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

Optimal. Leaf size=615 \[ -\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left (a-\frac{b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{32 b^{7/2}}-\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \cos \left (3 a-\frac{3 b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{576 b^{7/2}}+\frac{3 \sqrt{\frac{\pi }{10}} d^{5/2} \cos \left (5 a-\frac{5 b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{10}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{1600 b^{7/2}}-\frac{3 \sqrt{\frac{\pi }{10}} d^{5/2} \sin \left (5 a-\frac{5 b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{1600 b^{7/2}}+\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \sin \left (3 a-\frac{3 b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{576 b^{7/2}}+\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left (a-\frac{b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{32 b^{7/2}}+\frac{15 d^2 \sqrt{c+d x} \cos (a+b x)}{32 b^3}+\frac{5 d^2 \sqrt{c+d x} \cos (3 a+3 b x)}{576 b^3}-\frac{3 d^2 \sqrt{c+d x} \cos (5 a+5 b x)}{1600 b^3}+\frac{5 d (c+d x)^{3/2} \sin (a+b x)}{16 b^2}+\frac{5 d (c+d x)^{3/2} \sin (3 a+3 b x)}{288 b^2}-\frac{d (c+d x)^{3/2} \sin (5 a+5 b x)}{160 b^2}-\frac{(c+d x)^{5/2} \cos (a+b x)}{8 b}-\frac{(c+d x)^{5/2} \cos (3 a+3 b x)}{48 b}+\frac{(c+d x)^{5/2} \cos (5 a+5 b x)}{80 b} \]

[Out]

(15*d^2*Sqrt[c + d*x]*Cos[a + b*x])/(32*b^3) - ((c + d*x)^(5/2)*Cos[a + b*x])/(8*b) + (5*d^2*Sqrt[c + d*x]*Cos
[3*a + 3*b*x])/(576*b^3) - ((c + d*x)^(5/2)*Cos[3*a + 3*b*x])/(48*b) - (3*d^2*Sqrt[c + d*x]*Cos[5*a + 5*b*x])/
(1600*b^3) + ((c + d*x)^(5/2)*Cos[5*a + 5*b*x])/(80*b) - (15*d^(5/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqr
t[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(32*b^(7/2)) - (5*d^(5/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelC[(S
qrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(576*b^(7/2)) + (3*d^(5/2)*Sqrt[Pi/10]*Cos[5*a - (5*b*c)/d]*Fresnel
C[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(1600*b^(7/2)) - (3*d^(5/2)*Sqrt[Pi/10]*FresnelS[(Sqrt[b]*Sqrt
[10/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[5*a - (5*b*c)/d])/(1600*b^(7/2)) + (5*d^(5/2)*Sqrt[Pi/6]*FresnelS[(Sqrt[b]
*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(576*b^(7/2)) + (15*d^(5/2)*Sqrt[Pi/2]*FresnelS[(Sqr
t[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(32*b^(7/2)) + (5*d*(c + d*x)^(3/2)*Sin[a + b*x])/(1
6*b^2) + (5*d*(c + d*x)^(3/2)*Sin[3*a + 3*b*x])/(288*b^2) - (d*(c + d*x)^(3/2)*Sin[5*a + 5*b*x])/(160*b^2)

________________________________________________________________________________________

Rubi [A]  time = 0.952804, antiderivative size = 615, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {4406, 3296, 3306, 3305, 3351, 3304, 3352} \[ -\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left (a-\frac{b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{32 b^{7/2}}-\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \cos \left (3 a-\frac{3 b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{576 b^{7/2}}+\frac{3 \sqrt{\frac{\pi }{10}} d^{5/2} \cos \left (5 a-\frac{5 b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{10}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{1600 b^{7/2}}-\frac{3 \sqrt{\frac{\pi }{10}} d^{5/2} \sin \left (5 a-\frac{5 b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{1600 b^{7/2}}+\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \sin \left (3 a-\frac{3 b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{576 b^{7/2}}+\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left (a-\frac{b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{32 b^{7/2}}+\frac{15 d^2 \sqrt{c+d x} \cos (a+b x)}{32 b^3}+\frac{5 d^2 \sqrt{c+d x} \cos (3 a+3 b x)}{576 b^3}-\frac{3 d^2 \sqrt{c+d x} \cos (5 a+5 b x)}{1600 b^3}+\frac{5 d (c+d x)^{3/2} \sin (a+b x)}{16 b^2}+\frac{5 d (c+d x)^{3/2} \sin (3 a+3 b x)}{288 b^2}-\frac{d (c+d x)^{3/2} \sin (5 a+5 b x)}{160 b^2}-\frac{(c+d x)^{5/2} \cos (a+b x)}{8 b}-\frac{(c+d x)^{5/2} \cos (3 a+3 b x)}{48 b}+\frac{(c+d x)^{5/2} \cos (5 a+5 b x)}{80 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(15*d^2*Sqrt[c + d*x]*Cos[a + b*x])/(32*b^3) - ((c + d*x)^(5/2)*Cos[a + b*x])/(8*b) + (5*d^2*Sqrt[c + d*x]*Cos
[3*a + 3*b*x])/(576*b^3) - ((c + d*x)^(5/2)*Cos[3*a + 3*b*x])/(48*b) - (3*d^2*Sqrt[c + d*x]*Cos[5*a + 5*b*x])/
(1600*b^3) + ((c + d*x)^(5/2)*Cos[5*a + 5*b*x])/(80*b) - (15*d^(5/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqr
t[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(32*b^(7/2)) - (5*d^(5/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelC[(S
qrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(576*b^(7/2)) + (3*d^(5/2)*Sqrt[Pi/10]*Cos[5*a - (5*b*c)/d]*Fresnel
C[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(1600*b^(7/2)) - (3*d^(5/2)*Sqrt[Pi/10]*FresnelS[(Sqrt[b]*Sqrt
[10/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[5*a - (5*b*c)/d])/(1600*b^(7/2)) + (5*d^(5/2)*Sqrt[Pi/6]*FresnelS[(Sqrt[b]
*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(576*b^(7/2)) + (15*d^(5/2)*Sqrt[Pi/2]*FresnelS[(Sqr
t[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(32*b^(7/2)) + (5*d*(c + d*x)^(3/2)*Sin[a + b*x])/(1
6*b^2) + (5*d*(c + d*x)^(3/2)*Sin[3*a + 3*b*x])/(288*b^2) - (d*(c + d*x)^(3/2)*Sin[5*a + 5*b*x])/(160*b^2)

