Optimal. Leaf size=291 \[ \frac{i e^{i a} (c+d x) (d e-c f) \text{Gamma}\left (\frac{2}{3},-i b (c+d x)^{3/2}\right )}{3 d^2 \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac{i e^{-i a} (c+d x) (d e-c f) \text{Gamma}\left (\frac{2}{3},i b (c+d x)^{3/2}\right )}{3 d^2 \left (i b (c+d x)^{3/2}\right )^{2/3}}-\frac{e^{i a} f \sqrt{c+d x} \text{Gamma}\left (\frac{1}{3},-i b (c+d x)^{3/2}\right )}{9 b d^2 \sqrt [3]{-i b (c+d x)^{3/2}}}-\frac{e^{-i a} f \sqrt{c+d x} \text{Gamma}\left (\frac{1}{3},i b (c+d x)^{3/2}\right )}{9 b d^2 \sqrt [3]{i b (c+d x)^{3/2}}}-\frac{2 f \sqrt{c+d x} \cos \left (a+b (c+d x)^{3/2}\right )}{3 b d^2} \]
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Rubi [A] time = 0.1999, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3433, 3389, 2218, 3385, 3356, 2208} \[ \frac{i e^{i a} (c+d x) (d e-c f) \text{Gamma}\left (\frac{2}{3},-i b (c+d x)^{3/2}\right )}{3 d^2 \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac{i e^{-i a} (c+d x) (d e-c f) \text{Gamma}\left (\frac{2}{3},i b (c+d x)^{3/2}\right )}{3 d^2 \left (i b (c+d x)^{3/2}\right )^{2/3}}-\frac{e^{i a} f \sqrt{c+d x} \text{Gamma}\left (\frac{1}{3},-i b (c+d x)^{3/2}\right )}{9 b d^2 \sqrt [3]{-i b (c+d x)^{3/2}}}-\frac{e^{-i a} f \sqrt{c+d x} \text{Gamma}\left (\frac{1}{3},i b (c+d x)^{3/2}\right )}{9 b d^2 \sqrt [3]{i b (c+d x)^{3/2}}}-\frac{2 f \sqrt{c+d x} \cos \left (a+b (c+d x)^{3/2}\right )}{3 b d^2} \]
Antiderivative was successfully verified.
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Rule 3433
Rule 3389
Rule 2218
Rule 3385
Rule 3356
Rule 2208
Rubi steps
\begin{align*} \int (e+f x) \sin \left (a+b (c+d x)^{3/2}\right ) \, dx &=\frac{2 \operatorname{Subst}\left (\int \left ((d e-c f) x \sin \left (a+b x^3\right )+f x^3 \sin \left (a+b x^3\right )\right ) \, dx,x,\sqrt{c+d x}\right )}{d^2}\\ &=\frac{(2 f) \operatorname{Subst}\left (\int x^3 \sin \left (a+b x^3\right ) \, dx,x,\sqrt{c+d x}\right )}{d^2}+\frac{(2 (d e-c f)) \operatorname{Subst}\left (\int x \sin \left (a+b x^3\right ) \, dx,x,\sqrt{c+d x}\right )}{d^2}\\ &=-\frac{2 f \sqrt{c+d x} \cos \left (a+b (c+d x)^{3/2}\right )}{3 b d^2}+\frac{(2 f) \operatorname{Subst}\left (\int \cos \left (a+b x^3\right ) \, dx,x,\sqrt{c+d x}\right )}{3 b d^2}+\frac{(i (d e-c f)) \operatorname{Subst}\left (\int e^{-i a-i b x^3} x \, dx,x,\sqrt{c+d x}\right )}{d^2}-\frac{(i (d e-c f)) \operatorname{Subst}\left (\int e^{i a+i b x^3} x \, dx,x,\sqrt{c+d x}\right )}{d^2}\\ &=-\frac{2 f \sqrt{c+d x} \cos \left (a+b (c+d x)^{3/2}\right )}{3 b d^2}+\frac{i e^{i a} (d e-c f) (c+d x) \Gamma \left (\frac{2}{3},-i b (c+d x)^{3/2}\right )}{3 d^2 \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac{i e^{-i a} (d e-c f) (c+d x) \Gamma \left (\frac{2}{3},i b (c+d x)^{3/2}\right )}{3 d^2 \left (i b (c+d x)^{3/2}\right )^{2/3}}+\frac{f \operatorname{Subst}\left (\int e^{-i a-i b x^3} \, dx,x,\sqrt{c+d x}\right )}{3 b d^2}+\frac{f \operatorname{Subst}\left (\int e^{i a+i b x^3} \, dx,x,\sqrt{c+d x}\right )}{3 b d^2}\\ &=-\frac{2 f \sqrt{c+d x} \cos \left (a+b (c+d x)^{3/2}\right )}{3 b d^2}-\frac{e^{i a} f \sqrt{c+d x} \Gamma \left (\frac{1}{3},-i b (c+d x)^{3/2}\right )}{9 b d^2 \sqrt [3]{-i b (c+d x)^{3/2}}}-\frac{e^{-i a} f \sqrt{c+d x} \Gamma \left (\frac{1}{3},i b (c+d x)^{3/2}\right )}{9 b d^2 \sqrt [3]{i b (c+d x)^{3/2}}}+\frac{i e^{i a} (d e-c f) (c+d x) \Gamma \left (\frac{2}{3},-i b (c+d x)^{3/2}\right )}{3 d^2 \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac{i e^{-i a} (d e-c f) (c+d x) \Gamma \left (\frac{2}{3},i b (c+d x)^{3/2}\right )}{3 d^2 \left (i b (c+d x)^{3/2}\right )^{2/3}}\\ \end{align*}
