3.2.97 \(\int (e+f x)^2 \sin (a+\frac {b}{\sqrt {c+d x}}) \, dx\) [197]

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

[Out]

-1/180*b^3*f^2*(d*x+c)^(3/2)*cos(a+b/(d*x+c)^(1/2))/d^3+1/3*b*f*(-c*f+d*e)*(d*x+c)^(3/2)*cos(a+b/(d*x+c)^(1/2)
)/d^3+1/15*b*f^2*(d*x+c)^(5/2)*cos(a+b/(d*x+c)^(1/2))/d^3+1/360*b^6*f^2*cos(a)*Si(b/(d*x+c)^(1/2))/d^3-1/6*b^4
*f*(-c*f+d*e)*cos(a)*Si(b/(d*x+c)^(1/2))/d^3+b^2*(-c*f+d*e)^2*cos(a)*Si(b/(d*x+c)^(1/2))/d^3+1/360*b^6*f^2*Ci(
b/(d*x+c)^(1/2))*sin(a)/d^3-1/6*b^4*f*(-c*f+d*e)*Ci(b/(d*x+c)^(1/2))*sin(a)/d^3+b^2*(-c*f+d*e)^2*Ci(b/(d*x+c)^
(1/2))*sin(a)/d^3+1/360*b^4*f^2*(d*x+c)*sin(a+b/(d*x+c)^(1/2))/d^3-1/6*b^2*f*(-c*f+d*e)*(d*x+c)*sin(a+b/(d*x+c
)^(1/2))/d^3+(-c*f+d*e)^2*(d*x+c)*sin(a+b/(d*x+c)^(1/2))/d^3-1/60*b^2*f^2*(d*x+c)^2*sin(a+b/(d*x+c)^(1/2))/d^3
+f*(-c*f+d*e)*(d*x+c)^2*sin(a+b/(d*x+c)^(1/2))/d^3+1/3*f^2*(d*x+c)^3*sin(a+b/(d*x+c)^(1/2))/d^3+1/360*b^5*f^2*
cos(a+b/(d*x+c)^(1/2))*(d*x+c)^(1/2)/d^3-1/6*b^3*f*(-c*f+d*e)*cos(a+b/(d*x+c)^(1/2))*(d*x+c)^(1/2)/d^3+b*(-c*f
+d*e)^2*cos(a+b/(d*x+c)^(1/2))*(d*x+c)^(1/2)/d^3

________________________________________________________________________________________

Rubi [A]
time = 0.55, antiderivative size = 611, normalized size of antiderivative = 1.00, 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 = {3512, 3378, 3384, 3380, 3383} \begin {gather*} \frac {b^6 f^2 \sin (a) \text {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right )}{360 d^3}+\frac {b^6 f^2 \cos (a) \text {Si}\left (\frac {b}{\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^4 f \sin (a) (d e-c f) \text {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right )}{6 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^4 f^2 (c+d x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{360 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^3 f^2 (c+d x)^{3/2} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{180 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^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^2 f^2 (c+d x)^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{60 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} \end {gather*}

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 3378

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 3380

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 3383

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]

Rule 3384

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 3512

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]

Rubi steps

\begin {align*} \int (e+f x)^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx &=-\frac {2 \text {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 ) \text {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)) \text {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 ) \text {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 ) \text {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)) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] Result contains complex when optimal does not.
time = 1.36, size = 557, normalized size = 0.91 \begin {gather*} \frac {i e^{-i a} \left (e^{-\frac {i b}{\sqrt {c+d x}}} \sqrt {c+d x} \left (-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} \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 )-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 )\right )-e^{i \left (2 a+\frac {b}{\sqrt {c+d x}}\right )} \sqrt {c+d x} \left (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} \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 )+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 )\right )+b^2 \left (360 d^2 e^2-60 \left (b^2+12 c\right ) d e f+\left (b^4+60 b^2 c+360 c^2\right ) f^2\right ) \text {Ei}\left (-\frac {i b}{\sqrt {c+d x}}\right )-b^2 e^{2 i a} \left (360 d^2 e^2-60 \left (b^2+12 c\right ) d e f+\left (b^4+60 b^2 c+360 c^2\right ) f^2\right ) \text {Ei}\left (\frac {i b}{\sqrt {c+d x}}\right )\right )}{720 d^3} \end {gather*}