Rule 4406

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]

Rule 3296

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 3306

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 3305

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 3351

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

Rule 3304

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 3352

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

Rubi steps

\begin{align*} \int (c+d x)^{5/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx &=\int \left (\frac{1}{8} (c+d x)^{5/2} \sin (a+b x)+\frac{1}{16} (c+d x)^{5/2} \sin (3 a+3 b x)-\frac{1}{16} (c+d x)^{5/2} \sin (5 a+5 b x)\right ) \, dx\\ &=\frac{1}{16} \int (c+d x)^{5/2} \sin (3 a+3 b x) \, dx-\frac{1}{16} \int (c+d x)^{5/2} \sin (5 a+5 b x) \, dx+\frac{1}{8} \int (c+d x)^{5/2} \sin (a+b x) \, dx\\ &=-\frac{(c+d x)^{5/2} \cos (a+b x)}{8 b}-\frac{(c+d x)^{5/2} \cos (3 a+3 b x)}{48 b}+\frac{(c+d x)^{5/2} \cos (5 a+5 b x)}{80 b}-\frac{d \int (c+d x)^{3/2} \cos (5 a+5 b x) \, dx}{32 b}+\frac{(5 d) \int (c+d x)^{3/2} \cos (3 a+3 b x) \, dx}{96 b}+\frac{(5 d) \int (c+d x)^{3/2} \cos (a+b x) \, dx}{16 b}\\ &=-\frac{(c+d x)^{5/2} \cos (a+b x)}{8 b}-\frac{(c+d x)^{5/2} \cos (3 a+3 b x)}{48 b}+\frac{(c+d x)^{5/2} \cos (5 a+5 b x)}{80 b}+\frac{5 d (c+d x)^{3/2} \sin (a+b x)}{16 b^2}+\frac{5 d (c+d x)^{3/2} \sin (3 a+3 b x)}{288 b^2}-\frac{d (c+d x)^{3/2} \sin (5 a+5 b x)}{160 b^2}+\frac{\left (3 d^2\right ) \int \sqrt{c+d x} \sin (5 a+5 b x) \, dx}{320 b^2}-\frac{\left (5 d^2\right ) \int \sqrt{c+d x} \sin (3 a+3 b x) \, dx}{192 b^2}-\frac{\left (15 d^2\right ) \int \sqrt{c+d x} \sin (a+b x) \, dx}{32 b^2}\\ &=\frac{15 d^2 \sqrt{c+d x} \cos (a+b x)}{32 b^3}-\frac{(c+d x)^{5/2} \cos (a+b x)}{8 b}+\frac{5 d^2 \sqrt{c+d x} \cos (3 a+3 b x)}{576 b^3}-\frac{(c+d x)^{5/2} \cos (3 a+3 b x)}{48 b}-\frac{3 d^2 \sqrt{c+d x} \cos (5 a+5 b x)}{1600 b^3}+\frac{(c+d x)^{5/2} \cos (5 a+5 b x)}{80 b}+\frac{5 d (c+d x)^{3/2} \sin (a+b x)}{16 b^2}+\frac{5 d (c+d x)^{3/2} \sin (3 a+3 b x)}{288 b^2}-\frac{d (c+d x)^{3/2} \sin (5 a+5 b x)}{160 b^2}+\frac{\left (3 d^3\right ) \int \frac{\cos (5 a+5 b x)}{\sqrt{c+d x}} \, dx}{3200 b^3}-\frac{\left (5 d^3\right ) \int \frac{\cos (3 a+3 b x)}{\sqrt{c+d x}} \, dx}{1152 b^3}-\frac{\left (15 d^3\right ) \int \frac{\cos (a+b x)}{\sqrt{c+d x}} \, dx}{64 b^3}\\ &=\frac{15 d^2 \sqrt{c+d x} \cos (a+b x)}{32 b^3}-\frac{(c+d x)^{5/2} \cos (a+b x)}{8 b}+\frac{5 d^2 \sqrt{c+d x} \cos (3 a+3 b x)}{576 b^3}-\frac{(c+d x)^{5/2} \cos (3 a+3 b x)}{48 b}-\frac{3 d^2 \sqrt{c+d x} \cos (5 a+5 b x)}{1600 b^3}+\frac{(c+d x)^{5/2} \cos (5 a+5 b x)}{80 b}+\frac{5 d (c+d x)^{3/2} \sin (a+b x)}{16 b^2}+\frac{5 d (c+d x)^{3/2} \sin (3 a+3 b x)}{288 b^2}-\frac{d (c+d x)^{3/2} \sin (5 a+5 b x)}{160 b^2}+\frac{\left (3 d^3 \cos \left (5 a-\frac{5 b c}{d}\right )\right ) \int \frac{\cos \left (\frac{5 b c}{d}+5 b x\right )}{\sqrt{c+d x}} \, dx}{3200 b^3}-\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}{1152 b^3}-\frac{\left (15 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}{64 b^3}-\frac{\left (3 d^3 \sin \left (5 a-\frac{5 b c}{d}\right )\right ) \int \frac{\sin \left (\frac{5 b c}{d}+5 b x\right )}{\sqrt{c+d x}} \, dx}{3200 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}{1152 b^3}+\frac{\left (15 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}{64 b^3}\\ &=\frac{15 d^2 \sqrt{c+d x} \cos (a+b x)}{32 b^3}-\frac{(c+d x)^{5/2} \cos (a+b x)}{8 b}+\frac{5 d^2 \sqrt{c+d x} \cos (3 a+3 b x)}{576 b^3}-\frac{(c+d x)^{5/2} \cos (3 a+3 b x)}{48 b}-\frac{3 d^2 \sqrt{c+d x} \cos (5 a+5 b x)}{1600 b^3}+\frac{(c+d x)^{5/2} \cos (5 a+5 b x)}{80 b}+\frac{5 d (c+d x)^{3/2} \sin (a+b x)}{16 b^2}+\frac{5 d (c+d x)^{3/2} \sin (3 a+3 b x)}{288 b^2}-\frac{d (c+d x)^{3/2} \sin (5 a+5 b x)}{160 b^2}+\frac{\left (3 d^2 \cos \left (5 a-\frac{5 b c}{d}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{5 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{1600 b^3}-\frac{\left (5 d^2 \cos \left (3 a-\frac{3 b c}{d}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{3 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{576 b^3}-\frac{\left (15 d^2 \cos \left (a-\frac{b c}{d}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{32 b^3}-\frac{\left (3 d^2 \sin \left (5 a-\frac{5 b c}{d}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{5 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{1600 b^3}+\frac{\left (5 d^2 \sin \left (3 a-\frac{3 b c}{d}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{3 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{576 b^3}+\frac{\left (15 d^2 \sin \left (a-\frac{b c}{d}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{32 b^3}\\ &=\frac{15 d^2 \sqrt{c+d x} \cos (a+b x)}{32 b^3}-\frac{(c+d x)^{5/2} \cos (a+b x)}{8 b}+\frac{5 d^2 \sqrt{c+d x} \cos (3 a+3 b x)}{576 b^3}-\frac{(c+d x)^{5/2} \cos (3 a+3 b x)}{48 b}-\frac{3 d^2 \sqrt{c+d x} \cos (5 a+5 b x)}{1600 b^3}+\frac{(c+d x)^{5/2} \cos (5 a+5 b x)}{80 b}-\frac{15 d^{5/2} \sqrt{\frac{\pi }{2}} \cos \left (a-\frac{b c}{d}\right ) C\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{32 b^{7/2}}-\frac{5 d^{5/2} \sqrt{\frac{\pi }{6}} \cos \left (3 a-\frac{3 b c}{d}\right ) C\left (\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{576 b^{7/2}}+\frac{3 d^{5/2} \sqrt{\frac{\pi }{10}} \cos \left (5 a-\frac{5 b c}{d}\right ) C\left (\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{1600 b^{7/2}}-\frac{3 d^{5/2} \sqrt{\frac{\pi }{10}} S\left (\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right ) \sin \left (5 a-\frac{5 b c}{d}\right )}{1600 b^{7/2}}+\frac{5 d^{5/2} \sqrt{\frac{\pi }{6}} S\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 )}{576 b^{7/2}}+\frac{15 d^{5/2} \sqrt{\frac{\pi }{2}} S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right ) \sin \left (a-\frac{b c}{d}\right )}{32 b^{7/2}}+\frac{5 d (c+d x)^{3/2} \sin (a+b x)}{16 b^2}+\frac{5 d (c+d x)^{3/2} \sin (3 a+3 b x)}{288 b^2}-\frac{d (c+d x)^{3/2} \sin (5 a+5 b x)}{160 b^2}\\ \end{align*}

