Optimal. Leaf size=104 \[ -\frac{i e^{i a} b \sqrt{x} \text{Gamma}\left (\frac{1}{4},-i b x^2\right )}{3 \sqrt [4]{-i b x^2}}+\frac{i e^{-i a} b \sqrt{x} \text{Gamma}\left (\frac{1}{4},i b x^2\right )}{3 \sqrt [4]{i b x^2}}-\frac{2 \cos \left (a+b x^2\right )}{3 x^{3/2}} \]
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Rubi [A] time = 0.0748721, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3388, 3389, 2218} \[ -\frac{i e^{i a} b \sqrt{x} \text{Gamma}\left (\frac{1}{4},-i b x^2\right )}{3 \sqrt [4]{-i b x^2}}+\frac{i e^{-i a} b \sqrt{x} \text{Gamma}\left (\frac{1}{4},i b x^2\right )}{3 \sqrt [4]{i b x^2}}-\frac{2 \cos \left (a+b x^2\right )}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3388
Rule 3389
Rule 2218
Rubi steps
\begin{align*} \int \frac{\cos \left (a+b x^2\right )}{x^{5/2}} \, dx &=-\frac{2 \cos \left (a+b x^2\right )}{3 x^{3/2}}-\frac{1}{3} (4 b) \int \frac{\sin \left (a+b x^2\right )}{\sqrt{x}} \, dx\\ &=-\frac{2 \cos \left (a+b x^2\right )}{3 x^{3/2}}-\frac{1}{3} (2 i b) \int \frac{e^{-i a-i b x^2}}{\sqrt{x}} \, dx+\frac{1}{3} (2 i b) \int \frac{e^{i a+i b x^2}}{\sqrt{x}} \, dx\\ &=-\frac{2 \cos \left (a+b x^2\right )}{3 x^{3/2}}-\frac{i b e^{i a} \sqrt{x} \Gamma \left (\frac{1}{4},-i b x^2\right )}{3 \sqrt [4]{-i b x^2}}+\frac{i b e^{-i a} \sqrt{x} \Gamma \left (\frac{1}{4},i b x^2\right )}{3 \sqrt [4]{i b x^2}}\\ \end{align*}
Mathematica [A] time = 0.156479, size = 117, normalized size = 1.12 \[ \frac{b x^2 \sqrt [4]{i b x^2} (\sin (a)-i \cos (a)) \text{Gamma}\left (\frac{1}{4},-i b x^2\right )+i \left (-i b x^2\right )^{5/4} (\sin (a)+i \cos (a)) \text{Gamma}\left (\frac{1}{4},i b x^2\right )-2 \sqrt [4]{b^2 x^4} \cos \left (a+b x^2\right )}{3 x^{3/2} \sqrt [4]{b^2 x^4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.069, size = 358, normalized size = 3.4 \begin{align*}{\frac{\cos \left ( a \right ) \sqrt{\pi }\sqrt [4]{2}}{8} \left ({b}^{2} \right ) ^{{\frac{3}{8}}} \left ( -4\,{\frac{{2}^{3/4}\sin \left ( b{x}^{2} \right ) }{\sqrt{\pi }{x}^{7/2} \left ({b}^{2} \right ) ^{3/8}b} \left ({\frac{8\,{b}^{2}{x}^{4}}{15}}+2/3 \right ) }-{\frac{8\,{2}^{3/4} \left ( -16\,{b}^{2}{x}^{4}+5 \right ) \left ( \cos \left ( b{x}^{2} \right ) b{x}^{2}-\sin \left ( b{x}^{2} \right ) \right ) }{15\,\sqrt{\pi }b}{x}^{-{\frac{7}{2}}} \left ({b}^{2} \right ) ^{-{\frac{3}{8}}}}+{\frac{32\,{2}^{3/4}{b}^{3}\sin \left ( b{x}^{2} \right ) }{15\,\sqrt{\pi }}{x}^{{\frac{9}{2}}}{\it LommelS1} \left ({\frac{3}{4}},{\frac{3}{2}},b{x}^{2} \right ) \left ({b}^{2} \right ) ^{-{\frac{3}{8}}} \left ( b{x}^{2} \right ) ^{-{\frac{7}{4}}}}-{\frac{128\,{2}^{3/4}{b}^{3} \left ( \cos \left ( b{x}^{2} \right ) b{x}^{2}-\sin \left ( b{x}^{2} \right ) \right ) }{15\,\sqrt{\pi }}{x}^{{\frac{9}{2}}}{\it LommelS1} \left ({\frac{7}{4}},{\frac{1}{2}},b{x}^{2} \right ) \left ({b}^{2} \right ) ^{-{\frac{3}{8}}} \left ( b{x}^{2} \right ) ^{-{\frac{11}{4}}}} \right ) }-{\frac{\sin \left ( a \right ) \sqrt{\pi }\sqrt [4]{2}}{8}{b}^{{\frac{3}{4}}} \left ( 12\,{\frac{{2}^{3/4}\sin \left ( b{x}^{2} \right ) }{\sqrt{\pi }{x}^{3/2}{b}^{3/4}} \left ({\frac{32\,{b}^{2}{x}^{4}}{81}}+2/3 \right ) }+{\frac{32\,{2}^{3/4} \left ( \cos \left ( b{x}^{2} \right ) b{x}^{2}-\sin \left ( b{x}^{2} \right ) \right ) }{3\,\sqrt{\pi }}{x}^{-{\frac{3}{2}}}{b}^{-{\frac{3}{4}}}}-{\frac{128\,{2}^{3/4}\sin \left ( b{x}^{2} \right ) }{27\,\sqrt{\pi }}{x}^{{\frac{9}{2}}}{b}^{{\frac{9}{4}}}{\it LommelS1} \left ({\frac{7}{4}},{\frac{3}{2}},b{x}^{2} \right ) \left ( b{x}^{2} \right ) ^{-{\frac{7}{4}}}}-{\frac{32\,{2}^{3/4} \left ( \cos \left ( b{x}^{2} \right ) b{x}^{2}-\sin \left ( b{x}^{2} \right ) \right ) }{3\,\sqrt{\pi }}{x}^{{\frac{9}{2}}}{b}^{{\frac{9}{4}}}{\it LommelS1} \left ({\frac{3}{4}},{\frac{1}{2}},b{x}^{2} \right ) \left ( b{x}^{2} \right ) ^{-{\frac{11}{4}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.44541, size = 355, normalized size = 3.41 \begin{align*} -\frac{\left (x^{2}{\left | b \right |}\right )^{\frac{3}{4}}{\left ({\left ({\left (\Gamma \left (-\frac{3}{4}, i \, b x^{2}\right ) + \Gamma \left (-\frac{3}{4}, -i \, b x^{2}\right )\right )} \cos \left (\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right ) +{\left (\Gamma \left (-\frac{3}{4}, i \, b x^{2}\right ) + \Gamma \left (-\frac{3}{4}, -i \, b x^{2}\right )\right )} \cos \left (-\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right ) +{\left (i \, \Gamma \left (-\frac{3}{4}, i \, b x^{2}\right ) - i \, \Gamma \left (-\frac{3}{4}, -i \, b x^{2}\right )\right )} \sin \left (\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right ) +{\left (-i \, \Gamma \left (-\frac{3}{4}, i \, b x^{2}\right ) + i \, \Gamma \left (-\frac{3}{4}, -i \, b x^{2}\right )\right )} \sin \left (-\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right )\right )} \cos \left (a\right ) +{\left ({\left (-i \, \Gamma \left (-\frac{3}{4}, i \, b x^{2}\right ) + i \, \Gamma \left (-\frac{3}{4}, -i \, b x^{2}\right )\right )} \cos \left (\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right ) +{\left (-i \, \Gamma \left (-\frac{3}{4}, i \, b x^{2}\right ) + i \, \Gamma \left (-\frac{3}{4}, -i \, b x^{2}\right )\right )} \cos \left (-\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right ) +{\left (\Gamma \left (-\frac{3}{4}, i \, b x^{2}\right ) + \Gamma \left (-\frac{3}{4}, -i \, b x^{2}\right )\right )} \sin \left (\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right ) -{\left (\Gamma \left (-\frac{3}{4}, i \, b x^{2}\right ) + \Gamma \left (-\frac{3}{4}, -i \, b x^{2}\right )\right )} \sin \left (-\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right )\right )} \sin \left (a\right )\right )}}{8 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78771, size = 177, normalized size = 1.7 \begin{align*} \frac{\left (i \, b\right )^{\frac{3}{4}} x^{2} e^{\left (-i \, a\right )} \Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) + \left (-i \, b\right )^{\frac{3}{4}} x^{2} e^{\left (i \, a\right )} \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right ) - 2 \, \sqrt{x} \cos \left (b x^{2} + a\right )}{3 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos{\left (a + b x^{2} \right )}}{x^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (b x^{2} + a\right )}{x^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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