\(\int (c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x) \, dx\) [63]

Optimal result

Integrand size = 24, antiderivative size = 406 \[ \int (c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x) \, dx=\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{8 b^2}-\frac {5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{72 b^2}+\frac {15 d^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{7/2}}-\frac {5 d^{5/2} \sqrt {\frac {\pi }{6}} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{144 b^{7/2}}-\frac {5 d^{5/2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{144 b^{7/2}}+\frac {15 d^{5/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{16 b^{7/2}}-\frac {15 d^2 \sqrt {c+d x} \sin (a+b x)}{16 b^3}+\frac {(c+d x)^{5/2} \sin (a+b x)}{4 b}+\frac {5 d^2 \sqrt {c+d x} \sin (3 a+3 b x)}{144 b^3}-\frac {(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b} \]

[Out]

5/8*d*(d*x+c)^(3/2)*cos(b*x+a)/b^2-5/72*d*(d*x+c)^(3/2)*cos(3*b*x+3*a)/b^2+1/4*(d*x+c)^(5/2)*sin(b*x+a)/b-1/12
*(d*x+c)^(5/2)*sin(3*b*x+3*a)/b-5/864*d^(5/2)*cos(3*a-3*b*c/d)*FresnelS(b^(1/2)*6^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)
/d^(1/2))*6^(1/2)*Pi^(1/2)/b^(7/2)-5/864*d^(5/2)*FresnelC(b^(1/2)*6^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*sin(
3*a-3*b*c/d)*6^(1/2)*Pi^(1/2)/b^(7/2)+15/32*d^(5/2)*cos(a-b*c/d)*FresnelS(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c)^(1/
2)/d^(1/2))*2^(1/2)*Pi^(1/2)/b^(7/2)+15/32*d^(5/2)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*si
n(a-b*c/d)*2^(1/2)*Pi^(1/2)/b^(7/2)-15/16*d^2*sin(b*x+a)*(d*x+c)^(1/2)/b^3+5/144*d^2*sin(3*b*x+3*a)*(d*x+c)^(1
/2)/b^3

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {4491, 3377, 3387, 3386, 3432, 3385, 3433} \[ \int (c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x) \, dx=-\frac {5 \sqrt {\frac {\pi }{6}} d^{5/2} \sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{144 b^{7/2}}+\frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{7/2}}+\frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{7/2}}-\frac {5 \sqrt {\frac {\pi }{6}} d^{5/2} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{144 b^{7/2}}-\frac {15 d^2 \sqrt {c+d x} \sin (a+b x)}{16 b^3}+\frac {5 d^2 \sqrt {c+d x} \sin (3 a+3 b x)}{144 b^3}+\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{8 b^2}-\frac {5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{72 b^2}+\frac {(c+d x)^{5/2} \sin (a+b x)}{4 b}-\frac {(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b} \]

[In]

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

[Out]

(5*d*(c + d*x)^(3/2)*Cos[a + b*x])/(8*b^2) - (5*d*(c + d*x)^(3/2)*Cos[3*a + 3*b*x])/(72*b^2) + (15*d^(5/2)*Sqr
t[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(7/2)) - (5*d^(5/2)*Sqrt[
Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(144*b^(7/2)) - (5*d^(5/2)*Sq
rt[Pi/6]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(144*b^(7/2)) + (15*d^(5/2
)*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(16*b^(7/2)) - (15*d^2*Sqr
t[c + d*x]*Sin[a + b*x])/(16*b^3) + ((c + d*x)^(5/2)*Sin[a + b*x])/(4*b) + (5*d^2*Sqrt[c + d*x]*Sin[3*a + 3*b*
x])/(144*b^3) - ((c + d*x)^(5/2)*Sin[3*a + 3*b*x])/(12*b)

Rule 3377

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

Rule 3385

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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 3386

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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 3387

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) +
f*x]/Sqrt[c + d*x], x], x] + Dist[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 3432

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 3433

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]

