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

Optimal. Leaf size=407 \[ \frac{5 \sqrt{\frac{\pi }{3}} d^{5/2} \cos \left (6 a-\frac{6 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{3}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{18432 b^{7/2}}-\frac{45 \sqrt{\pi } d^{5/2} \cos \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{\pi } \sqrt{d}}\right )}{2048 b^{7/2}}-\frac{5 \sqrt{\frac{\pi }{3}} d^{5/2} \sin \left (6 a-\frac{6 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{18432 b^{7/2}}+\frac{45 \sqrt{\pi } d^{5/2} \sin \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right )}{2048 b^{7/2}}+\frac{45 d^2 \sqrt{c+d x} \cos (2 a+2 b x)}{1024 b^3}-\frac{5 d^2 \sqrt{c+d x} \cos (6 a+6 b x)}{9216 b^3}+\frac{15 d (c+d x)^{3/2} \sin (2 a+2 b x)}{256 b^2}-\frac{5 d (c+d x)^{3/2} \sin (6 a+6 b x)}{2304 b^2}-\frac{3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}+\frac{(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b} \]

[Out]

(45*d^2*Sqrt[c + d*x]*Cos[2*a + 2*b*x])/(1024*b^3) - (3*(c + d*x)^(5/2)*Cos[2*a + 2*b*x])/(64*b) - (5*d^2*Sqrt
[c + d*x]*Cos[6*a + 6*b*x])/(9216*b^3) + ((c + d*x)^(5/2)*Cos[6*a + 6*b*x])/(192*b) + (5*d^(5/2)*Sqrt[Pi/3]*Co
s[6*a - (6*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(18432*b^(7/2)) - (45*d^(5/2)*Sqrt[
Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(2048*b^(7/2)) - (5*d^(5/2)*S
qrt[Pi/3]*FresnelS[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[6*a - (6*b*c)/d])/(18432*b^(7/2)) + (45*d
^(5/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(2048*b^(7/2)) +
(15*d*(c + d*x)^(3/2)*Sin[2*a + 2*b*x])/(256*b^2) - (5*d*(c + d*x)^(3/2)*Sin[6*a + 6*b*x])/(2304*b^2)