Mathematica [C]  time = 25.3137, size = 3348, normalized size = 5.44 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^2*Sqrt[c + d*x]*(-((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d]) - (E^(((2*I)*
b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(16*b*E^((I*(b*c + a*d))/d)) + (c^2*(2*Sqrt[5
]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[5*(a + b*x)] - Sqrt[2*Pi]*Cos[5*a - (5*b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sq
rt[c + d*x]] + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*Sin[5*a - (5*b*c)/d]))/(160*Sqrt[5]*b*
Sqrt[b/d]) - (c^2*(2*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[3*(a + b*x)] - Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*Fresne
lC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]] + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*
b*c)/d]))/(96*Sqrt[3]*b*Sqrt[b/d]) - (c*Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(
3*d*Cos[a - (b*c)/d] - 2*b*c*Sin[a - (b*c)/d]) + Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(2*b*
c*Cos[a - (b*c)/d] + 3*d*Sin[a - (b*c)/d]) + 2*Sqrt[b/d]*d*Sqrt[c + d*x]*(2*b*x*Cos[a + b*x] - 3*Sin[a + b*x])
))/(16*b^3) + ((b/d)^(3/2)*d^2*(Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*((4*b^2*c^2 - 15*d^2)*
Cos[a - (b*c)/d] + 12*b*c*d*Sin[a - (b*c)/d]) - Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(-12*b
*c*d*Cos[a - (b*c)/d] + (4*b^2*c^2 - 15*d^2)*Sin[a - (b*c)/d]) - 2*Sqrt[b/d]*d*Sqrt[c + d*x]*(d*(-15 + 4*b^2*x
^2)*Cos[a + b*x] + 2*b*(c - 5*d*x)*Sin[a + b*x])))/(64*b^5) - (c*Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sq
rt[6/Pi]*Sqrt[c + d*x]]*(d*Cos[3*a - (3*b*c)/d] - 2*b*c*Sin[3*a - (3*b*c)/d]) + Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*
Sqrt[6/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[3*a - (3*b*c)/d] + d*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*Sqrt[b/d]*d*Sqrt[c
+ d*x]*(2*b*x*Cos[3*(a + b*x)] - Sin[3*(a + b*x)])))/(96*Sqrt[3]*b^3) + ((b/d)^(3/2)*d^2*(Sqrt[2*Pi]*FresnelC
[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*((12*b^2*c^2 - 5*d^2)*Cos[3*a - (3*b*c)/d] + 12*b*c*d*Sin[3*a - (3*b*c)/d
]) - Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(-12*b*c*d*Cos[3*a - (3*b*c)/d] + (12*b^2*c^2 - 5
*d^2)*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*Sqrt[b/d]*d*Sqrt[c + d*x]*(d*(5 - 12*b^2*x^2)*Cos[3*(a + b*x)] - 2*b*(
c - 5*d*x)*Sin[3*(a + b*x)])))/(1152*Sqrt[3]*b^5) + (c*Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*
Sqrt[c + d*x]]*(3*d*Cos[5*a - (5*b*c)/d] - 10*b*c*Sin[5*a - (5*b*c)/d]) + Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[1
0/Pi]*Sqrt[c + d*x]]*(10*b*c*Cos[5*a - (5*b*c)/d] + 3*d*Sin[5*a - (5*b*c)/d]) + 2*Sqrt[5]*Sqrt[b/d]*d*Sqrt[c +
d*x]*(10*b*x*Cos[5*(a + b*x)] - 3*Sin[5*(a + b*x)])))/(800*Sqrt[5]*b^3) - (d^2*(Sin[5*a]*((c^2*(-(Sqrt[5]*Sqr
t[b/d]*Sqrt[c + d*x]*Cos[(5*b*(c + d*x))/d]) + Sqrt[Pi/2]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]])*Sin[(
5*b*c)/d])/(5*Sqrt[5]*(b/d)^(3/2)*d^3) + (c^2*Cos[(5*b*c)/d]*(-(Sqrt[Pi/2]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt
[c + d*x]]) + Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[(5*b*(c + d*x))/d]))/(5*Sqrt[5]*(b/d)^(3/2)*d^3) - (2*c*Cos[
(5*b*c)/d]*((-3*(-(Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[(5*b*(c + d*x))/d]) + Sqrt[Pi/2]*FresnelC[Sqrt[b/d]*Sqr
t[10/Pi]*Sqrt[c + d*x]]))/2 + 5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Sin[(5*b*(c + d*x))/d]))/(25*Sqrt[5]*(b/d)
^(5/2)*d^3) - (2*c*Sin[(5*b*c)/d]*(-5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Cos[(5*b*(c + d*x))/d] + (3*(-(Sqrt[
Pi/2]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]) + Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[(5*b*(c + d*x))/d])
)/2))/(25*Sqrt[5]*(b/d)^(5/2)*d^3) + (Sin[(5*b*c)/d]*(-25*Sqrt[5]*(b/d)^(5/2)*(c + d*x)^(5/2)*Cos[(5*b*(c + d*
x))/d] + (5*((-3*(-(Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[(5*b*(c + d*x))/d]) + Sqrt[Pi/2]*FresnelC[Sqrt[b/d]*Sq
rt[10/Pi]*Sqrt[c + d*x]]))/2 + 5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Sin[(5*b*(c + d*x))/d]))/2))/(125*Sqrt[5]
*(b/d)^(7/2)*d^3) + (Cos[(5*b*c)/d]*(25*Sqrt[5]*(b/d)^(5/2)*(c + d*x)^(5/2)*Sin[(5*b*(c + d*x))/d] - (5*(-5*Sq
rt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Cos[(5*b*(c + d*x))/d] + (3*(-(Sqrt[Pi/2]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqr
t[c + d*x]]) + Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[(5*b*(c + d*x))/d]))/2))/2))/(125*Sqrt[5]*(b/d)^(7/2)*d^3))
+ Cos[5*a]*((c^2*Cos[(5*b*c)/d]*(-(Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[(5*b*(c + d*x))/d]) + Sqrt[Pi/2]*Fresn
elC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]))/(5*Sqrt[5]*(b/d)^(3/2)*d^3) - (c^2*Sin[(5*b*c)/d]*(-(Sqrt[Pi/2]*Fre
snelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]) + Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[(5*b*(c + d*x))/d]))/(5*Sqrt
[5]*(b/d)^(3/2)*d^3) + (2*c*Sin[(5*b*c)/d]*((-3*(-(Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[(5*b*(c + d*x))/d]) + S
qrt[Pi/2]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]))/2 + 5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Sin[(5*b*(
c + d*x))/d]))/(25*Sqrt[5]*(b/d)^(5/2)*d^3) - (2*c*Cos[(5*b*c)/d]*(-5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Cos[
(5*b*(c + d*x))/d] + (3*(-(Sqrt[Pi/2]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]) + Sqrt[5]*Sqrt[b/d]*Sqrt[
c + d*x]*Sin[(5*b*(c + d*x))/d]))/2))/(25*Sqrt[5]*(b/d)^(5/2)*d^3) + (Cos[(5*b*c)/d]*(-25*Sqrt[5]*(b/d)^(5/2)*
(c + d*x)^(5/2)*Cos[(5*b*(c + d*x))/d] + (5*((-3*(-(Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[(5*b*(c + d*x))/d]) +
Sqrt[Pi/2]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]))/2 + 5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Sin[(5*b*
(c + d*x))/d]))/2))/(125*Sqrt[5]*(b/d)^(7/2)*d^3) - (Sin[(5*b*c)/d]*(25*Sqrt[5]*(b/d)^(5/2)*(c + d*x)^(5/2)*Si
n[(5*b*(c + d*x))/d] - (5*(-5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Cos[(5*b*(c + d*x))/d] + (3*(-(Sqrt[Pi/2]*Fr
esnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]) + Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[(5*b*(c + d*x))/d]))/2))/2)
)/(125*Sqrt[5]*(b/d)^(7/2)*d^3))))/16

