3.197 \(\int (e+f x)^2 \sin (a+\frac{b}{\sqrt{c+d x}}) \, dx\)

Optimal. Leaf size=611 \[ -\frac{b^4 f \sin (a) (d e-c f) \text{CosIntegral}\left (\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b^2 \sin (a) (d e-c f)^2 \text{CosIntegral}\left (\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{b^6 f^2 \sin (a) \text{CosIntegral}\left (\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^4 f \cos (a) (d e-c f) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b^2 \cos (a) (d e-c f)^2 \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{b^2 f (c+d x) (d e-c f) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}-\frac{b^3 f \sqrt{c+d x} (d e-c f) \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b^6 f^2 \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}+\frac{b^4 f^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^2 f^2 (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{60 d^3}+\frac{b^5 f^2 \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^3 f^2 (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{180 d^3}+\frac{f (c+d x)^2 (d e-c f) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{(c+d x) (d e-c f)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{b f (c+d x)^{3/2} (d e-c f) \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b \sqrt{c+d x} (d e-c f)^2 \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b f^2 (c+d x)^{5/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{15 d^3} \]

[Out]

(b^5*f^2*Sqrt[c + d*x]*Cos[a + b/Sqrt[c + d*x]])/(360*d^3) - (b^3*f*(d*e - c*f)*Sqrt[c + d*x]*Cos[a + b/Sqrt[c
+ d*x]])/(6*d^3) + (b*(d*e - c*f)^2*Sqrt[c + d*x]*Cos[a + b/Sqrt[c + d*x]])/d^3 - (b^3*f^2*(c + d*x)^(3/2)*Co
s[a + b/Sqrt[c + d*x]])/(180*d^3) + (b*f*(d*e - c*f)*(c + d*x)^(3/2)*Cos[a + b/Sqrt[c + d*x]])/(3*d^3) + (b*f^
2*(c + d*x)^(5/2)*Cos[a + b/Sqrt[c + d*x]])/(15*d^3) + (b^6*f^2*CosIntegral[b/Sqrt[c + d*x]]*Sin[a])/(360*d^3)
- (b^4*f*(d*e - c*f)*CosIntegral[b/Sqrt[c + d*x]]*Sin[a])/(6*d^3) + (b^2*(d*e - c*f)^2*CosIntegral[b/Sqrt[c +
d*x]]*Sin[a])/d^3 + (b^4*f^2*(c + d*x)*Sin[a + b/Sqrt[c + d*x]])/(360*d^3) - (b^2*f*(d*e - c*f)*(c + d*x)*Sin
[a + b/Sqrt[c + d*x]])/(6*d^3) + ((d*e - c*f)^2*(c + d*x)*Sin[a + b/Sqrt[c + d*x]])/d^3 - (b^2*f^2*(c + d*x)^2
*Sin[a + b/Sqrt[c + d*x]])/(60*d^3) + (f*(d*e - c*f)*(c + d*x)^2*Sin[a + b/Sqrt[c + d*x]])/d^3 + (f^2*(c + d*x
)^3*Sin[a + b/Sqrt[c + d*x]])/(3*d^3) + (b^6*f^2*Cos[a]*SinIntegral[b/Sqrt[c + d*x]])/(360*d^3) - (b^4*f*(d*e
- c*f)*Cos[a]*SinIntegral[b/Sqrt[c + d*x]])/(6*d^3) + (b^2*(d*e - c*f)^2*Cos[a]*SinIntegral[b/Sqrt[c + d*x]])/
d^3

________________________________________________________________________________________

Rubi [A]  time = 0.790798, antiderivative size = 611, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {3431, 3297, 3303, 3299, 3302} \[ -\frac{b^4 f \sin (a) (d e-c f) \text{CosIntegral}\left (\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b^2 \sin (a) (d e-c f)^2 \text{CosIntegral}\left (\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{b^6 f^2 \sin (a) \text{CosIntegral}\left (\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^4 f \cos (a) (d e-c f) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b^2 \cos (a) (d e-c f)^2 \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{b^2 f (c+d x) (d e-c f) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}-\frac{b^3 f \sqrt{c+d x} (d e-c f) \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b^6 f^2 \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}+\frac{b^4 f^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^2 f^2 (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{60 d^3}+\frac{b^5 f^2 \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^3 f^2 (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{180 d^3}+\frac{f (c+d x)^2 (d e-c f) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{(c+d x) (d e-c f)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{b f (c+d x)^{3/2} (d e-c f) \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b \sqrt{c+d x} (d e-c f)^2 \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b f^2 (c+d x)^{5/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{15 d^3} \]

Antiderivative was successfully verified.

