Integrand size = 24, antiderivative size = 351 \[ \int (c+d x)^{3/2} \cos (a+b x) \sin ^3(a+b x) \, dx=-\frac {(c+d x)^{3/2} \cos (2 a+2 b x)}{8 b}+\frac {(c+d x)^{3/2} \cos (4 a+4 b x)}{32 b}+\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (4 a-\frac {4 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{512 b^{5/2}}-\frac {3 d^{3/2} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{64 b^{5/2}}+\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (4 a-\frac {4 b c}{d}\right )}{512 b^{5/2}}-\frac {3 d^{3/2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{64 b^{5/2}}+\frac {3 d \sqrt {c+d x} \sin (2 a+2 b x)}{32 b^2}-\frac {3 d \sqrt {c+d x} \sin (4 a+4 b x)}{256 b^2} \]
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Time = 0.76 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {4491, 3377, 3387, 3386, 3432, 3385, 3433} \[ \int (c+d x)^{3/2} \cos (a+b x) \sin ^3(a+b x) \, dx=\frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} \sin \left (4 a-\frac {4 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{512 b^{5/2}}-\frac {3 \sqrt {\pi } d^{3/2} \sin \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{64 b^{5/2}}+\frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} \cos \left (4 a-\frac {4 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{512 b^{5/2}}-\frac {3 \sqrt {\pi } d^{3/2} \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{64 b^{5/2}}+\frac {3 d \sqrt {c+d x} \sin (2 a+2 b x)}{32 b^2}-\frac {3 d \sqrt {c+d x} \sin (4 a+4 b x)}{256 b^2}-\frac {(c+d x)^{3/2} \cos (2 a+2 b x)}{8 b}+\frac {(c+d x)^{3/2} \cos (4 a+4 b x)}{32 b} \]
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Rule 3377
Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4491
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{4} (c+d x)^{3/2} \sin (2 a+2 b x)-\frac {1}{8} (c+d x)^{3/2} \sin (4 a+4 b x)\right ) \, dx \\ & = -\left (\frac {1}{8} \int (c+d x)^{3/2} \sin (4 a+4 b x) \, dx\right )+\frac {1}{4} \int (c+d x)^{3/2} \sin (2 a+2 b x) \, dx \\ & = -\frac {(c+d x)^{3/2} \cos (2 a+2 b x)}{8 b}+\frac {(c+d x)^{3/2} \cos (4 a+4 b x)}{32 b}-\frac {(3 d) \int \sqrt {c+d x} \cos (4 a+4 b x) \, dx}{64 b}+\frac {(3 d) \int \sqrt {c+d x} \cos (2 a+2 b x) \, dx}{16 b} \\ & = -\frac {(c+d x)^{3/2} \cos (2 a+2 b x)}{8 b}+\frac {(c+d x)^{3/2} \cos (4 a+4 b x)}{32 b}+\frac {3 d \sqrt {c+d x} \sin (2 a+2 b x)}{32 b^2}-\frac {3 d \sqrt {c+d x} \sin (4 a+4 b x)}{256 b^2}+\frac {\left (3 d^2\right ) \int \frac {\sin (4 a+4 b x)}{\sqrt {c+d x}} \, dx}{512 b^2}-\frac {\left (3 d^2\right ) \int \frac {\sin (2 a+2 b x)}{\sqrt {c+d x}} \, dx}{64 b^2} \\ & = -\frac {(c+d x)^{3/2} \cos (2 a+2 b x)}{8 b}+\frac {(c+d x)^{3/2} \cos (4 a+4 b x)}{32 b}+\frac {3 d \sqrt {c+d x} \sin (2 a+2 b x)}{32 b^2}-\frac {3 d \sqrt {c+d x} \sin (4 a+4 b x)}{256 b^2}+\frac {\left (3 d^2 \cos \left (4 a-\frac {4 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {4 b c}{d}+4 b x\right )}{\sqrt {c+d x}} \, dx}{512 b^2}-\frac {\left (3 d^2 \cos \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}{64 b^2}+\frac {\left (3 d^2 \sin \left (4 a-\frac {4 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {4 b c}{d}+4 b x\right )}{\sqrt {c+d x}} \, dx}{512 b^2}-\frac {\left (3 d^2 \sin \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}{64 b^2} \\ & = -\frac {(c+d x)^{3/2} \cos (2 a+2 b x)}{8 b}+\frac {(c+d x)^{3/2} \cos (4 a+4 b x)}{32 b}+\frac {3 d \sqrt {c+d x} \sin (2 a+2 b x)}{32 b^2}-\frac {3 d \sqrt {c+d x} \sin (4 a+4 b x)}{256 b^2}+\frac {\left (3 d \cos \left (4 a-\frac {4 b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {4 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{256 b^2}-\frac {\left (3 d \cos \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 )}{32 b^2}+\frac {\left (3 d \sin \left (4 a-\frac {4 b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {4 