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

Optimal. Leaf size=407 \[ -\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left (4 a-\frac{4 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4096 b^{7/2}}-\frac{15 \sqrt{\pi } d^{5/2} \cos \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{\pi } \sqrt{d}}\right )}{256 b^{7/2}}+\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left (4 a-\frac{4 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{4096 b^{7/2}}+\frac{15 \sqrt{\pi } d^{5/2} \sin \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right )}{256 b^{7/2}}+\frac{15 d^2 \sqrt{c+d x} \cos (2 a+2 b x)}{128 b^3}+\frac{15 d^2 \sqrt{c+d x} \cos (4 a+4 b x)}{2048 b^3}+\frac{5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{32 b^2}+\frac{5 d (c+d x)^{3/2} \sin (4 a+4 b x)}{256 b^2}-\frac{(c+d x)^{5/2} \cos (2 a+2 b x)}{8 b}-\frac{(c+d x)^{5/2} \cos (4 a+4 b x)}{32 b} \]

[Out]

(15*d^2*Sqrt[c + d*x]*Cos[2*a + 2*b*x])/(128*b^3) - ((c + d*x)^(5/2)*Cos[2*a + 2*b*x])/(8*b) + (15*d^2*Sqrt[c
+ d*x]*Cos[4*a + 4*b*x])/(2048*b^3) - ((c + d*x)^(5/2)*Cos[4*a + 4*b*x])/(32*b) - (15*d^(5/2)*Sqrt[Pi/2]*Cos[4
*a - (4*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(4096*b^(7/2)) - (15*d^(5/2)*Sqrt[Pi]*
Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(256*b^(7/2)) + (15*d^(5/2)*Sqrt[
Pi/2]*FresnelS[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[4*a - (4*b*c)/d])/(4096*b^(7/2)) + (15*d^(5/2
)*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(256*b^(7/2)) + (5*d*(
c + d*x)^(3/2)*Sin[2*a + 2*b*x])/(32*b^2) + (5*d*(c + d*x)^(3/2)*Sin[4*a + 4*b*x])/(256*b^2)

