\(\int (c+d x)^{5/2} \sin ^3(a+b x) \, dx\) [53]

Optimal result

Integrand size = 18, antiderivative size = 410 \[ \int (c+d x)^{5/2} \sin ^3(a+b x) \, dx=\frac {45 d^2 \sqrt {c+d x} \cos (a+b x)}{16 b^3}-\frac {2 (c+d x)^{5/2} \cos (a+b x)}{3 b}-\frac {5 d^2 \sqrt {c+d x} \cos (3 a+3 b x)}{144 b^3}-\frac {45 d^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{7/2}}+\frac {5 d^{5/2} \sqrt {\frac {\pi }{6}} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{144 b^{7/2}}-\frac {5 d^{5/2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{144 b^{7/2}}+\frac {45 d^{5/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{16 b^{7/2}}+\frac {5 d (c+d x)^{3/2} \sin (a+b x)}{3 b^2}-\frac {(c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2} \] Output:

45/16*d^2*(d*x+c)^(1/2)*cos(b*x+a)/b^3-2/3*(d*x+c)^(5/2)*cos(b*x+a)/b-5/14 
4*d^2*(d*x+c)^(1/2)*cos(3*b*x+3*a)/b^3-45/32*d^(5/2)*2^(1/2)*Pi^(1/2)*cos( 
a-b*c/d)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))/b^(7/2)+ 
5/864*d^(5/2)*6^(1/2)*Pi^(1/2)*cos(3*a-3*b*c/d)*FresnelC(b^(1/2)*6^(1/2)/P 
i^(1/2)*(d*x+c)^(1/2)/d^(1/2))/b^(7/2)-5/864*d^(5/2)*6^(1/2)*Pi^(1/2)*Fres 
nelS(b^(1/2)*6^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*sin(3*a-3*b*c/d)/b^(7 
/2)+45/32*d^(5/2)*2^(1/2)*Pi^(1/2)*FresnelS(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+ 
c)^(1/2)/d^(1/2))*sin(a-b*c/d)/b^(7/2)+5/3*d*(d*x+c)^(3/2)*sin(b*x+a)/b^2- 
1/3*(d*x+c)^(5/2)*cos(b*x+a)*sin(b*x+a)^2/b+5/18*d*(d*x+c)^(3/2)*sin(b*x+a 
)^3/b^2
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.79 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.61 \[ \int (c+d x)^{5/2} \sin ^3(a+b x) \, dx=\frac {e^{-\frac {3 i (b c+a d)}{d}} (c+d x)^{5/2} \left (243 e^{2 i \left (2 a+\frac {b c}{d}\right )} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {7}{2},-\frac {i b (c+d x)}{d}\right )+243 e^{2 i a+\frac {4 i b c}{d}} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {7}{2},\frac {i b (c+d x)}{d}\right )-\sqrt {3} \left (e^{6 i a} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {7}{2},-\frac {3 i b (c+d x)}{d}\right )+e^{\frac {6 i b c}{d}} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {7}{2},\frac {3 i b (c+d x)}{d}\right )\right )\right )}{648 b \left (\frac {b^2 (c+d x)^2}{d^2}\right )^{3/2}} \] Input:

Integrate[(c + d*x)^(5/2)*Sin[a + b*x]^3,x]
 

Output:

((c + d*x)^(5/2)*(243*E^((2*I)*(2*a + (b*c)/d))*Sqrt[(I*b*(c + d*x))/d]*Ga 
mma[7/2, ((-I)*b*(c + d*x))/d] + 243*E^((2*I)*a + ((4*I)*b*c)/d)*Sqrt[((-I 
)*b*(c + d*x))/d]*Gamma[7/2, (I*b*(c + d*x))/d] - Sqrt[3]*(E^((6*I)*a)*Sqr 
t[(I*b*(c + d*x))/d]*Gamma[7/2, ((-3*I)*b*(c + d*x))/d] + E^(((6*I)*b*c)/d 
)*Sqrt[((-I)*b*(c + d*x))/d]*Gamma[7/2, ((3*I)*b*(c + d*x))/d])))/(648*b*E 
^(((3*I)*(b*c + a*d))/d)*((b^2*(c + d*x)^2)/d^2)^(3/2))
 

Rubi [A] (verified)