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Rubi [A]  time = 0.896867, antiderivative size = 407, normalized size of antiderivative = 1., number of steps used = 18, 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} \[ \frac{5 \sqrt{\frac{\pi }{3}} d^{5/2} \cos \left (6 a-\frac{6 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{3}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{18432 b^{7/2}}-\frac{45 \sqrt{\pi } d^{5/2} \cos \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{\pi } \sqrt{d}}\right )}{2048 b^{7/2}}-\frac{5 \sqrt{\frac{\pi }{3}} d^{5/2} \sin \left (6 a-\frac{6 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{18432 b^{7/2}}+\frac{45 \sqrt{\pi } d^{5/2} \sin \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right )}{2048 b^{7/2}}+\frac{45 d^2 \sqrt{c+d x} \cos (2 a+2 b x)}{1024 b^3}-\frac{5 d^2 \sqrt{c+d x} \cos (6 a+6 b x)}{9216 b^3}+\frac{15 d (c+d x)^{3/2} \sin (2 a+2 b x)}{256 b^2}-\frac{5 d (c+d x)^{3/2} \sin (6 a+6 b x)}{2304 b^2}-\frac{3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}+\frac{(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(45*d^2*Sqrt[c + d*x]*Cos[2*a + 2*b*x])/(1024*b^3) - (3*(c + d*x)^(5/2)*Cos[2*a + 2*b*x])/(64*b) - (5*d^2*Sqrt
[c + d*x]*Cos[6*a + 6*b*x])/(9216*b^3) + ((c + d*x)^(5/2)*Cos[6*a + 6*b*x])/(192*b) + (5*d^(5/2)*Sqrt[Pi/3]*Co
s[6*a - (6*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(18432*b^(7/2)) - (45*d^(5/2)*Sqrt[
Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(2048*b^(7/2)) - (5*d^(5/2)*S
qrt[Pi/3]*FresnelS[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[6*a - (6*b*c)/d])/(18432*b^(7/2)) + (45*d
^(5/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(2048*b^(7/2)) +
(15*d*(c + d*x)^(3/2)*Sin[2*a + 2*b*x])/(256*b^2) - (5*d*(c + d*x)^(3/2)*Sin[6*a + 6*b*x])/(2304*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 ^3(a+b x) \, dx &=\int \left (\frac{3}{32} (c+d x)^{5/2} \sin (2 a+2 b x)-\frac{1}{32} (c+d x)^{5/2} \sin (6 a+6 b x)\right ) \, dx\\ &=-\left (\frac{1}{32} \int (c+d x)^{5/2} \sin (6 a+6 b x) \, dx\right )+\frac{3}{32} \int (c+d x)^{5/2} \sin (2 a+2 b x) \, dx\\ &=-\frac{3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}+\frac{(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b}-\frac{(5 d) \int (c+d x)^{3/2} \cos (6 a+6 b x) \, dx}{384 b}+\frac{(15 d) \int (c+d x)^{3/2} \cos (2 a+2 b x) \, dx}{128 b}\\ &=-\frac{3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}+\frac{(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b}+\frac{15 d (c+d x)^{3/2} \sin (2 a+2 b x)}{256 b^2}-\frac{5 d (c+d x)^{3/2} \sin (6 a+6 b x)}{2304 b^2}+\frac{\left (5 d^2\right ) \int \sqrt{c+d x} \sin (6 a+6 b x) \, dx}{1536 b^2}-\frac{\left (45 d^2\right ) \int \sqrt{c+d x} \sin (2 a+2 b x) \, dx}{512 b^2}\\ &=\frac{45 d^2 \sqrt{c+d x} \cos (2 a+2 b x)}{1024 b^3}-\frac{3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}-\frac{5 d^2 \sqrt{c+d x} \cos (6 a+6 b x)}{9216 b^3}+\frac{(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b}+\frac{15 d (c+d x)^{3/2} \sin (2 a+2 b x)}{256 b^2}-\frac{5 d (c+d x)^{3/2} \sin (6 a+6 b x)}{2304 b^2}+\frac{\left (5 d^3\right ) \int \frac{\cos (6 a+6 b x)}{\sqrt{c+d x}} \, dx}{18432 b^3}-\frac{\left (45 d^3\right ) \int \frac{\cos (2 a+2 b x)}{\sqrt{c+d x}} \, dx}{2048 b^3}\\ &=\frac{45 d^2 \sqrt{c+d x} \cos (2 a+2 b x)}{1024 b^3}-\frac{3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}-\frac{5 d^2 \sqrt{c+d x} \cos (6 a+6 b x)}{9216 b^3}+\frac{(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b}+\frac{15 d (c+d x)^{3/2} \sin (2 a+2 b x)}{256 b^2}-\frac{5 d (c+d x)^{3/2} \sin (6 a+6 b x)}{2304 b^2}+\frac{\left (5 d^3 \cos \left (6 a-\frac{6 b c}{d}\right )\right ) \int \frac{\cos \left (\frac{6 b c}{d}+6 b x\right )}{\sqrt{c+d x}} \, dx}{18432 b^3}-\frac{\left (45 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}{2048 b^3}-\frac{\left (5 d^3 \sin \left (6 a-\frac{6 b c}{d}\right )\right ) \int \frac{\sin \left (\frac{6 b c}{d}+6 b x\right )}{\sqrt{c+d x}} \, dx}{18432 b^3}+\frac{\left (45 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}{2048 b^3}\\ &=\frac{45 d^2 \sqrt{c+d x} \cos (2 a+2 b x)}{1024 b^3}-\frac{3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}-\frac{5 d^2 \sqrt{c+d x} \cos (6 a+6 b x)}{9216 b^3}+\frac{(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b}+\frac{15 d (c+d x)^{3/2} \sin (2 a+2 b x)}{256 b^2}-\frac{5 d (c+d x)^{3/2} \sin (6 a+6 b x)}{2304 b^2}+\frac{\left (5 d^2 \cos \left (6 a-\frac{6 b c}{d}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{6 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{9216 b^3}-\frac{\left (45 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 )}{1024 b^3}-\frac{\left (5 d^2 \sin \left (6 a-\frac{6 b c}{d}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{6 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{9216 b^3}+\frac{\left (45 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 )}{1024 b^3}\\ &=\frac{45 d^2 \sqrt{c+d x} \cos (2 a+2 b x)}{1024 b^3}-\frac{3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}-\frac{5 d^2 \sqrt{c+d x} \cos (6 a+6 b x)}{9216 b^3}+\frac{(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b}+\frac{5 d^{5/2} \sqrt{\frac{\pi }{3}} \cos \left (6 a-\frac{6 b c}{d}\right ) C\left (\frac{2 \sqrt{b} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{18432 b^{7/2}}-\frac{45 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 )}{2048 b^{7/2}}-\frac{5 d^{5/2} \sqrt{\frac{\pi }{3}} S\left (\frac{2 \sqrt{b} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right ) \sin \left (6 a-\frac{6 b c}{d}\right )}{18432 b^{7/2}}+\frac{45 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 )}{2048 b^{7/2}}+\frac{15 d (c+d x)^{3/2} \sin (2 a+2 b x)}{256 b^2}-\frac{5 d (c+d x)^{3/2} \sin (6 a+6 b x)}{2304 b^2}\\ \end{align*}

