Optimal. Leaf size=115 \[ \frac {i e^{i a} (c+d x) \Gamma \left (\frac {2}{3},-i b (c+d x)^{3/2}\right )}{3 d \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac {i e^{-i a} (c+d x) \Gamma \left (\frac {2}{3},i b (c+d x)^{3/2}\right )}{3 d \left (i b (c+d x)^{3/2}\right )^{2/3}} \]
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Rubi [A] time = 0.08, antiderivative size = 115, 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 = {3363, 3389, 2218} \[ \frac {i e^{i a} (c+d x) \text {Gamma}\left (\frac {2}{3},-i b (c+d x)^{3/2}\right )}{3 d \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac {i e^{-i a} (c+d x) \text {Gamma}\left (\frac {2}{3},i b (c+d x)^{3/2}\right )}{3 d \left (i b (c+d x)^{3/2}\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 2218
Rule 3363
Rule 3389
Rubi steps
\begin {align*} \int \sin \left (a+b (c+d x)^{3/2}\right ) \, dx &=\frac {2 \operatorname {Subst}\left (\int x \sin \left (a+b x^3\right ) \, dx,x,\sqrt {c+d x}\right )}{d}\\ &=\frac {i \operatorname {Subst}\left (\int e^{-i a-i b x^3} x \, dx,x,\sqrt {c+d x}\right )}{d}-\frac {i \operatorname {Subst}\left (\int e^{i a+i b x^3} x \, dx,x,\sqrt {c+d x}\right )}{d}\\ &=\frac {i e^{i a} (c+d x) \Gamma \left (\frac {2}{3},-i b (c+d x)^{3/2}\right )}{3 d \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac {i e^{-i a} (c+d x) \Gamma \left (\frac {2}{3},i b (c+d x)^{3/2}\right )}{3 d \left (i b (c+d x)^{3/2}\right )^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 123, normalized size = 1.07 \[ \frac {i (c+d x) \left ((\cos (a)+i \sin (a)) \left (i b (c+d x)^{3/2}\right )^{2/3} \Gamma \left (\frac {2}{3},-i b (c+d x)^{3/2}\right )-(\cos (a)-i \sin (a)) \left (-i b (c+d x)^{3/2}\right )^{2/3} \Gamma \left (\frac {2}{3},i b (c+d x)^{3/2}\right )\right )}{3 d \left (b^2 (c+d x)^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 69, normalized size = 0.60 \[ -\frac {\left (i \, b\right )^{\frac {1}{3}} e^{\left (-i \, a\right )} \Gamma \left (\frac {2}{3}, {\left (i \, b d x + i \, b c\right )} \sqrt {d x + c}\right ) + \left (-i \, b\right )^{\frac {1}{3}} e^{\left (i \, a\right )} \Gamma \left (\frac {2}{3}, {\left (-i \, b d x - i \, b c\right )} \sqrt {d x + c}\right )}{3 \, b d} \]
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 \[ \int \sin \left ({\left (d x + c\right )}^{\frac {3}{2}} b + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int \sin \left (a +b \left (d x +c \right )^{\frac {3}{2}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 112, normalized size = 0.97 \[ -\frac {\left ({\left (d x + c\right )}^{\frac {3}{2}} b\right )^{\frac {1}{3}} {\left ({\left ({\left (\sqrt {3} + i\right )} \Gamma \left (\frac {2}{3}, i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right ) + {\left (\sqrt {3} - i\right )} \Gamma \left (\frac {2}{3}, -i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right )\right )} \cos \relax (a) - {\left ({\left (i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {2}{3}, i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right ) + {\left (-i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {2}{3}, -i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right )\right )} \sin \relax (a)\right )}}{6 \, \sqrt {d x + c} b d} \]
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 \[ \int \sin \left (a+b\,{\left (c+d\,x\right )}^{3/2}\right ) \,d x \]
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 \[ \int \sin {\left (a + b \left (c + d x\right )^{\frac {3}{2}} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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