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))

________________________________________________________________________________________

Maple [A]
time = 0.51, size = 696, normalized size = 1.14 Too large to display

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,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [C] Result contains complex when optimal does not.
time = 0.78, size = 878, normalized size = 1.44 \begin {gather*} \frac {\frac {360 \, {\left ({\left ({\left (-i \, {\rm Ei}\left (\frac {i \, b}{\sqrt {d x + c}}\right ) + i \, {\rm Ei}\left (-\frac {i \, b}{\sqrt {d x + c}}\right )\right )} \cos \left (a\right ) + {\left ({\rm Ei}\left (\frac {i \, b}{\sqrt {d x + c}}\right ) + {\rm Ei}\left (-\frac {i \, b}{\sqrt {d x + c}}\right )\right )} \sin \left (a\right )\right )} b^{2} + 2 \, \sqrt {d x + c} b \cos \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right ) + 2 \, {\left (d x + c\right )} \sin \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right )\right )} c^{2} f^{2}}{d^{2}} - \frac {720 \, {\left ({\left ({\left (-i \, {\rm Ei}\left (\frac {i \, b}{\sqrt {d x + c}}\right ) + i \, {\rm Ei}\left (-\frac {i \, b}{\sqrt {d x + c}}\right )\right )} \cos \left (a\right ) + {\left ({\rm Ei}\left (\frac {i \, b}{\sqrt {d x + c}}\right ) + {\rm Ei}\left (-\frac {i \, b}{\sqrt {d x + c}}\right )\right )} \sin \left (a\right )\right )} b^{2} + 2 \, \sqrt {d x + c} b \cos \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right ) + 2 \, {\left (d x + c\right )} \sin \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right )\right )} c f e}{d} - \frac {60 \, {\left ({\left ({\left (i \, {\rm Ei}\left (\frac {i \, b}{\sqrt {d x + c}}\right ) - i \, {\rm Ei}\left (-\frac {i \, b}{\sqrt {d x + c}}\right )\right )} \cos \left (a\right ) - {\left ({\rm Ei}\left (\frac {i \, b}{\sqrt {d x + c}}\right ) + {\rm Ei}\left (-\frac {i \, b}{\sqrt {d x + c}}\right )\right )} \sin \left (a\right )\right )} b^{4} - 2 \, {\left (\sqrt {d x + c} b^{3} - 2 \, {\left (d x + c\right )}^{\frac {3}{2}} b\right )} \cos \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right ) - 2 \, {\left ({\left (d x + c\right )} b^{2} - 6 \, {\left (d x + c\right )}^{2}\right )} \sin \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right )\right )} c f^{2}}{d^{2}} + 360 \, {\left ({\left ({\left (-i \, {\rm Ei}\left (\frac {i \, b}{\sqrt {d x + c}}\right ) + i \, {\rm Ei}\left (-\frac {i \, b}{\sqrt {d x + c}}\right )\right )} \cos \left (a\right ) + {\left ({\rm Ei}\left (\frac {i \, b}{\sqrt {d x + c}}\right ) + {\rm Ei}\left (-\frac {i \, b}{\sqrt {d x + c}}\right )\right )} \sin \left (a\right )\right )} b^{2} + 2 \, \sqrt {d x + c} b \cos \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right ) + 2 \, {\left (d x + c\right )} \sin \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right )\right )} e^{2} + \frac {60 \, {\left ({\left ({\left (i \, {\rm Ei}\left (\frac {i \, b}{\sqrt {d x + c}}\right ) - i \, {\rm Ei}\left (-\frac {i \, b}{\sqrt {d x + c}}\right )\right )} \cos \left (a\right ) - {\left ({\rm Ei}\left (\frac {i \, b}{\sqrt {d x + c}}\right ) + {\rm Ei}\left (-\frac {i \, b}{\sqrt {d x + c}}\right )\right )} \sin \left (a\right )\right )} b^{4} - 2 \, {\left (\sqrt {d x + c} b^{3} - 2 \, {\left (d x + c\right )}^{\frac {3}{2}} b\right )} \cos \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right ) - 2 \, {\left ({\left (d x + c\right )} b^{2} - 6 \, {\left (d x + c\right )}^{2}\right )} \sin \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right )\right )} f e}{d} + \frac {{\left ({\left ({\left (-i \, {\rm Ei}\left (\frac {i \, b}{\sqrt {d x + c}}\right ) + i \, {\rm Ei}\left (-\frac {i \, b}{\sqrt {d x + c}}\right )\right )} \cos \left (a\right ) + {\left ({\rm Ei}\left (\frac {i \, b}{\sqrt {d x + c}}\right ) + {\rm Ei}\left (-\frac {i \, b}{\sqrt {d x + c}}\right )\right )} \sin \left (a\right )\right )} b^{6} + 2 \, {\left (\sqrt {d x + c} b^{5} - 2 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} + 24 \, {\left (d x + c\right )}^{\frac {5}{2}} b\right )} \cos \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right ) + 2 \, {\left ({\left (d x + c\right )} b^{4} - 6 \, {\left (d x + c\right )}^{2} b^{2} + 120 \, {\left (d x + c\right )}^{3}\right )} \sin \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right )\right )} f^{2}}{d^{2}}}{720 \, d} \end {gather*}