[In]

Int[(e + f*x)^2*Sin[a + b/Sqrt[c + d*x]],x]

[Out]

(b^5*f^2*Sqrt[c + d*x]*Cos[a + b/Sqrt[c + d*x]])/(360*d^3) - (b^3*f*(d*e - c*f)*Sqrt[c + d*x]*Cos[a + b/Sqrt[c
+ d*x]])/(6*d^3) + (b*(d*e - c*f)^2*Sqrt[c + d*x]*Cos[a + b/Sqrt[c + d*x]])/d^3 - (b^3*f^2*(c + d*x)^(3/2)*Co
s[a + b/Sqrt[c + d*x]])/(180*d^3) + (b*f*(d*e - c*f)*(c + d*x)^(3/2)*Cos[a + b/Sqrt[c + d*x]])/(3*d^3) + (b*f^
2*(c + d*x)^(5/2)*Cos[a + b/Sqrt[c + d*x]])/(15*d^3) + (b^6*f^2*CosIntegral[b/Sqrt[c + d*x]]*Sin[a])/(360*d^3)
- (b^4*f*(d*e - c*f)*CosIntegral[b/Sqrt[c + d*x]]*Sin[a])/(6*d^3) + (b^2*(d*e - c*f)^2*CosIntegral[b/Sqrt[c +
d*x]]*Sin[a])/d^3 + (b^4*f^2*(c + d*x)*Sin[a + b/Sqrt[c + d*x]])/(360*d^3) - (b^2*f*(d*e - c*f)*(c + d*x)*Sin
[a + b/Sqrt[c + d*x]])/(6*d^3) + ((d*e - c*f)^2*(c + d*x)*Sin[a + b/Sqrt[c + d*x]])/d^3 - (b^2*f^2*(c + d*x)^2
*Sin[a + b/Sqrt[c + d*x]])/(60*d^3) + (f*(d*e - c*f)*(c + d*x)^2*Sin[a + b/Sqrt[c + d*x]])/d^3 + (f^2*(c + d*x
)^3*Sin[a + b/Sqrt[c + d*x]])/(3*d^3) + (b^6*f^2*Cos[a]*SinIntegral[b/Sqrt[c + d*x]])/(360*d^3) - (b^4*f*(d*e
- c*f)*Cos[a]*SinIntegral[b/Sqrt[c + d*x]])/(6*d^3) + (b^2*(d*e - c*f)^2*Cos[a]*SinIntegral[b/Sqrt[c + d*x]])/
d^3

Rule 3431

Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_Symbol] :
> Dist[1/(n*f), Subst[Int[ExpandIntegrand[(a + b*Sin[c + d*x])^p, x^(1/n - 1)*(g - (e*h)/f + (h*x^(1/n))/f)^m,
x], x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p, 0] && IntegerQ[1/n]