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Rubi [A]  time = 0.796576, antiderivative size = 407, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {4406, 3296, 3306, 3305, 3351, 3304, 3352} \[ -\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left (4 a-\frac{4 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4096 b^{7/2}}-\frac{15 \sqrt{\pi } d^{5/2} \cos \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{\pi } \sqrt{d}}\right )}{256 b^{7/2}}+\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left (4 a-\frac{4 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{4096 b^{7/2}}+\frac{15 \sqrt{\pi } d^{5/2} \sin \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right )}{256 b^{7/2}}+\frac{15 d^2 \sqrt{c+d x} \cos (2 a+2 b x)}{128 b^3}+\frac{15 d^2 \sqrt{c+d x} \cos (4 a+4 b x)}{2048 b^3}+\frac{5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{32 b^2}+\frac{5 d (c+d x)^{3/2} \sin (4 a+4 b x)}{256 b^2}-\frac{(c+d x)^{5/2} \cos (2 a+2 b x)}{8 b}-\frac{(c+d x)^{5/2} \cos (4 a+4 b x)}{32 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(15*d^2*Sqrt[c + d*x]*Cos[2*a + 2*b*x])/(128*b^3) - ((c + d*x)^(5/2)*Cos[2*a + 2*b*x])/(8*b) + (15*d^2*Sqrt[c
+ d*x]*Cos[4*a + 4*b*x])/(2048*b^3) - ((c + d*x)^(5/2)*Cos[4*a + 4*b*x])/(32*b) - (15*d^(5/2)*Sqrt[Pi/2]*Cos[4
*a - (4*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(4096*b^(7/2)) - (15*d^(5/2)*Sqrt[Pi]*
Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(256*b^(7/2)) + (15*d^(5/2)*Sqrt[
Pi/2]*FresnelS[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[4*a - (4*b*c)/d])/(4096*b^(7/2)) + (15*d^(5/2
)*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(256*b^(7/2)) + (5*d*(
c + d*x)^(3/2)*Sin[2*a + 2*b*x])/(32*b^2) + (5*d*(c + d*x)^(3/2)*Sin[4*a + 4*b*x])/(256*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 ^3(a+b x) \sin (a+b x) \, dx &=\int \left (\frac{1}{4} (c+d x)^{5/2} \sin (2 a+2 b x)+\frac{1}{8} (c+d x)^{5/2} \sin (4 a+4 b x)\right ) \, dx\\ &=\frac{1}{8} \int (c+d x)^{5/2} \sin (4 a+4 b x) \, dx+\frac{1}{4} \int (c+d x)^{5/2} \sin (2 a+2 b x) \, dx\\ &=-\frac{(c+d x)^{5/2} \cos (2 a+2 b x)}{8 b}-\frac{(c+d x)^{5/2} \cos (4 a+4 b x)}{32 b}+\frac{(5 d) \int (c+d x)^{3/2} \cos (4 a+4 b x) \, dx}{64 b}+\frac{(5 d) \int (c+d x)^{3/2} \cos (2 a+2 b x) \, dx}{16 b}\\ &=-\frac{(c+d x)^{5/2} \cos (2 a+2 b x)}{8 b}-\frac{(c+d x)^{5/2} \cos (4 a+4 b x)}{32 b}+\frac{5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{32 b^2}+\frac{5 d (c+d x)^{3/2} \sin (4 a+4 b x)}{256 b^2}-\frac{\left (15 d^2\right ) \int \sqrt{c+d x} \sin (4 a+4 b x) \, dx}{512 b^2}-\frac{\left (15 d^2\right ) \int \sqrt{c+d x} \sin (2 a+2 b x) \, dx}{64 b^2}\\ &=\frac{15 d^2 \sqrt{c+d x} \cos (2 a+2 b x)}{128 b^3}-\frac{(c+d x)^{5/2} \cos (2 a+2 b x)}{8 b}+\frac{15 d^2 \sqrt{c+d x} \cos (4 a+4 b x)}{2048 b^3}-\frac{(c+d x)^{5/2} \cos (4 a+4 b x)}{32 b}+\frac{5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{32 b^2}+\frac{5 d (c+d x)^{3/2} \sin (4 a+4 b x)}{256 b^2}-\frac{\left (15 d^3\right ) \int \frac{\cos (4 a+4 b x)}{\sqrt{c+d x}} \, dx}{4096 b^3}-\frac{\left (15 d^3\right ) \int \frac{\cos (2 a+2 b x)}{\sqrt{c+d x}} \, dx}{256 b^3}\\ &=\frac{15 d^2 \sqrt{c+d x} \cos (2 a+2 b x)}{128 b^3}-\frac{(c+d x)^{5/2} \cos (2 a+2 b x)}{8 b}+\frac{15 d^2 \sqrt{c+d x} \cos (4 a+4 b x)}{2048 b^3}-\frac{(c+d x)^{5/2} \cos (4 a+4 b x)}{32 b}+\frac{5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{32 b^2}+\frac{5 d (c+d x)^{3/2} \sin (4 a+4 b x)}{256 b^2}-\frac{\left (15 d^3 \cos \left (4 a-\frac{4 b c}{d}\right )\right ) \int \frac{\cos \left (\frac{4 b c}{d}+4 b x\right )}{\sqrt{c+d x}} \, dx}{4096 b^3}-\frac{\left (15 d^3 \cos \left (2 a-\frac{2 b c}{d}\right )\right ) \int \frac{\cos \left (\frac{2 b c}{d}+2 b x\right )}{\sqrt{c+d x}} \, dx}{256 b^3}+\frac{\left (15 d^3 \sin \left (4 a-\frac{4 b c}{d}\right )\right ) \int \frac{\sin \left (\frac{4 b c}{d}+4 b x\right )}{\sqrt{c+d x}} \, dx}{4096 b^3}+\frac{\left (15 d^3 \sin \left (2 a-\frac{2 b c}{d}\right )\right ) \int \frac{\sin \left (\frac{2 b c}{d}+2 b x\right )}{\sqrt{c+d x}} \, dx}{256 b^3}\\ &=\frac{15 d^2 \sqrt{c+d x} \cos (2 a+2 b x)}{128 b^3}-\frac{(c+d x)^{5/2} \cos (2 a+2 b x)}{8 b}+\frac{15 d^2 \sqrt{c+d x} \cos (4 a+4 b x)}{2048 b^3}-\frac{(c+d x)^{5/2} \cos (4 a+4 b x)}{32 b}+\frac{5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{32 b^2}+\frac{5 d (c+d x)^{3/2} \sin (4 a+4 b x)}{256 b^2}-\frac{\left (15 d^2 \cos \left (4 a-\frac{4 b c}{d}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{4 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{2048 b^3}-\frac{\left (15 d^2 \cos \left (2 a-\frac{2 b c}{d}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{2 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{128 b^3}+\frac{\left (15 d^2 \sin \left (4 a-\frac{4 b c}{d}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{4 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{2048 b^3}+\frac{\left (15 d^2 \sin \left (2 a-\frac{2 b c}{d}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{2 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{128 b^3}\\ &=\frac{15 d^2 \sqrt{c+d x} \cos (2 a+2 b x)}{128 b^3}-\frac{(c+d x)^{5/2} \cos (2 a+2 b x)}{8 b}+\frac{15 d^2 \sqrt{c+d x} \cos (4 a+4 b x)}{2048 b^3}-\frac{(c+d x)^{5/2} \cos (4 a+4 b x)}{32 b}-\frac{15 d^{5/2} \sqrt{\frac{\pi }{2}} \cos \left (4 a-\frac{4 b c}{d}\right ) C\left (\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{4096 b^{7/2}}-\frac{15 d^{5/2} \sqrt{\pi } \cos \left (2 a-\frac{2 b c}{d}\right ) C\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right )}{256 b^{7/2}}+\frac{15 d^{5/2} \sqrt{\frac{\pi }{2}} S\left (\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right ) \sin \left (4 a-\frac{4 b c}{d}\right )}{4096 b^{7/2}}+\frac{15 d^{5/2} \sqrt{\pi } S\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right ) \sin \left (2 a-\frac{2 b c}{d}\right )}{256 b^{7/2}}+\frac{5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{32 b^2}+\frac{5 d (c+d x)^{3/2} \sin (4 a+4 b x)}{256 b^2}\\ \end{align*}

