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

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

[Out]

(5*d*(c + d*x)^(3/2)*Cos[a + b*x])/(16*b^2) - (5*d*(c + d*x)^(3/2)*Cos[3*a + 3*b*x])/(288*b^2) - (d*(c + d*x)^
(3/2)*Cos[5*a + 5*b*x])/(160*b^2) + (15*d^(5/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[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]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqr
t[c + d*x])/Sqrt[d]])/(576*b^(7/2)) - (3*d^(5/2)*Sqrt[Pi/10]*Cos[5*a - (5*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[10/Pi
]*Sqrt[c + d*x])/Sqrt[d]])/(1600*b^(7/2)) - (3*d^(5/2)*Sqrt[Pi/10]*FresnelC[(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]*FresnelC[(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]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[
c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(32*b^(7/2)) - (15*d^2*Sqrt[c + d*x]*Sin[a + b*x])/(32*b^3) + ((c + d*x)^
(5/2)*Sin[a + b*x])/(8*b) + (5*d^2*Sqrt[c + d*x]*Sin[3*a + 3*b*x])/(576*b^3) - ((c + d*x)^(5/2)*Sin[3*a + 3*b*
x])/(48*b) + (3*d^2*Sqrt[c + d*x]*Sin[5*a + 5*b*x])/(1600*b^3) - ((c + d*x)^(5/2)*Sin[5*a + 5*b*x])/(80*b)

________________________________________________________________________________________

Rubi [A]  time = 1.01548, 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} \[ -\frac{3 \sqrt{\frac{\pi }{10}} d^{5/2} \sin \left (5 a-\frac{5 b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{10}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{1600 b^{7/2}}-\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \sin \left (3 a-\frac{3 b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{576 b^{7/2}}+\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left (a-\frac{b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{32 b^{7/2}}+\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left (a-\frac{b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{32 b^{7/2}}-\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \cos \left (3 a-\frac{3 b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{576 b^{7/2}}-\frac{3 \sqrt{\frac{\pi }{10}} d^{5/2} \cos \left (5 a-\frac{5 b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{1600 b^{7/2}}-\frac{15 d^2 \sqrt{c+d x} \sin (a+b x)}{32 b^3}+\frac{5 d^2 \sqrt{c+d x} \sin (3 a+3 b x)}{576 b^3}+\frac{3 d^2 \sqrt{c+d x} \sin (5 a+5 b x)}{1600 b^3}+\frac{5 d (c+d x)^{3/2} \cos (a+b x)}{16 b^2}-\frac{5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{288 b^2}-\frac{d (c+d x)^{3/2} \cos (5 a+5 b x)}{160 b^2}+\frac{(c+d x)^{5/2} \sin (a+b x)}{8 b}-\frac{(c+d x)^{5/2} \sin (3 a+3 b x)}{48 b}-\frac{(c+d x)^{5/2} \sin (5 a+5 b x)}{80 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*d*(c + d*x)^(3/2)*Cos[a + b*x])/(16*b^2) - (5*d*(c + d*x)^(3/2)*Cos[3*a + 3*b*x])/(288*b^2) - (d*(c + d*x)^
(3/2)*Cos[5*a + 5*b*x])/(160*b^2) + (15*d^(5/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[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]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqr
t[c + d*x])/Sqrt[d]])/(576*b^(7/2)) - (3*d^(5/2)*Sqrt[Pi/10]*Cos[5*a - (5*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[10/Pi
]*Sqrt[c + d*x])/Sqrt[d]])/(1600*b^(7/2)) - (3*d^(5/2)*Sqrt[Pi/10]*FresnelC[(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]*FresnelC[(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]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[
c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(32*b^(7/2)) - (15*d^2*Sqrt[c + d*x]*Sin[a + b*x])/(32*b^3) + ((c + d*x)^
(5/2)*Sin[a + b*x])/(8*b) + (5*d^2*Sqrt[c + d*x]*Sin[3*a + 3*b*x])/(576*b^3) - ((c + d*x)^(5/2)*Sin[3*a + 3*b*
x])/(48*b) + (3*d^2*Sqrt[c + d*x]*Sin[5*a + 5*b*x])/(1600*b^3) - ((c + d*x)^(5/2)*Sin[5*a + 5*b*x])/(80*b)

