Optimal. Leaf size=170 \[ -\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \sin \left (a-\frac{b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{2 b^{5/2}}-\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \cos \left (a-\frac{b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{2 b^{5/2}}+\frac{3 d \sqrt{c+d x} \sin (a+b x)}{2 b^2}-\frac{(c+d x)^{3/2} \cos (a+b x)}{b} \]
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Rubi [A] time = 0.242042, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {3296, 3306, 3305, 3351, 3304, 3352} \[ -\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \sin \left (a-\frac{b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{2 b^{5/2}}-\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \cos \left (a-\frac{b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{2 b^{5/2}}+\frac{3 d \sqrt{c+d x} \sin (a+b x)}{2 b^2}-\frac{(c+d x)^{3/2} \cos (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 3296
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int (c+d x)^{3/2} \sin (a+b x) \, dx &=-\frac{(c+d x)^{3/2} \cos (a+b x)}{b}+\frac{(3 d) \int \sqrt{c+d x} \cos (a+b x) \, dx}{2 b}\\ &=-\frac{(c+d x)^{3/2} \cos (a+b x)}{b}+\frac{3 d \sqrt{c+d x} \sin (a+b x)}{2 b^2}-\frac{\left (3 d^2\right ) \int \frac{\sin (a+b x)}{\sqrt{c+d x}} \, dx}{4 b^2}\\ &=-\frac{(c+d x)^{3/2} \cos (a+b x)}{b}+\frac{3 d \sqrt{c+d x} \sin (a+b x)}{2 b^2}-\frac{\left (3 d^2 \cos \left (a-\frac{b c}{d}\right )\right ) \int \frac{\sin \left (\frac{b c}{d}+b x\right )}{\sqrt{c+d x}} \, dx}{4 b^2}-\frac{\left (3 d^2 \sin \left (a-\frac{b c}{d}\right )\right ) \int \frac{\cos \left (\frac{b c}{d}+b x\right )}{\sqrt{c+d x}} \, dx}{4 b^2}\\ &=-\frac{(c+d x)^{3/2} \cos (a+b x)}{b}+\frac{3 d \sqrt{c+d x} \sin (a+b x)}{2 b^2}-\frac{\left (3 d \cos \left (a-\frac{b c}{d}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{2 b^2}-\frac{\left (3 d \sin \left (a-\frac{b c}{d}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{2 b^2}\\ &=-\frac{(c+d x)^{3/2} \cos (a+b x)}{b}-\frac{3 d^{3/2} \sqrt{\frac{\pi }{2}} \cos \left (a-\frac{b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{2 b^{5/2}}-\frac{3 d^{3/2} \sqrt{\frac{\pi }{2}} C\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right ) \sin \left (a-\frac{b c}{d}\right )}{2 b^{5/2}}+\frac{3 d \sqrt{c+d x} \sin (a+b x)}{2 b^2}\\ \end{align*}
Mathematica [C] time = 0.100351, size = 125, normalized size = 0.74 \[ -\frac{i d \sqrt{c+d x} e^{-\frac{i (a d+b c)}{d}} \left (\frac{e^{2 i a} \text{Gamma}\left (\frac{5}{2},-\frac{i b (c+d x)}{d}\right )}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \text{Gamma}\left (\frac{5}{2},\frac{i b (c+d x)}{d}\right )}{\sqrt{\frac{i b (c+d x)}{d}}}\right )}{2 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 188, normalized size = 1.1 \begin{align*} 2\,{\frac{1}{d} \left ( -1/2\,{\frac{d \left ( dx+c \right ) ^{3/2}}{b}\cos \left ({\frac{ \left ( dx+c \right ) b}{d}}+{\frac{da-cb}{d}} \right ) }+3/2\,{\frac{d}{b} \left ( 1/2\,{\frac{d\sqrt{dx+c}}{b}\sin \left ({\frac{ \left ( dx+c \right ) b}{d}}+{\frac{da-cb}{d}} \right ) }-1/4\,{\frac{d\sqrt{2}\sqrt{\pi }}{b} \left ( \cos \left ({\frac{da-cb}{d}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{dx+c}b}{\sqrt{\pi }d}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) +\sin \left ({\frac{da-cb}{d}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{dx+c}b}{\sqrt{\pi }d}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.82981, size = 855, normalized size = 5.03 \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 = 1.79597, size = 394, normalized size = 2.32 \begin{align*} -\frac{3 \, \sqrt{2} \pi d^{2} \sqrt{\frac{b}{\pi d}} \cos \left (-\frac{b c - a d}{d}\right ) \operatorname{S}\left (\sqrt{2} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) + 3 \, \sqrt{2} \pi d^{2} \sqrt{\frac{b}{\pi d}} \operatorname{C}\left (\sqrt{2} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) \sin \left (-\frac{b c - a d}{d}\right ) - 2 \,{\left (3 \, b d \sin \left (b x + a\right ) - 2 \,{\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )\right )} \sqrt{d x + c}}{4 \, b^{3}} \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 (c + d x\right )^{\frac{3}{2}} \sin{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.25184, size = 764, normalized size = 4.49 \begin{align*} -\frac{2 \,{\left (\frac{\sqrt{2} \sqrt{\pi } d^{2} \operatorname{erf}\left (-\frac{\sqrt{2} \sqrt{b d} \sqrt{d x + c}{\left (\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac{i \, b c - i \, a d}{d}\right )}}{\sqrt{b d}{\left (\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )} b} + \frac{\sqrt{2} \sqrt{\pi } d^{2} \operatorname{erf}\left (-\frac{\sqrt{2} \sqrt{b d} \sqrt{d x + c}{\left (-\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac{-i \, b c + i \, a d}{d}\right )}}{\sqrt{b d}{\left (-\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )} b} + \frac{2 \, \sqrt{d x + c} d e^{\left (\frac{i \,{\left (d x + c\right )} b - i \, b c + i \, a d}{d}\right )}}{b} + \frac{2 \, \sqrt{d x + c} d e^{\left (\frac{-i \,{\left (d x + c\right )} b + i \, b c - i \, a d}{d}\right )}}{b}\right )} c + \frac{i \, \sqrt{2} \sqrt{\pi }{\left (2 i \, b c d - 3 \, d^{2}\right )} d \operatorname{erf}\left (-\frac{\sqrt{2} \sqrt{b d} \sqrt{d x + c}{\left (\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac{i \, b c - i \, a d}{d}\right )}}{\sqrt{b d}{\left (\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )} b^{2}} + \frac{i \, \sqrt{2} \sqrt{\pi }{\left (2 i \, b c d + 3 \, d^{2}\right )} d \operatorname{erf}\left (-\frac{\sqrt{2} \sqrt{b d} \sqrt{d x + c}{\left (-\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac{-i \, b c + i \, a d}{d}\right )}}{\sqrt{b d}{\left (-\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )} b^{2}} - \frac{2 i \,{\left (2 i \,{\left (d x + c\right )}^{\frac{3}{2}} b d - 2 i \, \sqrt{d x + c} b c d - 3 \, \sqrt{d x + c} d^{2}\right )} e^{\left (\frac{i \,{\left (d x + c\right )} b - i \, b c + i \, a d}{d}\right )}}{b^{2}} - \frac{2 i \,{\left (2 i \,{\left (d x + c\right )}^{\frac{3}{2}} b d - 2 i \, \sqrt{d x + c} b c d + 3 \, \sqrt{d x + c} d^{2}\right )} e^{\left (\frac{-i \,{\left (d x + c\right )} b + i \, b c - i \, a d}{d}\right )}}{b^{2}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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