Mathematica [A]  time = 13.6521, size = 550, normalized size = 1.35 \[ \frac{-1024 b^3 c^2 \sqrt{c+d x} \cos (2 (a+b x))-256 b^3 c^2 \sqrt{c+d x} \cos (4 (a+b x))-1024 b^3 d^2 x^2 \sqrt{c+d x} \cos (2 (a+b x))-256 b^3 d^2 x^2 \sqrt{c+d x} \cos (4 (a+b x))+1280 b^2 d^2 x \sqrt{c+d x} \sin (2 (a+b x))+160 b^2 d^2 x \sqrt{c+d x} \sin (4 (a+b x))+1280 b^2 c d \sqrt{c+d x} \sin (2 (a+b x))+160 b^2 c d \sqrt{c+d x} \sin (4 (a+b x))-2048 b^3 c d x \sqrt{c+d x} \cos (2 (a+b x))-512 b^3 c d x \sqrt{c+d x} \cos (4 (a+b x))-15 \sqrt{2 \pi } d^3 \sqrt{\frac{b}{d}} \cos \left (4 a-\frac{4 b c}{d}\right ) \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\frac{b}{d}} \sqrt{c+d x}\right )-480 \sqrt{\pi } d^3 \sqrt{\frac{b}{d}} \cos \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right )+15 \sqrt{2 \pi } d^3 \sqrt{\frac{b}{d}} \sin \left (4 a-\frac{4 b c}{d}\right ) S\left (2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right )+480 \sqrt{\pi } d^3 \sqrt{\frac{b}{d}} \sin \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right )+960 b d^2 \sqrt{c+d x} \cos (2 (a+b x))+60 b d^2 \sqrt{c+d x} \cos (4 (a+b x))}{8192 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1024*b^3*c^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] + 960*b*d^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 2048*b^3*c*d*x*Sqrt
[c + d*x]*Cos[2*(a + b*x)] - 1024*b^3*d^2*x^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 256*b^3*c^2*Sqrt[c + d*x]*Cos[4
*(a + b*x)] + 60*b*d^2*Sqrt[c + d*x]*Cos[4*(a + b*x)] - 512*b^3*c*d*x*Sqrt[c + d*x]*Cos[4*(a + b*x)] - 256*b^3
*d^2*x^2*Sqrt[c + d*x]*Cos[4*(a + b*x)] - 15*Sqrt[b/d]*d^3*Sqrt[2*Pi]*Cos[4*a - (4*b*c)/d]*FresnelC[2*Sqrt[b/d
]*Sqrt[2/Pi]*Sqrt[c + d*x]] - 480*Sqrt[b/d]*d^3*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d
*x])/Sqrt[Pi]] + 15*Sqrt[b/d]*d^3*Sqrt[2*Pi]*FresnelS[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[4*a - (4*b*c)/
d] + 480*Sqrt[b/d]*d^3*Sqrt[Pi]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d] + 1280*b^2
*c*d*Sqrt[c + d*x]*Sin[2*(a + b*x)] + 1280*b^2*d^2*x*Sqrt[c + d*x]*Sin[2*(a + b*x)] + 160*b^2*c*d*Sqrt[c + d*x
]*Sin[4*(a + b*x)] + 160*b^2*d^2*x*Sqrt[c + d*x]*Sin[4*(a + b*x)])/(8192*b^4)

