Optimal. Leaf size=60 \[ \frac {12 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{25 b^2}+\frac {4 \sin (a+b x) \cos ^{\frac {3}{2}}(a+b x)}{25 b^2}-\frac {2 x \cos ^{\frac {5}{2}}(a+b x)}{5 b} \]
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Rubi [A] time = 0.04, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3444, 2635, 2639} \[ \frac {12 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{25 b^2}+\frac {4 \sin (a+b x) \cos ^{\frac {3}{2}}(a+b x)}{25 b^2}-\frac {2 x \cos ^{\frac {5}{2}}(a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 3444
Rubi steps
\begin {align*} \int x \cos ^{\frac {3}{2}}(a+b x) \sin (a+b x) \, dx &=-\frac {2 x \cos ^{\frac {5}{2}}(a+b x)}{5 b}+\frac {2 \int \cos ^{\frac {5}{2}}(a+b x) \, dx}{5 b}\\ &=-\frac {2 x \cos ^{\frac {5}{2}}(a+b x)}{5 b}+\frac {4 \cos ^{\frac {3}{2}}(a+b x) \sin (a+b x)}{25 b^2}+\frac {6 \int \sqrt {\cos (a+b x)} \, dx}{25 b}\\ &=-\frac {2 x \cos ^{\frac {5}{2}}(a+b x)}{5 b}+\frac {12 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{25 b^2}+\frac {4 \cos ^{\frac {3}{2}}(a+b x) \sin (a+b x)}{25 b^2}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 51, normalized size = 0.85 \[ -\frac {2 \left (\cos ^{\frac {3}{2}}(a+b x) (5 b x \cos (a+b x)-2 \sin (a+b x))-6 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )\right )}{25 b^2} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int x \cos \left (b x + a\right )^{\frac {3}{2}} \sin \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int x \left (\cos ^{\frac {3}{2}}\left (b x +a \right )\right ) \sin \left (b x +a \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \cos \left (b x + a\right )^{\frac {3}{2}} \sin \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,{\cos \left (a+b\,x\right )}^{3/2}\,\sin \left (a+b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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