Mathematica [A]  time = 5.34667, size = 550, normalized size = 1.35 \[ \frac{-2592 b^3 c^2 \sqrt{c+d x} \cos (2 (a+b x))+288 b^3 c^2 \sqrt{c+d x} \cos (6 (a+b x))-2592 b^3 d^2 x^2 \sqrt{c+d x} \cos (2 (a+b x))+288 b^3 d^2 x^2 \sqrt{c+d x} \cos (6 (a+b x))+3240 b^2 d^2 x \sqrt{c+d x} \sin (2 (a+b x))-120 b^2 d^2 x \sqrt{c+d x} \sin (6 (a+b x))+3240 b^2 c d \sqrt{c+d x} \sin (2 (a+b x))-120 b^2 c d \sqrt{c+d x} \sin (6 (a+b x))-5184 b^3 c d x \sqrt{c+d x} \cos (2 (a+b x))+576 b^3 c d x \sqrt{c+d x} \cos (6 (a+b x))+5 \sqrt{3 \pi } d^3 \sqrt{\frac{b}{d}} \cos \left (6 a-\frac{6 b c}{d}\right ) \text{FresnelC}\left (2 \sqrt{\frac{3}{\pi }} \sqrt{\frac{b}{d}} \sqrt{c+d x}\right )-1215 \sqrt{\pi } d^3 \sqrt{\frac{b}{d}} \cos \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right )-5 \sqrt{3 \pi } d^3 \sqrt{\frac{b}{d}} \sin \left (6 a-\frac{6 b c}{d}\right ) S\left (2 \sqrt{\frac{b}{d}} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}\right )+1215 \sqrt{\pi } d^3 \sqrt{\frac{b}{d}} \sin \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right )+2430 b d^2 \sqrt{c+d x} \cos (2 (a+b x))-30 b d^2 \sqrt{c+d x} \cos (6 (a+b x))}{55296 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2592*b^3*c^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] + 2430*b*d^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 5184*b^3*c*d*x*Sqr
t[c + d*x]*Cos[2*(a + b*x)] - 2592*b^3*d^2*x^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] + 288*b^3*c^2*Sqrt[c + d*x]*Cos[
6*(a + b*x)] - 30*b*d^2*Sqrt[c + d*x]*Cos[6*(a + b*x)] + 576*b^3*c*d*x*Sqrt[c + d*x]*Cos[6*(a + b*x)] + 288*b^
3*d^2*x^2*Sqrt[c + d*x]*Cos[6*(a + b*x)] + 5*Sqrt[b/d]*d^3*Sqrt[3*Pi]*Cos[6*a - (6*b*c)/d]*FresnelC[2*Sqrt[b/d
]*Sqrt[3/Pi]*Sqrt[c + d*x]] - 1215*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]] - 5*Sqrt[b/d]*d^3*Sqrt[3*Pi]*FresnelS[2*Sqrt[b/d]*Sqrt[3/Pi]*Sqrt[c + d*x]]*Sin[6*a - (6*b*c)/
d] + 1215*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] + 3240*b^
2*c*d*Sqrt[c + d*x]*Sin[2*(a + b*x)] + 3240*b^2*d^2*x*Sqrt[c + d*x]*Sin[2*(a + b*x)] - 120*b^2*c*d*Sqrt[c + d*
x]*Sin[6*(a + b*x)] - 120*b^2*d^2*x*Sqrt[c + d*x]*Sin[6*(a + b*x)])/(55296*b^4)