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Maple [A]  time = 0.035, size = 470, normalized size = 1.2 \begin{align*} 2\,{\frac{1}{d} \left ( -1/16\,{\frac{d \left ( dx+c \right ) ^{5/2}}{b}\cos \left ( 2\,{\frac{ \left ( dx+c \right ) b}{d}}+2\,{\frac{ad-bc}{d}} \right ) }+{\frac{5\,d}{16\,b} \left ( 1/4\,{\frac{d \left ( dx+c \right ) ^{3/2}}{b}\sin \left ( 2\,{\frac{ \left ( dx+c \right ) b}{d}}+2\,{\frac{ad-bc}{d}} \right ) }-3/4\,{\frac{d}{b} \left ( -1/4\,{\frac{d\sqrt{dx+c}}{b}\cos \left ( 2\,{\frac{ \left ( dx+c \right ) b}{d}}+2\,{\frac{ad-bc}{d}} \right ) }+1/8\,{\frac{d\sqrt{\pi }}{b} \left ( \cos \left ( 2\,{\frac{ad-bc}{d}} \right ){\it FresnelC} \left ( 2\,{\frac{\sqrt{dx+c}b}{d\sqrt{\pi }}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) -\sin \left ( 2\,{\frac{ad-bc}{d}} \right ){\it FresnelS} \left ( 2\,{\frac{\sqrt{dx+c}b}{d\sqrt{\pi }}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) } \right ) }-{\frac{d \left ( dx+c \right ) ^{5/2}}{64\,b}\cos \left ( 4\,{\frac{ \left ( dx+c \right ) b}{d}}+4\,{\frac{ad-bc}{d}} \right ) }+{\frac{5\,d}{64\,b} \left ( 1/8\,{\frac{d \left ( dx+c \right ) ^{3/2}}{b}\sin \left ( 4\,{\frac{ \left ( dx+c \right ) b}{d}}+4\,{\frac{ad-bc}{d}} \right ) }-3/8\,{\frac{d}{b} \left ( -1/8\,{\frac{d\sqrt{dx+c}}{b}\cos \left ( 4\,{\frac{ \left ( dx+c \right ) b}{d}}+4\,{\frac{ad-bc}{d}} \right ) }+1/32\,{\frac{d\sqrt{2}\sqrt{\pi }}{b} \left ( \cos \left ( 4\,{\frac{ad-bc}{d}} \right ){\it FresnelC} \left ( 2\,{\frac{\sqrt{2}\sqrt{dx+c}b}{d\sqrt{\pi }}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) -\sin \left ( 4\,{\frac{ad-bc}{d}} \right ){\it FresnelS} \left ( 2\,{\frac{\sqrt{2}\sqrt{dx+c}b}{d\sqrt{\pi }}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) } \right ) } \right ) } \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),x)