Rule 3297

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

Rule 3303

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

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rubi steps

\begin{align*} \int (e+f x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right ) \, dx &=-\frac{2 \operatorname{Subst}\left (\int \left (\frac{f^2 \sin (a+b x)}{d^2 x^7}+\frac{2 f (d e-c f) \sin (a+b x)}{d^2 x^5}+\frac{(d e-c f)^2 \sin (a+b x)}{d^2 x^3}\right ) \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d}\\ &=-\frac{\left (2 f^2\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^7} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d^3}-\frac{(4 f (d e-c f)) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^5} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d^3}-\frac{\left (2 (d e-c f)^2\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^3} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d^3}\\ &=\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}-\frac{\left (b f^2\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^6} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{3 d^3}-\frac{(b f (d e-c f)) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^4} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d^3}-\frac{\left (b (d e-c f)^2\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^2} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d^3}\\ &=\frac{b (d e-c f)^2 \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{b f (d e-c f) (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b f^2 (c+d x)^{5/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{15 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{\left (b^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^5} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{15 d^3}+\frac{\left (b^2 f (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^3} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{\left (b^2 (d e-c f)^2\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d^3}\\ &=\frac{b (d e-c f)^2 \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{b f (d e-c f) (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b f^2 (c+d x)^{5/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{15 d^3}-\frac{b^2 f (d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{b^2 f^2 (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{60 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{\left (b^3 f^2\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^4} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{60 d^3}+\frac{\left (b^3 f (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^2} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{\left (b^2 (d e-c f)^2 \cos (a)\right ) \operatorname{Subst}\left (\int \frac{\sin (b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d^3}+\frac{\left (b^2 (d e-c f)^2 \sin (a)\right ) \operatorname{Subst}\left (\int \frac{\cos (b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d^3}\\ &=-\frac{b^3 f (d e-c f) \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b (d e-c f)^2 \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{b^3 f^2 (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{180 d^3}+\frac{b f (d e-c f) (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b f^2 (c+d x)^{5/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{15 d^3}+\frac{b^2 (d e-c f)^2 \text{Ci}\left (\frac{b}{\sqrt{c+d x}}\right ) \sin (a)}{d^3}-\frac{b^2 f (d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{b^2 f^2 (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{60 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b^2 (d e-c f)^2 \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{\left (b^4 f^2\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^3} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{180 d^3}-\frac{\left (b^4 f (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{6 d^3}\\ &=-\frac{b^3 f (d e-c f) \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b (d e-c f)^2 \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{b^3 f^2 (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{180 d^3}+\frac{b f (d e-c f) (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b f^2 (c+d x)^{5/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{15 d^3}+\frac{b^2 (d e-c f)^2 \text{Ci}\left (\frac{b}{\sqrt{c+d x}}\right ) \sin (a)}{d^3}+\frac{b^4 f^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^2 f (d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{b^2 f^2 (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{60 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b^2 (d e-c f)^2 \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{\left (b^5 f^2\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^2} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{\left (b^4 f (d e-c f) \cos (a)\right ) \operatorname{Subst}\left (\int \frac{\sin (b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{6 d^3}-\frac{\left (b^4 f (d e-c f) \sin (a)\right ) \operatorname{Subst}\left (\int \frac{\cos (b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{6 d^3}\\ &=\frac{b^5 f^2 \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^3 f (d e-c f) \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b (d e-c f)^2 \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{b^3 f^2 (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{180 d^3}+\frac{b f (d e-c f) (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b f^2 (c+d x)^{5/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{15 d^3}-\frac{b^4 f (d e-c f) \text{Ci}\left (\frac{b}{\sqrt{c+d x}}\right ) \sin (a)}{6 d^3}+\frac{b^2 (d e-c f)^2 \text{Ci}\left (\frac{b}{\sqrt{c+d x}}\right ) \sin (a)}{d^3}+\frac{b^4 f^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^2 f (d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{b^2 f^2 (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{60 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}-\frac{b^4 f (d e-c f) \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b^2 (d e-c f)^2 \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{\left (b^6 f^2\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{360 d^3}\\ &=\frac{b^5 f^2 \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^3 f (d e-c f) \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b (d e-c f)^2 \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{b^3 f^2 (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{180 d^3}+\frac{b f (d e-c f) (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b f^2 (c+d x)^{5/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{15 d^3}-\frac{b^4 f (d e-c f) \text{Ci}\left (\frac{b}{\sqrt{c+d x}}\right ) \sin (a)}{6 d^3}+\frac{b^2 (d e-c f)^2 \text{Ci}\left (\frac{b}{\sqrt{c+d x}}\right ) \sin (a)}{d^3}+\frac{b^4 f^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^2 f (d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{b^2 f^2 (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{60 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}-\frac{b^4 f (d e-c f) \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b^2 (d e-c f)^2 \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{\left (b^6 f^2 \cos (a)\right ) \operatorname{Subst}\left (\int \frac{\sin (b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{360 d^3}+\frac{\left (b^6 f^2 \sin (a)\right ) \operatorname{Subst}\left (\int \frac{\cos (b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{360 d^3}\\ &=\frac{b^5 f^2 \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^3 f (d e-c f) \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b (d e-c f)^2 \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{b^3 f^2 (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{180 d^3}+\frac{b f (d e-c f) (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b f^2 (c+d x)^{5/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{15 d^3}+\frac{b^6 f^2 \text{Ci}\left (\frac{b}{\sqrt{c+d x}}\right ) \sin (a)}{360 d^3}-\frac{b^4 f (d e-c f) \text{Ci}\left (\frac{b}{\sqrt{c+d x}}\right ) \sin (a)}{6 d^3}+\frac{b^2 (d e-c f)^2 \text{Ci}\left (\frac{b}{\sqrt{c+d x}}\right ) \sin (a)}{d^3}+\frac{b^4 f^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^2 f (d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{b^2 f^2 (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{60 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b^6 f^2 \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^4 f (d e-c f) \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b^2 (d e-c f)^2 \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{d^3}\\ \end{align*}

