Optimal. Leaf size=179 \[ -\frac{9 b^7 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2048 a^{5/2}}+\frac{9 b^6 \sqrt{a+b x^2}}{2048 a^2 x^2}-\frac{3 b^5 \sqrt{a+b x^2}}{1024 a x^4}-\frac{3 b^4 \sqrt{a+b x^2}}{256 x^6}-\frac{3 b^3 \left (a+b x^2\right )^{3/2}}{128 x^8}-\frac{3 b^2 \left (a+b x^2\right )^{5/2}}{80 x^{10}}-\frac{\left (a+b x^2\right )^{9/2}}{14 x^{14}}-\frac{3 b \left (a+b x^2\right )^{7/2}}{56 x^{12}} \]
[Out]
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Rubi [A] time = 0.313151, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{9 b^7 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2048 a^{5/2}}+\frac{9 b^6 \sqrt{a+b x^2}}{2048 a^2 x^2}-\frac{3 b^5 \sqrt{a+b x^2}}{1024 a x^4}-\frac{3 b^4 \sqrt{a+b x^2}}{256 x^6}-\frac{3 b^3 \left (a+b x^2\right )^{3/2}}{128 x^8}-\frac{3 b^2 \left (a+b x^2\right )^{5/2}}{80 x^{10}}-\frac{\left (a+b x^2\right )^{9/2}}{14 x^{14}}-\frac{3 b \left (a+b x^2\right )^{7/2}}{56 x^{12}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^(9/2)/x^15,x]
[Out]
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Rubi in Sympy [A] time = 31.688, size = 168, normalized size = 0.94 \[ - \frac{3 b^{4} \sqrt{a + b x^{2}}}{256 x^{6}} - \frac{3 b^{3} \left (a + b x^{2}\right )^{\frac{3}{2}}}{128 x^{8}} - \frac{3 b^{2} \left (a + b x^{2}\right )^{\frac{5}{2}}}{80 x^{10}} - \frac{3 b \left (a + b x^{2}\right )^{\frac{7}{2}}}{56 x^{12}} - \frac{\left (a + b x^{2}\right )^{\frac{9}{2}}}{14 x^{14}} - \frac{3 b^{5} \sqrt{a + b x^{2}}}{1024 a x^{4}} + \frac{9 b^{6} \sqrt{a + b x^{2}}}{2048 a^{2} x^{2}} - \frac{9 b^{7} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{2048 a^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**(9/2)/x**15,x)
[Out]
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Mathematica [A] time = 0.201484, size = 134, normalized size = 0.75 \[ \frac{-\sqrt{a} \sqrt{a+b x^2} \left (5120 a^6+24320 a^5 b x^2+44928 a^4 b^2 x^4+39056 a^3 b^3 x^6+14168 a^2 b^4 x^8+210 a b^5 x^{10}-315 b^6 x^{12}\right )-315 b^7 x^{14} \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+315 b^7 x^{14} \log (x)}{71680 a^{5/2} x^{14}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^(9/2)/x^15,x]
[Out]
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Maple [A] time = 0.437, size = 253, normalized size = 1.4 \[ -{\frac{1}{14\,a{x}^{14}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{b}{56\,{a}^{2}{x}^{12}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{{b}^{2}}{560\,{a}^{3}{x}^{10}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{{b}^{3}}{4480\,{a}^{4}{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{{b}^{4}}{8960\,{a}^{5}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{{b}^{5}}{7168\,{a}^{6}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{{b}^{6}}{2048\,{a}^{7}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{{b}^{7}}{2048\,{a}^{7}} \left ( b{x}^{2}+a \right ) ^{{\frac{9}{2}}}}+{\frac{9\,{b}^{7}}{14336\,{a}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{9\,{b}^{7}}{10240\,{a}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{3\,{b}^{7}}{2048\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{9\,{b}^{7}}{2048}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{9\,{b}^{7}}{2048\,{a}^{3}}\sqrt{b{x}^{2}+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^(9/2)/x^15,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(9/2)/x^15,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.492759, size = 1, normalized size = 0.01 \[ \left [\frac{315 \, b^{7} x^{14} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (315 \, b^{6} x^{12} - 210 \, a b^{5} x^{10} - 14168 \, a^{2} b^{4} x^{8} - 39056 \, a^{3} b^{3} x^{6} - 44928 \, a^{4} b^{2} x^{4} - 24320 \, a^{5} b x^{2} - 5120 \, a^{6}\right )} \sqrt{b x^{2} + a} \sqrt{a}}{143360 \, a^{\frac{5}{2}} x^{14}}, -\frac{315 \, b^{7} x^{14} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (315 \, b^{6} x^{12} - 210 \, a b^{5} x^{10} - 14168 \, a^{2} b^{4} x^{8} - 39056 \, a^{3} b^{3} x^{6} - 44928 \, a^{4} b^{2} x^{4} - 24320 \, a^{5} b x^{2} - 5120 \, a^{6}\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{71680 \, \sqrt{-a} a^{2} x^{14}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(9/2)/x^15,x, algorithm="fricas")
[Out]
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Sympy [A] time = 92.3171, size = 231, normalized size = 1.29 \[ - \frac{a^{5}}{14 \sqrt{b} x^{15} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{23 a^{4} \sqrt{b}}{56 x^{13} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{541 a^{3} b^{\frac{3}{2}}}{560 x^{11} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{5249 a^{2} b^{\frac{5}{2}}}{4480 x^{9} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{6653 a b^{\frac{7}{2}}}{8960 x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{1027 b^{\frac{9}{2}}}{5120 x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{3 b^{\frac{11}{2}}}{2048 a x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{9 b^{\frac{13}{2}}}{2048 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{9 b^{7} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2048 a^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**(9/2)/x**15,x)
[Out]
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GIAC/XCAS [A] time = 0.213348, size = 184, normalized size = 1.03 \[ \frac{1}{71680} \, b^{7}{\left (\frac{315 \, \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{315 \,{\left (b x^{2} + a\right )}^{\frac{13}{2}} - 2100 \,{\left (b x^{2} + a\right )}^{\frac{11}{2}} a - 8393 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} a^{2} + 9216 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a^{3} - 5943 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{4} + 2100 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{5} - 315 \, \sqrt{b x^{2} + a} a^{6}}{a^{2} b^{7} x^{14}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(9/2)/x^15,x, algorithm="giac")
[Out]