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Maple [A]  time = 0.043, size = 719, normalized size = 1.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d*(-1/16/b*d*(d*x+c)^(5/2)*cos(1/d*(d*x+c)*b+(a*d-b*c)/d)+5/16/b*d*(1/2/b*d*(d*x+c)^(3/2)*sin(1/d*(d*x+c)*b+
(a*d-b*c)/d)-3/2/b*d*(-1/2/b*d*(d*x+c)^(1/2)*cos(1/d*(d*x+c)*b+(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)*(d*x+c)^(1/2)*b/d)-sin((a*d-b*c)/d)*FresnelS(2^(1/2
)/Pi^(1/2)/(b/d)^(1/2)*(d*x+c)^(1/2)*b/d))))-1/96/b*d*(d*x+c)^(5/2)*cos(3/d*(d*x+c)*b+3*(a*d-b*c)/d)+5/96/b*d*
(1/6/b*d*(d*x+c)^(3/2)*sin(3/d*(d*x+c)*b+3*(a*d-b*c)/d)-1/2/b*d*(-1/6/b*d*(d*x+c)^(1/2)*cos(3/d*(d*x+c)*b+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)*(d*x+c)^(1/2)*b/d)-sin(3*(a*d-b*c)/d)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)/(b/d)^(1/2)*(d*x+c)^(1/2)
*b/d))))+1/160/b*d*(d*x+c)^(5/2)*cos(5/d*(d*x+c)*b+5*(a*d-b*c)/d)-1/32/b*d*(1/10/b*d*(d*x+c)^(3/2)*sin(5/d*(d*
x+c)*b+5*(a*d-b*c)/d)-3/10/b*d*(-1/10/b*d*(d*x+c)^(1/2)*cos(5/d*(d*x+c)*b+5*(a*d-b*c)/d)+1/100/b*d*2^(1/2)*Pi^
(1/2)*5^(1/2)/(b/d)^(1/2)*(cos(5*(a*d-b*c)/d)*FresnelC(2^(1/2)/Pi^(1/2)*5^(1/2)/(b/d)^(1/2)*(d*x+c)^(1/2)*b/d)
-sin(5*(a*d-b*c)/d)*FresnelS(2^(1/2)/Pi^(1/2)*5^(1/2)/(b/d)^(1/2)*(d*x+c)^(1/2)*b/d)))))

