Optimal. Leaf size=126 \[ -\frac {63}{8} a^{5/2} b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )+\frac {63}{8} a^2 b^2 \sqrt {a+b x^2}+\frac {63}{40} b^2 \left (a+b x^2\right )^{5/2}+\frac {21}{8} a b^2 \left (a+b x^2\right )^{3/2}-\frac {9 b \left (a+b x^2\right )^{7/2}}{8 x^2}-\frac {\left (a+b x^2\right )^{9/2}}{4 x^4} \]
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Rubi [A] time = 0.08, antiderivative size = 126, 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, 50, 63, 208} \[ \frac {63}{8} a^2 b^2 \sqrt {a+b x^2}-\frac {63}{8} a^{5/2} b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )+\frac {63}{40} b^2 \left (a+b x^2\right )^{5/2}+\frac {21}{8} a b^2 \left (a+b x^2\right )^{3/2}-\frac {\left (a+b x^2\right )^{9/2}}{4 x^4}-\frac {9 b \left (a+b x^2\right )^{7/2}}{8 x^2} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{9/2}}{x^5} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^{9/2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2\right )^{9/2}}{4 x^4}+\frac {1}{8} (9 b) \operatorname {Subst}\left (\int \frac {(a+b x)^{7/2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {9 b \left (a+b x^2\right )^{7/2}}{8 x^2}-\frac {\left (a+b x^2\right )^{9/2}}{4 x^4}+\frac {1}{16} \left (63 b^2\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{x} \, dx,x,x^2\right )\\ &=\frac {63}{40} b^2 \left (a+b x^2\right )^{5/2}-\frac {9 b \left (a+b x^2\right )^{7/2}}{8 x^2}-\frac {\left (a+b x^2\right )^{9/2}}{4 x^4}+\frac {1}{16} \left (63 a b^2\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x} \, dx,x,x^2\right )\\ &=\frac {21}{8} a b^2 \left (a+b x^2\right )^{3/2}+\frac {63}{40} b^2 \left (a+b x^2\right )^{5/2}-\frac {9 b \left (a+b x^2\right )^{7/2}}{8 x^2}-\frac {\left (a+b x^2\right )^{9/2}}{4 x^4}+\frac {1}{16} \left (63 a^2 b^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,x^2\right )\\ &=\frac {63}{8} a^2 b^2 \sqrt {a+b x^2}+\frac {21}{8} a b^2 \left (a+b x^2\right )^{3/2}+\frac {63}{40} b^2 \left (a+b x^2\right )^{5/2}-\frac {9 b \left (a+b x^2\right )^{7/2}}{8 x^2}-\frac {\left (a+b x^2\right )^{9/2}}{4 x^4}+\frac {1}{16} \left (63 a^3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )\\ &=\frac {63}{8} a^2 b^2 \sqrt {a+b x^2}+\frac {21}{8} a b^2 \left (a+b x^2\right )^{3/2}+\frac {63}{40} b^2 \left (a+b x^2\right )^{5/2}-\frac {9 b \left (a+b x^2\right )^{7/2}}{8 x^2}-\frac {\left (a+b x^2\right )^{9/2}}{4 x^4}+\frac {1}{8} \left (63 a^3 b\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )\\ &=\frac {63}{8} a^2 b^2 \sqrt {a+b x^2}+\frac {21}{8} a b^2 \left (a+b x^2\right )^{3/2}+\frac {63}{40} b^2 \left (a+b x^2\right )^{5/2}-\frac {9 b \left (a+b x^2\right )^{7/2}}{8 x^2}-\frac {\left (a+b x^2\right )^{9/2}}{4 x^4}-\frac {63}{8} a^{5/2} b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 39, normalized size = 0.31 \[ -\frac {b^2 \left (a+b x^2\right )^{11/2} \, _2F_1\left (3,\frac {11}{2};\frac {13}{2};\frac {b x^2}{a}+1\right )}{11 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 192, normalized size = 1.52 \[ \left [\frac {315 \, a^{\frac {5}{2}} b^{2} x^{4} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (8 \, b^{4} x^{8} + 56 \, a b^{3} x^{6} + 288 \, a^{2} b^{2} x^{4} - 85 \, a^{3} b x^{2} - 10 \, a^{4}\right )} \sqrt {b x^{2} + a}}{80 \, x^{4}}, \frac {315 \, \sqrt {-a} a^{2} b^{2} x^{4} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (8 \, b^{4} x^{8} + 56 \, a b^{3} x^{6} + 288 \, a^{2} b^{2} x^{4} - 85 \, a^{3} b x^{2} - 10 \, a^{4}\right )} \sqrt {b x^{2} + a}}{40 \, x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.15, size = 124, normalized size = 0.98 \[ \frac {\frac {315 \, a^{3} b^{3} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + 8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3} + 40 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a b^{3} + 240 \, \sqrt {b x^{2} + a} a^{2} b^{3} - \frac {5 \, {\left (17 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} b^{3} - 15 \, \sqrt {b x^{2} + a} a^{4} b^{3}\right )}}{b^{2} x^{4}}}{40 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 148, normalized size = 1.17 \[ -\frac {63 a^{\frac {5}{2}} b^{2} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{8}+\frac {63 \sqrt {b \,x^{2}+a}\, a^{2} b^{2}}{8}+\frac {21 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a \,b^{2}}{8}+\frac {63 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{2}}{40}+\frac {9 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{2}}{8 a}+\frac {7 \left (b \,x^{2}+a \right )^{\frac {9}{2}} b^{2}}{8 a^{2}}-\frac {7 \left (b \,x^{2}+a \right )^{\frac {11}{2}} b}{8 a^{2} x^{2}}-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{4 a \,x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.46, size = 136, normalized size = 1.08 \[ -\frac {63}{8} \, a^{\frac {5}{2}} b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \frac {63}{40} \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2} + \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} b^{2}}{8 \, a^{2}} + \frac {9 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}}{8 \, a} + \frac {21}{8} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a b^{2} + \frac {63}{8} \, \sqrt {b x^{2} + a} a^{2} b^{2} - \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b}{8 \, a^{2} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{4 \, a x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.65, size = 132, normalized size = 1.05 \[ \frac {\frac {15\,a^4\,b^2\,\sqrt {b\,x^2+a}}{8}-\frac {17\,a^3\,b^2\,{\left (b\,x^2+a\right )}^{3/2}}{8}}{{\left (b\,x^2+a\right )}^2-2\,a\,\left (b\,x^2+a\right )+a^2}+\frac {b^2\,{\left (b\,x^2+a\right )}^{5/2}}{5}+a\,b^2\,{\left (b\,x^2+a\right )}^{3/2}+6\,a^2\,b^2\,\sqrt {b\,x^2+a}+\frac {a^{5/2}\,b^2\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,63{}\mathrm {i}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.73, size = 175, normalized size = 1.39 \[ - \frac {63 a^{\frac {5}{2}} b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8} - \frac {a^{5}}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {19 a^{4} \sqrt {b}}{8 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {203 a^{3} b^{\frac {3}{2}}}{40 x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {43 a^{2} b^{\frac {5}{2}} x}{5 \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {8 a b^{\frac {7}{2}} x^{3}}{5 \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {b^{\frac {9}{2}} x^{5}}{5 \sqrt {\frac {a}{b x^{2}} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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