Optimal. Leaf size=354 \[ -\frac{9 \sqrt{\frac{\pi }{2}} d^{3/2} \cos \left (a-\frac{b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{8 b^{5/2}}-\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \cos \left (3 a-\frac{3 b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{24 b^{5/2}}+\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \sin \left (3 a-\frac{3 b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{24 b^{5/2}}+\frac{9 \sqrt{\frac{\pi }{2}} d^{3/2} \sin \left (a-\frac{b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{8 b^{5/2}}+\frac{d \sqrt{c+d x} \cos ^3(a+b x)}{6 b^2}+\frac{d \sqrt{c+d x} \cos (a+b x)}{b^2}+\frac{2 (c+d x)^{3/2} \sin (a+b x)}{3 b}+\frac{(c+d x)^{3/2} \sin (a+b x) \cos ^2(a+b x)}{3 b} \]
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Rubi [A] time = 0.99067, antiderivative size = 354, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {3311, 3296, 3306, 3305, 3351, 3304, 3352, 3312} \[ -\frac{9 \sqrt{\frac{\pi }{2}} d^{3/2} \cos \left (a-\frac{b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{8 b^{5/2}}-\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \cos \left (3 a-\frac{3 b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{24 b^{5/2}}+\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \sin \left (3 a-\frac{3 b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{24 b^{5/2}}+\frac{9 \sqrt{\frac{\pi }{2}} d^{3/2} \sin \left (a-\frac{b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{8 b^{5/2}}+\frac{d \sqrt{c+d x} \cos ^3(a+b x)}{6 b^2}+\frac{d \sqrt{c+d x} \cos (a+b x)}{b^2}+\frac{2 (c+d x)^{3/2} \sin (a+b x)}{3 b}+\frac{(c+d x)^{3/2} \sin (a+b x) \cos ^2(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 3311
Rule 3296
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rule 3312
Rubi steps
\begin{align*} \int (c+d x)^{3/2} \cos ^3(a+b x) \, dx &=\frac{d \sqrt{c+d x} \cos ^3(a+b x)}{6 b^2}+\frac{(c+d x)^{3/2} \cos ^2(a+b x) \sin (a+b x)}{3 b}+\frac{2}{3} \int (c+d x)^{3/2} \cos (a+b x) \, dx-\frac{d^2 \int \frac{\cos ^3(a+b x)}{\sqrt{c+d x}} \, dx}{12 b^2}\\ &=\frac{d \sqrt{c+d x} \cos ^3(a+b x)}{6 b^2}+\frac{2 (c+d x)^{3/2} \sin (a+b x)}{3 b}+\frac{(c+d x)^{3/2} \cos ^2(a+b x) \sin (a+b x)}{3 b}-\frac{d \int \sqrt{c+d x} \sin (a+b x) \, dx}{b}-\frac{d^2 \int \left (\frac{3 \cos (a+b x)}{4 \sqrt{c+d x}}+\frac{\cos (3 a+3 b x)}{4 \sqrt{c+d x}}\right ) \, dx}{12 b^2}\\ &=\frac{d \sqrt{c+d x} \cos (a+b x)}{b^2}+\frac{d \sqrt{c+d x} \cos ^3(a+b x)}{6 b^2}+\frac{2 (c+d x)^{3/2} \sin (a+b x)}{3 b}+\frac{(c+d x)^{3/2} \cos ^2(a+b x) \sin (a+b x)}{3 b}-\frac{d^2 \int \frac{\cos (3 a+3 b x)}{\sqrt{c+d x}} \, dx}{48 b^2}-\frac{d^2 \int \frac{\cos (a+b x)}{\sqrt{c+d x}} \, dx}{16 b^2}-\frac{d^2 \int \frac{\cos (a+b x)}{\sqrt{c+d x}} \, dx}{2 b^2}\\ &=\frac{d \sqrt{c+d x} \cos (a+b x)}{b^2}+\frac{d \sqrt{c+d x} \cos ^3(a+b x)}{6 b^2}+\frac{2 (c+d x)^{3/2} \sin (a+b x)}{3 b}+\frac{(c+d x)^{3/2} \cos ^2(a+b x) \sin (a+b x)}{3 b}-\frac{\left (d^2 \cos \left (3 a-\frac{3 b c}{d}\right )\right ) \int \frac{\cos \left (\frac{3 b c}{d}+3 b x\right )}{\sqrt{c+d x}} \, dx}{48 b^2}-\frac{\left (d^2 \cos \left (a-\frac{b c}{d}\right )\right ) \int \frac{\cos \left (\frac{b c}{d}+b x\right )}{\sqrt{c+d x}} \, dx}{16 b^2}-\frac{\left (d^2 \cos \left (a-\frac{b c}{d}\right )\right ) \int \frac{\cos \left (\frac{b c}{d}+b x\right )}{\sqrt{c+d x}} \, dx}{2 b^2}+\frac{\left (d^2 \sin \left (3 a-\frac{3 b c}{d}\right )\right ) \int \frac{\sin \left (\frac{3 b c}{d}+3 b x\right )}{\sqrt{c+d x}} \, dx}{48 b^2}+\frac{\left (d^2 \sin \left (a-\frac{b c}{d}\right )\right ) \int \frac{\sin \left (\frac{b c}{d}+b x\right )}{\sqrt{c+d x}} \, dx}{16 b^2}+\frac{\left (d^2 \sin \left (a-\frac{b c}{d}\right )\right ) \int \frac{\sin \left (\frac{b c}{d}+b x\right )}{\sqrt{c+d x}} \, dx}{2 b^2}\\ &=\frac{d \sqrt{c+d x} \cos (a+b x)}{b^2}+\frac{d \sqrt{c+d x} \cos ^3(a+b x)}{6 b^2}+\frac{2 (c+d x)^{3/2} \sin (a+b x)}{3 b}+\frac{(c+d x)^{3/2} \cos ^2(a+b x) \sin (a+b x)}{3 b}-\frac{\left (d \cos \left (3 a-\frac{3 b c}{d}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{3 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{24 b^2}-\frac{\left (d \cos \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 )}{8 b^2}-\frac{\left (d \cos \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 )}{b^2}+\frac{\left (d \sin \left (3 a-\frac{3 b c}{d}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{3 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{24 b^2}+\frac{\left (d \sin \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 )}{8 b^2}+\frac{\left (d \sin \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 )}{b^2}\\ &=\frac{d \sqrt{c+d x} \cos (a+b x)}{b^2}+\frac{d \sqrt{c+d x} \cos ^3(a+b x)}{6 b^2}-\frac{9 d^{3/2} \sqrt{\frac{\pi }{2}} \cos \left (a-\frac{b c}{d}\right ) C\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{8 b^{5/2}}-\frac{d^{3/2} \sqrt{\frac{\pi }{6}} \cos \left (3 a-\frac{3 b c}{d}\right ) C\left (\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{24 b^{5/2}}+\frac{d^{3/2} \sqrt{\frac{\pi }{6}} S\left (\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right ) \sin \left (3 a-\frac{3 b c}{d}\right )}{24 b^{5/2}}+\frac{9 d^{3/2} \sqrt{\frac{\pi }{2}} S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right ) \sin \left (a-\frac{b c}{d}\right )}{8 b^{5/2}}+\frac{2 (c+d x)^{3/2} \sin (a+b x)}{3 b}+\frac{(c+d x)^{3/2} \cos ^2(a+b x) \sin (a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 1.6959, size = 390, normalized size = 1.1 \[ \frac{-81 \sqrt{2 \pi } d \cos \left (a-\frac{b c}{d}\right ) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\frac{b}{d}} \sqrt{c+d x}\right )-\sqrt{6 \pi } d \cos \left (3 a-\frac{3 b c}{d}\right ) \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\frac{b}{d}} \sqrt{c+d x}\right )+\sqrt{6 \pi } d \sin \left (3 a-\frac{3 b c}{d}\right ) S\left (\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right )+81 \sqrt{2 \pi } d \sin \left (a-\frac{b c}{d}\right ) S\left (\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right )+108 b d x \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (a+b x)+108 b c \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (a+b x)+12 b d x \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (3 (a+b x))+12 b c \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (3 (a+b x))+162 d \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (a+b x)+6 d \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (3 (a+b x))}{144 b^2 \sqrt{\frac{b}{d}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 386, normalized size = 1.1 \begin{align*} 2\,{\frac{1}{d} \left ( 3/8\,{\frac{d \left ( dx+c \right ) ^{3/2}}{b}\sin \left ({\frac{ \left ( dx+c \right ) b}{d}}+{\frac{da-cb}{d}} \right ) }-{\frac{9\,d}{8\,b} \left ( -1/2\,{\frac{d\sqrt{dx+c}}{b}\cos \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 FresnelC} \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 FresnelS} \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 ) }+1/24\,{\frac{d \left ( dx+c \right ) ^{3/2}}{b}\sin \left ( 3\,{\frac{ \left ( dx+c \right ) b}{d}}+3\,{\frac{da-cb}{d}} \right ) }-1/8\,{\frac{d}{b} \left ( -1/6\,{\frac{d\sqrt{dx+c}}{b}\cos \left ( 3\,{\frac{ \left ( dx+c \right ) b}{d}}+3\,{\frac{da-cb}{d}} \right ) }+1/36\,{\frac{d\sqrt{2}\sqrt{\pi }\sqrt{3}}{b} \left ( \cos \left ( 3\,{\frac{da-cb}{d}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{3}\sqrt{dx+c}b}{\sqrt{\pi }d}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) -\sin \left ( 3\,{\frac{da-cb}{d}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{3}\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 = 2.70164, size = 1793, normalized size = 5.06 \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 = 2.06816, size = 763, normalized size = 2.16 \begin{align*} -\frac{\sqrt{6} \pi d^{2} \sqrt{\frac{b}{\pi d}} \cos \left (-\frac{3 \,{\left (b c - a d\right )}}{d}\right ) \operatorname{C}\left (\sqrt{6} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) + 81 \, \sqrt{2} \pi d^{2} \sqrt{\frac{b}{\pi d}} \cos \left (-\frac{b c - a d}{d}\right ) \operatorname{C}\left (\sqrt{2} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) - 81 \, \sqrt{2} \pi d^{2} \sqrt{\frac{b}{\pi d}} \operatorname{S}\left (\sqrt{2} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) \sin \left (-\frac{b c - a d}{d}\right ) - \sqrt{6} \pi d^{2} \sqrt{\frac{b}{\pi d}} \operatorname{S}\left (\sqrt{6} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) \sin \left (-\frac{3 \,{\left (b c - a d\right )}}{d}\right ) - 24 \,{\left (b d \cos \left (b x + a\right )^{3} + 6 \, b d \cos \left (b x + a\right ) + 2 \,{\left (2 \, b^{2} d x + 2 \, b^{2} c +{\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )\right )} \sqrt{d x + c}}{144 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.44153, size = 1515, normalized size = 4.28 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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