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)))*c^2*f^2/d^2 - 720*(((-I*Ei(I*b/sqrt(d*x + c)) + I*Ei(-I*b/sqrt(d*x + c)))*co
s(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*f*e/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*(sq
rt(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 + 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)))*e^2 + 60*(((I*Ei(I*b/sq
rt(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)))*f*e/d + (((-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

________________________________________________________________________________________

Fricas [A]
time = 0.43, size = 457, normalized size = 0.75 \begin {gather*} \frac {{\left (360 \, b^{2} d^{2} e^{2} - 60 \, {\left (b^{4} + 12 \, b^{2} c\right )} d f e + {\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 f e + {\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 f e + {\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} - 2 \, {\left (b^{3} + 36 \, b c\right )} d f^{2} x + 360 \, b d^{2} e^{2} + {\left (b^{5} + 58 \, b^{3} c + 264 \, b c^{2}\right )} f^{2} + 60 \, {\left (2 \, b d^{2} f x - {\left (b^{3} + 10 \, b c\right )} d f\right )} e\right )} \sqrt {d x + c} \cos \left (\frac {a d x + a c + \sqrt {d x + c} b}{d x + c}\right ) - 2 \, {\left (6 \, b^{2} d^{2} f^{2} x^{2} - 120 \, d^{3} f^{2} x^{3} - {\left (b^{4} + 48 \, b^{2} c\right )} d f^{2} x - {\left (b^{4} c + 54 \, b^{2} c^{2} + 120 \, c^{3}\right )} f^{2} - 360 \, {\left (d^{3} x + c d^{2}\right )} e^{2} + 60 \, {\left (b^{2} d^{2} f x - 6 \, d^{3} f x^{2} + {\left (b^{2} c + 6 \, c^{2}\right )} d f\right )} e\right )} \sin \left (\frac {a d x + a c + \sqrt {d x + c} b}{d x + c}\right )}{720 \, d^{3}} \end {gather*}

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*f*e + (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*f*e + (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*f*e + (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 - 2*(b^3 + 36*b*c)*d*f^2*x + 360*b*d^2*e^2 +
(b^5 + 58*b^3*c + 264*b*c^2)*f^2 + 60*(2*b*d^2*f*x - (b^3 + 10*b*c)*d*f)*e)*sqrt(d*x + c)*cos((a*d*x + a*c +
sqrt(d*x + c)*b)/(d*x + c)) - 2*(6*b^2*d^2*f^2*x^2 - 120*d^3*f^2*x^3 - (b^4 + 48*b^2*c)*d*f^2*x - (b^4*c + 54*
b^2*c^2 + 120*c^3)*f^2 - 360*(d^3*x + c*d^2)*e^2 + 60*(b^2*d^2*f*x - 6*d^3*f*x^2 + (b^2*c + 6*c^2)*d*f)*e)*sin
((a*d*x + a*c + sqrt(d*x + c)*b)/(d*x + c)))/d^3