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Maxima [C]  time = 2.95017, size = 2943, normalized size = 4.79 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1728000*sqrt(5)*sqrt(3)*(720*sqrt(5)*sqrt(3)*(d*x + c)^(3/2)*b*d^2*sqrt(abs(b)/abs(d))*abs(b)*sin(5*((d*x +
c)*b - b*c + a*d)/d)/abs(d) - 2000*sqrt(5)*sqrt(3)*(d*x + c)^(3/2)*b*d^2*sqrt(abs(b)/abs(d))*abs(b)*sin(3*((d
*x + c)*b - b*c + a*d)/d)/abs(d) - 36000*sqrt(5)*sqrt(3)*(d*x + c)^(3/2)*b*d^2*sqrt(abs(b)/abs(d))*abs(b)*sin(
((d*x + c)*b - b*c + a*d)/d)/abs(d) - 72*(20*sqrt(5)*sqrt(3)*(d*x + c)^(5/2)*b^2*d*sqrt(abs(b)/abs(d))*abs(b)/
abs(d) - 3*sqrt(5)*sqrt(3)*sqrt(d*x + c)*d^3*sqrt(abs(b)/abs(d))*abs(b)/abs(d))*cos(5*((d*x + c)*b - b*c + a*d
)/d) + 200*(12*sqrt(5)*sqrt(3)*(d*x + c)^(5/2)*b^2*d*sqrt(abs(b)/abs(d))*abs(b)/abs(d) - 5*sqrt(5)*sqrt(3)*sqr
t(d*x + c)*d^3*sqrt(abs(b)/abs(d))*abs(b)/abs(d))*cos(3*((d*x + c)*b - b*c + a*d)/d) + 3600*(4*sqrt(5)*sqrt(3)
*(d*x + c)^(5/2)*b^2*d*sqrt(abs(b)/abs(d))*abs(b)/abs(d) - 15*sqrt(5)*sqrt(3)*sqrt(d*x + c)*d^3*sqrt(abs(b)/ab
s(d))*abs(b)/abs(d))*cos(((d*x + c)*b - b*c + a*d)/d) - (sqrt(3)*(27*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) +
1/2*arctan2(0, d/sqrt(d^2))) + 27*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 2
7*I*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 27*I*sqrt(pi)*sin(-1/4*pi + 1/2*a
rctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^3*abs(b)*cos(-5*(b*c - a*d)/d)/abs(d) - sqrt(3)*(27*I*sqrt(pi)*
cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 27*I*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b)
+ 1/2*arctan2(0, d/sqrt(d^2))) + 27*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 2
7*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^3*abs(b)*sin(-5*(b*c - a*d)/d)/ab
s(d))*erf(sqrt(d*x + c)*sqrt(5*I*b/d)) + (sqrt(5)*(125*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0
, d/sqrt(d^2))) + 125*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 125*I*sqrt(pi)
*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 125*I*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b
) + 1/2*arctan2(0, d/sqrt(d^2))))*d^3*abs(b)*cos(-3*(b*c - a*d)/d)/abs(d) + sqrt(5)*(-125*I*sqrt(pi)*cos(1/4*p
i + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 125*I*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*ar
ctan2(0, d/sqrt(d^2))) - 125*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 125*sqrt
(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^3*abs(b)*sin(-3*(b*c - a*d)/d)/abs(d))*
erf(sqrt(d*x + c)*sqrt(3*I*b/d)) + (sqrt(5)*sqrt(3)*(6750*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan
2(0, d/sqrt(d^2))) + 6750*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 6750*I*sqr
t(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 6750*I*sqrt(pi)*sin(-1/4*pi + 1/2*arctan
2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^3*abs(b)*cos(-(b*c - a*d)/d)/abs(d) + sqrt(5)*sqrt(3)*(-6750*I*sqrt(
pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 6750*I*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(
0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 6750*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^
2))) + 6750*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^3*abs(b)*sin(-(b*c - a*
d)/d)/abs(d))*erf(sqrt(d*x + c)*sqrt(I*b/d)) + (sqrt(5)*sqrt(3)*(6750*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b)
+ 1/2*arctan2(0, d/sqrt(d^2))) + 6750*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2)))
+ 6750*I*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 6750*I*sqrt(pi)*sin(-1/4*pi
+ 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^3*abs(b)*cos(-(b*c - a*d)/d)/abs(d) + sqrt(5)*sqrt(3)*(6
750*I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 6750*I*sqrt(pi)*cos(-1/4*pi + 1
/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 6750*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0
, d/sqrt(d^2))) + 6750*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^3*abs(b)*sin
(-(b*c - a*d)/d)/abs(d))*erf(sqrt(d*x + c)*sqrt(-I*b/d)) + (sqrt(5)*(125*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0,
b) + 1/2*arctan2(0, d/sqrt(d^2))) + 125*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))
) + 125*I*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 125*I*sqrt(pi)*sin(-1/4*pi
+ 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^3*abs(b)*cos(-3*(b*c - a*d)/d)/abs(d) + sqrt(5)*(125*I*s
qrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 125*I*sqrt(pi)*cos(-1/4*pi + 1/2*arcta
n2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 125*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(
d^2))) + 125*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^3*abs(b)*sin(-3*(b*c -
a*d)/d)/abs(d))*erf(sqrt(d*x + c)*sqrt(-3*I*b/d)) - (sqrt(3)*(27*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/
2*arctan2(0, d/sqrt(d^2))) + 27*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 27*I
*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 27*I*sqrt(pi)*sin(-1/4*pi + 1/2*arct
an2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^3*abs(b)*cos(-5*(b*c - a*d)/d)/abs(d) - sqrt(3)*(-27*I*sqrt(pi)*co
s(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 27*I*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) +
1/2*arctan2(0, d/sqrt(d^2))) + 27*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 27*
sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^3*abs(b)*sin(-5*(b*c - a*d)/d)/abs(
d))*erf(sqrt(d*x + c)*sqrt(-5*I*b/d)))*abs(d)/(b^3*d*sqrt(abs(b)/abs(d))*abs(b))