[Out]

2/d*(-1/16/b*d*(d*x+c)^(5/2)*cos(2/d*(d*x+c)*b+2*(a*d-b*c)/d)+5/16/b*d*(1/4/b*d*(d*x+c)^(3/2)*sin(2/d*(d*x+c)*
b+2*(a*d-b*c)/d)-3/4/b*d*(-1/4/b*d*(d*x+c)^(1/2)*cos(2/d*(d*x+c)*b+2*(a*d-b*c)/d)+1/8/b*d*Pi^(1/2)/(b/d)^(1/2)
*(cos(2*(a*d-b*c)/d)*FresnelC(2/Pi^(1/2)/(b/d)^(1/2)*(d*x+c)^(1/2)*b/d)-sin(2*(a*d-b*c)/d)*FresnelS(2/Pi^(1/2)
/(b/d)^(1/2)*(d*x+c)^(1/2)*b/d))))-1/64/b*d*(d*x+c)^(5/2)*cos(4/d*(d*x+c)*b+4*(a*d-b*c)/d)+5/64/b*d*(1/8/b*d*(
d*x+c)^(3/2)*sin(4/d*(d*x+c)*b+4*(a*d-b*c)/d)-3/8/b*d*(-1/8/b*d*(d*x+c)^(1/2)*cos(4/d*(d*x+c)*b+4*(a*d-b*c)/d)
+1/32/b*d*2^(1/2)*Pi^(1/2)/(b/d)^(1/2)*(cos(4*(a*d-b*c)/d)*FresnelC(2*2^(1/2)/Pi^(1/2)/(b/d)^(1/2)*(d*x+c)^(1/
2)*b/d)-sin(4*(a*d-b*c)/d)*FresnelS(2*2^(1/2)/Pi^(1/2)/(b/d)^(1/2)*(d*x+c)^(1/2)*b/d)))))

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Maxima [C]  time = 2.31125, size = 1879, normalized size = 4.62 \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),x, algorithm="maxima")

[Out]