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (e + f x\right )^{2} \sin {\left (a + \frac {b}{\sqrt {c + d x}} \right )}\, dx \end {gather*}

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)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 6606 vs. \(2 (548) = 1096\).
time = 5.78, size = 6606, normalized size = 10.81 \begin {gather*} \text {Too large to display} \end {gather*}

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]

1/360*((a^6*b^7*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(a) - a^6*b^7*cos(a)*sin_integral(a
- (sqrt(d*x + c)*a + b)/sqrt(d*x + c)) - 6*(sqrt(d*x + c)*a + b)*a^5*b^7*cos_integral(-a + (sqrt(d*x + c)*a +
b)/sqrt(d*x + c))*sin(a)/sqrt(d*x + c) + 6*(sqrt(d*x + c)*a + b)*a^5*b^7*cos(a)*sin_integral(a - (sqrt(d*x + c
)*a + b)/sqrt(d*x + c))/sqrt(d*x + c) + 15*(sqrt(d*x + c)*a + b)^2*a^4*b^7*cos_integral(-a + (sqrt(d*x + c)*a
+ b)/sqrt(d*x + c))*sin(a)/(d*x + c) + 60*a^6*b^5*c*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin
(a) - 15*(sqrt(d*x + c)*a + b)^2*a^4*b^7*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c))/(d*x + c
) - 60*a^6*b^5*c*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c)) - 20*(sqrt(d*x + c)*a + b)^3*a^3
*b^7*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(a)/(d*x + c)^(3/2) - 360*(sqrt(d*x + c)*a + b)
*a^5*b^5*c*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(a)/sqrt(d*x + c) + 20*(sqrt(d*x + c)*a +
b)^3*a^3*b^7*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c))/(d*x + c)^(3/2) + 360*(sqrt(d*x + c
)*a + b)*a^5*b^5*c*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c))/sqrt(d*x + c) - a^5*b^7*cos((s
qrt(d*x + c)*a + b)/sqrt(d*x + c)) + 15*(sqrt(d*x + c)*a + b)^4*a^2*b^7*cos_integral(-a + (sqrt(d*x + c)*a + b
)/sqrt(d*x + c))*sin(a)/(d*x + c)^2 + 900*(sqrt(d*x + c)*a + b)^2*a^4*b^5*c*cos_integral(-a + (sqrt(d*x + c)*a
+ b)/sqrt(d*x + c))*sin(a)/(d*x + c) + 360*a^6*b^3*c^2*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))
*sin(a) - 15*(sqrt(d*x + c)*a + b)^4*a^2*b^7*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c))/(d*x
+ c)^2 - 900*(sqrt(d*x + c)*a + b)^2*a^4*b^5*c*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c))/(
d*x + c) - 360*a^6*b^3*c^2*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c)) + 5*(sqrt(d*x + c)*a +
b)*a^4*b^7*cos((sqrt(d*x + c)*a + b)/sqrt(d*x + c))/sqrt(d*x + c) - 6*(sqrt(d*x + c)*a + b)^5*a*b^7*cos_integ
ral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(a)/(d*x + c)^(5/2) - 1200*(sqrt(d*x + c)*a + b)^3*a^3*b^5*c*
cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(a)/(d*x + c)^(3/2) - 2160*(sqrt(d*x + c)*a + b)*a^5
*b^3*c^2*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(a)/sqrt(d*x + c) + 6*(sqrt(d*x + c)*a + b)
^5*a*b^7*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c))/(d*x + c)^(5/2) + 1200*(sqrt(d*x + c)*a
