Optimal. Leaf size=111 \[ \frac {\sqrt {x} \sin \left (a+b x^2\right )}{2 b}-\frac {i e^{i a} \sqrt {x} \Gamma \left (\frac {1}{4},-i b x^2\right )}{16 b \sqrt [4]{-i b x^2}}+\frac {i e^{-i a} \sqrt {x} \Gamma \left (\frac {1}{4},i b x^2\right )}{16 b \sqrt [4]{i b x^2}} \]
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Rubi [A] time = 0.08, antiderivative size = 111, normalized size of antiderivative = 1.00, 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 2218
Rule 3386
Rule 3389
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*}
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Mathematica [A] time = 0.15, size = 111, normalized size = 1.00 \[ \frac {b x^{9/2} \left (8 \sqrt [4]{b^2 x^4} \sin \left (a+b x^2\right )+\sqrt [4]{i b x^2} (\sin (a)-i \cos (a)) \Gamma \left (\frac {1}{4},-i b x^2\right )+\sqrt [4]{-i b x^2} (\sin (a)+i \cos (a)) \Gamma \left (\frac {1}{4},i b x^2\right )\right )}{16 \left (b^2 x^4\right )^{5/4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 56, normalized size = 0.50 \[ \frac {\left (i \, b\right )^{\frac {3}{4}} e^{\left (-i \, a\right )} \Gamma \left (\frac {1}{4}, i \, b x^{2}\right ) + \left (-i \, b\right )^{\frac {3}{4}} e^{\left (i \, a\right )} \Gamma \left (\frac {1}{4}, -i \, b x^{2}\right ) + 8 \, b \sqrt {x} \sin \left (b x^{2} + a\right )}{16 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{\frac {3}{2}} \cos \left (b x^{2} + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.08, size = 290, normalized size = 2.61 \[ \frac {2^{\frac {1}{4}} \cos \relax (a ) \sqrt {\pi }\, \left (\frac {2 \sqrt {x}\, 2^{\frac {3}{4}} \left (b^{2}\right )^{\frac {5}{8}} \sin \left (b \,x^{2}\right )}{5 \sqrt {\pi }\, b}+\frac {2 \sqrt {x}\, 2^{\frac {3}{4}} \left (b^{2}\right )^{\frac {5}{8}} \left (\cos \left (b \,x^{2}\right ) x^{2} b -\sin \left (b \,x^{2}\right )\right )}{5 \sqrt {\pi }\, b}+\frac {x^{\frac {9}{2}} \left (b^{2}\right )^{\frac {5}{8}} 2^{\frac {3}{4}} b \sin \left (b \,x^{2}\right ) \LommelS 1 \left (\frac {3}{4}, \frac {3}{2}, b \,x^{2}\right )}{10 \sqrt {\pi }\, \left (b \,x^{2}\right )^{\frac {7}{4}}}-\frac {2 x^{\frac {9}{2}} \left (b^{2}\right )^{\frac {5}{8}} 2^{\frac {3}{4}} b \left (\cos \left (b \,x^{2}\right ) x^{2} b -\sin \left (b \,x^{2}\right )\right ) \LommelS 1 \left (\frac {7}{4}, \frac {1}{2}, b \,x^{2}\right )}{5 \sqrt {\pi }\, \left (b \,x^{2}\right )^{\frac {11}{4}}}\right )}{2 \left (b^{2}\right )^{\frac {5}{8}}}-\frac {2^{\frac {1}{4}} \sin \relax (a ) \sqrt {\pi }\, \left (\frac {2 x^{\frac {5}{2}} 2^{\frac {3}{4}} b^{\frac {5}{4}} \sin \left (b \,x^{2}\right )}{9 \sqrt {\pi }}-\frac {2 x^{\frac {9}{2}} b^{\frac {9}{4}} 2^{\frac {3}{4}} \sin \left (b \,x^{2}\right ) \LommelS 1 \left (\frac {7}{4}, \frac {3}{2}, b \,x^{2}\right )}{9 \sqrt {\pi }\, \left (b \,x^{2}\right )^{\frac {7}{4}}}-\frac {x^{\frac {9}{2}} b^{\frac {9}{4}} 2^{\frac {3}{4}} \left (\cos \left (b \,x^{2}\right ) x^{2} b -\sin \left (b \,x^{2}\right )\right ) \LommelS 1 \left (\frac {3}{4}, \frac {1}{2}, b \,x^{2}\right )}{2 \sqrt {\pi }\, \left (b \,x^{2}\right )^{\frac {11}{4}}}\right )}{2 b^{\frac {5}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.34, size = 158, normalized size = 1.42 \[ \frac {16 \, \left (b x^{2}\right )^{\frac {1}{4}} \sqrt {x} \sin \left (b x^{2} + a\right ) + {\left ({\left (\sqrt {-\sqrt {2} + 2} {\left (\Gamma \left (\frac {1}{4}, i \, b x^{2}\right ) + \Gamma \left (\frac {1}{4}, -i \, b x^{2}\right )\right )} + \sqrt {\sqrt {2} + 2} {\left (i \, \Gamma \left (\frac {1}{4}, i \, b x^{2}\right ) - i \, \Gamma \left (\frac {1}{4}, -i \, b x^{2}\right )\right )}\right )} \cos \relax (a) + {\left (\sqrt {\sqrt {2} + 2} {\left (\Gamma \left (\frac {1}{4}, i \, b x^{2}\right ) + \Gamma \left (\frac {1}{4}, -i \, b x^{2}\right )\right )} + \sqrt {-\sqrt {2} + 2} {\left (-i \, \Gamma \left (\frac {1}{4}, i \, b x^{2}\right ) + i \, \Gamma \left (\frac {1}{4}, -i \, b x^{2}\right )\right )}\right )} \sin \relax (a)\right )} \sqrt {x}}{32 \, \left (b x^{2}\right )^{\frac {1}{4}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^{3/2}\,\cos \left (b\,x^2+a\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{\frac {3}{2}} \cos {\left (a + b x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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