Optimal. Leaf size=137 \[ -\frac {3 b^5 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{256 a^{5/2}}+\frac {3 b^4 \sqrt {a+b x^2}}{256 a^2 x^2}-\frac {b^3 \sqrt {a+b x^2}}{128 a x^4}-\frac {b^2 \sqrt {a+b x^2}}{32 x^6}-\frac {\left (a+b x^2\right )^{5/2}}{10 x^{10}}-\frac {b \left (a+b x^2\right )^{3/2}}{16 x^8} \]
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Rubi [A] time = 0.08, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {266, 47, 51, 63, 208} \[ \frac {3 b^4 \sqrt {a+b x^2}}{256 a^2 x^2}-\frac {3 b^5 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{256 a^{5/2}}-\frac {b^3 \sqrt {a+b x^2}}{128 a x^4}-\frac {b^2 \sqrt {a+b x^2}}{32 x^6}-\frac {b \left (a+b x^2\right )^{3/2}}{16 x^8}-\frac {\left (a+b x^2\right )^{5/2}}{10 x^{10}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{5/2}}{x^{11}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{x^6} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2\right )^{5/2}}{10 x^{10}}+\frac {1}{4} b \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^5} \, dx,x,x^2\right )\\ &=-\frac {b \left (a+b x^2\right )^{3/2}}{16 x^8}-\frac {\left (a+b x^2\right )^{5/2}}{10 x^{10}}+\frac {1}{32} \left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^4} \, dx,x,x^2\right )\\ &=-\frac {b^2 \sqrt {a+b x^2}}{32 x^6}-\frac {b \left (a+b x^2\right )^{3/2}}{16 x^8}-\frac {\left (a+b x^2\right )^{5/2}}{10 x^{10}}+\frac {1}{64} b^3 \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x}} \, dx,x,x^2\right )\\ &=-\frac {b^2 \sqrt {a+b x^2}}{32 x^6}-\frac {b^3 \sqrt {a+b x^2}}{128 a x^4}-\frac {b \left (a+b x^2\right )^{3/2}}{16 x^8}-\frac {\left (a+b x^2\right )^{5/2}}{10 x^{10}}-\frac {\left (3 b^4\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^2\right )}{256 a}\\ &=-\frac {b^2 \sqrt {a+b x^2}}{32 x^6}-\frac {b^3 \sqrt {a+b x^2}}{128 a x^4}+\frac {3 b^4 \sqrt {a+b x^2}}{256 a^2 x^2}-\frac {b \left (a+b x^2\right )^{3/2}}{16 x^8}-\frac {\left (a+b x^2\right )^{5/2}}{10 x^{10}}+\frac {\left (3 b^5\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{512 a^2}\\ &=-\frac {b^2 \sqrt {a+b x^2}}{32 x^6}-\frac {b^3 \sqrt {a+b x^2}}{128 a x^4}+\frac {3 b^4 \sqrt {a+b x^2}}{256 a^2 x^2}-\frac {b \left (a+b x^2\right )^{3/2}}{16 x^8}-\frac {\left (a+b x^2\right )^{5/2}}{10 x^{10}}+\frac {\left (3 b^4\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{256 a^2}\\ &=-\frac {b^2 \sqrt {a+b x^2}}{32 x^6}-\frac {b^3 \sqrt {a+b x^2}}{128 a x^4}+\frac {3 b^4 \sqrt {a+b x^2}}{256 a^2 x^2}-\frac {b \left (a+b x^2\right )^{3/2}}{16 x^8}-\frac {\left (a+b x^2\right )^{5/2}}{10 x^{10}}-\frac {3 b^5 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{256 a^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 39, normalized size = 0.28 \[ \frac {b^5 \left (a+b x^2\right )^{7/2} \, _2F_1\left (\frac {7}{2},6;\frac {9}{2};\frac {b x^2}{a}+1\right )}{7 a^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 201, normalized size = 1.47 \[ \left [\frac {15 \, \sqrt {a} b^{5} x^{10} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (15 \, a b^{4} x^{8} - 10 \, a^{2} b^{3} x^{6} - 248 \, a^{3} b^{2} x^{4} - 336 \, a^{4} b x^{2} - 128 \, a^{5}\right )} \sqrt {b x^{2} + a}}{2560 \, a^{3} x^{10}}, \frac {15 \, \sqrt {-a} b^{5} x^{10} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (15 \, a b^{4} x^{8} - 10 \, a^{2} b^{3} x^{6} - 248 \, a^{3} b^{2} x^{4} - 336 \, a^{4} b x^{2} - 128 \, a^{5}\right )} \sqrt {b x^{2} + a}}{1280 \, a^{3} x^{10}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.16, size = 126, normalized size = 0.92 \[ \frac {\frac {15 \, b^{6} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {15 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} b^{6} - 70 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a b^{6} - 128 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} b^{6} + 70 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} b^{6} - 15 \, \sqrt {b x^{2} + a} a^{4} b^{6}}{a^{2} b^{5} x^{10}}}{1280 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 179, normalized size = 1.31 \[ -\frac {3 b^{5} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{256 a^{\frac {5}{2}}}+\frac {3 \sqrt {b \,x^{2}+a}\, b^{5}}{256 a^{3}}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{5}}{256 a^{4}}+\frac {3 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{5}}{1280 a^{5}}-\frac {3 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{4}}{1280 a^{5} x^{2}}-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{3}}{640 a^{4} x^{4}}-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{2}}{160 a^{3} x^{6}}+\frac {3 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b}{80 a^{2} x^{8}}-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{10 a \,x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.44, size = 167, normalized size = 1.22 \[ -\frac {3 \, b^{5} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{256 \, a^{\frac {5}{2}}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{5}}{1280 \, a^{5}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{5}}{256 \, a^{4}} + \frac {3 \, \sqrt {b x^{2} + a} b^{5}}{256 \, a^{3}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}}{1280 \, a^{5} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}}{640 \, a^{4} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}}{160 \, a^{3} x^{6}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b}{80 \, a^{2} x^{8}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}}}{10 \, a x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.68, size = 106, normalized size = 0.77 \[ \frac {7\,a\,{\left (b\,x^2+a\right )}^{3/2}}{128\,x^{10}}-\frac {{\left (b\,x^2+a\right )}^{5/2}}{10\,x^{10}}-\frac {3\,a^2\,\sqrt {b\,x^2+a}}{256\,x^{10}}-\frac {7\,{\left (b\,x^2+a\right )}^{7/2}}{128\,a\,x^{10}}+\frac {3\,{\left (b\,x^2+a\right )}^{9/2}}{256\,a^2\,x^{10}}+\frac {b^5\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,3{}\mathrm {i}}{256\,a^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.54, size = 175, normalized size = 1.28 \[ - \frac {a^{3}}{10 \sqrt {b} x^{11} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {29 a^{2} \sqrt {b}}{80 x^{9} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {73 a b^{\frac {3}{2}}}{160 x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {129 b^{\frac {5}{2}}}{640 x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {b^{\frac {7}{2}}}{256 a x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {3 b^{\frac {9}{2}}}{256 a^{2} x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 b^{5} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{256 a^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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