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{256 b^2}-\frac {\left (3 d \sin \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 )}{32 b^2} \\ & = -\frac {(c+d x)^{3/2} \cos (2 a+2 b x)}{8 b}+\frac {(c+d x)^{3/2} \cos (4 a+4 b x)}{32 b}+\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (4 a-\frac {4 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{512 b^{5/2}}-\frac {3 d^{3/2} \sqrt {\pi } \cos \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 )}{64 b^{5/2}}+\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (4 a-\frac {4 b c}{d}\right )}{512 b^{5/2}}-\frac {3 d^{3/2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{64 b^{5/2}}+\frac {3 d \sqrt {c+d x} \sin (2 a+2 b x)}{32 b^2}-\frac {3 d \sqrt {c+d x} \sin (4 a+4 b x)}{256 b^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.94 (sec) , antiderivative size = 693, normalized size of antiderivative = 1.97 \[ \int (c+d x)^{3/2} \cos (a+b x) \sin ^3(a+b x) \, dx=\frac {e^{-4 i a} \left (8 e^{2 i \left (a-\frac {b (c+d x)}{d}\right )} \left (-4 \sqrt {b} d e^{\frac {2 i b c}{d}} \sqrt {c+d x} \left (-3 i+4 b x+e^{4 i (a+b x)} (3 i+4 b x)\right )-(1-i) (4 b c+3 i d) \sqrt {d} e^{\frac {2 i b (2 c+d x)}{d}} \sqrt {\pi } \text {erf}\left (\frac {(1+i) \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )+(1+i) \sqrt {d} (4 i b c+3 d) e^{2 i (2 a+b x)} \sqrt {\pi } \text {erfi}\left (\frac {(1+i) \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )\right )-\sqrt {d} e^{-\frac {4 i b (c+d x)}{d}} \left (-4 \sqrt {b} \sqrt {d} e^{\frac {4 i b c}{d}} \sqrt {c+d x} \left (-3 i+8 b x+e^{8 i (a+b x)} (3 i+8 b x)\right )+(-1)^{3/4} (8 b c+3 i d) e^{\frac {4 i b (2 c+d x)}{d}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt [4]{-1} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )+\sqrt [4]{-1} (8 i b c+3 d) e^{4 i (2 a+b x)} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt [4]{-1} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )\right )-\frac {64 \sqrt {2} b^{3/2} c e^{2 i \left (a-\frac {b c}{d}\right )} \sqrt {c+d x} \left (e^{4 i a} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {3}{2},-\frac {2 i b (c+d x)}{d}\right )+e^{\frac {4 i b c}{d}} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {3}{2},\frac {2 i b (c+d x)}{d}\right )\right )}{\sqrt {\frac {b^2 (c+d x)^2}{d^2}}}-16 b^{3/2} c e^{-\frac {4 i b c}{d}} \sqrt {c+d x} \left (-\frac {e^{8 i a} \Gamma \left (\frac {3}{2},-\frac {4 i b (c+d x)}{d}\right )}{\sqrt {-\frac {i b (c+d x)}{d}}}-\frac {e^{\frac {8 i b c}{d}} \Gamma \left (\frac {3}{2},\frac {4 i b (c+d x)}{d}\right )}{\sqrt {\frac {i b (c+d x)}{d}}}\right )\right )}{2048 b^{5/2}} \]
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Time = 0.76 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.07
method | result | size |
derivativedivides | \(\frac {-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{8 b}+\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}-\frac {d \sqrt {\pi }\, \left (\cos \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{8 b \sqrt {\frac {b}{d}}}\right )}{8 b}+\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {4 b \left (d x +c \right )}{d}+\frac {4 a d -4 c b}{d}\right )}{32 b}-\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {4 b \left (d x +c \right )}{d}+\frac {4 a d -4 c b}{d}\right )}{8 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {4 a d -4 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {4 a d -4 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{32 b \sqrt {\frac {b}{d}}}\right )}{32 b}}{d}\) | \(376\) |