Mathematica [B] time = 2.61017, size = 705, normalized size = 2.42 \[ \frac{f \cos (a) \left (-\frac{2 \sqrt{c+d x} \text{Gamma}\left (\frac{1}{3},-i b (c+d x)^{3/2}\right )}{3 \sqrt [3]{-i b (c+d x)^{3/2}}}-\frac{2 \sqrt{c+d x} \text{Gamma}\left (\frac{1}{3},i b (c+d x)^{3/2}\right )}{3 \sqrt [3]{i b (c+d x)^{3/2}}}\right )}{6 b d^2}+\frac{i c f \cos (a) \left (\frac{2 (c+d x) \text{Gamma}\left (\frac{2}{3},i b (c+d x)^{3/2}\right )}{3 \left (i b (c+d x)^{3/2}\right )^{2/3}}-\frac{2 (c+d x) \text{Gamma}\left (\frac{2}{3},-i b (c+d x)^{3/2}\right )}{3 \left (-i b (c+d x)^{3/2}\right )^{2/3}}\right )}{2 d^2}+\frac{i f \sin (a) \left (\frac{2 \sqrt{c+d x} \text{Gamma}\left (\frac{1}{3},i b (c+d x)^{3/2}\right )}{3 \sqrt [3]{i b (c+d x)^{3/2}}}-\frac{2 \sqrt{c+d x} \text{Gamma}\left (\frac{1}{3},-i b (c+d x)^{3/2}\right )}{3 \sqrt [3]{-i b (c+d x)^{3/2}}}\right )}{6 b d^2}-\frac{c f \sin (a) \left (-\frac{2 (c+d x) \text{Gamma}\left (\frac{2}{3},-i b (c+d x)^{3/2}\right )}{3 \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac{2 (c+d x) \text{Gamma}\left (\frac{2}{3},i b (c+d x)^{3/2}\right )}{3 \left (i b (c+d x)^{3/2}\right )^{2/3}}\right )}{2 d^2}-\frac{i e \cos (a) \left (\frac{2 (c+d x) \text{Gamma}\left (\frac{2}{3},i b (c+d x)^{3/2}\right )}{3 \left (i b (c+d x)^{3/2}\right )^{2/3}}-\frac{2 (c+d x) \text{Gamma}\left (\frac{2}{3},-i b (c+d x)^{3/2}\right )}{3 \left (-i b (c+d x)^{3/2}\right )^{2/3}}\right )}{2 d}+\frac{e \sin (a) \left (-\frac{2 (c+d x) \text{Gamma}\left (\frac{2}{3},-i b (c+d x)^{3/2}\right )}{3 \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac{2 (c+d x) \text{Gamma}\left (\frac{2}{3},i b (c+d x)^{3/2}\right )}{3 \left (i b (c+d x)^{3/2}\right )^{2/3}}\right )}{2 d}+\frac{2 f \sin (a) \sqrt{c+d x} \sin \left (b (c+d x)^{3/2}\right )}{3 b d^2}-\frac{2 f \cos (a) \sqrt{c+d x} \cos \left (b (c+d x)^{3/2}\right )}{3 b d^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.02, size = 0, normalized size = 0. \begin{align*} \int \left ( fx+e \right ) \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 = 1.98336, size = 1449, normalized size = 4.98 \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 = 2.33272, size = 524, normalized size = 1.8 \begin{align*} \frac{i \, \left (i \, b\right )^{\frac{2}{3}} f e^{\left (-i \, a\right )} \Gamma \left (\frac{1}{3},{\left (i \, b d x + i \, b c\right )} \sqrt{d x + c}\right ) - i \, \left (-i \, b\right )^{\frac{2}{3}} f e^{\left (i \, a\right )} \Gamma \left (\frac{1}{3},{\left (-i \, b d x - i \, b c\right )} \sqrt{d x + c}\right ) - 6 \, \sqrt{d x + c} b f \cos \left ({\left (b d x + b c\right )} \sqrt{d x + c} + a\right ) - 3 \,{\left (b d e - b c f\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 \,{\left (b d e - b c f\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 )}{9 \, b^{2} d^{2}} \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 \left (e + f x\right ) \sin{\left (a + b c \sqrt{c + d x} + b d x \sqrt{c + d x} \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{\left (f x + e\right )} \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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