Time = 2.63 (sec) , antiderivative size = 578, normalized size of antiderivative = 1.41, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3792, 3042, 3777, 3042, 3777, 25, 3042, 3777, 3042, 3787, 3042, 3785, 3786, 3793, 2009, 3832, 3833}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (c+d x)^{5/2} \sin ^3(a+b x) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (c+d x)^{5/2} \sin (a+b x)^3dx\)

\(\Big \downarrow \) 3792

\(\displaystyle -\frac {5 d^2 \int \sqrt {c+d x} \sin ^3(a+b x)dx}{12 b^2}+\frac {2}{3} \int (c+d x)^{5/2} \sin (a+b x)dx+\frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}-\frac {(c+d x)^{5/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {5 d^2 \int \sqrt {c+d x} \sin (a+b x)^3dx}{12 b^2}+\frac {2}{3} \int (c+d x)^{5/2} \sin (a+b x)dx+\frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}-\frac {(c+d x)^{5/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3777

\(\displaystyle -\frac {5 d^2 \int \sqrt {c+d x} \sin (a+b x)^3dx}{12 b^2}+\frac {2}{3} \left (\frac {5 d \int (c+d x)^{3/2} \cos (a+b x)dx}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\right )+\frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}-\frac {(c+d x)^{5/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {5 d^2 \int \sqrt {c+d x} \sin (a+b x)^3dx}{12 b^2}+\frac {2}{3} \left (\frac {5 d \int (c+d x)^{3/2} \sin \left (a+b x+\frac {\pi }{2}\right )dx}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\right )+\frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}-\frac {(c+d x)^{5/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3777

\(\displaystyle -\frac {5 d^2 \int \sqrt {c+d x} \sin (a+b x)^3dx}{12 b^2}+\frac {2}{3} \left (\frac {5 d \left (\frac {3 d \int -\sqrt {c+d x} \sin (a+b x)dx}{2 b}+\frac {(c+d x)^{3/2} \sin (a+b x)}{b}\right )}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\right )+\frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}-\frac {(c+d x)^{5/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {5 d^2 \int \sqrt {c+d x} \sin (a+b x)^3dx}{12 b^2}+\frac {2}{3} \left (\frac {5 d \left (\frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \int \sqrt {c+d x} \sin (a+b x)dx}{2 b}\right )}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\right )+\frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}-\frac {(c+d x)^{5/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {5 d^2 \int \sqrt {c+d x} \sin (a+b x)^3dx}{12 b^2}+\frac {2}{3} \left (\frac {5 d \left (\frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \int \sqrt {c+d x} \sin (a+b x)dx}{2 b}\right )}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\right )+\frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}-\frac {(c+d x)^{5/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3777

\(\displaystyle -\frac {5 d^2 \int \sqrt {c+d x} \sin (a+b x)^3dx}{12 b^2}+\frac {2}{3} \left (\frac {5 d \left (\frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \left (\frac {d \int \frac {\cos (a+b x)}{\sqrt {c+d x}}dx}{2 b}-\frac {\sqrt {c+d x} \cos (a+b x)}{b}\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\right )+\frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}-\frac {(c+d x)^{5/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {5 d^2 \int \sqrt {c+d x} \sin (a+b x)^3dx}{12 b^2}+\frac {2}{3} \left (\frac {5 d \left (\frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \left (\frac {d \int \frac {\sin \left (a+b x+\frac {\pi }{2}\right )}{\sqrt {c+d x}}dx}{2 b}-\frac {\sqrt {c+d x} \cos (a+b x)}{b}\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\right )+\frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}-\frac {(c+d x)^{5/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3787

\(\displaystyle -\frac {5 d^2 \int \sqrt {c+d x} \sin (a+b x)^3dx}{12 b^2}+\frac {2}{3} \left (\frac {5 d \left (\frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \left (\frac {d \left (\cos \left (a-\frac {b c}{d}\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx-\sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx\right )}{2 b}-\frac {\sqrt {c+d x} \cos (a+b x)}{b}\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\right )+\frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}-\frac {(c+d x)^{5/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {5 d^2 \int \sqrt {c+d x} \sin (a+b x)^3dx}{12 b^2}+\frac {2}{3} \left (\frac {5 d \left (\frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \left (\frac {d \left (\cos \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x+\frac {\pi }{2}\right )}{\sqrt {c+d x}}dx-\sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx\right )}{2 b}-\frac {\sqrt {c+d x} \cos (a+b x)}{b}\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\right )+\frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}-\frac {(c+d x)^{5/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3785