Mathematica [C]  time = 2.20019, size = 557, normalized size = 0.91 \[ \frac{i e^{-i a} \left (-e^{2 i a} b^2 \left (f^2 \left (60 b^2 c+b^4+360 c^2\right )-60 d e f \left (b^2+12 c\right )+360 d^2 e^2\right ) \text{Ei}\left (\frac{i b}{\sqrt{c+d x}}\right )-\sqrt{c+d x} e^{i \left (2 a+\frac{b}{\sqrt{c+d x}}\right )} \left (-2 i b^3 f (-29 c f+30 d e+d f x)-6 b^2 f \sqrt{c+d x} (-9 c f+10 d e+d f x)+b^4 f^2 \sqrt{c+d x}+i b^5 f^2+24 i b \left (11 c^2 f^2-c d f (25 e+3 f x)+d^2 \left (15 e^2+5 e f x+f^2 x^2\right )\right )+120 \sqrt{c+d x} \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )\right )+b^2 \left (f^2 \left (60 b^2 c+b^4+360 c^2\right )-60 d e f \left (b^2+12 c\right )+360 d^2 e^2\right ) \text{Ei}\left (-\frac{i b}{\sqrt{c+d x}}\right )+\sqrt{c+d x} e^{-\frac{i b}{\sqrt{c+d x}}} \left (2 i b^3 f (-29 c f+30 d e+d f x)-6 b^2 f \sqrt{c+d x} (-9 c f+10 d e+d f x)+b^4 f^2 \sqrt{c+d x}-i b^5 f^2-24 i b \left (11 c^2 f^2-c d f (25 e+3 f x)+d^2 \left (15 e^2+5 e f x+f^2 x^2\right )\right )+120 \sqrt{c+d x} \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )\right )\right )}{720 d^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(e + f*x)^2*Sin[a + b/Sqrt[c + d*x]],x]

[Out]

((I/720)*((Sqrt[c + d*x]*((-I)*b^5*f^2 + b^4*f^2*Sqrt[c + d*x] + (2*I)*b^3*f*(30*d*e - 29*c*f + d*f*x) - 6*b^2
*f*Sqrt[c + d*x]*(10*d*e - 9*c*f + d*f*x) + 120*Sqrt[c + d*x]*(c^2*f^2 - c*d*f*(3*e + f*x) + d^2*(3*e^2 + 3*e*
f*x + f^2*x^2)) - (24*I)*b*(11*c^2*f^2 - c*d*f*(25*e + 3*f*x) + d^2*(15*e^2 + 5*e*f*x + f^2*x^2))))/E^((I*b)/S
qrt[c + d*x]) - E^(I*(2*a + b/Sqrt[c + d*x]))*Sqrt[c + d*x]*(I*b^5*f^2 + b^4*f^2*Sqrt[c + d*x] - (2*I)*b^3*f*(
30*d*e - 29*c*f + d*f*x) - 6*b^2*f*Sqrt[c + d*x]*(10*d*e - 9*c*f + d*f*x) + 120*Sqrt[c + d*x]*(c^2*f^2 - c*d*f
*(3*e + f*x) + d^2*(3*e^2 + 3*e*f*x + f^2*x^2)) + (24*I)*b*(11*c^2*f^2 - c*d*f*(25*e + 3*f*x) + d^2*(15*e^2 +
5*e*f*x + f^2*x^2))) + b^2*(360*d^2*e^2 - 60*(b^2 + 12*c)*d*e*f + (b^4 + 60*b^2*c + 360*c^2)*f^2)*ExpIntegralE
i[((-I)*b)/Sqrt[c + d*x]] - b^2*E^((2*I)*a)*(360*d^2*e^2 - 60*(b^2 + 12*c)*d*e*f + (b^4 + 60*b^2*c + 360*c^2)*
f^2)*ExpIntegralEi[(I*b)/Sqrt[c + d*x]]))/(d^3*E^(I*a))