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

Mathematica [C]  time = 23.4094, size = 1795, normalized size = 2.92 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-I/16)*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]))/(b*E^((I*(b*c + a*d))/d)) + (c*d*(Sqrt[b
/d]*Sqrt[2*Pi]*FresnelC[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[b/d]*Sqrt[2*Pi]*FresnelS[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*b*Sqrt[c + d*x]*(3*Cos[a + b*x] + 2*b*x*Sin[a + b*x])))/(16*b^3) + ((b/d)^(3/2)*d^2*(-(Sqrt[2*Pi]*Fre
snelS[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]*FresnelC[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]*(-2*b*(c - 5*d*x)*Cos[a + b*x] + d*(-15 + 4*b^2*x^2)*Sin[a + b
*x])))/(64*b^5) - (c^2*(-(Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]) - Sqrt
[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d] + 2*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]*S
in[3*(a + b*x)]))/(96*Sqrt[3]*b*Sqrt[b/d]) - (c*d*(Sqrt[b/d]*Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[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[b/d]*Sqrt[2*Pi]*FresnelS[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]*b*Sqrt[c + d*x]*(Cos[3
*(a + b*x)] + 2*b*x*Sin[3*(a + b*x)])))/(96*Sqrt[3]*b^3) - ((b/d)^(3/2)*d^2*(-(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*S
qrt[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]*FresnelC[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]*(-2*b*(c - 5*d*x)*Cos[3*(a + b*x)] + d*(-5 + 12*b^2*x^2
)*Sin[3*(a + b*x)])))/(1152*Sqrt[3]*b^5) - (c^2*(-(Sqrt[2*Pi]*Cos[5*a - (5*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[10/
Pi]*Sqrt[c + d*x]]) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*Sin[5*a - (5*b*c)/d] + 2*Sqrt[5
]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[5*(a + b*x)]))/(160*Sqrt[5]*b*Sqrt[b/d]) - (c*d*(Sqrt[b/d]*Sqrt[2*Pi]*FresnelC[S
qrt[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[b/d]*Sqrt
[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[10/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]*b*Sqrt[c + d*x]*(3*Cos[5*(a + b*x)] + 10*b*x*Sin[5*(a + b*x)])))/(800*Sqrt[5]*b^3) - ((b/d)^(3/2)*
d^2*(-(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*((20*b^2*c^2 - 3*d^2)*Cos[5*a - (5*b*c)/d] + 1
2*b*c*d*Sin[5*a - (5*b*c)/d])) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*(-12*b*c*d*Cos[5*a -
(5*b*c)/d] + (20*b^2*c^2 - 3*d^2)*Sin[5*a - (5*b*c)/d]) + 2*Sqrt[5]*Sqrt[b/d]*d*Sqrt[c + d*x]*(-2*b*(c - 5*d*
x)*Cos[5*(a + b*x)] + d*(-3 + 20*b^2*x^2)*Sin[5*(a + b*x)])))/(3200*Sqrt[5]*b^5)

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Maple [A]  time = 0.051, size = 716, 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)^3*sin(b*x+a)^2,x)

[Out]

2/d*(1/16/b*d*(d*x+c)^(5/2)*sin(1/d*(d*x+c)*b+(a*d-b*c)/d)-5/16/b*d*(-1/2/b*d*(d*x+c)^(3/2)*cos(1/d*(d*x+c)*b+
(a*d-b*c)/d)+3/2/b*d*(1/2/b*d*(d*x+c)^(1/2)*sin(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)*FresnelS(2^(1/2)/Pi^(1/2)/(b/d)^(1/2)*(d*x+c)^(1/2)*b/d)+sin((a*d-b*c)/d)*FresnelC(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)*sin(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)*cos(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)*sin(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)*FresnelS(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)*FresnelC(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)*sin(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)*cos(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)*sin(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)*FresnelS(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)*FresnelC(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 = 2.69643, size = 2940, normalized size = 4.78 \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)^3*sin(b*x+a)^2,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)*cos(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)*cos(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)*cos(
((d*x + c)*b - 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))) - 27*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(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*sqrt(pi)*cos(1/4*pi + 1/2*arc
tan2(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/sqr
t(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*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)) + (sqrt(5)*(125*I*sqrt(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*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))) - 125*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*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)*sin(-3*(b*c - a*d)/d)/abs(d))*erf(sqrt(d*x + c)*s
qrt(3*I*b/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*arct
an2(0, d/sqrt(d^2))))*d^3*abs(b)*cos(-(b*c - a*d)/d)/abs(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)*sin(-(b*c - a*d)/d)/abs(d))*erf(s
qrt(d*x + c)*sqrt(I*b/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)*cos(-(b*c - a*d)/d)/abs(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))) - 675
0*I*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)/ab
s(d))*erf(sqrt(d*x + c)*sqrt(-I*b/d)) + (sqrt(5)*(-125*I*sqrt(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*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 125*sqrt(p
i)*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)*cos(-3*(b*c - a*d)/d)/abs(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)*sin(-3*(b*c - a*d)/d)/abs(d))*e
rf(sqrt(d*x + c)*sqrt(-3*I*b/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))) - 27*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*a
rctan2(0, d/sqrt(d^2))))*d^3*abs(b)*cos(-5*(b*c - a*d)/d)/abs(d) + sqrt(3)*(27*sqrt(pi)*cos(1/4*pi + 1/2*arcta
n2(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*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)) + 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))*sin(5*((d*x + c)*b - b*c + a*d)/d) + 200*(12*s
qrt(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)*sqrt(d*x + c)*d^3*s
qrt(abs(b)/abs(d))*abs(b)/abs(d))*sin(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)/abs(d))*abs(b)/abs
(d))*sin(((d*x + c)*b - b*c + a*d)/d))*abs(d)/(b^3*d*sqrt(abs(b)/abs(d))*abs(b))