\(\displaystyle -\frac {5 d^2 \int \sqrt {c+d x} \sin (a+b x)^3dx}{12 b^2}+\frac {2}{3} \left (\frac {5 d \left (\frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \left (\frac {d \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx\right )}{2 b}-\frac {\sqrt {c+d x} \cos (a+b x)}{b}\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\right )+\frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}-\frac {(c+d x)^{5/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3786

\(\displaystyle -\frac {5 d^2 \int \sqrt {c+d x} \sin (a+b x)^3dx}{12 b^2}+\frac {2}{3} \left (\frac {5 d \left (\frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \left (\frac {d \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\frac {2 \sin \left (a-\frac {b c}{d}\right ) \int \sin \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}\right )}{2 b}-\frac {\sqrt {c+d x} \cos (a+b x)}{b}\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\right )+\frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}-\frac {(c+d x)^{5/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3793

\(\displaystyle -\frac {5 d^2 \int \left (\frac {3}{4} \sqrt {c+d x} \sin (a+b x)-\frac {1}{4} \sqrt {c+d x} \sin (3 a+3 b x)\right )dx}{12 b^2}+\frac {2}{3} \left (\frac {5 d \left (\frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \left (\frac {d \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\frac {2 \sin \left (a-\frac {b c}{d}\right ) \int \sin \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}\right )}{2 b}-\frac {\sqrt {c+d x} \cos (a+b x)}{b}\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\right )+\frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}-\frac {(c+d x)^{5/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2}{3} \left (\frac {5 d \left (\frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \left (\frac {d \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\frac {2 \sin \left (a-\frac {b c}{d}\right ) \int \sin \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}\right )}{2 b}-\frac {\sqrt {c+d x} \cos (a+b x)}{b}\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\right )+\frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}-\frac {5 d^2 \left (\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {d} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 b^{3/2}}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {d} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{12 b^{3/2}}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {d} \sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{12 b^{3/2}}-\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {d} \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 b^{3/2}}-\frac {3 \sqrt {c+d x} \cos (a+b x)}{4 b}+\frac {\sqrt {c+d x} \cos (3 a+3 b x)}{12 b}\right )}{12 b^2}-\frac {(c+d x)^{5/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {2}{3} \left (\frac {5 d \left (\frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \left (\frac {d \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\frac {\sqrt {2 \pi } \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}\right )}{2 b}-\frac {\sqrt {c+d x} \cos (a+b x)}{b}\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\right )+\frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}-\frac {5 d^2 \left (\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {d} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 b^{3/2}}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {d} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{12 b^{3/2}}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {d} \sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{12 b^{3/2}}-\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {d} \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 b^{3/2}}-\frac {3 \sqrt {c+d x} \cos (a+b x)}{4 b}+\frac {\sqrt {c+d x} \cos (3 a+3 b x)}{12 b}\right )}{12 b^2}-\frac {(c+d x)^{5/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3833

\(\displaystyle \frac {5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}-\frac {5 d^2 \left (\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {d} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 b^{3/2}}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {d} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{12 b^{3/2}}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {d} \sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{12 b^{3/2}}-\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {d} \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 b^{3/2}}-\frac {3 \sqrt {c+d x} \cos (a+b x)}{4 b}+\frac {\sqrt {c+d x} \cos (3 a+3 b x)}{12 b}\right )}{12 b^2}+\frac {2}{3} \left (\frac {5 d \left (\frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \left (\frac {d \left (\frac {\sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}-\frac {\sqrt {2 \pi } \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}\right )}{2 b}-\frac {\sqrt {c+d x} \cos (a+b x)}{b}\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\right )-\frac {(c+d x)^{5/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

Input:

Int[(c + d*x)^(5/2)*Sin[a + b*x]^3,x]
 

Output:

(-5*d^2*((-3*Sqrt[c + d*x]*Cos[a + b*x])/(4*b) + (Sqrt[c + d*x]*Cos[3*a + 
3*b*x])/(12*b) + (3*Sqrt[d]*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]* 
Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(4*b^(3/2)) - (Sqrt[d]*Sqrt[Pi/6]*Cos[ 
3*a - (3*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(12 
*b^(3/2)) + (Sqrt[d]*Sqrt[Pi/6]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x] 
)/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(12*b^(3/2)) - (3*Sqrt[d]*Sqrt[Pi/2]*Fres 
nelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(4*b^(3 
/2))))/(12*b^2) - ((c + d*x)^(5/2)*Cos[a + b*x]*Sin[a + b*x]^2)/(3*b) + (5 
*d*(c + d*x)^(3/2)*Sin[a + b*x]^3)/(18*b^2) + (2*(-(((c + d*x)^(5/2)*Cos[a 
 + b*x])/b) + (5*d*((-3*d*(-((Sqrt[c + d*x]*Cos[a + b*x])/b) + (d*((Sqrt[2 
*Pi]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]] 
)/(Sqrt[b]*Sqrt[d]) - (Sqrt[2*Pi]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d* 
x])/Sqrt[d]]*Sin[a - (b*c)/d])/(Sqrt[b]*Sqrt[d])))/(2*b)))/(2*b) + ((c + d 
*x)^(3/2)*Sin[a + b*x])/b))/(2*b)))/3
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 3785

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S 
imp[2/d   Subst[Int[Cos[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, 
d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
 

rule 3786

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[2/d 
   Subst[Int[Sin[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f 
}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
 

rule 3787

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cos 
[(d*e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] + Simp[Sin[( 
d*e - c*f)/d]   Int[Cos[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c, d 
, e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]
 

rule 3792

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim 
p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 
2*((n - 1)/n)   Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 
*m*((m - 1)/(f^2*n^2))   Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) 
/; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 3832

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ 
d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
 

rule 3833

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ 
d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
 

Maple [A] (verified)

Time = 2.01 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.16

method	result	size
derivativedivides	\(\frac {-\frac {3 d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{4 b}+\frac {15 d \left (\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{2 b}-\frac {3 d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{2 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {a d -b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {a d -b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{2 b}\right )}{4 b}+\frac {d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 b c}{d}\right )}{12 b}-\frac {5 d \left (\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 b c}{d}\right )}{6 b}-\frac {d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 b c}{d}\right )}{6 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 a d -3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {3 a d -3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{36 b \sqrt {\frac {b}{d}}}\right )}{2 b}\right )}{12 b}}{d}\)	\(476\)
default	\(\frac {-\frac {3 d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{4 b}+\frac {15 d \left (\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{2 b}-\frac {3 d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{2 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {a d -b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {a d -b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{2 b}\right )}{4 b}+\frac {d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 b c}{d}\right )}{12 b}-\frac {5 d \left (\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 b c}{d}\right )}{6 b}-\frac {d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 b c}{d}\right )}{6 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 a d -3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {3 a d -3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{36 b \sqrt {\frac {b}{d}}}\right )}{2 b}\right )}{12 b}}{d}\)	\(476\)

Input:

int((d*x+c)^(5/2)*sin(b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

2/d*(-3/8/b*d*(d*x+c)^(5/2)*cos(b*(d*x+c)/d+(a*d-b*c)/d)+15/8/b*d*(1/2/b*d 
*(d*x+c)^(3/2)*sin(b*(d*x+c)/d+(a*d-b*c)/d)-3/2/b*d*(-1/2/b*d*(d*x+c)^(1/2 
)*cos(b*(d*x+c)/d+(a*d-b*c)/d)+1/4/b*d*2^(1/2)*Pi^(1/2)/(b/d)^(1/2)*(cos(( 
a*d-b*c)/d)*FresnelC(2^(1/2)/Pi^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d)-sin(( 
a*d-b*c)/d)*FresnelS(2^(1/2)/Pi^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d))))+1/ 
24/b*d*(d*x+c)^(5/2)*cos(3*b*(d*x+c)/d+3*(a*d-b*c)/d)-5/24/b*d*(1/6/b*d*(d 
*x+c)^(3/2)*sin(3*b*(d*x+c)/d+3*(a*d-b*c)/d)-1/2/b*d*(-1/6/b*d*(d*x+c)^(1/ 
2)*cos(3*b*(d*x+c)/d+3*(a*d-b*c)/d)+1/36/b*d*2^(1/2)*Pi^(1/2)*3^(1/2)/(b/d 
)^(1/2)*(cos(3*(a*d-b*c)/d)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)/(b/d)^(1/2)* 
b*(d*x+c)^(1/2)/d)-sin(3*(a*d-b*c)/d)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)/(b 
/d)^(1/2)*b*(d*x+c)^(1/2)/d)))))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 371, normalized size of antiderivative = 0.90 \[ \int (c+d x)^{5/2} \sin ^3(a+b x) \, dx=\frac {5 \, \sqrt {6} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 1215 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 1215 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) - 5 \, \sqrt {6} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 24 \, {\left ({\left (12 \, b^{3} d^{2} x^{2} + 24 \, b^{3} c d x + 12 \, b^{3} c^{2} - 5 \, b d^{2}\right )} \cos \left (b x + a\right )^{3} - 3 \, {\left (12 \, b^{3} d^{2} x^{2} + 24 \, b^{3} c d x + 12 \, b^{3} c^{2} - 35 \, b d^{2}\right )} \cos \left (b x + a\right ) + 10 \, {\left (7 \, b^{2} d^{2} x + 7 \, b^{2} c d - {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{864 \, b^{4}} \] Input:

