Optimal. Leaf size=104 \[ -\frac {i e^{i a} b \sqrt {x} \Gamma \left (\frac {1}{4},-i b x^2\right )}{3 \sqrt [4]{-i b x^2}}+\frac {i e^{-i a} b \sqrt {x} \Gamma \left (\frac {1}{4},i b x^2\right )}{3 \sqrt [4]{i b x^2}}-\frac {2 \cos \left (a+b x^2\right )}{3 x^{3/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 104, 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 = {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 2218
Rule 3388
Rule 3389
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*}
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Mathematica [A] time = 0.15, size = 117, normalized size = 1.12 \[ \frac {-2 \sqrt [4]{b^2 x^4} \cos \left (a+b x^2\right )+b x^2 \sqrt [4]{i b x^2} (\sin (a)-i \cos (a)) \Gamma \left (\frac {1}{4},-i b x^2\right )+i \left (-i b x^2\right )^{5/4} (\sin (a)+i \cos (a)) \Gamma \left (\frac {1}{4},i b x^2\right )}{3 x^{3/2} \sqrt [4]{b^2 x^4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 61, normalized size = 0.59 \[ \frac {\left (i \, b\right )^{\frac {3}{4}} x^{2} e^{\left (-i \, a\right )} \Gamma \left (\frac {1}{4}, i \, b x^{2}\right ) + \left (-i \, b\right )^{\frac {3}{4}} x^{2} e^{\left (i \, a\right )} \Gamma \left (\frac {1}{4}, -i \, b x^{2}\right ) - 2 \, \sqrt {x} \cos \left (b x^{2} + a\right )}{3 \, x^{2}} \]
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 \[ \int \frac {\cos \left (b x^{2} + a\right )}{x^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.06, size = 358, normalized size = 3.44 \[ \frac {\cos \relax (a ) \sqrt {\pi }\, 2^{\frac {1}{4}} \left (b^{2}\right )^{\frac {3}{8}} \left (-\frac {4 \,2^{\frac {3}{4}} \left (\frac {8 x^{4} b^{2}}{15}+\frac {2}{3}\right ) \sin \left (b \,x^{2}\right )}{\sqrt {\pi }\, x^{\frac {7}{2}} \left (b^{2}\right )^{\frac {3}{8}} b}-\frac {8 \,2^{\frac {3}{4}} \left (-16 x^{4} b^{2}+5\right ) \left (\cos \left (b \,x^{2}\right ) x^{2} b -\sin \left (b \,x^{2}\right )\right )}{15 \sqrt {\pi }\, x^{\frac {7}{2}} \left (b^{2}\right )^{\frac {3}{8}} b}+\frac {32 x^{\frac {9}{2}} 2^{\frac {3}{4}} b^{3} \sin \left (b \,x^{2}\right ) \LommelS 1 \left (\frac {3}{4}, \frac {3}{2}, b \,x^{2}\right )}{15 \sqrt {\pi }\, \left (b^{2}\right )^{\frac {3}{8}} \left (b \,x^{2}\right )^{\frac {7}{4}}}-\frac {128 x^{\frac {9}{2}} 2^{\frac {3}{4}} b^{3} \left (\cos \left (b \,x^{2}\right ) x^{2} b -\sin \left (b \,x^{2}\right )\right ) \LommelS 1 \left (\frac {7}{4}, \frac {1}{2}, b \,x^{2}\right )}{15 \sqrt {\pi }\, \left (b^{2}\right )^{\frac {3}{8}} \left (b \,x^{2}\right )^{\frac {11}{4}}}\right )}{8}-\frac {\sin \relax (a ) \sqrt {\pi }\, 2^{\frac {1}{4}} b^{\frac {3}{4}} \left (\frac {12 \,2^{\frac {3}{4}} \left (\frac {32 x^{4} b^{2}}{81}+\frac {2}{3}\right ) \sin \left (b \,x^{2}\right )}{\sqrt {\pi }\, x^{\frac {3}{2}} b^{\frac {3}{4}}}+\frac {32 \,2^{\frac {3}{4}} \left (\cos \left (b \,x^{2}\right ) x^{2} b -\sin \left (b \,x^{2}\right )\right )}{3 \sqrt {\pi }\, x^{\frac {3}{2}} b^{\frac {3}{4}}}-\frac {128 x^{\frac {9}{2}} b^{\frac {9}{4}} 2^{\frac {3}{4}} \sin \left (b \,x^{2}\right ) \LommelS 1 \left (\frac {7}{4}, \frac {3}{2}, b \,x^{2}\right )}{27 \sqrt {\pi }\, \left (b \,x^{2}\right )^{\frac {7}{4}}}-\frac {32 x^{\frac {9}{2}} b^{\frac {9}{4}} 2^{\frac {3}{4}} \left (\cos \left (b \,x^{2}\right ) x^{2} b -\sin \left (b \,x^{2}\right )\right ) \LommelS 1 \left (\frac {3}{4}, \frac {1}{2}, b \,x^{2}\right )}{3 \sqrt {\pi }\, \left (b \,x^{2}\right )^{\frac {11}{4}}}\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.80, size = 133, normalized size = 1.28 \[ -\frac {\left (b x^{2}\right )^{\frac {3}{4}} {\left ({\left (\sqrt {-\sqrt {2} + 2} {\left (\Gamma \left (-\frac {3}{4}, i \, b x^{2}\right ) + \Gamma \left (-\frac {3}{4}, -i \, b x^{2}\right )\right )} + \sqrt {\sqrt {2} + 2} {\left (i \, \Gamma \left (-\frac {3}{4}, i \, b x^{2}\right ) - i \, \Gamma \left (-\frac {3}{4}, -i \, b x^{2}\right )\right )}\right )} \cos \relax (a) + {\left (\sqrt {\sqrt {2} + 2} {\left (\Gamma \left (-\frac {3}{4}, i \, b x^{2}\right ) + \Gamma \left (-\frac {3}{4}, -i \, b x^{2}\right )\right )} + \sqrt {-\sqrt {2} + 2} {\left (-i \, \Gamma \left (-\frac {3}{4}, i \, b x^{2}\right ) + i \, \Gamma \left (-\frac {3}{4}, -i \, b x^{2}\right )\right )}\right )} \sin \relax (a)\right )}}{8 \, x^{\frac {3}{2}}} \]
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 \[ \int \frac {\cos \left (b\,x^2+a\right )}{x^{5/2}} \,d x \]
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 \[ \int \frac {\cos {\left (a + b x^{2} \right )}}{x^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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