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Fricas [A]  time = 0.830458, size = 1397, normalized size = 2.27 \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{S}\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{S}\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{S}\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{C}\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{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 ) + 81 \, \sqrt{10} \pi d^{3} \sqrt{\frac{b}{\pi d}} \operatorname{C}\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 (90 \,{\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )^{5} - 50 \,{\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )^{3} - 300 \,{\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right ) -{\left (120 \, b^{3} d^{2} x^{2} + 240 \, b^{3} c d x + 120 \, b^{3} c^{2} - 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 )^{4} - 428 \, b d^{2} +{\left (60 \, b^{3} d^{2} x^{2} + 120 \, b^{3} c d x + 60 \, b^{3} c^{2} + 11 \, b d^{2}\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)^3*sin(b*x+a)^2,x, algorithm="fricas")

[Out]

-1/432000*(81*sqrt(10)*pi*d^3*sqrt(b/(pi*d))*cos(-5*(b*c - a*d)/d)*fresnel_sin(sqrt(10)*sqrt(d*x + c)*sqrt(b/(
pi*d))) + 625*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))) - 101250*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))) - 101250*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) + 625*sqrt(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
) + 81*sqrt(10)*pi*d^3*sqrt(b/(pi*d))*fresnel_cos(sqrt(10)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-5*(b*c - a*d)/d)
+ 480*(90*(b^2*d^2*x + b^2*c*d)*cos(b*x + a)^5 - 50*(b^2*d^2*x + b^2*c*d)*cos(b*x + a)^3 - 300*(b^2*d^2*x + b
^2*c*d)*cos(b*x + a) - (120*b^3*d^2*x^2 + 240*b^3*c*d*x + 120*b^3*c^2 - 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)^4 - 428*b*d^2 + (60*b^3*d^2*x^2 + 120*b^3*c*d*x + 60*b^3*c^2 + 11*b*d^2)*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)**3*sin(b*x+a)**2,x)

[Out]

Timed out

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Giac [C]  time = 2.12865, size = 4077, normalized size = 6.63 \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)^3*sin(b*x+a)^2,x, algorithm="giac")

[Out]

-1/864000*(60*(9*I*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*I*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) - 450*I*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*I*sqrt(2)*sqrt(pi)*d^2*erf(-1/2*sqrt(2)*s
qrt(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*I*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*I*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) - 90*I*sqrt(d*x + c)*d*e^((5*I*(d*x + c)*b - 5*I*b*c + 5*I*a*d)/d)/b - 150*I*sqrt(d*x + c)*d*e^((3
*I*(d*x + c)*b - 3*I*b*c + 3*I*a*d)/d)/b + 900*I*sqrt(d*x + c)*d*e^((I*(d*x + c)*b - I*b*c + I*a*d)/d)/b - 900
*I*sqrt(d*x + c)*d*e^((-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b + 150*I*sqrt(d*x + c)*d*e^((-3*I*(d*x + c)*b + 3*I
*b*c - 3*I*a*d)/d)/b + 90*I*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*(sqr
t(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/sqr
t(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*(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*(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*(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 + 6750*(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*(-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*(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*(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*(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*(-12*I*(d*x + c)^(5/2)*b^2*d + 24*I*(d*x + c)^(3/2)*b^2*c*d - 12*I*s
qrt(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*(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*(-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*sqrt(10)*sqrt(pi)*(10*I*b*c*d - 3*d^2)*d*erf(-1/2*sqrt(10)*s
qrt(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*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*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*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*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*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*(-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*(-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*(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*(-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*(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*(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