integrate((d*x+c)^(5/2)*sin(b*x+a)^3,x, algorithm="fricas")
 

Output:

1/864*(5*sqrt(6)*pi*d^3*sqrt(b/(pi*d))*cos(-3*(b*c - a*d)/d)*fresnel_cos(s 
qrt(6)*sqrt(d*x + c)*sqrt(b/(pi*d))) - 1215*sqrt(2)*pi*d^3*sqrt(b/(pi*d))* 
cos(-(b*c - a*d)/d)*fresnel_cos(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d))) + 12 
15*sqrt(2)*pi*d^3*sqrt(b/(pi*d))*fresnel_sin(sqrt(2)*sqrt(d*x + c)*sqrt(b/ 
(pi*d)))*sin(-(b*c - a*d)/d) - 5*sqrt(6)*pi*d^3*sqrt(b/(pi*d))*fresnel_sin 
(sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-3*(b*c - a*d)/d) + 24*((12*b^3 
*d^2*x^2 + 24*b^3*c*d*x + 12*b^3*c^2 - 5*b*d^2)*cos(b*x + a)^3 - 3*(12*b^3 
*d^2*x^2 + 24*b^3*c*d*x + 12*b^3*c^2 - 35*b*d^2)*cos(b*x + a) + 10*(7*b^2* 
d^2*x + 7*b^2*c*d - (b^2*d^2*x + b^2*c*d)*cos(b*x + a)^2)*sin(b*x + a))*sq 
rt(d*x + c))/b^4
 

Sympy [F(-1)]

Timed out. \[ \int (c+d x)^{5/2} \sin ^3(a+b x) \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**(5/2)*sin(b*x+a)**3,x)
 

Output:

Timed out
 

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 547, normalized size of antiderivative = 1.33 \[ \int (c+d x)^{5/2} \sin ^3(a+b x) \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^(5/2)*sin(b*x+a)^3,x, algorithm="maxima")
 

Output:

-1/3456*(240*(d*x + c)^(3/2)*b^3*sin(3*((d*x + c)*b - b*c + a*d)/d) - 6480 
*(d*x + c)^(3/2)*b^3*sin(((d*x + c)*b - b*c + a*d)/d) - 24*(12*(d*x + c)^( 
5/2)*b^4/d - 5*sqrt(d*x + c)*b^2*d)*cos(3*((d*x + c)*b - b*c + a*d)/d) + 6 
48*(4*(d*x + c)^(5/2)*b^4/d - 15*sqrt(d*x + c)*b^2*d)*cos(((d*x + c)*b - b 
*c + a*d)/d) - 5*(-(I - 1)*9^(1/4)*sqrt(2)*sqrt(pi)*b*d^2*(b^2/d^2)^(1/4)* 
cos(-3*(b*c - a*d)/d) - (I + 1)*9^(1/4)*sqrt(2)*sqrt(pi)*b*d^2*(b^2/d^2)^( 
1/4)*sin(-3*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(3*I*b/d)) - 1215*((I - 
1)*sqrt(2)*sqrt(pi)*b*d^2*(b^2/d^2)^(1/4)*cos(-(b*c - a*d)/d) + (I + 1)*sq 
rt(2)*sqrt(pi)*b*d^2*(b^2/d^2)^(1/4)*sin(-(b*c - a*d)/d))*erf(sqrt(d*x + c 
)*sqrt(I*b/d)) - 1215*(-(I + 1)*sqrt(2)*sqrt(pi)*b*d^2*(b^2/d^2)^(1/4)*cos 
(-(b*c - a*d)/d) - (I - 1)*sqrt(2)*sqrt(pi)*b*d^2*(b^2/d^2)^(1/4)*sin(-(b* 
c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-I*b/d)) - 5*((I + 1)*9^(1/4)*sqrt(2)* 
sqrt(pi)*b*d^2*(b^2/d^2)^(1/4)*cos(-3*(b*c - a*d)/d) + (I - 1)*9^(1/4)*sqr 
t(2)*sqrt(pi)*b*d^2*(b^2/d^2)^(1/4)*sin(-3*(b*c - a*d)/d))*erf(sqrt(d*x + 
c)*sqrt(-3*I*b/d)))*d/b^5
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.04 (sec) , antiderivative size = 2453, normalized size of antiderivative = 5.98 \[ \int (c+d x)^{5/2} \sin ^3(a+b x) \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^(5/2)*sin(b*x+a)^3,x, algorithm="giac")
 