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Maple [A]  time = 0.059, size = 696, normalized size = 1.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^2*sin(a+b/(d*x+c)^(1/2)),x)

[Out]

-2/d^3*b^2*(c^2*f^2*(-1/2*sin(a+b/(d*x+c)^(1/2))*(d*x+c)/b^2-1/2*cos(a+b/(d*x+c)^(1/2))*(d*x+c)^(1/2)/b-1/2*Si
(b/(d*x+c)^(1/2))*cos(a)-1/2*Ci(b/(d*x+c)^(1/2))*sin(a))-2*c*d*e*f*(-1/2*sin(a+b/(d*x+c)^(1/2))*(d*x+c)/b^2-1/
2*cos(a+b/(d*x+c)^(1/2))*(d*x+c)^(1/2)/b-1/2*Si(b/(d*x+c)^(1/2))*cos(a)-1/2*Ci(b/(d*x+c)^(1/2))*sin(a))+d^2*e^
2*(-1/2*sin(a+b/(d*x+c)^(1/2))*(d*x+c)/b^2-1/2*cos(a+b/(d*x+c)^(1/2))*(d*x+c)^(1/2)/b-1/2*Si(b/(d*x+c)^(1/2))*
cos(a)-1/2*Ci(b/(d*x+c)^(1/2))*sin(a))-2*b^2*c*f^2*(-1/4*sin(a+b/(d*x+c)^(1/2))*(d*x+c)^2/b^4-1/12*cos(a+b/(d*
x+c)^(1/2))*(d*x+c)^(3/2)/b^3+1/24*sin(a+b/(d*x+c)^(1/2))*(d*x+c)/b^2+1/24*cos(a+b/(d*x+c)^(1/2))*(d*x+c)^(1/2
)/b+1/24*Si(b/(d*x+c)^(1/2))*cos(a)+1/24*Ci(b/(d*x+c)^(1/2))*sin(a))+2*b^2*d*e*f*(-1/4*sin(a+b/(d*x+c)^(1/2))*
(d*x+c)^2/b^4-1/12*cos(a+b/(d*x+c)^(1/2))*(d*x+c)^(3/2)/b^3+1/24*sin(a+b/(d*x+c)^(1/2))*(d*x+c)/b^2+1/24*cos(a
+b/(d*x+c)^(1/2))*(d*x+c)^(1/2)/b+1/24*Si(b/(d*x+c)^(1/2))*cos(a)+1/24*Ci(b/(d*x+c)^(1/2))*sin(a))+b^4*f^2*(-1
/6*sin(a+b/(d*x+c)^(1/2))*(d*x+c)^3/b^6-1/30*cos(a+b/(d*x+c)^(1/2))*(d*x+c)^(5/2)/b^5+1/120*sin(a+b/(d*x+c)^(1
/2))*(d*x+c)^2/b^4+1/360*cos(a+b/(d*x+c)^(1/2))*(d*x+c)^(3/2)/b^3-1/720*sin(a+b/(d*x+c)^(1/2))*(d*x+c)/b^2-1/7
20*cos(a+b/(d*x+c)^(1/2))*(d*x+c)^(1/2)/b-1/720*Si(b/(d*x+c)^(1/2))*cos(a)-1/720*Ci(b/(d*x+c)^(1/2))*sin(a)))

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Maxima [C]  time = 2.04818, size = 1184, normalized size = 1.94 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^2*sin(a+b/(d*x+c)^(1/2)),x, algorithm="maxima")

[Out]