default | \(\frac {-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{8 b}+\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}-\frac {d \sqrt {\pi }\, \left (\cos \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{8 b \sqrt {\frac {b}{d}}}\right )}{8 b}+\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {4 b \left (d x +c \right )}{d}+\frac {4 a d -4 c b}{d}\right )}{32 b}-\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {4 b \left (d x +c \right )}{d}+\frac {4 a d -4 c b}{d}\right )}{8 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {4 a d -4 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {4 a d -4 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{32 b \sqrt {\frac {b}{d}}}\right )}{32 b}}{d}\) | \(376\) |
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Time = 0.27 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.90 \[ \int (c+d x)^{3/2} \cos (a+b x) \sin ^3(a+b x) \, dx=\frac {3 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (2 \, \sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 3 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (2 \, \sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) - 48 \, \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 48 \, \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + 16 \, {\left (16 \, {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{4} + 10 \, b^{2} d x + 10 \, b^{2} c - 32 \, {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{2} - 3 \, {\left (2 \, b d \cos \left (b x + a\right )^{3} - 5 \, b d \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{1024 \, b^{3}} \]
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\[ \int (c+d x)^{3/2} \cos (a+b x) \sin ^3(a+b x) \, dx=\int \left (c + d x\right )^{\frac {3}{2}} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}\, dx \]
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Result contains complex when optimal does not.
Time = 0.38 (sec) , antiderivative size = 503, normalized size of antiderivative = 1.43 \[ \int (c+d x)^{3/2} \cos (a+b x) \sin ^3(a+b x) \, dx=\frac {{\left (\frac {128 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} \cos \left (\frac {4 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right )}{d} - \frac {512 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} \cos \left (\frac {2 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right )}{d} - 48 \, \sqrt {d x + c} b^{2} \sin \left (\frac {4 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) + 384 \, \sqrt {d x + c} b^{2} \sin \left (\frac {2 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) + 24 \, {\left (-\left (i + 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d \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 {2} \sqrt {\pi } b d \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 ) + 3 \, {\left (\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } b d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) - \left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } b d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (2 \, \sqrt {d x + c} \sqrt {\frac {i \, b}{d}}\right ) + 3 \, {\left (-\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } b d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) + \left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } b d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (2 \, \sqrt {d x + c} \sqrt {-\frac {i \, b}{d}}\right ) + 24 \, {\left (\left (i - 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d \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 {2} \sqrt {\pi } b d \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 )} d}{4096 \, b^{4}} \]
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Result contains complex when optimal does not.
Time = 0.90 (sec) , antiderivative size = 1515, normalized size of antiderivative = 4.32 \[ \int (c+d x)^{3/2} \cos (a+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]
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Timed out. \[ \int (c+d x)^{3/2} \cos (a+b x) \sin ^3(a+b x) \, dx=\int \cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^{3/2} \,d x \]
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