1/65536*sqrt(2)*(640*sqrt(2)*(d*x + c)^(3/2)*b*d^2*abs(b)*sin(4*((d*x + c)*b - b*c + a*d)/d)/abs(d) + 5120*sqr
t(2)*(d*x + c)^(3/2)*b*d^2*abs(b)*sin(2*((d*x + c)*b - b*c + a*d)/d)/abs(d) - 16*(64*sqrt(2)*(d*x + c)^(5/2)*b
^2*d*abs(b)/abs(d) - 15*sqrt(2)*sqrt(d*x + c)*d^3*abs(b)/abs(d))*cos(4*((d*x + c)*b - b*c + a*d)/d) - 256*(16*
sqrt(2)*(d*x + c)^(5/2)*b^2*d*abs(b)/abs(d) - 15*sqrt(2)*sqrt(d*x + c)*d^3*abs(b)/abs(d))*cos(2*((d*x + c)*b -
b*c + a*d)/d) - ((480*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 480*sqrt(pi)*c
os(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 480*I*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b)
+ 1/2*arctan2(0, d/sqrt(d^2))) + 480*I*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2)))
)*d^3*sqrt(abs(b)/abs(d))*cos(-2*(b*c - a*d)/d) - (480*I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2
(0, d/sqrt(d^2))) + 480*I*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 480*sqrt(p
i)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 480*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b
) + 1/2*arctan2(0, d/sqrt(d^2))))*d^3*sqrt(abs(b)/abs(d))*sin(-2*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(2*I*b/
d)) - (sqrt(2)*(15*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 15*sqrt(pi)*cos(-1
/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 15*I*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*
arctan2(0, d/sqrt(d^2))) + 15*I*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^3*s
qrt(abs(b)/abs(d))*cos(-4*(b*c - a*d)/d) - sqrt(2)*(15*I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2
(0, d/sqrt(d^2))) + 15*I*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 15*sqrt(pi)
*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 15*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) +
1/2*arctan2(0, d/sqrt(d^2))))*d^3*sqrt(abs(b)/abs(d))*sin(-4*(b*c - a*d)/d))*erf(2*sqrt(d*x + c)*sqrt(I*b/d))
- (sqrt(2)*(15*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 15*sqrt(pi)*cos(-1/4*
pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 15*I*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arc
tan2(0, d/sqrt(d^2))) - 15*I*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^3*sqrt
(abs(b)/abs(d))*cos(-4*(b*c - a*d)/d) - sqrt(2)*(-15*I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0
, d/sqrt(d^2))) - 15*I*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 15*sqrt(pi)*s
in(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 15*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1
/2*arctan2(0, d/sqrt(d^2))))*d^3*sqrt(abs(b)/abs(d))*sin(-4*(b*c - a*d)/d))*erf(2*sqrt(d*x + c)*sqrt(-I*b/d))
- ((480*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 480*sqrt(pi)*cos(-1/4*pi + 1/
2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 480*I*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0
, d/sqrt(d^2))) - 480*I*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^3*sqrt(abs(
b)/abs(d))*cos(-2*(b*c - a*d)/d) - (-480*I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2
))) - 480*I*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 480*sqrt(pi)*sin(1/4*pi
+ 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 480*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan
2(0, d/sqrt(d^2))))*d^3*sqrt(abs(b)/abs(d))*sin(-2*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-2*I*b/d)))*abs(d)/(
b^3*d*abs(b))

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Fricas [A]  time = 0.698365, size = 944, normalized size = 2.32 \begin{align*} -\frac{15 \, \sqrt{2} \pi d^{3} \sqrt{\frac{b}{\pi d}} \cos \left (-\frac{4 \,{\left (b c - a d\right )}}{d}\right ) \operatorname{C}\left (2 \, \sqrt{2} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) - 15 \, \sqrt{2} \pi d^{3} \sqrt{\frac{b}{\pi d}} \operatorname{S}\left (2 \, \sqrt{2} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) \sin \left (-\frac{4 \,{\left (b c - a d\right )}}{d}\right ) + 480 \, \pi d^{3} \sqrt{\frac{b}{\pi d}} \cos \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) \operatorname{C}\left (2 \, \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) - 480 \, \pi d^{3} \sqrt{\frac{b}{\pi d}} \operatorname{S}\left (2 \, \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) \sin \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) - 4 \,{\left (192 \, b^{3} d^{2} x^{2} + 384 \, b^{3} c d x + 192 \, b^{3} c^{2} + 360 \, b d^{2} \cos \left (b x + a\right )^{2} - 8 \,{\left (64 \, b^{3} d^{2} x^{2} + 128 \, b^{3} c d x + 64 \, b^{3} c^{2} - 15 \, b d^{2}\right )} \cos \left (b x + a\right )^{4} - 225 \, b d^{2} + 160 \,{\left (2 \,{\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )^{3} + 3 \,{\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )\right )} \sqrt{d x + c}}{8192 \, 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),x, algorithm="fricas")

