Integrand size = 18, antiderivative size = 410 \[ \int (c+d x)^{5/2} \sin ^3(a+b x) \, dx=\frac {45 d^2 \sqrt {c+d x} \cos (a+b x)}{16 b^3}-\frac {2 (c+d x)^{5/2} \cos (a+b x)}{3 b}-\frac {5 d^2 \sqrt {c+d x} \cos (3 a+3 b x)}{144 b^3}-\frac {45 d^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{7/2}}+\frac {5 d^{5/2} \sqrt {\frac {\pi }{6}} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{144 b^{7/2}}-\frac {5 d^{5/2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{144 b^{7/2}}+\frac {45 d^{5/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{16 b^{7/2}}+\frac {5 d (c+d x)^{3/2} \sin (a+b x)}{3 b^2}-\frac {(c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2} \]
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Time = 0.70 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3392, 3377, 3387, 3386, 3432, 3385, 3433, 3393} \[ \int (c+d x)^{5/2} \sin ^3(a+b x) \, dx=-\frac {45 \sqrt {\frac {\pi }{2}} d^{5/2} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{7/2}}+\frac {5 \sqrt {\frac {\pi }{6}} d^{5/2} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{144 b^{7/2}}-\frac {5 \sqrt {\frac {\pi }{6}} d^{5/2} \sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{144 b^{7/2}}+\frac {45 \sqrt {\frac {\pi }{2}} d^{5/2} \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{7/2}}+\frac {45 d^2 \sqrt {c+d x} \cos (a+b x)}{16 b^3}-\frac {5 d^2 \sqrt {c+d x} \cos (3 a+3 b x)}{144 b^3}+\frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}+\frac {5 d (c+d x)^{3/2} \sin (a+b x)}{3 b^2}-\frac {2 (c+d x)^{5/2} \cos (a+b x)}{3 b}-\frac {(c+d x)^{5/2} \sin ^2(a+b x) \cos (a+b x)}{3 b} \]
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Rule 3377
Rule 3385
Rule 3386
Rule 3387
Rule 3392
Rule 3393
Rule 3432
Rule 3433
Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}+\frac {2}{3} \int (c+d x)^{5/2} \sin (a+b x) \, dx-\frac {\left (5 d^2\right ) \int \sqrt {c+d x} \sin ^3(a+b x) \, dx}{12 b^2} \\ & = -\frac {2 (c+d x)^{5/2} \cos (a+b x)}{3 b}-\frac {(c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}+\frac {(5 d) \int (c+d x)^{3/2} \cos (a+b x) \, dx}{3 b}-\frac {\left (5 d^2\right ) \int \left (\frac {3}{4} \sqrt {c+d x} \sin (a+b x)-\frac {1}{4} \sqrt {c+d x} \sin (3 a+3 b x)\right ) \, dx}{12 b^2} \\ & = -\frac {2 (c+d x)^{5/2} \cos (a+b x)}{3 b}+\frac {5 d (c+d x)^{3/2} \sin (a+b x)}{3 b^2}-\frac {(c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}+\frac {\left (5 d^2\right ) \int \sqrt {c+d x} \sin (3 a+3 b x) \, dx}{48 b^2}-\frac {\left (5 d^2\right ) \int \sqrt {c+d x} \sin (a+b x) \, dx}{16 b^2}-\frac {\left (5 d^2\right ) \int \sqrt {c+d x} \sin (a+b x) \, dx}{2 b^2} \\ & = \frac {45 d^2 \sqrt {c+d x} \cos (a+b x)}{16 b^3}-\frac {2 (c+d x)^{5/2} \cos (a+b x)}{3 b}-\frac {5 d^2 \sqrt {c+d x} \cos (3 a+3 b x)}{144 b^3}+\frac {5 d (c+d x)^{3/2} \sin (a+b x)}{3 b^2}-\frac {(c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}+\frac {\left (5 d^3\right ) \int \frac {\cos (3 a+3 b x)}{\sqrt {c+d x}} \, dx}{288 b^3}-\frac {\left (5 d^3\right ) \int \frac {\cos (a+b x)}{\sqrt {c+d x}} \, dx}{32 b^3}-\frac {\left (5 d^3\right ) \int \frac {\cos (a+b x)}{\sqrt {c+d x}} \, dx}{4 b^3} \\ & = \frac {45 d^2 \sqrt {c+d x} \cos (a+b x)}{16 b^3}-\frac {2 (c+d x)^{5/2} \cos (a+b x)}{3 b}-\frac {5 d^2 \sqrt {c+d x} \cos (3 a+3 b x)}{144 b^3}+\frac {5 d (c+d x)^{3/2} \sin (a+b x)}{3 b^2}-\frac {(c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}+\frac {\left (5 d^3 \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}{288 b^3}-\frac {\left (5 d^3 \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}{32 b^3}-\frac {\left (5 d^3 \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}{4 b^3}-\frac {\left (5 d^3 \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}{288 b^3}+\frac {\left (5 d^3 \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}{32 b^3}+\frac {\left (5 d^3 \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}{4 b^3} \\ & = \frac {45 d^2 \sqrt {c+d x} \cos (a+b x)}{16 b^3}-\frac {2 (c+d x)^{5/2} \cos (a+b x)}{3 b}-\frac {5 d^2 \sqrt {c+d x} \cos (3 a+3 b x)}{144 b^3}+\frac {5 d (c+d x)^{3/2} \sin (a+b x)}{3 b^2}-\frac {(c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}+\frac {\left (5 d^2 \cos \left (3 a-\frac {3 b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {3 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{144 b^3}-\frac {\left (5 d^2 \cos \left (a-\frac {b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{16 b^3}-\frac {\left (5 d^2 \cos \left (a-\frac {b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{2 b^3}-\frac {\left (5 d^2 \sin \left (3 a-\frac {3 b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {3 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{144 b^3}+\frac {\left (5 d^2 \sin \left (a-\frac {b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{16 b^3}+\frac {\left (5 d^2 \sin \left (a-\frac {b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{2 b^3} \\ & = \frac {45 d^2 \sqrt {c+d x} \cos (a+b x)}{16 b^3}-\frac {2 (c+d x)^{5/2} \cos (a+b x)}{3 b}-\frac {5 d^2 \sqrt {c+d x} \cos (3 a+3 b x)}{144 b^3}-\frac {45 d^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{7/2}}+\frac {5 d^{5/2} \sqrt {\frac {\pi }{6}} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{144 b^{7/2}}-\frac {5 d^{5/2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\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 )}{144 b^{7/2}}+\frac {45 d^{5/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{16 b^{7/2}}+\frac {5 d (c+d x)^{3/2} \sin (a+b x)}{3 b^2}-\frac {(c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.11 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.61 \[ \int (c+d x)^{5/2} \sin ^3(a+b x) \, dx=\frac {e^{-\frac {3 i (b c+a d)}{d}} (c+d x)^{5/2} \left (243 e^{2 i \left (2 a+\frac {b c}{d}\right )} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {7}{2},-\frac {i b (c+d x)}{d}\right )+243 e^{2 i a+\frac {4 i b c}{d}} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {7}{2},\frac {i b (c+d x)}{d}\right )-\sqrt {3} \left (e^{6 i a} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {7}{2},-\frac {3 i b (c+d x)}{d}\right )+e^{\frac {6 i b c}{d}} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {7}{2},\frac {3 i b (c+d x)}{d}\right )\right )\right )}{648 b \left (\frac {b^2 (c+d x)^2}{d^2}\right )^{3/2}} \]