+ b)^3*a^3*b^5*c*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c))/(d*x + c)^(3/2) + 2160*(sqrt(d*x
+ c)*a + b)*a^5*b^3*c^2*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c))/sqrt(d*x + c) - 10*(sqrt
(d*x + c)*a + b)^2*a^3*b^7*cos((sqrt(d*x + c)*a + b)/sqrt(d*x + c))/(d*x + c) - 60*a^5*b^5*c*cos((sqrt(d*x + c
)*a + b)/sqrt(d*x + c)) + (sqrt(d*x + c)*a + b)^6*b^7*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*s
in(a)/(d*x + c)^3 + 900*(sqrt(d*x + c)*a + b)^4*a^2*b^5*c*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c
))*sin(a)/(d*x + c)^2 + 5400*(sqrt(d*x + c)*a + b)^2*a^4*b^3*c^2*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(
d*x + c))*sin(a)/(d*x + c) + a^4*b^7*sin((sqrt(d*x + c)*a + b)/sqrt(d*x + c)) - (sqrt(d*x + c)*a + b)^6*b^7*co
s(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c))/(d*x + c)^3 - 900*(sqrt(d*x + c)*a + b)^4*a^2*b^5*c
*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c))/(d*x + c)^2 - 5400*(sqrt(d*x + c)*a + b)^2*a^4*b
^3*c^2*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c))/(d*x + c) + 10*(sqrt(d*x + c)*a + b)^3*a^2
*b^7*cos((sqrt(d*x + c)*a + b)/sqrt(d*x + c))/(d*x + c)^(3/2) + 300*(sqrt(d*x + c)*a + b)*a^4*b^5*c*cos((sqrt(
d*x + c)*a + b)/sqrt(d*x + c))/sqrt(d*x + c) - 360*(sqrt(d*x + c)*a + b)^5*a*b^5*c*cos_integral(-a + (sqrt(d*x
+ c)*a + b)/sqrt(d*x + c))*sin(a)/(d*x + c)^(5/2) - 7200*(sqrt(d*x + c)*a + b)^3*a^3*b^3*c^2*cos_integral(-a
+ (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(a)/(d*x + c)^(3/2) - 4*(sqrt(d*x + c)*a + b)*a^3*b^7*sin((sqrt(d*x
+ c)*a + b)/sqrt(d*x + c))/sqrt(d*x + c) + 360*(sqrt(d*x + c)*a + b)^5*a*b^5*c*cos(a)*sin_integral(a - (sqrt(d
*x + c)*a + b)/sqrt(d*x + c))/(d*x + c)^(5/2) + 7200*(sqrt(d*x + c)*a + b)^3*a^3*b^3*c^2*cos(a)*sin_integral(a
- (sqrt(d*x + c)*a + b)/sqrt(d*x + c))/(d*x + c)^(3/2) - 5*(sqrt(d*x + c)*a + b)^4*a*b^7*cos((sqrt(d*x + c)*a
+ b)/sqrt(d*x + c))/(d*x + c)^2 + 2*a^3*b^7*cos((sqrt(d*x + c)*a + b)/sqrt(d*x + c)) - 600*(sqrt(d*x + c)*a +
b)^2*a^3*b^5*c*cos((sqrt(d*x + c)*a + b)/sqrt(d*x + c))/(d*x + c) - 360*a^5*b^3*c^2*cos((sqrt(d*x + c)*a + b)
/sqrt(d*x + c)) + 60*(sqrt(d*x + c)*a + b)^6*b^5*c*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(
a)/(d*x + c)^3 + 5400*(sqrt(d*x + c)*a + b)^4*a^2*b^3*c^2*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c
))*sin(a)/(d*x + c)^2 + 6*(sqrt(d*x + c)*a + b)^2*a^2*b^7*sin((sqrt(d*x + c)*a + b)/sqrt(d*x + c))/(d*x + c) +
60*a^4*b^5*c*sin((sqrt(d*x + c)*a + b)/sqrt(d*x + c)) - 60*(sqrt(d*x + c)*a + b)^6*b^5*c*cos(a)*sin_integral(
a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c))/(d*x +...

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \sin \left (a+\frac {b}{\sqrt {c+d\,x}}\right )\,{\left (e+f\,x\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________