4} x^{8}-1120 a^{2} b^{3} x^{6}-560 a^{3} b^{2} x^{4}-70 a^{4} b \,x^{2}+7 a^{5}}{21 x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{6}}\) | \(72\) |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-14 b \,x^{2}+a \right )}{3 a^{6} x^{3}}+\frac {\sqrt {b \,x^{2}+a}\, x \left (158 b^{3} x^{6}+511 a \,b^{2} x^{4}+560 a^{2} b \,x^{2}+210 a^{3}\right ) b^{2}}{21 a^{6} \left (b^{4} x^{8}+4 a \,b^{3} x^{6}+6 a^{2} b^{2} x^{4}+4 a^{3} b \,x^{2}+a^{4}\right )}\) | \(119\) |
default | \(-\frac {1}{3 a \,x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {10 b \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {8 b \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{a}\right )}{3 a}\) | \(122\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 108, normalized size = 0.82 \begin {gather*} \frac {256 \, b^{2} x}{21 \, \sqrt {b x^{2} + a} a^{6}} + \frac {128 \, b^{2} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{5}} + \frac {32 \, b^{2} x}{7 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{4}} + \frac {80 \, b^{2} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3}} + \frac {10 \, b}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x} - \frac {1}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.42, size = 116, normalized size = 0.88 \begin {gather*} \frac {{\left (256 \, b^{5} x^{10} + 896 \, a b^{4} x^{8} + 1120 \, a^{2} b^{3} x^{6} + 560 \, a^{3} b^{2} x^{4} + 70 \, a^{4} b x^{2} - 7 \, a^{5}\right )} \sqrt {b x^{2} + a}}{21 \, {\left (a^{6} b^{4} x^{11} + 4 \, a^{7} b^{3} x^{9} + 6 \, a^{8} b^{2} x^{7} + 4 \, a^{9} b x^{5} + a^{10} x^{3}\right )}} \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. 668 vs.
\(2 (126) = 252\).
time = 1.97, size = 668, normalized size = 5.06 \begin {gather*} - \frac {7 a^{6} b^{\frac {51}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{21 a^{11} b^{25} x^{2} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{6} + 210 a^{8} b^{28} x^{8} + 105 a^{7} b^{29} x^{10} + 21 a^{6} b^{30} x^{12}} + \frac {63 a^{5} b^{\frac {53}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{21 a^{11} b^{25} x^{2} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{6} + 210 a^{8} b^{28} x^{8} + 105 a^{7} b^{29} x^{10} + 21 a^{6} b^{30} x^{12}} + \frac {630 a^{4} b^{\frac {55}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{21 a^{11} b^{25} x^{2} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{6} + 210 a^{8} b^{28} x^{8} + 105 a^{7} b^{29} x^{10} + 21 a^{6} b^{30} x^{12}} + \frac {1680 a^{3} b^{\frac {57}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{21 a^{11} b^{25} x^{2} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{6} + 210 a^{8} b^{28} x^{8} + 105 a^{7} b^{29} x^{10} + 21 a^{6} b^{30} x^{12}} + \frac {2016 a^{2} b^{\frac {59}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{21 a^{11} b^{25} x^{2} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{6} + 210 a^{8} b^{28} x^{8} + 105 a^{7} b^{29} x^{10} + 21 a^{6} b^{30} x^{12}} + \frac {1152 a b^{\frac {61}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{21 a^{11} b^{25} x^{2} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{6} + 210 a^{8} b^{28} x^{8} + 105 a^{7} b^{29} x^{10} + 21 a^{6} b^{30} x^{12}} + \frac {256 b^{\frac {63}{2}} x^{12} \sqrt {\frac {a}{b x^{2}} + 1}}{21 a^{11} b^{25} x^{2} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{6} + 210 a^{8} b^{28} x^{8} + 105 a^{7} b^{29} x^{10} + 21 a^{6} b^{30} x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.63, size = 147, normalized size = 1.11 \begin {gather*} \frac {{\left ({\left (x^{2} {\left (\frac {158 \, b^{5} x^{2}}{a^{6}} + \frac {511 \, b^{4}}{a^{5}}\right )} + \frac {560 \, b^{3}}{a^{4}}\right )} x^{2} + \frac {210 \, b^{2}}{a^{3}}\right )} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} - \frac {4 \, {\left (6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} b^{\frac {3}{2}} - 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b^{\frac {3}{2}} + 7 \, a^{2} b^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3} a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.80, size = 97, normalized size = 0.73 \begin {gather*} \frac {\frac {128\,b}{21\,a^5}+\frac {256\,b^2\,x^2}{21\,a^6}}{x\,\sqrt {b\,x^2+a}}-\frac {\frac {1}{3\,a^2}+\frac {19\,b\,x^2}{21\,a^3}}{x^3\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {32\,b}{21\,a^4\,x\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {b^2\,x}{7\,a^3\,{\left (b\,x^2+a\right )}^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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