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Maple [A]  time = 0.043, size = 477, normalized size = 1.2 \begin{align*} 2\,{\frac{1}{d} \left ( -{\frac{3\,d \left ( dx+c \right ) ^{5/2}}{128\,b}\cos \left ( 2\,{\frac{ \left ( dx+c \right ) b}{d}}+2\,{\frac{ad-bc}{d}} \right ) }+{\frac{15\,d}{128\,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}}{384\,b}\cos \left ( 6\,{\frac{ \left ( dx+c \right ) b}{d}}+6\,{\frac{ad-bc}{d}} \right ) }-{\frac{5\,d}{384\,b} \left ( 1/12\,{\frac{d \left ( dx+c \right ) ^{3/2}}{b}\sin \left ( 6\,{\frac{ \left ( dx+c \right ) b}{d}}+6\,{\frac{ad-bc}{d}} \right ) }-1/4\,{\frac{d}{b} \left ( -1/12\,{\frac{d\sqrt{dx+c}}{b}\cos \left ( 6\,{\frac{ \left ( dx+c \right ) b}{d}}+6\,{\frac{ad-bc}{d}} \right ) }+{\frac{d\sqrt{2}\sqrt{\pi }\sqrt{6}}{144\,b} \left ( \cos \left ( 6\,{\frac{ad-bc}{d}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{6}\sqrt{dx+c}b}{d\sqrt{\pi }}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) -\sin \left ( 6\,{\frac{ad-bc}{d}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{6}\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)^3,x)

[Out]

2/d*(-3/128/b*d*(d*x+c)^(5/2)*cos(2/d*(d*x+c)*b+2*(a*d-b*c)/d)+15/128/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))))+1/384/b*d*(d*x+c)^(5/2)*cos(6/d*(d*x+c)*b+6*(a*d-b*c)/d)-5/384/b*d*(1/12
/b*d*(d*x+c)^(3/2)*sin(6/d*(d*x+c)*b+6*(a*d-b*c)/d)-1/4/b*d*(-1/12/b*d*(d*x+c)^(1/2)*cos(6/d*(d*x+c)*b+6*(a*d-
b*c)/d)+1/144/b*d*2^(1/2)*Pi^(1/2)*6^(1/2)/(b/d)^(1/2)*(cos(6*(a*d-b*c)/d)*FresnelC(2^(1/2)/Pi^(1/2)*6^(1/2)/(
b/d)^(1/2)*(d*x+c)^(1/2)*b/d)-sin(6*(a*d-b*c)/d)*FresnelS(2^(1/2)/Pi^(1/2)*6^(1/2)/(b/d)^(1/2)*(d*x+c)^(1/2)*b
/d)))))

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Maxima [C]  time = 2.24806, size = 1916, normalized size = 4.71 \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)^3,x, algorithm="maxima")

[Out]

-1/884736*sqrt(6)*sqrt(2)*(160*sqrt(6)*sqrt(2)*(d*x + c)^(3/2)*b*d^2*abs(b)*sin(6*((d*x + c)*b - b*c + a*d)/d)
/abs(d) - 4320*sqrt(6)*sqrt(2)*(d*x + c)^(3/2)*b*d^2*abs(b)*sin(2*((d*x + c)*b - b*c + a*d)/d)/abs(d) - 8*(48*
sqrt(6)*sqrt(2)*(d*x + c)^(5/2)*b^2*d*abs(b)/abs(d) - 5*sqrt(6)*sqrt(2)*sqrt(d*x + c)*d^3*abs(b)/abs(d))*cos(6
*((d*x + c)*b - b*c + a*d)/d) + 216*(16*sqrt(6)*sqrt(2)*(d*x + c)^(5/2)*b^2*d*abs(b)/abs(d) - 15*sqrt(6)*sqrt(
2)*sqrt(d*x + c)*d^3*abs(b)/abs(d))*cos(2*((d*x + c)*b - b*c + a*d)/d) - (sqrt(2)*(5*sqrt(pi)*cos(1/4*pi + 1/2
*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 5*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/
sqrt(d^2))) - 5*I*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 5*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(-6*(b*c - a*d)/d) - sqrt
(2)*(5*I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 5*I*sqrt(pi)*cos(-1/4*pi + 1
/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 5*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d
/sqrt(d^2))) - 5*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(-6*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(6*I*b/d)) + (sqrt(6)*(405*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(
0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 405*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^
2))) - 405*I*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 405*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) + sqrt(6)
*(-405*I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 405*I*sqrt(pi)*cos(-1/4*pi +
1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 405*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(
0, d/sqrt(d^2))) + 405*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(6)*(405*sqrt(pi)*cos(1/4*pi + 1/2*ar
ctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 405*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/s
qrt(d^2))) + 405*I*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 405*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) + s
qrt(6)*(405*I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 405*I*sqrt(pi)*cos(-1/4
*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 405*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arc
tan2(0, d/sqrt(d^2))) + 405*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)*(5*sqrt(pi)*cos(1/4*pi + 1/
2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 5*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d
/sqrt(d^2))) + 5*I*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 5*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(-6*(b*c - a*d)/d) - sqr
t(2)*(-5*I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 5*I*sqrt(pi)*cos(-1/4*pi +
1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 5*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0,
d/sqrt(d^2))) - 5*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^3*sqrt(abs(b)/ab
s(d))*sin(-6*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-6*I*b/d)))*abs(d)/(b^3*d*abs(b))

