Optimal. Leaf size=132 \[ \frac {x^{7/2}}{7}-\frac {3 i e^{2 i a} x^{3/2} \text {Gamma}\left (\frac {3}{4},-2 i b x^2\right )}{64\ 2^{3/4} b \left (-i b x^2\right )^{3/4}}+\frac {3 i e^{-2 i a} x^{3/2} \text {Gamma}\left (\frac {3}{4},2 i b x^2\right )}{64\ 2^{3/4} b \left (i b x^2\right )^{3/4}}+\frac {x^{3/2} \sin \left (2 \left (a+b x^2\right )\right )}{8 b} \]
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Rubi [A]
time = 0.12, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3483, 3485,
3467, 3470, 2250} \begin {gather*} -\frac {3 i e^{2 i a} x^{3/2} \text {Gamma}\left (\frac {3}{4},-2 i b x^2\right )}{64\ 2^{3/4} b \left (-i b x^2\right )^{3/4}}+\frac {3 i e^{-2 i a} x^{3/2} \text {Gamma}\left (\frac {3}{4},2 i b x^2\right )}{64\ 2^{3/4} b \left (i b x^2\right )^{3/4}}+\frac {x^{3/2} \sin \left (2 \left (a+b x^2\right )\right )}{8 b}+\frac {x^{7/2}}{7} \end {gather*}
Antiderivative was successfully verified.
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Rule 2250
Rule 3467
Rule 3470
Rule 3483
Rule 3485
Rubi steps
\begin {align*} \int x^{5/2} \cos ^2\left (a+b x^2\right ) \, dx &=2 \text {Subst}\left (\int x^6 \cos ^2\left (a+b x^4\right ) \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (\frac {x^6}{2}+\frac {1}{2} x^6 \cos \left (2 a+2 b x^4\right )\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {x^{7/2}}{7}+\text {Subst}\left (\int x^6 \cos \left (2 a+2 b x^4\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {x^{7/2}}{7}+\frac {x^{3/2} \sin \left (2 \left (a+b x^2\right )\right )}{8 b}-\frac {3 \text {Subst}\left (\int x^2 \sin \left (2 a+2 b x^4\right ) \, dx,x,\sqrt {x}\right )}{8 b}\\ &=\frac {x^{7/2}}{7}+\frac {x^{3/2} \sin \left (2 \left (a+b x^2\right )\right )}{8 b}-\frac {(3 i) \text {Subst}\left (\int e^{-2 i a-2 i b x^4} x^2 \, dx,x,\sqrt {x}\right )}{16 b}+\frac {(3 i) \text {Subst}\left (\int e^{2 i a+2 i b x^4} x^2 \, dx,x,\sqrt {x}\right )}{16 b}\\ &=\frac {x^{7/2}}{7}-\frac {3 i e^{2 i a} x^{3/2} \Gamma \left (\frac {3}{4},-2 i b x^2\right )}{64\ 2^{3/4} b \left (-i b x^2\right )^{3/4}}+\frac {3 i e^{-2 i a} x^{3/2} \Gamma \left (\frac {3}{4},2 i b x^2\right )}{64\ 2^{3/4} b \left (i b x^2\right )^{3/4}}+\frac {x^{3/2} \sin \left (2 \left (a+b x^2\right )\right )}{8 b}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 142, normalized size = 1.08 \begin {gather*} \frac {b x^{11/2} \left (21 \sqrt [4]{2} \left (i b x^2\right )^{3/4} \text {Gamma}\left (\frac {3}{4},-2 i b x^2\right ) (-i \cos (2 a)+\sin (2 a))+21 \sqrt [4]{2} \left (-i b x^2\right )^{3/4} \text {Gamma}\left (\frac {3}{4},2 i b x^2\right ) (i \cos (2 a)+\sin (2 a))+16 \left (b^2 x^4\right )^{3/4} \left (8 b x^2+7 \sin \left (2 \left (a+b x^2\right )\right )\right )\right )}{896 \left (b^2 x^4\right )^{7/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int x^{\frac {5}{2}} \left (\cos ^{2}\left (b \,x^{2}+a \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.12, size = 171, normalized size = 1.30 \begin {gather*} \frac {256 \, b^{2} x^{4} + 224 \, b x^{2} \sin \left (2 \, b x^{2} + 2 \, a\right ) + 21 \cdot 2^{\frac {1}{4}} \left (b x^{2}\right )^{\frac {1}{4}} {\left ({\left (\sqrt {\sqrt {2} + 2} {\left (\Gamma \left (\frac {3}{4}, 2 i \, b x^{2}\right ) + \Gamma \left (\frac {3}{4}, -2 i \, b x^{2}\right )\right )} + \sqrt {-\sqrt {2} + 2} {\left (i \, \Gamma \left (\frac {3}{4}, 2 i \, b x^{2}\right ) - i \, \Gamma \left (\frac {3}{4}, -2 i \, b x^{2}\right )\right )}\right )} \cos \left (2 \, a\right ) + {\left (\sqrt {-\sqrt {2} + 2} {\left (\Gamma \left (\frac {3}{4}, 2 i \, b x^{2}\right ) + \Gamma \left (\frac {3}{4}, -2 i \, b x^{2}\right )\right )} + \sqrt {\sqrt {2} + 2} {\left (-i \, \Gamma \left (\frac {3}{4}, 2 i \, b x^{2}\right ) + i \, \Gamma \left (\frac {3}{4}, -2 i \, b x^{2}\right )\right )}\right )} \sin \left (2 \, a\right )\right )}}{1792 \, b^{2} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.13, size = 78, normalized size = 0.59 \begin {gather*} \frac {21 \, \left (2 i \, b\right )^{\frac {1}{4}} e^{\left (-2 i \, a\right )} \Gamma \left (\frac {3}{4}, 2 i \, b x^{2}\right ) + 21 \, \left (-2 i \, b\right )^{\frac {1}{4}} e^{\left (2 i \, a\right )} \Gamma \left (\frac {3}{4}, -2 i \, b x^{2}\right ) + 32 \, {\left (4 \, b^{2} x^{3} + 7 \, b x \cos \left (b x^{2} + a\right ) \sin \left (b x^{2} + a\right )\right )} \sqrt {x}}{896 \, b^{2}} \end {gather*}
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 \begin {gather*} \int x^{\frac {5}{2}} \cos ^{2}{\left (a + b x^{2} \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^{5/2}\,{\cos \left (b\,x^2+a\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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