Optimal. Leaf size=111 \[ -\frac{i e^{i a} \sqrt{x} \text{Gamma}\left (\frac{1}{4},-i b x^2\right )}{16 b \sqrt [4]{-i b x^2}}+\frac{i e^{-i a} \sqrt{x} \text{Gamma}\left (\frac{1}{4},i b x^2\right )}{16 b \sqrt [4]{i b x^2}}+\frac{\sqrt{x} \sin \left (a+b x^2\right )}{2 b} \]
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Rubi [A] time = 0.0786466, antiderivative size = 111, 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 = {3386, 3389, 2218} \[ -\frac{i e^{i a} \sqrt{x} \text{Gamma}\left (\frac{1}{4},-i b x^2\right )}{16 b \sqrt [4]{-i b x^2}}+\frac{i e^{-i a} \sqrt{x} \text{Gamma}\left (\frac{1}{4},i b x^2\right )}{16 b \sqrt [4]{i b x^2}}+\frac{\sqrt{x} \sin \left (a+b x^2\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 3386
Rule 3389
Rule 2218
Rubi steps
\begin{align*} \int x^{3/2} \cos \left (a+b x^2\right ) \, dx &=\frac{\sqrt{x} \sin \left (a+b x^2\right )}{2 b}-\frac{\int \frac{\sin \left (a+b x^2\right )}{\sqrt{x}} \, dx}{4 b}\\ &=\frac{\sqrt{x} \sin \left (a+b x^2\right )}{2 b}-\frac{i \int \frac{e^{-i a-i b x^2}}{\sqrt{x}} \, dx}{8 b}+\frac{i \int \frac{e^{i a+i b x^2}}{\sqrt{x}} \, dx}{8 b}\\ &=-\frac{i e^{i a} \sqrt{x} \Gamma \left (\frac{1}{4},-i b x^2\right )}{16 b \sqrt [4]{-i b x^2}}+\frac{i e^{-i a} \sqrt{x} \Gamma \left (\frac{1}{4},i b x^2\right )}{16 b \sqrt [4]{i b x^2}}+\frac{\sqrt{x} \sin \left (a+b x^2\right )}{2 b}\\ \end{align*}
Mathematica [A] time = 0.157765, size = 111, normalized size = 1. \[ \frac{b x^{9/2} \left (\sqrt [4]{i b x^2} (\sin (a)-i \cos (a)) \text{Gamma}\left (\frac{1}{4},-i b x^2\right )+\sqrt [4]{-i b x^2} (\sin (a)+i \cos (a)) \text{Gamma}\left (\frac{1}{4},i b x^2\right )+8 \sqrt [4]{b^2 x^4} \sin \left (a+b x^2\right )\right )}{16 \left (b^2 x^4\right )^{5/4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.077, size = 290, normalized size = 2.6 \begin{align*}{\frac{\sqrt [4]{2}\cos \left ( a \right ) \sqrt{\pi }}{2} \left ({\frac{2\,{2}^{3/4}\sin \left ( b{x}^{2} \right ) }{5\,\sqrt{\pi }b}\sqrt{x} \left ({b}^{2} \right ) ^{{\frac{5}{8}}}}+{\frac{2\,{2}^{3/4} \left ( \cos \left ( b{x}^{2} \right ) b{x}^{2}-\sin \left ( b{x}^{2} \right ) \right ) }{5\,\sqrt{\pi }b}\sqrt{x} \left ({b}^{2} \right ) ^{{\frac{5}{8}}}}+{\frac{{2}^{{\frac{3}{4}}}b\sin \left ( b{x}^{2} \right ) }{10\,\sqrt{\pi }}{x}^{{\frac{9}{2}}} \left ({b}^{2} \right ) ^{{\frac{5}{8}}}{\it LommelS1} \left ({\frac{3}{4}},{\frac{3}{2}},b{x}^{2} \right ) \left ( b{x}^{2} \right ) ^{-{\frac{7}{4}}}}-{\frac{2\,{2}^{3/4}b \left ( \cos \left ( b{x}^{2} \right ) b{x}^{2}-\sin \left ( b{x}^{2} \right ) \right ) }{5\,\sqrt{\pi }}{x}^{{\frac{9}{2}}} \left ({b}^{2} \right ) ^{{\frac{5}{8}}}{\it LommelS1} \left ({\frac{7}{4}},{\frac{1}{2}},b{x}^{2} \right ) \left ( b{x}^{2} \right ) ^{-{\frac{11}{4}}}} \right ) \left ({b}^{2} \right ) ^{-{\frac{5}{8}}}}-{\frac{\sqrt [4]{2}\sin \left ( a \right ) \sqrt{\pi }}{2} \left ({\frac{2\,{2}^{3/4}\sin \left ( b{x}^{2} \right ) }{9\,\sqrt{\pi }}{x}^{{\frac{5}{2}}}{b}^{{\frac{5}{4}}}}-{\frac{2\,{2}^{3/4}\sin \left ( b{x}^{2} \right ) }{9\,\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{{2}^{{\frac{3}{4}}} \left ( \cos \left ( b{x}^{2} \right ) b{x}^{2}-\sin \left ( b{x}^{2} \right ) \right ) }{2\,\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 ){b}^{-{\frac{5}{4}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.52483, size = 390, normalized size = 3.51 \begin{align*} \frac{16 \, \left (x^{2}{\left | b \right |}\right )^{\frac{1}{4}} \sqrt{x} \sin \left (b x^{2} + a\right ) +{\left ({\left ({\left (i \, \Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) - i \, \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right )\right )} \cos \left (\frac{1}{8} \, \pi + \frac{1}{4} \, \arctan \left (0, b\right )\right ) +{\left (i \, \Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) - i \, \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right )\right )} \cos \left (-\frac{1}{8} \, \pi + \frac{1}{4} \, \arctan \left (0, b\right )\right ) +{\left (\Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) + \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right )\right )} \sin \left (\frac{1}{8} \, \pi + \frac{1}{4} \, \arctan \left (0, b\right )\right ) -{\left (\Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) + \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right )\right )} \sin \left (-\frac{1}{8} \, \pi + \frac{1}{4} \, \arctan \left (0, b\right )\right )\right )} \cos \left (a\right ) +{\left ({\left (\Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) + \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right )\right )} \cos \left (\frac{1}{8} \, \pi + \frac{1}{4} \, \arctan \left (0, b\right )\right ) +{\left (\Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) + \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right )\right )} \cos \left (-\frac{1}{8} \, \pi + \frac{1}{4} \, \arctan \left (0, b\right )\right ) +{\left (-i \, \Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) + i \, \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right )\right )} \sin \left (\frac{1}{8} \, \pi + \frac{1}{4} \, \arctan \left (0, b\right )\right ) +{\left (i \, \Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) - i \, \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right )\right )} \sin \left (-\frac{1}{8} \, \pi + \frac{1}{4} \, \arctan \left (0, b\right )\right )\right )} \sin \left (a\right )\right )} \sqrt{x}}{32 \, \left (x^{2}{\left | b \right |}\right )^{\frac{1}{4}} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75403, size = 170, normalized size = 1.53 \begin{align*} \frac{\left (i \, b\right )^{\frac{3}{4}} e^{\left (-i \, a\right )} \Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) + \left (-i \, b\right )^{\frac{3}{4}} e^{\left (i \, a\right )} \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right ) + 8 \, b \sqrt{x} \sin \left (b x^{2} + a\right )}{16 \, b^{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 x^{\frac{3}{2}} \cos{\left (a + b x^{2} \right )}\, 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 x^{\frac{3}{2}} \cos \left (b x^{2} + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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