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{4} (c+d x)^{5/2} \cos (a+b x)-\frac {1}{4} (c+d x)^{5/2} \cos (3 a+3 b x)\right ) \, dx \\ & = \frac {1}{4} \int (c+d x)^{5/2} \cos (a+b x) \, dx-\frac {1}{4} \int (c+d x)^{5/2} \cos (3 a+3 b x) \, dx \\ & = \frac {(c+d x)^{5/2} \sin (a+b x)}{4 b}-\frac {(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b}+\frac {(5 d) \int (c+d x)^{3/2} \sin (3 a+3 b x) \, dx}{24 b}-\frac {(5 d) \int (c+d x)^{3/2} \sin (a+b x) \, dx}{8 b} \\ & = \frac {5 d (c+d x)^{3/2} \cos (a+b x)}{8 b^2}-\frac {5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{72 b^2}+\frac {(c+d x)^{5/2} \sin (a+b x)}{4 b}-\frac {(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b}+\frac {\left (5 d^2\right ) \int \sqrt {c+d x} \cos (3 a+3 b x) \, dx}{48 b^2}-\frac {\left (15 d^2\right ) \int \sqrt {c+d x} \cos (a+b x) \, dx}{16 b^2} \\ & = \frac {5 d (c+d x)^{3/2} \cos (a+b x)}{8 b^2}-\frac {5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{72 b^2}-\frac {15 d^2 \sqrt {c+d x} \sin (a+b x)}{16 b^3}+\frac {(c+d x)^{5/2} \sin (a+b x)}{4 b}+\frac {5 d^2 \sqrt {c+d x} \sin (3 a+3 b x)}{144 b^3}-\frac {(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b}-\frac {\left (5 d^3\right ) \int \frac {\sin (3 a+3 b x)}{\sqrt {c+d x}} \, dx}{288 b^3}+\frac {\left (15 d^3\right ) \int \frac {\sin (a+b x)}{\sqrt {c+d x}} \, dx}{32 b^3} \\ & = \frac {5 d (c+d x)^{3/2} \cos (a+b x)}{8 b^2}-\frac {5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{72 b^2}-\frac {15 d^2 \sqrt {c+d x} \sin (a+b x)}{16 b^3}+\frac {(c+d x)^{5/2} \sin (a+b x)}{4 b}+\frac {5 d^2 \sqrt {c+d x} \sin (3 a+3 b x)}{144 b^3}-\frac {(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b}-\frac {\left (5 d^3 \cos \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 (15 d^3 \cos \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 (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 (15 d^3 \sin \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 {5 d (c+d x)^{3/2} \cos (a+b x)}{8 b^2}-\frac {5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{72 b^2}-\frac {15 d^2 \sqrt {c+d x} \sin (a+b x)}{16 b^3}+\frac {(c+d x)^{5/2} \sin (a+b x)}{4 b}+\frac {5 d^2 \sqrt {c+d x} \sin (3 a+3 b x)}{144 b^3}-\frac {(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b}-\frac {\left (5 d^2 \cos \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 (15 d^2 \cos \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 (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 (15 d^2 \sin \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 {5 d (c+d x)^{3/2} \cos (a+b x)}{8 b^2}-\frac {5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{72 b^2}+\frac {15 d^{5/2} \sqrt {\frac {\pi }{2}} \cos \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 {5 d^{5/2} \sqrt {\frac {\pi }{6}} \cos \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 {5 d^{5/2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\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 {15 d^{5/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\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 {15 d^2 \sqrt {c+d x} \sin (a+b x)}{16 b^3}+\frac {(c+d x)^{5/2} \sin (a+b x)}{4 b}+\frac {5 d^2 \sqrt {c+d x} \sin (3 a+3 b x)}{144 b^3}-\frac {(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.01 (sec) , antiderivative size = 1418, normalized size of antiderivative = 3.49 \[ \int (c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x) \, dx=-\frac {c \sqrt {d} e^{-\frac {3 i (a d+b (c+d x))}{d}} \left (12 \sqrt {b} \sqrt {d} e^{\frac {3 i b c}{d}} \sqrt {c+d x} \left (1+2 i b x+e^{6 