Optimal. Leaf size=168 \[ -\frac{4 \sqrt{2 \pi } b^{3/2} \cos \left (a-\frac{b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{3 d^{5/2}}+\frac{4 \sqrt{2 \pi } b^{3/2} \sin \left (a-\frac{b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{3 d^{5/2}}+\frac{4 b \sin (a+b x)}{3 d^2 \sqrt{c+d x}}-\frac{2 \cos (a+b x)}{3 d (c+d x)^{3/2}} \]
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Rubi [A] time = 0.262334, antiderivative size = 168, 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 = {3297, 3306, 3305, 3351, 3304, 3352} \[ -\frac{4 \sqrt{2 \pi } b^{3/2} \cos \left (a-\frac{b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{3 d^{5/2}}+\frac{4 \sqrt{2 \pi } b^{3/2} \sin \left (a-\frac{b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{3 d^{5/2}}+\frac{4 b \sin (a+b x)}{3 d^2 \sqrt{c+d x}}-\frac{2 \cos (a+b x)}{3 d (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{\cos (a+b x)}{(c+d x)^{5/2}} \, dx &=-\frac{2 \cos (a+b x)}{3 d (c+d x)^{3/2}}-\frac{(2 b) \int \frac{\sin (a+b x)}{(c+d x)^{3/2}} \, dx}{3 d}\\ &=-\frac{2 \cos (a+b x)}{3 d (c+d x)^{3/2}}+\frac{4 b \sin (a+b x)}{3 d^2 \sqrt{c+d x}}-\frac{\left (4 b^2\right ) \int \frac{\cos (a+b x)}{\sqrt{c+d x}} \, dx}{3 d^2}\\ &=-\frac{2 \cos (a+b x)}{3 d (c+d x)^{3/2}}+\frac{4 b \sin (a+b x)}{3 d^2 \sqrt{c+d x}}-\frac{\left (4 b^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}{3 d^2}+\frac{\left (4 b^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}{3 d^2}\\ &=-\frac{2 \cos (a+b x)}{3 d (c+d x)^{3/2}}+\frac{4 b \sin (a+b x)}{3 d^2 \sqrt{c+d x}}-\frac{\left (8 b^2 \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 )}{3 d^3}+\frac{\left (8 b^2 \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 )}{3 d^3}\\ &=-\frac{2 \cos (a+b x)}{3 d (c+d x)^{3/2}}-\frac{4 b^{3/2} \sqrt{2 \pi } \cos \left (a-\frac{b c}{d}\right ) C\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{3 d^{5/2}}+\frac{4 b^{3/2} \sqrt{2 \pi } S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right ) \sin \left (a-\frac{b c}{d}\right )}{3 d^{5/2}}+\frac{4 b \sin (a+b x)}{3 d^2 \sqrt{c+d x}}\\ \end{align*}
Mathematica [C] time = 0.302547, size = 190, normalized size = 1.13 \[ \frac{e^{-i a} \left (e^{-i b x} \left (-4 d e^{\frac{i b (c+d x)}{d}} \left (\frac{i b (c+d x)}{d}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},\frac{i b (c+d x)}{d}\right )+4 i b (c+d x)-2 d\right )-2 i e^{2 i a-\frac{i b c}{d}} \left (e^{\frac{i b (c+d x)}{d}} (2 b (c+d x)-i d)-2 i d \left (-\frac{i b (c+d x)}{d}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{i b (c+d x)}{d}\right )\right )\right )}{6 d^2 (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 180, normalized size = 1.1 \begin{align*} 2\,{\frac{1}{d} \left ( -1/3\,{\frac{1}{ \left ( dx+c \right ) ^{3/2}}\cos \left ({\frac{ \left ( dx+c \right ) b}{d}}+{\frac{da-cb}{d}} \right ) }-2/3\,{\frac{b}{d} \left ( -{\frac{1}{\sqrt{dx+c}}\sin \left ({\frac{ \left ( dx+c \right ) b}{d}}+{\frac{da-cb}{d}} \right ) }+{\frac{\sqrt{2}b\sqrt{\pi }}{d} \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 ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.5105, size = 632, normalized size = 3.76 \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 = 1.47589, size = 510, normalized size = 3.04 \begin{align*} -\frac{2 \,{\left (2 \, \sqrt{2}{\left (\pi b d^{2} x^{2} + 2 \, \pi b c d x + \pi b c^{2}\right )} \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 ) - 2 \, \sqrt{2}{\left (\pi b d^{2} x^{2} + 2 \, \pi b c d x + \pi b c^{2}\right )} \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{d x + c}{\left (d \cos \left (b x + a\right ) - 2 \,{\left (b d x + b c\right )} \sin \left (b x + a\right )\right )}\right )}}{3 \,{\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \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 \frac{\cos{\left (a + b x \right )}}{\left (c + d x\right )^{\frac{5}{2}}}\, 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 \frac{\cos \left (b x + a\right )}{{\left (d x + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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