Optimal. Leaf size=168 \[ -\frac {4 \sqrt {2 \pi } b^{3/2} \cos \left (a-\frac {b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{3 d^{5/2}}+\frac {4 \sqrt {2 \pi } b^{3/2} \sin \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{3 d^{5/2}}+\frac {4 b \sin (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}} \]
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Rubi [A] time = 0.26, antiderivative size = 168, normalized size of antiderivative = 1.00, 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 3304
Rule 3305
Rule 3306
Rule 3351
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*}
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Mathematica [C] time = 0.29, size = 190, normalized size = 1.13 \[ \frac {e^{-i a} \left (e^{-i b x} \left (4 i b (c+d x)-4 d e^{\frac {i b (c+d x)}{d}} \left (\frac {i b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {i b (c+d x)}{d}\right )-2 d\right )-2 i e^{2 i a-\frac {i b c}{d}} \left (e^{\frac {i b (c+d x)}{d}} (2 b (c+d x)-i d)-2 i d \left (-\frac {i b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {i b (c+d x)}{d}\right )\right )\right )}{6 d^2 (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 208, normalized size = 1.24 \[ -\frac {2 \, {\left (2 \, \sqrt {2} {\left (\pi b d^{2} x^{2} + 2 \, \pi b c d x + \pi b c^{2}\right )} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 2 \, \sqrt {2} {\left (\pi b d^{2} x^{2} + 2 \, \pi b c d x + \pi b c^{2}\right )} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + \sqrt {d x + c} {\left (d \cos \left (b x + a\right ) - 2 \, {\left (b d x + b c\right )} \sin \left (b x + a\right )\right )}\right )}}{3 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (b x + a\right )}{{\left (d x + c\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 180, normalized size = 1.07 \[ \frac {-\frac {2 \cos \left (\frac {\left (d x +c \right ) b}{d}+\frac {d a -c b}{d}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {4 b \left (-\frac {\sin \left (\frac {\left (d x +c \right ) b}{d}+\frac {d a -c b}{d}\right )}{\sqrt {d x +c}}+\frac {b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {d a -c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{3 d}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.72, size = 129, normalized size = 0.77 \[ -\frac {{\left ({\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) + {\left (\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) - \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right )\right )} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {3}{2}}}{4 \, {\left (d x + c\right )}^{\frac {3}{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\cos \left (a+b\,x\right )}{{\left (c+d\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos {\left (a + b x \right )}}{\left (c + d x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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