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Fricas [A]  time = 0.767491, size = 1089, normalized size = 2.68 \begin{align*} \frac{5 \, \sqrt{3} \pi d^{3} \sqrt{\frac{b}{\pi d}} \cos \left (-\frac{6 \,{\left (b c - a d\right )}}{d}\right ) \operatorname{C}\left (2 \, \sqrt{3} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) - 5 \, \sqrt{3} \pi d^{3} \sqrt{\frac{b}{\pi d}} \operatorname{S}\left (2 \, \sqrt{3} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) \sin \left (-\frac{6 \,{\left (b c - a d\right )}}{d}\right ) - 1215 \, \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 ) + 1215 \, \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 ) + 96 \,{\left (24 \, b^{3} d^{2} x^{2} + 2 \,{\left (48 \, b^{3} d^{2} x^{2} + 96 \, b^{3} c d x + 48 \, b^{3} c^{2} - 5 \, b d^{2}\right )} \cos \left (b x + a\right )^{6} + 48 \, b^{3} c d x + 24 \, b^{3} c^{2} + 45 \, b d^{2} \cos \left (b x + a\right )^{2} - 3 \,{\left (48 \, b^{3} d^{2} x^{2} + 96 \, b^{3} c d x + 48 \, b^{3} c^{2} - 5 \, b d^{2}\right )} \cos \left (b x + a\right )^{4} - 25 \, b d^{2} - 20 \,{\left (2 \,{\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )^{5} - 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}}{55296 \, 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)^3,x, algorithm="fricas")

[Out]

1/55296*(5*sqrt(3)*pi*d^3*sqrt(b/(pi*d))*cos(-6*(b*c - a*d)/d)*fresnel_cos(2*sqrt(3)*sqrt(d*x + c)*sqrt(b/(pi*
d))) - 5*sqrt(3)*pi*d^3*sqrt(b/(pi*d))*fresnel_sin(2*sqrt(3)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-6*(b*c - a*d)/
d) - 1215*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))) + 1215*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) + 96*(24*b^3*d^2*x^2 + 2*(4
8*b^3*d^2*x^2 + 96*b^3*c*d*x + 48*b^3*c^2 - 5*b*d^2)*cos(b*x + a)^6 + 48*b^3*c*d*x + 24*b^3*c^2 + 45*b*d^2*cos
(b*x + a)^2 - 3*(48*b^3*d^2*x^2 + 96*b^3*c*d*x + 48*b^3*c^2 - 5*b*d^2)*cos(b*x + a)^4 - 25*b*d^2 - 20*(2*(b^2*
d^2*x + b^2*c*d)*cos(b*x + a)^5 - 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)**3,x)

[Out]

Timed out

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Giac [C]  time = 2.70005, size = 2677, 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)^3,x, algorithm="giac")

[Out]