i (a+b x)} (1-2 i b x)\right )+(1+i) (2 b c+i d) e^{\frac {3 i b (2 c+d x)}{d}} \sqrt {6 \pi } \text {erf}\left (\frac {(1+i) \sqrt {\frac {3}{2}} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )-(1+i) (2 b c-i d) e^{3 i (2 a+b x)} \sqrt {6 \pi } \text {erfi}\left (\frac {(1+i) \sqrt {\frac {3}{2}} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )\right )}{288 b^{5/2}}+\frac {c^2 d e^{-\frac {i (b c+a d)}{d}} \left (e^{2 i a} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {3}{2},-\frac {i b (c+d x)}{d}\right )+e^{\frac {2 i b c}{d}} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {3}{2},\frac {i b (c+d x)}{d}\right )\right )}{8 b^2 \sqrt {c+d x}}-\frac {c^2 e^{-\frac {3 i (b c+a d)}{d}} (c+d x)^{3/2} \left (-\frac {e^{6 i a} \Gamma \left (\frac {3}{2},-\frac {3 i b (c+d x)}{d}\right )}{\left (-\frac {i b (c+d x)}{d}\right )^{3/2}}-\frac {e^{\frac {6 i b c}{d}} \Gamma \left (\frac {3}{2},\frac {3 i b (c+d x)}{d}\right )}{\left (\frac {i b (c+d x)}{d}\right )^{3/2}}\right )}{24 \sqrt {3} d}+\frac {c \sqrt {d} \left (e^{i \left (a-\frac {b c}{d}\right )} \left (2 \sqrt {b} \sqrt {d} e^{\frac {i b (c+d x)}{d}} (3-2 i b x) \sqrt {c+d x}+\sqrt [4]{-1} (-2 b c+3 i d) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt [4]{-1} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )\right )+\left (2 \sqrt {b} \sqrt {d} (3+2 i b x) \sqrt {c+d x}+(1+i) (2 b c+3 i d) \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {(1+i) \sqrt {b} \sqrt {c+d x}}{\sqrt {2} \sqrt {d}}\right ) \left (\cos \left (b \left (\frac {c}{d}+x\right )\right )+i \sin \left (b \left (\frac {c}{d}+x\right )\right )\right )\right ) (\cos (a+b x)-i \sin (a+b x))\right )}{16 b^{5/2}}+\frac {\sqrt {d} \left (\left (\cos \left (a-\frac {b c}{d}\right )+i \sin \left (a-\frac {b c}{d}\right )\right ) \left ((1+i) \left (4 b^2 c^2-12 i b c d-15 d^2\right ) \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {(1+i) \sqrt {b} \sqrt {c+d x}}{\sqrt {2} \sqrt {d}}\right )+2 \sqrt {b} \sqrt {d} \sqrt {c+d x} \left (15 i d-4 i b^2 d x^2-2 b (c-5 d x)\right ) \left (\cos \left (b \left (\frac {c}{d}+x\right )\right )+i \sin \left (b \left (\frac {c}{d}+x\right )\right )\right )\right )+\left (2 \sqrt {b} \sqrt {d} \sqrt {c+d x} \left (-15 i d+4 i b^2 d x^2-2 b (c-5 d x)\right )-(1+i) \left (4 b^2 c^2+12 i b c d-15 d^2\right ) \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {(1+i) \sqrt {b} \sqrt {c+d x}}{\sqrt {2} \sqrt {d}}\right ) \left (\cos \left (b \left (\frac {c}{d}+x\right )\right )+i \sin \left (b \left (\frac {c}{d}+x\right )\right )\right )\right ) (\cos (a+b x)-i \sin (a+b x))\right )}{64 b^{7/2}}-\frac {\sqrt {d} \left (\left (\cos \left (3 a-\frac {3 b c}{d}\right )+i \sin \left (3 a-\frac {3 b c}{d}\right )\right ) \left ((1+i) \left (12 b^2 c^2-12 i b c d-5 d^2\right ) \sqrt {\frac {3 \pi }{2}} \text {erfi}\left (\frac {(1+i) \sqrt {\frac {3}{2}} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )+6 \sqrt {b} \sqrt {d} \sqrt {c+d x} \left (5 i d-12 i b^2 d x^2-2 b (c-5 d x)\right ) \left (\cos \left (\frac {3 b (c+d x)}{d}\right )+i \sin \left (\frac {3 b (c+d x)}{d}\right )\right )\right )+(\cos (3 (a+b x))-i \sin (3 (a+b x))) \left (6 \sqrt {b} \sqrt {d} \sqrt {c+d x} \left (-5 i d+12 i b^2 d x^2-2 b (c-5 d x)\right )-(1+i) \left (12 b^2 c^2+12 i b c d-5 d^2\right ) \sqrt {\frac {3 \pi }{2}} \text {erf}\left (\frac {(1+i) \sqrt {\frac {3}{2}} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right ) \left (\cos \left (\frac {3 b (c+d x)}{d}\right )+i \sin \left (\frac {3 b (c+d x)}{d}\right )\right )\right )\right )}{1728 b^{7/2}} \]