[Out]

-1/8192*(15*sqrt(2)*pi*d^3*sqrt(b/(pi*d))*cos(-4*(b*c - a*d)/d)*fresnel_cos(2*sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi
*d))) - 15*sqrt(2)*pi*d^3*sqrt(b/(pi*d))*fresnel_sin(2*sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-4*(b*c - a*d
)/d) + 480*pi*d^3*sqrt(b/(pi*d))*cos(-2*(b*c - a*d)/d)*fresnel_cos(2*sqrt(d*x + c)*sqrt(b/(pi*d))) - 480*pi*d^
3*sqrt(b/(pi*d))*fresnel_sin(2*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-2*(b*c - a*d)/d) - 4*(192*b^3*d^2*x^2 + 384*
b^3*c*d*x + 192*b^3*c^2 + 360*b*d^2*cos(b*x + a)^2 - 8*(64*b^3*d^2*x^2 + 128*b^3*c*d*x + 64*b^3*c^2 - 15*b*d^2
)*cos(b*x + a)^4 - 225*b*d^2 + 160*(2*(b^2*d^2*x + b^2*c*d)*cos(b*x + a)^3 + 3*(b^2*d^2*x + b^2*c*d)*cos(b*x +
a))*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),x)

[Out]

Timed out

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Giac [C]  time = 1.77497, size = 2680, normalized size = 6.58 \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),x, algorithm="giac")

[Out]