1/720*(360*(((-I*Ei(I*b/sqrt(d*x + c)) + I*Ei(-I*b/sqrt(d*x + c)))*cos(a) + (Ei(I*b/sqrt(d*x + c)) + Ei(-I*b/s
qrt(d*x + c)))*sin(a))*b^2 + 2*sqrt(d*x + c)*b*cos((sqrt(d*x + c)*a + b)/sqrt(d*x + c)) + 2*(d*x + c)*sin((sqr
t(d*x + c)*a + b)/sqrt(d*x + c)))*e^2 - 720*(((-I*Ei(I*b/sqrt(d*x + c)) + I*Ei(-I*b/sqrt(d*x + c)))*cos(a) + (
Ei(I*b/sqrt(d*x + c)) + Ei(-I*b/sqrt(d*x + c)))*sin(a))*b^2 + 2*sqrt(d*x + c)*b*cos((sqrt(d*x + c)*a + b)/sqrt
(d*x + c)) + 2*(d*x + c)*sin((sqrt(d*x + c)*a + b)/sqrt(d*x + c)))*c*e*f/d + 360*(((-I*Ei(I*b/sqrt(d*x + c)) +
I*Ei(-I*b/sqrt(d*x + c)))*cos(a) + (Ei(I*b/sqrt(d*x + c)) + Ei(-I*b/sqrt(d*x + c)))*sin(a))*b^2 + 2*sqrt(d*x
+ c)*b*cos((sqrt(d*x + c)*a + b)/sqrt(d*x + c)) + 2*(d*x + c)*sin((sqrt(d*x + c)*a + b)/sqrt(d*x + c)))*c^2*f^
2/d^2 + 60*(((I*Ei(I*b/sqrt(d*x + c)) - I*Ei(-I*b/sqrt(d*x + c)))*cos(a) - (Ei(I*b/sqrt(d*x + c)) + Ei(-I*b/sq
rt(d*x + c)))*sin(a))*b^4 - 2*(sqrt(d*x + c)*b^3 - 2*(d*x + c)^(3/2)*b)*cos((sqrt(d*x + c)*a + b)/sqrt(d*x + c
)) - 2*((d*x + c)*b^2 - 6*(d*x + c)^2)*sin((sqrt(d*x + c)*a + b)/sqrt(d*x + c)))*e*f/d - 60*(((I*Ei(I*b/sqrt(d
*x + c)) - I*Ei(-I*b/sqrt(d*x + c)))*cos(a) - (Ei(I*b/sqrt(d*x + c)) + Ei(-I*b/sqrt(d*x + c)))*sin(a))*b^4 - 2
*(sqrt(d*x + c)*b^3 - 2*(d*x + c)^(3/2)*b)*cos((sqrt(d*x + c)*a + b)/sqrt(d*x + c)) - 2*((d*x + c)*b^2 - 6*(d*
x + c)^2)*sin((sqrt(d*x + c)*a + b)/sqrt(d*x + c)))*c*f^2/d^2 + (((-I*Ei(I*b/sqrt(d*x + c)) + I*Ei(-I*b/sqrt(d
*x + c)))*cos(a) + (Ei(I*b/sqrt(d*x + c)) + Ei(-I*b/sqrt(d*x + c)))*sin(a))*b^6 + 2*(sqrt(d*x + c)*b^5 - 2*(d*
x + c)^(3/2)*b^3 + 24*(d*x + c)^(5/2)*b)*cos((sqrt(d*x + c)*a + b)/sqrt(d*x + c)) + 2*((d*x + c)*b^4 - 6*(d*x
+ c)^2*b^2 + 120*(d*x + c)^3)*sin((sqrt(d*x + c)*a + b)/sqrt(d*x + c)))*f^2/d^2)/d

