Integrand size = 22, antiderivative size = 196 \[ \int (c+d x)^{5/2} \cos (a+b x) \sin (a+b x) \, dx=\frac {15 d^2 \sqrt {c+d x} \cos (2 a+2 b x)}{64 b^3}-\frac {(c+d x)^{5/2} \cos (2 a+2 b x)}{4 b}-\frac {15 d^{5/2} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{128 b^{7/2}}+\frac {15 d^{5/2} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{128 b^{7/2}}+\frac {5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{16 b^2} \]
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Time = 0.39 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4491, 12, 3377, 3387, 3386, 3432, 3385, 3433} \[ \int (c+d x)^{5/2} \cos (a+b x) \sin (a+b x) \, dx=-\frac {15 \sqrt {\pi } d^{5/2} \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{128 b^{7/2}}+\frac {15 \sqrt {\pi } d^{5/2} \sin \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{128 b^{7/2}}+\frac {15 d^2 \sqrt {c+d x} \cos (2 a+2 b x)}{64 b^3}+\frac {5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{16 b^2}-\frac {(c+d x)^{5/2} \cos (2 a+2 b x)}{4 b} \]
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Rule 12
Rule 3377
Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4491
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{2} (c+d x)^{5/2} \sin (2 a+2 b x) \, dx \\ & = \frac {1}{2} \int (c+d x)^{5/2} \sin (2 a+2 b x) \, dx \\ & = -\frac {(c+d x)^{5/2} \cos (2 a+2 b x)}{4 b}+\frac {(5 d) \int (c+d x)^{3/2} \cos (2 a+2 b x) \, dx}{8 b} \\ & = -\frac {(c+d x)^{5/2} \cos (2 a+2 b x)}{4 b}+\frac {5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{16 b^2}-\frac {\left (15 d^2\right ) \int \sqrt {c+d x} \sin (2 a+2 b x) \, dx}{32 b^2} \\ & = \frac {15 d^2 \sqrt {c+d x} \cos (2 a+2 b x)}{64 b^3}-\frac {(c+d x)^{5/2} \cos (2 a+2 b x)}{4 b}+\frac {5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{16 b^2}-\frac {\left (15 d^3\right ) \int \frac {\cos (2 a+2 b x)}{\sqrt {c+d x}} \, dx}{128 b^3} \\ & = \frac {15 d^2 \sqrt {c+d x} \cos (2 a+2 b x)}{64 b^3}-\frac {(c+d x)^{5/2} \cos (2 a+2 b x)}{4 b}+\frac {5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{16 b^2}-\frac {\left (15 d^3 \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{\sqrt {c+d x}} \, dx}{128 b^3}+\frac {\left (15 d^3 \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{\sqrt {c+d x}} \, dx}{128 b^3} \\ & = \frac {15 d^2 \sqrt {c+d x} \cos (2 a+2 b x)}{64 b^3}-\frac {(c+d x)^{5/2} \cos (2 a+2 b x)}{4 b}+\frac {5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{16 b^2}-\frac {\left (15 d^2 \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{64 b^3}+\frac {\left (15 d^2 \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{64 b^3} \\ & = \frac {15 d^2 \sqrt {c+d x} \cos (2 a+2 b x)}{64 b^3}-\frac {(c+d x)^{5/2} \cos (2 a+2 b x)}{4 b}-\frac {15 d^{5/2} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{128 b^{7/2}}+\frac {15 d^{5/2} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{128 b^{7/2}}+\frac {5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{16 b^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.73 \[ \int (c+d x)^{5/2} \cos (a+b x) \sin (a+b x) \, dx=\frac {e^{-\frac {2 i (b c+a d)}{d}} (c+d x)^{5/2} \left (e^{4 i a} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {7}{2},-\frac {2 i b (c+d x)}{d}\right )+e^{\frac {4 i b c}{d}} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {7}{2},\frac {2 i b (c+d x)}{d}\right )\right )}{32 \sqrt {2} b \left (\frac {b^2 (c+d x)^2}{d^2}\right )^{3/2}} \]
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Time = 0.75 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.19
method | result | size |
derivativedivides | \(\frac {-\frac {d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}+\frac {5 d \left (\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}-\frac {3 d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}+\frac {d \sqrt {\pi }\, \left (\cos \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{8 b \sqrt {\frac {b}{d}}}\right )}{4 b}\right )}{4 b}}{d}\) | \(234\) |
default | \(\frac {-\frac {d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}+\frac {5 d \left (\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}-\frac {3 d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}+\frac {d \sqrt {\pi }\, \left (\cos \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{8 b \sqrt {\frac {b}{d}}}\right )}{4 b}\right )}{4 b}}{d}\) | \(234\) |
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Time = 0.25 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.13 \[ \int (c+d x)^{5/2} \cos (a+b x) \sin (a+b x) \, dx=-\frac {15 \, \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 15 \, \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - 2 \, {\left (16 \, b^{3} d^{2} x^{2} + 32 \, b^{3} c d x + 16 \, b^{3} c^{2} - 15 \, b d^{2} - 2 \, {\left (16 \, b^{3} d^{2} x^{2} + 32 \, b^{3} c d x + 16 \, b^{3} c^{2} - 15 \, b d^{2}\right )} \cos \left (b x + a\right )^{2} + 40 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{128 \, b^{4}} \]
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\[ \int (c+d x)^{5/2} \cos (a+b x) \sin (a+b x) \, dx=\int \left (c + d x\right )^{\frac {5}{2}} \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}\, dx \]
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Result contains complex when optimal does not.
Time = 0.38 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.41 \[ \int (c+d x)^{5/2} \cos (a+b x) \sin (a+b x) \, dx=\frac {\sqrt {2} {\left (160 \, \sqrt {2} {\left (d x + c\right )}^{\frac {3}{2}} b^{2} d \sin \left (\frac {2 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) - 8 \, {\left (16 \, \sqrt {2} {\left (d x + c\right )}^{\frac {5}{2}} b^{3} - 15 \, \sqrt {2} \sqrt {d x + c} b d^{2}\right )} \cos \left (\frac {2 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) - 15 \, {\left (-\left (i - 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {\pi } d^{3} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - \left (i + 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {\pi } d^{3} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {2 i \, b}{d}}\right ) - 15 \, {\left (\left (i + 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {\pi } d^{3} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + \left (i - 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {\pi } d^{3} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {2 i \, b}{d}}\right )\right )}}{1024 \, b^{4}} \]
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Result contains complex when optimal does not.
Time = 0.61 (sec) , antiderivative size = 1205, normalized size of antiderivative = 6.15 \[ \int (c+d x)^{5/2} \cos (a+b x) \sin (a+b x) \, dx=\text {Too large to display} \]
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Timed out. \[ \int (c+d x)^{5/2} \cos (a+b x) \sin (a+b x) \, dx=\int \cos \left (a+b\,x\right )\,\sin \left (a+b\,x\right )\,{\left (c+d\,x\right )}^{5/2} \,d x \]
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