[In]

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

[Out]

-1/288*(c*Sqrt[d]*(12*Sqrt[b]*Sqrt[d]*E^(((3*I)*b*c)/d)*Sqrt[c + d*x]*(1 + (2*I)*b*x + E^((6*I)*(a + b*x))*(1
- (2*I)*b*x)) + (1 + I)*(2*b*c + I*d)*E^(((3*I)*b*(2*c + d*x))/d)*Sqrt[6*Pi]*Erf[((1 + I)*Sqrt[3/2]*Sqrt[b]*Sq
rt[c + d*x])/Sqrt[d]] - (1 + I)*(2*b*c - I*d)*E^((3*I)*(2*a + b*x))*Sqrt[6*Pi]*Erfi[((1 + I)*Sqrt[3/2]*Sqrt[b]
*Sqrt[c + d*x])/Sqrt[d]]))/(b^(5/2)*E^(((3*I)*(a*d + b*(c + d*x)))/d)) + (c^2*d*(E^((2*I)*a)*Sqrt[((-I)*b*(c +
d*x))/d]*Gamma[3/2, ((-I)*b*(c + d*x))/d] + E^(((2*I)*b*c)/d)*Sqrt[(I*b*(c + d*x))/d]*Gamma[3/2, (I*b*(c + d*
x))/d]))/(8*b^2*E^((I*(b*c + a*d))/d)*Sqrt[c + d*x]) - (c^2*(c + d*x)^(3/2)*(-((E^((6*I)*a)*Gamma[3/2, ((-3*I)
*b*(c + d*x))/d])/(((-I)*b*(c + d*x))/d)^(3/2)) - (E^(((6*I)*b*c)/d)*Gamma[3/2, ((3*I)*b*(c + d*x))/d])/((I*b*
(c + d*x))/d)^(3/2)))/(24*Sqrt[3]*d*E^(((3*I)*(b*c + a*d))/d)) + (c*Sqrt[d]*(E^(I*(a - (b*c)/d))*(2*Sqrt[b]*Sq
rt[d]*E^((I*b*(c + d*x))/d)*(3 - (2*I)*b*x)*Sqrt[c + d*x] + (-1)^(1/4)*(-2*b*c + (3*I)*d)*Sqrt[Pi]*Erfi[((-1)^
(1/4)*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]]) + (2*Sqrt[b]*Sqrt[d]*(3 + (2*I)*b*x)*Sqrt[c + d*x] + (1 + I)*(2*b*c + (
3*I)*d)*Sqrt[Pi/2]*Erf[((1 + I)*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[2]*Sqrt[d])]*(Cos[b*(c/d + x)] + I*Sin[b*(c/d + x
)]))*(Cos[a + b*x] - I*Sin[a + b*x])))/(16*b^(5/2)) + (Sqrt[d]*((Cos[a - (b*c)/d] + I*Sin[a - (b*c)/d])*((1 +
I)*(4*b^2*c^2 - (12*I)*b*c*d - 15*d^2)*Sqrt[Pi/2]*Erfi[((1 + I)*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[2]*Sqrt[d])] + 2*
Sqrt[b]*Sqrt[d]*Sqrt[c + d*x]*((15*I)*d - (4*I)*b^2*d*x^2 - 2*b*(c - 5*d*x))*(Cos[b*(c/d + x)] + I*Sin[b*(c/d
+ x)])) + (2*Sqrt[b]*Sqrt[d]*Sqrt[c + d*x]*((-15*I)*d + (4*I)*b^2*d*x^2 - 2*b*(c - 5*d*x)) - (1 + I)*(4*b^2*c^
2 + (12*I)*b*c*d - 15*d^2)*Sqrt[Pi/2]*Erf[((1 + I)*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[2]*Sqrt[d])]*(Cos[b*(c/d + x)]
+ I*Sin[b*(c/d + x)]))*(Cos[a + b*x] - I*Sin[a + b*x])))/(64*b^(7/2)) - (Sqrt[d]*((Cos[3*a - (3*b*c)/d] + I*S
in[3*a - (3*b*c)/d])*((1 + I)*(12*b^2*c^2 - (12*I)*b*c*d - 5*d^2)*Sqrt[(3*Pi)/2]*Erfi[((1 + I)*Sqrt[3/2]*Sqrt[
b]*Sqrt[c + d*x])/Sqrt[d]] + 6*Sqrt[b]*Sqrt[d]*Sqrt[c + d*x]*((5*I)*d - (12*I)*b^2*d*x^2 - 2*b*(c - 5*d*x))*(C
os[(3*b*(c + d*x))/d] + I*Sin[(3*b*(c + d*x))/d])) + (Cos[3*(a + b*x)] - I*Sin[3*(a + b*x)])*(6*Sqrt[b]*Sqrt[d
]*Sqrt[c + d*x]*((-5*I)*d + (12*I)*b^2*d*x^2 - 2*b*(c - 5*d*x)) - (1 + I)*(12*b^2*c^2 + (12*I)*b*c*d - 5*d^2)*
Sqrt[(3*Pi)/2]*Erf[((1 + I)*Sqrt[3/2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]]*(Cos[(3*b*(c + d*x))/d] + I*Sin[(3*b*(c
+ d*x))/d]))))/(1728*b^(7/2))