-1/16384*(64*(sqrt(2)*sqrt(pi)*d^2*erf(-sqrt(2)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((4*I*b
*c - 4*I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b) + sqrt(2)*sqrt(pi)*d^2*erf(-sqrt(2)*sqrt(b*d)*sqrt(d*
x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-4*I*b*c + 4*I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b) + 8*s
qrt(pi)*d^2*erf(-sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((2*I*b*c - 2*I*a*d)/d)/(sqrt(b*d)*(I*
b*d/sqrt(b^2*d^2) + 1)*b) + 8*sqrt(pi)*d^2*erf(-sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-2*I
*b*c + 2*I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b) + 4*sqrt(d*x + c)*d*e^((4*I*(d*x + c)*b - 4*I*b*c
+ 4*I*a*d)/d)/b + 16*sqrt(d*x + c)*d*e^((2*I*(d*x + c)*b - 2*I*b*c + 2*I*a*d)/d)/b + 16*sqrt(d*x + c)*d*e^((-2
*I*(d*x + c)*b + 2*I*b*c - 2*I*a*d)/d)/b + 4*sqrt(d*x + c)*d*e^((-4*I*(d*x + c)*b + 4*I*b*c - 4*I*a*d)/d)/b)*c
^2 + d^2*((I*sqrt(2)*sqrt(pi)*(-64*I*b^2*c^2*d + 48*b*c*d^2 + 15*I*d^3)*d*erf(-sqrt(2)*sqrt(b*d)*sqrt(d*x + c)
*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((4*I*b*c - 4*I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b^3) - 4*I*(64*I*
(d*x + c)^(5/2)*b^2*d - 128*I*(d*x + c)^(3/2)*b^2*c*d + 64*I*sqrt(d*x + c)*b^2*c^2*d + 40*(d*x + c)^(3/2)*b*d^
2 - 48*sqrt(d*x + c)*b*c*d^2 - 15*I*sqrt(d*x + c)*d^3)*e^((-4*I*(d*x + c)*b + 4*I*b*c - 4*I*a*d)/d)/b^3)/d^2 +
(I*sqrt(2)*sqrt(pi)*(-64*I*b^2*c^2*d - 48*b*c*d^2 + 15*I*d^3)*d*erf(-sqrt(2)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/
sqrt(b^2*d^2) + 1)/d)*e^((-4*I*b*c + 4*I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b^3) - 4*I*(64*I*(d*x +
c)^(5/2)*b^2*d - 128*I*(d*x + c)^(3/2)*b^2*c*d + 64*I*sqrt(d*x + c)*b^2*c^2*d - 40*(d*x + c)^(3/2)*b*d^2 + 48
*sqrt(d*x + c)*b*c*d^2 - 15*I*sqrt(d*x + c)*d^3)*e^((4*I*(d*x + c)*b - 4*I*b*c + 4*I*a*d)/d)/b^3)/d^2 + 32*(I*
sqrt(pi)*(-16*I*b^2*c^2*d + 24*b*c*d^2 + 15*I*d^3)*d*erf(-sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)
*e^((2*I*b*c - 2*I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b^3) - 2*I*(16*I*(d*x + c)^(5/2)*b^2*d - 32*I*
(d*x + c)^(3/2)*b^2*c*d + 16*I*sqrt(d*x + c)*b^2*c^2*d + 20*(d*x + c)^(3/2)*b*d^2 - 24*sqrt(d*x + c)*b*c*d^2 -
15*I*sqrt(d*x + c)*d^3)*e^((-2*I*(d*x + c)*b + 2*I*b*c - 2*I*a*d)/d)/b^3)/d^2 + 32*(I*sqrt(pi)*(-16*I*b^2*c^2
*d - 24*b*c*d^2 + 15*I*d^3)*d*erf(-sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-2*I*b*c + 2*I*a*
d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b^3) - 2*I*(16*I*(d*x + c)^(5/2)*b^2*d - 32*I*(d*x + c)^(3/2)*b^2*
c*d + 16*I*sqrt(d*x + c)*b^2*c^2*d - 20*(d*x + c)^(3/2)*b*d^2 + 24*sqrt(d*x + c)*b*c*d^2 - 15*I*sqrt(d*x + c)*
d^3)*e^((2*I*(d*x + c)*b - 2*I*b*c + 2*I*a*d)/d)/b^3)/d^2) + 16*(I*sqrt(2)*sqrt(pi)*(8*I*b*c*d - 3*d^2)*d*erf(
-sqrt(2)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((4*I*b*c - 4*I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt
(b^2*d^2) + 1)*b^2) + I*sqrt(2)*sqrt(pi)*(8*I*b*c*d + 3*d^2)*d*erf(-sqrt(2)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sq
rt(b^2*d^2) + 1)/d)*e^((-4*I*b*c + 4*I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b^2) + 16*I*sqrt(pi)*(4*I
*b*c*d - 3*d^2)*d*erf(-sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((2*I*b*c - 2*I*a*d)/d)/(sqrt(b*
d)*(I*b*d/sqrt(b^2*d^2) + 1)*b^2) + 16*I*sqrt(pi)*(4*I*b*c*d + 3*d^2)*d*erf(-sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/s
qrt(b^2*d^2) + 1)/d)*e^((-2*I*b*c + 2*I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b^2) - 4*I*(8*I*(d*x + c
)^(3/2)*b*d - 8*I*sqrt(d*x + c)*b*c*d - 3*sqrt(d*x + c)*d^2)*e^((4*I*(d*x + c)*b - 4*I*b*c + 4*I*a*d)/d)/b^2 -
32*I*(4*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^((2*I*(d*x + c)*b - 2*I*b*c
+ 2*I*a*d)/d)/b^2 - 32*I*(4*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^((-2*I*(d
*x + c)*b + 2*I*b*c - 2*I*a*d)/d)/b^2 - 4*I*(8*I*(d*x + c)^(3/2)*b*d - 8*I*sqrt(d*x + c)*b*c*d + 3*sqrt(d*x +
c)*d^2)*e^((-4*I*(d*x + c)*b + 4*I*b*c - 4*I*a*d)/d)/b^2)*c)/d