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Time = 0.51 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.16
method | result | size |
derivativedivides | \(\frac {-\frac {3 d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{4 b}+\frac {15 d \left (\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{2 b}-\frac {3 d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{2 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -c b}{d}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {d a -c b}{d}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{2 b}\right )}{4 b}+\frac {d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 d a -3 c b}{d}\right )}{12 b}-\frac {5 d \left (\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 d a -3 c b}{d}\right )}{6 b}-\frac {d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 d a -3 c b}{d}\right )}{6 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 d a -3 c b}{d}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {3 d a -3 c b}{d}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{36 b \sqrt {\frac {b}{d}}}\right )}{2 b}\right )}{12 b}}{d}\) | \(476\) |
default | \(\frac {-\frac {3 d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{4 b}+\frac {15 d \left (\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{2 b}-\frac {3 d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{2 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -c b}{d}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {d a -c b}{d}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{2 b}\right )}{4 b}+\frac {d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 d a -3 c b}{d}\right )}{12 b}-\frac {5 d \left (\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 d a -3 c b}{d}\right )}{6 b}-\frac {d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 d a -3 c b}{d}\right )}{6 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 d a -3 c b}{d}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {3 d a -3 c b}{d}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{36 b \sqrt {\frac {b}{d}}}\right )}{2 b}\right )}{12 b}}{d}\) | \(476\) |
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Time = 0.31 (sec) , antiderivative size = 371, normalized size of antiderivative = 0.90 \[ \int (c+d x)^{5/2} \sin ^3(a+b x) \, dx=\frac {5 \, \sqrt {6} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 1215 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 1215 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) - 5 \, \sqrt {6} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 24 \, {\left ({\left (12 \, b^{3} d^{2} x^{2} + 24 \, b^{3} c d x + 12 \, b^{3} c^{2} - 5 \, b d^{2}\right )} \cos \left (b x + a\right )^{3} - 3 \, {\left (12 \, b^{3} d^{2} x^{2} + 24 \, b^{3} c d x + 12 \, b^{3} c^{2} - 35 \, b d^{2}\right )} \cos \left (b x + a\right ) + 10 \, {\left (7 \, b^{2} d^{2} x + 7 \, b^{2} c d - {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{864 \, b^{4}} \]
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Timed out. \[ \int (c+d x)^{5/2} \sin ^3(a+b x) \, dx=\text {Timed out} \]
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Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 547, normalized size of antiderivative = 1.33 \[ \int (c+d x)^{5/2} \sin ^3(a+b x) \, dx=-\frac {{\left (240 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} \sin \left (\frac {3 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) - 6480 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} \sin \left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right ) - 24 \, {\left (\frac {12 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{4}}{d} - 5 \, \sqrt {d x + c} b^{2} d\right )} \cos \left (\frac {3 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) + 648 \, {\left (\frac {4 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{4}}{d} - 15 \, \sqrt {d x + c} b^{2} d\right )} \cos \left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right ) - 5 \, {\left (-\left (i - 1\right ) \cdot 9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - \left (i + 1\right ) \cdot 9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {3 i \, b}{d}}\right ) - 1215 \, {\left (\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {b c - a d}{d}\right ) + \left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {i \, b}{d}}\right ) - 1215 \, {\left (-\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {b c - a d}{d}\right ) - \left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {i \, b}{d}}\right ) - 5 \, {\left (\left (i + 1\right ) \cdot 9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + \left (i - 1\right ) \cdot 9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {3 i \, b}{d}}\right )\right )} d}{3456 \, b^{5}} \]
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Result contains complex when optimal does not.
Time = 0.84 (sec) , antiderivative size = 2476, normalized size of antiderivative = 6.04 \[ \int (c+d x)^{5/2} \sin ^3(a+b x) \, dx=\text {Too large to display} \]
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Timed out. \[ \int (c+d x)^{5/2} \sin ^3(a+b x) \, dx=\int {\sin \left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^{5/2} \,d x \]
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