1/110592*(48*(sqrt(3)*sqrt(pi)*d^2*erf(-sqrt(3)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((6*I*b
*c - 6*I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b) + sqrt(3)*sqrt(pi)*d^2*erf(-sqrt(3)*sqrt(b*d)*sqrt(d*
x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-6*I*b*c + 6*I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b) - 27*
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) - 27*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) + 6*sqrt(d*x + c)*d*e^((6*I*(d*x + c)*b - 6*I*b*
c + 6*I*a*d)/d)/b - 54*sqrt(d*x + c)*d*e^((2*I*(d*x + c)*b - 2*I*b*c + 2*I*a*d)/d)/b - 54*sqrt(d*x + c)*d*e^((
-2*I*(d*x + c)*b + 2*I*b*c - 2*I*a*d)/d)/b + 6*sqrt(d*x + c)*d*e^((-6*I*(d*x + c)*b + 6*I*b*c - 6*I*a*d)/d)/b)
*c^2 - d^2*((I*sqrt(3)*sqrt(pi)*(48*I*b^2*c^2*d - 24*b*c*d^2 - 5*I*d^3)*d*erf(-sqrt(3)*sqrt(b*d)*sqrt(d*x + c)
*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((6*I*b*c - 6*I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b^3) - 6*I*(-48*I
*(d*x + c)^(5/2)*b^2*d + 96*I*(d*x + c)^(3/2)*b^2*c*d - 48*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 + 5*I*sqrt(d*x + c)*d^3)*e^((-6*I*(d*x + c)*b + 6*I*b*c - 6*I*a*d)/d)/b^3)/d^2 +
(I*sqrt(3)*sqrt(pi)*(48*I*b^2*c^2*d + 24*b*c*d^2 - 5*I*d^3)*d*erf(-sqrt(3)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqr
t(b^2*d^2) + 1)/d)*e^((-6*I*b*c + 6*I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b^3) - 6*I*(-48*I*(d*x + c
)^(5/2)*b^2*d + 96*I*(d*x + c)^(3/2)*b^2*c*d - 48*I*sqrt(d*x + c)*b^2*c^2*d + 20*(d*x + c)^(3/2)*b*d^2 - 24*sq
rt(d*x + c)*b*c*d^2 + 5*I*sqrt(d*x + c)*d^3)*e^((6*I*(d*x + c)*b - 6*I*b*c + 6*I*a*d)/d)/b^3)/d^2 + 27*(I*sqrt
(pi)*(-48*I*b^2*c^2*d + 72*b*c*d^2 + 45*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*(48*I*(d*x + c)^(5/2)*b^2*d - 96*I*(d*x
+ c)^(3/2)*b^2*c*d + 48*I*sqrt(d*x + c)*b^2*c^2*d + 60*(d*x + c)^(3/2)*b*d^2 - 72*sqrt(d*x + c)*b*c*d^2 - 45*
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 + 27*(I*sqrt(pi)*(-48*I*b^2*c^2*d -
72*b*c*d^2 + 45*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*(48*I*(d*x + c)^(5/2)*b^2*d - 96*I*(d*x + c)^(3/2)*b^2*c*d
+ 48*I*sqrt(d*x + c)*b^2*c^2*d - 60*(d*x + c)^(3/2)*b*d^2 + 72*sqrt(d*x + c)*b*c*d^2 - 45*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) - 24*(I*sqrt(3)*sqrt(pi)*(-4*I*b*c*d + d^2)*d*erf(-sqrt
(3)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((6*I*b*c - 6*I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*
d^2) + 1)*b^2) + I*sqrt(3)*sqrt(pi)*(-4*I*b*c*d - d^2)*d*erf(-sqrt(3)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2
*d^2) + 1)/d)*e^((-6*I*b*c + 6*I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b^2) + 9*I*sqrt(pi)*(12*I*b*c*d
- 9*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) + 9*I*sqrt(pi)*(12*I*b*c*d + 9*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) - 6*I*(-4*I*(d*x + c)^(3/
2)*b*d + 4*I*sqrt(d*x + c)*b*c*d + sqrt(d*x + c)*d^2)*e^((6*I*(d*x + c)*b - 6*I*b*c + 6*I*a*d)/d)/b^2 - 18*I*(
12*I*(d*x + c)^(3/2)*b*d - 12*I*sqrt(d*x + c)*b*c*d - 9*sqrt(d*x + c)*d^2)*e^((2*I*(d*x + c)*b - 2*I*b*c + 2*I
*a*d)/d)/b^2 - 18*I*(12*I*(d*x + c)^(3/2)*b*d - 12*I*sqrt(d*x + c)*b*c*d + 9*sqrt(d*x + c)*d^2)*e^((-2*I*(d*x
+ c)*b + 2*I*b*c - 2*I*a*d)/d)/b^2 - 6*I*(-4*I*(d*x + c)^(3/2)*b*d + 4*I*sqrt(d*x + c)*b*c*d - sqrt(d*x + c)*d
^2)*e^((-6*I*(d*x + c)*b + 6*I*b*c - 6*I*a*d)/d)/b^2)*c)/d