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Fricas [A]  time = 0.832086, size = 1337, normalized size = 2.17 \begin{align*} \frac{81 \, \sqrt{10} \pi d^{3} \sqrt{\frac{b}{\pi d}} \cos \left (-\frac{5 \,{\left (b c - a d\right )}}{d}\right ) \operatorname{C}\left (\sqrt{10} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) - 625 \, \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 ) - 101250 \, \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 ) + 101250 \, \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 ) + 625 \, \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 ) - 81 \, \sqrt{10} \pi d^{3} \sqrt{\frac{b}{\pi d}} \operatorname{S}\left (\sqrt{10} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) \sin \left (-\frac{5 \,{\left (b c - a d\right )}}{d}\right ) + 480 \,{\left (9 \,{\left (20 \, b^{3} d^{2} x^{2} + 40 \, b^{3} c d x + 20 \, b^{3} c^{2} - 3 \, b d^{2}\right )} \cos \left (b x + a\right )^{5} + 390 \, b d^{2} \cos \left (b x + a\right ) - 5 \,{\left (60 \, b^{3} d^{2} x^{2} + 120 \, b^{3} c d x + 60 \, b^{3} c^{2} - 13 \, b d^{2}\right )} \cos \left (b x + a\right )^{3} + 10 \,{\left (26 \, b^{2} d^{2} x - 9 \,{\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )^{4} + 26 \, b^{2} c d + 13 \,{\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}}{432000 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/432000*(81*sqrt(10)*pi*d^3*sqrt(b/(pi*d))*cos(-5*(b*c - a*d)/d)*fresnel_cos(sqrt(10)*sqrt(d*x + c)*sqrt(b/(p
i*d))) - 625*sqrt(6)*pi*d^3*sqrt(b/(pi*d))*cos(-3*(b*c - a*d)/d)*fresnel_cos(sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi*
d))) - 101250*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
))) + 101250*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) + 625*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)
- 81*sqrt(10)*pi*d^3*sqrt(b/(pi*d))*fresnel_sin(sqrt(10)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-5*(b*c - a*d)/d)
+ 480*(9*(20*b^3*d^2*x^2 + 40*b^3*c*d*x + 20*b^3*c^2 - 3*b*d^2)*cos(b*x + a)^5 + 390*b*d^2*cos(b*x + a) - 5*(6
0*b^3*d^2*x^2 + 120*b^3*c*d*x + 60*b^3*c^2 - 13*b*d^2)*cos(b*x + a)^3 + 10*(26*b^2*d^2*x - 9*(b^2*d^2*x + b^2*
c*d)*cos(b*x + a)^4 + 26*b^2*c*d + 13*(b^2*d^2*x + b^2*c*d)*cos(b*x + a)^2)*sin(b*x + a))*sqrt(d*x + c))/b^4