Maple [A] (verified)

Time = 1.23 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.17

method	result	size
derivativedivides	\(\frac {\frac {d \left (d x +c \right )^{\frac {5}{2}} \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -c b}{d}\right )}{4 b}-\frac {5 d \left (-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -c b}{d}\right )}{2 b}+\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -c b}{d}\right )}{2 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {a d -c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {a d -c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{2 b}\right )}{4 b}-\frac {d \left (d x +c \right )^{\frac {5}{2}} \sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 c b}{d}\right )}{12 b}+\frac {5 d \left (-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 c b}{d}\right )}{6 b}+\frac {d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 c b}{d}\right )}{6 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 a d -3 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {3 a d -3 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{36 b \sqrt {\frac {b}{d}}}\right )}{2 b}\right )}{12 b}}{d}\)	\(474\)
default	\(\frac {\frac {d \left (d x +c \right )^{\frac {5}{2}} \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -c b}{d}\right )}{4 b}-\frac {5 d \left (-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -c b}{d}\right )}{2 b}+\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -c b}{d}\right )}{2 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {a d -c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {a d -c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{2 b}\right )}{4 b}-\frac {d \left (d x +c \right )^{\frac {5}{2}} \sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 c b}{d}\right )}{12 b}+\frac {5 d \left (-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 c b}{d}\right )}{6 b}+\frac {d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 c b}{d}\right )}{6 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 a d -3 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {3 a d -3 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{36 b \sqrt {\frac {b}{d}}}\right )}{2 b}\right )}{12 b}}{d}\)	\(474\)

[In]

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

[Out]

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

none

Time = 0.26 (sec) , antiderivative size = 370, normalized size of antiderivative = 0.91 \[ \int (c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x) \, dx=-\frac {5 \, \sqrt {6} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 405 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 405 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + 5 \, \sqrt {6} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 24 \, {\left (10 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )^{3} - 30 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right ) - {\left (12 \, b^{3} d^{2} x^{2} + 24 \, b^{3} c d x + 12 \, b^{3} c^{2} - 35 \, b d^{2} - {\left (12 \, b^{3} d^{2} x^{2} + 24 \, b^{3} c d x + 12 \, b^{3} c^{2} - 5 \, b d^{2}\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{864 \, b^{4}} \]

[In]

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

[Out]

-1/864*(5*sqrt(6)*pi*d^3*sqrt(b/(pi*d))*cos(-3*(b*c - a*d)/d)*fresnel_sin(sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi*d))
) - 405*sqrt(2)*pi*d^3*sqrt(b/(pi*d))*cos(-(b*c - a*d)/d)*fresnel_sin(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d))) -
405*sqrt(2)*pi*d^3*sqrt(b/(pi*d))*fresnel_cos(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-(b*c - a*d)/d) + 5*sq
rt(6)*pi*d^3*sqrt(b/(pi*d))*fresnel_cos(sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-3*(b*c - a*d)/d) + 24*(10*(
b^2*d^2*x + b^2*c*d)*cos(b*x + a)^3 - 30*(b^2*d^2*x + b^2*c*d)*cos(b*x + a) - (12*b^3*d^2*x^2 + 24*b^3*c*d*x +
12*b^3*c^2 - 35*b*d^2 - (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)^2)*sin(b*x + a))*
sqrt(d*x + c))/b^4

