Optimal. Leaf size=115 \[ \frac{i e^{i a} (c+d x) \text{Gamma}\left (\frac{2}{3},-i b (c+d x)^{3/2}\right )}{3 d \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac{i e^{-i a} (c+d x) \text{Gamma}\left (\frac{2}{3},i b (c+d x)^{3/2}\right )}{3 d \left (i b (c+d x)^{3/2}\right )^{2/3}} \]
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Rubi [A] time = 0.0812329, antiderivative size = 115, normalized size of antiderivative = 1., 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 3363
Rule 3389
Rule 2218
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*}
Mathematica [A] time = 0.149353, size = 123, normalized size = 1.07 \[ \frac{i (c+d x) \left ((\cos (a)+i \sin (a)) \left (i b (c+d x)^{3/2}\right )^{2/3} \text{Gamma}\left (\frac{2}{3},-i b (c+d x)^{3/2}\right )-(\cos (a)-i \sin (a)) \left (-i b (c+d x)^{3/2}\right )^{2/3} \text{Gamma}\left (\frac{2}{3},i b (c+d x)^{3/2}\right )\right )}{3 d \left (b^2 (c+d x)^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.004, size = 0, normalized size = 0. \begin{align*} \int \sin \left ( a+b \left ( dx+c \right ) ^{{\frac{3}{2}}} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.3706, size = 466, normalized size = 4.05 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.14939, size = 198, normalized size = 1.72 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin{\left (a + b \left (c + d x\right )^{\frac{3}{2}} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin \left ({\left (d x + c\right )}^{\frac{3}{2}} b + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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