Output:

1/1728*(72*(9*sqrt(2)*sqrt(pi)*d*erf(1/2*I*sqrt(2)*sqrt(b*d)*sqrt(d*x + c) 
*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((I*b*c - I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sq 
rt(b^2*d^2) + 1)) - sqrt(6)*sqrt(pi)*d*erf(-1/2*I*sqrt(6)*sqrt(b*d)*sqrt(d 
*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^(-3*(I*b*c - I*a*d)/d)/(sqrt(b*d)*( 
I*b*d/sqrt(b^2*d^2) + 1)) + 9*sqrt(2)*sqrt(pi)*d*erf(-1/2*I*sqrt(2)*sqrt(b 
*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-I*b*c + I*a*d)/d)/(sqr 
t(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)) - sqrt(6)*sqrt(pi)*d*erf(1/2*I*sqrt(6)*s 
qrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^(-3*(-I*b*c + I*a*d 
)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)))*c^3 - d^3*(81*(sqrt(2)*sqrt(p 
i)*(8*b^3*c^3 + 12*I*b^2*c^2*d - 18*b*c*d^2 - 15*I*d^3)*d*erf(1/2*I*sqrt(2 
)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((I*b*c - I*a*d) 
/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b^3) - 2*I*(4*I*(d*x + c)^(5/2)* 
b^2*d - 12*I*(d*x + c)^(3/2)*b^2*c*d + 12*I*sqrt(d*x + c)*b^2*c^2*d + 10*( 
d*x + c)^(3/2)*b*d^2 - 18*sqrt(d*x + c)*b*c*d^2 - 15*I*sqrt(d*x + c)*d^3)* 
e^((-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b^3)/d^3 - (sqrt(6)*sqrt(pi)*(72*b^ 
3*c^3 - 36*I*b^2*c^2*d - 18*b*c*d^2 + 5*I*d^3)*d*erf(-1/2*I*sqrt(6)*sqrt(b 
*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^(-3*(I*b*c - I*a*d)/d)/(s 
qrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b^3) + 6*I*(-12*I*(d*x + c)^(5/2)*b^2*d 
 + 36*I*(d*x + c)^(3/2)*b^2*c*d - 36*I*sqrt(d*x + c)*b^2*c^2*d + 10*(d*x + 
 c)^(3/2)*b*d^2 - 18*sqrt(d*x + c)*b*c*d^2 + 5*I*sqrt(d*x + c)*d^3)*e^(...
 

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^{5/2} \sin ^3(a+b x) \, dx=\int {\sin \left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^{5/2} \,d x \] Input:

int(sin(a + b*x)^3*(c + d*x)^(5/2),x)
 

Output:

int(sin(a + b*x)^3*(c + d*x)^(5/2), x)
 

Reduce [F]

\[ \int (c+d x)^{5/2} \sin ^3(a+b x) \, dx=\left (\int \sqrt {d x +c}\, \sin \left (b x +a \right )^{3} x^{2}d x \right ) d^{2}+2 \left (\int \sqrt {d x +c}\, \sin \left (b x +a \right )^{3} x d x \right ) c d +\left (\int \sqrt {d x +c}\, \sin \left (b x +a \right )^{3}d x \right ) c^{2} \] Input:

int((d*x+c)^(5/2)*sin(b*x+a)^3,x)
 

Output:

int(sqrt(c + d*x)*sin(a + b*x)**3*x**2,x)*d**2 + 2*int(sqrt(c + d*x)*sin(a 
 + b*x)**3*x,x)*c*d + int(sqrt(c + d*x)*sin(a + b*x)**3,x)*c**2