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.37 (sec) , antiderivative size = 547, normalized size of antiderivative = 1.35 \[ \int (c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x) \, dx=-\frac {{\left (240 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} \cos \left (\frac {3 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) - 2160 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} \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 ) - 405 \, {\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 ) - 405 \, {\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 ) + 24 \, {\left (\frac {12 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{4}}{d} - 5 \, \sqrt {d x + c} b^{2} d\right )} \sin \left (\frac {3 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) - 216 \, {\left (\frac {4 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{4}}{d} - 15 \, \sqrt {d x + c} b^{2} d\right )} \sin \left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right )\right )} d}{3456 \, b^{5}} \]

[In]

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

[Out]

-1/3456*(240*(d*x + c)^(3/2)*b^3*cos(3*((d*x + c)*b - b*c + a*d)/d) - 2160*(d*x + c)^(3/2)*b^3*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)) - 405*
((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)) - 405*(-(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)*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)) + 24*(12
*(d*x + c)^(5/2)*b^4/d - 5*sqrt(d*x + c)*b^2*d)*sin(3*((d*x + c)*b - b*c + a*d)/d) - 216*(4*(d*x + c)^(5/2)*b^
4/d - 15*sqrt(d*x + c)*b^2*d)*sin(((d*x + c)*b - b*c + a*d)/d))*d/b^5

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.01 (sec) , antiderivative size = 2478, normalized size of antiderivative = 6.10 \[ \int (c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/1728*(72*(3*I*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/sqrt(b^2*d^2) + 1)) + I*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)) - 3
*I*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/sqrt(b^2*d^2) + 1)) - I*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)))*c^3 + 18*c*d^2
*(9*(I*sqrt(2)*sqrt(pi)*(4*b^2*c^2 + 4*I*b*c*d - 3*d^2)*d*erf(1/2*I*sqrt(2)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sq
rt(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^2) + 2*(-2*I*(d*x + c)^(3/2)
*b*d + 4*I*sqrt(d*x + c)*b*c*d - 3*sqrt(d*x + c)*d^2)*e^((-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b^2)/d^2 + (I*sqr
t(6)*sqrt(pi)*(12*b^2*c^2 - 4*I*b*c*d - d^2)*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)*b^2) + 6*(-2*I*(d*x + c)^(3/2)*b*d + 4*
I*sqrt(d*x + c)*b*c*d + sqrt(d*x + c)*d^2)*e^(-3*(-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b^2)/d^2 + 9*(-I*sqrt(2)*
sqrt(pi)*(4*b^2*c^2 - 4*I*b*c*d - 3*d^2)*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^2) + 2*(2*I*(d*x + c)^(3/2)*b*d - 4*I*sqrt(
d*x + c)*b*c*d - 3*sqrt(d*x + c)*d^2)*e^((I*(d*x + c)*b - I*b*c + I*a*d)/d)/b^2)/d^2 + (-I*sqrt(6)*sqrt(pi)*(1
2*b^2*c^2 + 4*I*b*c*d - d^2)*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)*b^2) + 6*(2*I*(d*x + c)^(3/2)*b*d - 4*I*sqrt(d*x + c)
*b*c*d + sqrt(d*x + c)*d^2)*e^(-3*(I*(d*x + c)*b - I*b*c + I*a*d)/d)/b^2)/d^2) + d^3*(27*(-I*sqrt(2)*sqrt(pi)*
(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*(-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 + (-I*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)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b^3) + 6*(-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^(-3*(-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b^3)/d^3 + 27*(I*sqrt(2)*sqrt(pi)
*(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*(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 + (I*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)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b^3) + 6*(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^(-3*(I*(d*x + c)*b - I*b*c + I*a*d)/d)/b^3)/d^3) + 36*(-9*I*sqrt(2)*sqrt(p
i)*(2*b*c + I*d)*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) - I*sqrt(6)*sqrt(pi)*(6*b*c - I*d)*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)*b) +
9*I*sqrt(2)*sqrt(pi)*(2*b*c - I*d)*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) + I*sqrt(6)*sqrt(pi)*(6*b*c + I*d)*d*erf(1/2*I*s
qrt(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)*b) + 18*I*sqrt(d*x + c)*d*e^((I*(d*x + c)*b - I*b*c + I*a*d)/d)/b + 6*I*sqrt(d*x + c)*d*e^(-3*(
I*(d*x + c)*b - I*b*c + I*a*d)/d)/b - 18*I*sqrt(d*x + c)*d*e^((-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b - 6*I*sqrt
(d*x + c)*d*e^(-3*(-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b)*c^2)/d

Mupad [F(-1)]

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

[In]

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

[Out]

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