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [C]  time = 2.21758, size = 4084, normalized size = 6.64 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/864000*(60*(9*sqrt(10)*sqrt(pi)*d^2*erf(-1/2*sqrt(10)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e
^((5*I*b*c - 5*I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b) - 25*sqrt(6)*sqrt(pi)*d^2*erf(-1/2*sqrt(6)*sq
rt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((3*I*b*c - 3*I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) +
1)*b) - 450*sqrt(2)*sqrt(pi)*d^2*erf(-1/2*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) - 450*sqrt(2)*sqrt(pi)*d^2*erf(-1/2*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) -
25*sqrt(6)*sqrt(pi)*d^2*erf(-1/2*sqrt(6)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-3*I*b*c +
3*I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b) + 9*sqrt(10)*sqrt(pi)*d^2*erf(-1/2*sqrt(10)*sqrt(b*d)*sqr
t(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-5*I*b*c + 5*I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b) +
90*sqrt(d*x + c)*d*e^((5*I*(d*x + c)*b - 5*I*b*c + 5*I*a*d)/d)/b - 150*sqrt(d*x + c)*d*e^((3*I*(d*x + c)*b -
3*I*b*c + 3*I*a*d)/d)/b - 900*sqrt(d*x + c)*d*e^((I*(d*x + c)*b - I*b*c + I*a*d)/d)/b - 900*sqrt(d*x + c)*d*e^
((-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b - 150*sqrt(d*x + c)*d*e^((-3*I*(d*x + c)*b + 3*I*b*c - 3*I*a*d)/d)/b +
90*sqrt(d*x + c)*d*e^((-5*I*(d*x + c)*b + 5*I*b*c - 5*I*a*d)/d)/b)*c^2 - d^2*(27*(I*sqrt(10)*sqrt(pi)*(20*I*b^
2*c^2*d - 12*b*c*d^2 - 3*I*d^3)*d*erf(-1/2*sqrt(10)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((5
*I*b*c - 5*I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b^3) - 10*I*(-20*I*(d*x + c)^(5/2)*b^2*d + 40*I*(d*x
+ c)^(3/2)*b^2*c*d - 20*I*sqrt(d*x + c)*b^2*c^2*d - 10*(d*x + c)^(3/2)*b*d^2 + 12*sqrt(d*x + c)*b*c*d^2 + 3*I
*sqrt(d*x + c)*d^3)*e^((-5*I*(d*x + c)*b + 5*I*b*c - 5*I*a*d)/d)/b^3)/d^2 + 125*(I*sqrt(6)*sqrt(pi)*(-12*I*b^2
*c^2*d + 12*b*c*d^2 + 5*I*d^3)*d*erf(-1/2*sqrt(6)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((3*I
*b*c - 3*I*a*d)/d)/(sqrt(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 - 24*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 - 12*sqrt(d*x + c)*b*c*d^2 - 5*I*sqr
t(d*x + c)*d^3)*e^((-3*I*(d*x + c)*b + 3*I*b*c - 3*I*a*d)/d)/b^3)/d^2 + 6750*(I*sqrt(2)*sqrt(pi)*(-4*I*b^2*c^2
*d + 12*b*c*d^2 + 15*I*d^3)*d*erf(-1/2*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 - 8*I*(d*x + c)^(3/2)*b
^2*c*d + 4*I*sqrt(d*x + c)*b^2*c^2*d + 10*(d*x + c)^(3/2)*b*d^2 - 12*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^2 + 6750*(I*sqrt(2)*sqrt(pi)*(-4*I*b^2*c^2*d - 12*b*c*d^2
+ 15*I*d^3)*d*erf(-1/2*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 - 8*I*(d*x + c)^(3/2)*b^2*c*d + 4*I
*sqrt(d*x + c)*b^2*c^2*d - 10*(d*x + c)^(3/2)*b*d^2 + 12*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^2 + 125*(I*sqrt(6)*sqrt(pi)*(-12*I*b^2*c^2*d - 12*b*c*d^2 + 5*I*d^3)*d
*erf(-1/2*sqrt(6)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-3*I*b*c + 3*I*a*d)/d)/(sqrt(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 - 24*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 + 12*sqrt(d*x + c)*b*c*d^2 - 5*I*sqrt(d*x + c)*d^3)*e^((3*I*(d*x
+ c)*b - 3*I*b*c + 3*I*a*d)/d)/b^3)/d^2 + 27*(I*sqrt(10)*sqrt(pi)*(20*I*b^2*c^2*d + 12*b*c*d^2 - 3*I*d^3)*d*er
f(-1/2*sqrt(10)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-5*I*b*c + 5*I*a*d)/d)/(sqrt(b*d)*(-
I*b*d/sqrt(b^2*d^2) + 1)*b^3) - 10*I*(-20*I*(d*x + c)^(5/2)*b^2*d + 40*I*(d*x + c)^(3/2)*b^2*c*d - 20*I*sqrt(d
*x + c)*b^2*c^2*d + 10*(d*x + c)^(3/2)*b*d^2 - 12*sqrt(d*x + c)*b*c*d^2 + 3*I*sqrt(d*x + c)*d^3)*e^((5*I*(d*x
+ c)*b - 5*I*b*c + 5*I*a*d)/d)/b^3)/d^2) - 12*(9*I*sqrt(10)*sqrt(pi)*(-10*I*b*c*d + 3*d^2)*d*erf(-1/2*sqrt(10)
*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((5*I*b*c - 5*I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2
) + 1)*b^2) + 125*I*sqrt(6)*sqrt(pi)*(2*I*b*c*d - d^2)*d*erf(-1/2*sqrt(6)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(
b^2*d^2) + 1)/d)*e^((3*I*b*c - 3*I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b^2) + 2250*I*sqrt(2)*sqrt(pi)
*(2*I*b*c*d - 3*d^2)*d*erf(-1/2*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) + 2250*I*sqrt(2)*sqrt(pi)*(2*I*b*c*d + 3*d^2)*d*erf(-1/2*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) + 125*I*sqrt(6)*sqrt(pi)*(2*I*b*c*d + d^2)*d*erf(-1/2*sqrt(6)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt
(b^2*d^2) + 1)/d)*e^((-3*I*b*c + 3*I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b^2) + 9*I*sqrt(10)*sqrt(pi
)*(-10*I*b*c*d - 3*d^2)*d*erf(-1/2*sqrt(10)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-5*I*b*c
+ 5*I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b^2) - 90*I*(-10*I*(d*x + c)^(3/2)*b*d + 10*I*sqrt(d*x +
c)*b*c*d + 3*sqrt(d*x + c)*d^2)*e^((5*I*(d*x + c)*b - 5*I*b*c + 5*I*a*d)/d)/b^2 - 750*I*(2*I*(d*x + c)^(3/2)*b
*d - 2*I*sqrt(d*x + c)*b*c*d - sqrt(d*x + c)*d^2)*e^((3*I*(d*x + c)*b - 3*I*b*c + 3*I*a*d)/d)/b^2 - 4500*I*(2*
I*(d*x + c)^(3/2)*b*d - 2*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 - 4500*I*(2*I*(d*x + c)^(3/2)*b*d - 2*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 - 750*I*(2*I*(d*x + c)^(3/2)*b*d - 2*I*sqrt(d*x + c)*b*c*d + sqrt(d*x + c)*d^2)*e^((-3*I*(d
*x + c)*b + 3*I*b*c - 3*I*a*d)/d)/b^2 - 90*I*(-10*I*(d*x + c)^(3/2)*b*d + 10*I*sqrt(d*x + c)*b*c*d - 3*sqrt(d*
x + c)*d^2)*e^((-5*I*(d*x + c)*b + 5*I*b*c - 5*I*a*d)/d)/b^2)*c)/d