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Fricas [A]  time = 2.38604, size = 1119, normalized size = 1.83 \begin{align*} \frac{{\left (360 \, b^{2} d^{2} e^{2} - 60 \,{\left (b^{4} + 12 \, b^{2} c\right )} d e f +{\left (b^{6} + 60 \, b^{4} c + 360 \, b^{2} c^{2}\right )} f^{2}\right )} \operatorname{Ci}\left (\frac{b}{\sqrt{d x + c}}\right ) \sin \left (a\right ) +{\left (360 \, b^{2} d^{2} e^{2} - 60 \,{\left (b^{4} + 12 \, b^{2} c\right )} d e f +{\left (b^{6} + 60 \, b^{4} c + 360 \, b^{2} c^{2}\right )} f^{2}\right )} \operatorname{Ci}\left (-\frac{b}{\sqrt{d x + c}}\right ) \sin \left (a\right ) + 2 \,{\left (360 \, b^{2} d^{2} e^{2} - 60 \,{\left (b^{4} + 12 \, b^{2} c\right )} d e f +{\left (b^{6} + 60 \, b^{4} c + 360 \, b^{2} c^{2}\right )} f^{2}\right )} \cos \left (a\right ) \operatorname{Si}\left (\frac{b}{\sqrt{d x + c}}\right ) + 2 \,{\left (24 \, b d^{2} f^{2} x^{2} + 360 \, b d^{2} e^{2} - 60 \,{\left (b^{3} + 10 \, b c\right )} d e f +{\left (b^{5} + 58 \, b^{3} c + 264 \, b c^{2}\right )} f^{2} + 2 \,{\left (60 \, b d^{2} e f -{\left (b^{3} + 36 \, b c\right )} d f^{2}\right )} x\right )} \sqrt{d x + c} \cos \left (\frac{a d x + a c + \sqrt{d x + c} b}{d x + c}\right ) + 2 \,{\left (120 \, d^{3} f^{2} x^{3} + 360 \, c d^{2} e^{2} - 60 \,{\left (b^{2} c + 6 \, c^{2}\right )} d e f +{\left (b^{4} c + 54 \, b^{2} c^{2} + 120 \, c^{3}\right )} f^{2} - 6 \,{\left (b^{2} d^{2} f^{2} - 60 \, d^{3} e f\right )} x^{2} -{\left (60 \, b^{2} d^{2} e f - 360 \, d^{3} e^{2} -{\left (b^{4} + 48 \, b^{2} c\right )} d f^{2}\right )} x\right )} \sin \left (\frac{a d x + a c + \sqrt{d x + c} b}{d x + c}\right )}{720 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^2*sin(a+b/(d*x+c)^(1/2)),x, algorithm="fricas")

[Out]

1/720*((360*b^2*d^2*e^2 - 60*(b^4 + 12*b^2*c)*d*e*f + (b^6 + 60*b^4*c + 360*b^2*c^2)*f^2)*cos_integral(b/sqrt(
d*x + c))*sin(a) + (360*b^2*d^2*e^2 - 60*(b^4 + 12*b^2*c)*d*e*f + (b^6 + 60*b^4*c + 360*b^2*c^2)*f^2)*cos_inte
gral(-b/sqrt(d*x + c))*sin(a) + 2*(360*b^2*d^2*e^2 - 60*(b^4 + 12*b^2*c)*d*e*f + (b^6 + 60*b^4*c + 360*b^2*c^2
)*f^2)*cos(a)*sin_integral(b/sqrt(d*x + c)) + 2*(24*b*d^2*f^2*x^2 + 360*b*d^2*e^2 - 60*(b^3 + 10*b*c)*d*e*f +
(b^5 + 58*b^3*c + 264*b*c^2)*f^2 + 2*(60*b*d^2*e*f - (b^3 + 36*b*c)*d*f^2)*x)*sqrt(d*x + c)*cos((a*d*x + a*c +
sqrt(d*x + c)*b)/(d*x + c)) + 2*(120*d^3*f^2*x^3 + 360*c*d^2*e^2 - 60*(b^2*c + 6*c^2)*d*e*f + (b^4*c + 54*b^2
*c^2 + 120*c^3)*f^2 - 6*(b^2*d^2*f^2 - 60*d^3*e*f)*x^2 - (60*b^2*d^2*e*f - 360*d^3*e^2 - (b^4 + 48*b^2*c)*d*f^
2)*x)*sin((a*d*x + a*c + sqrt(d*x + c)*b)/(d*x + c)))/d^3

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e + f x\right )^{2} \sin{\left (a + \frac{b}{\sqrt{c + d x}} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**2*sin(a+b/(d*x+c)**(1/2)),x)

[Out]

Integral((e + f*x)**2*sin(a + b/sqrt(c + d*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x + e\right )}^{2} \sin \left (a + \frac{b}{\sqrt{d x + c}}\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^2*sin(a+b/(d*x+c)^(1/2)),x, algorithm="giac")

[Out]

integrate((f*x + e)^2*sin(a + b/sqrt(d*x + c)), x)