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

Optimal. Leaf size=301 \[ \frac{b^2 \sin (a) (d e-c f) \text{CosIntegral}\left (\frac{b}{\sqrt{c+d x}}\right )}{d^2}-\frac{b^4 f \sin (a) \text{CosIntegral}\left (\frac{b}{\sqrt{c+d x}}\right )}{12 d^2}+\frac{b^2 \cos (a) (d e-c f) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{d^2}-\frac{b^4 f \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{12 d^2}-\frac{b^2 f (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{12 d^2}-\frac{b^3 f \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{12 d^2}+\frac{(c+d x) (d e-c f) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^2}+\frac{b \sqrt{c+d x} (d e-c f) \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{2 d^2}+\frac{b f (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^2} \]

[Out]

-(b^3*f*Sqrt[c + d*x]*Cos[a + b/Sqrt[c + d*x]])/(12*d^2) + (b*(d*e - c*f)*Sqrt[c + d*x]*Cos[a + b/Sqrt[c + d*x
]])/d^2 + (b*f*(c + d*x)^(3/2)*Cos[a + b/Sqrt[c + d*x]])/(6*d^2) - (b^4*f*CosIntegral[b/Sqrt[c + d*x]]*Sin[a])
/(12*d^2) + (b^2*(d*e - c*f)*CosIntegral[b/Sqrt[c + d*x]]*Sin[a])/d^2 - (b^2*f*(c + d*x)*Sin[a + b/Sqrt[c + d*
x]])/(12*d^2) + ((d*e - c*f)*(c + d*x)*Sin[a + b/Sqrt[c + d*x]])/d^2 + (f*(c + d*x)^2*Sin[a + b/Sqrt[c + d*x]]
)/(2*d^2) - (b^4*f*Cos[a]*SinIntegral[b/Sqrt[c + d*x]])/(12*d^2) + (b^2*(d*e - c*f)*Cos[a]*SinIntegral[b/Sqrt[
c + d*x]])/d^2

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Rubi [A]  time = 0.393377, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3431, 3297, 3303, 3299, 3302} \[ \frac{b^2 \sin (a) (d e-c f) \text{CosIntegral}\left (\frac{b}{\sqrt{c+d x}}\right )}{d^2}-\frac{b^4 f \sin (a) \text{CosIntegral}\left (\frac{b}{\sqrt{c+d x}}\right )}{12 d^2}+\frac{b^2 \cos (a) (d e-c f) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{d^2}-\frac{b^4 f \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{12 d^2}-\frac{b^2 f (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{12 d^2}-\frac{b^3 f \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{12 d^2}+\frac{(c+d x) (d e-c f) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^2}+\frac{b \sqrt{c+d x} (d e-c f) \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{2 d^2}+\frac{b f (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^3*f*Sqrt[c + d*x]*Cos[a + b/Sqrt[c + d*x]])/(12*d^2) + (b*(d*e - c*f)*Sqrt[c + d*x]*Cos[a + b/Sqrt[c + d*x
]])/d^2 + (b*f*(c + d*x)^(3/2)*Cos[a + b/Sqrt[c + d*x]])/(6*d^2) - (b^4*f*CosIntegral[b/Sqrt[c + d*x]]*Sin[a])
/(12*d^2) + (b^2*(d*e - c*f)*CosIntegral[b/Sqrt[c + d*x]]*Sin[a])/d^2 - (b^2*f*(c + d*x)*Sin[a + b/Sqrt[c + d*
x]])/(12*d^2) + ((d*e - c*f)*(c + d*x)*Sin[a + b/Sqrt[c + d*x]])/d^2 + (f*(c + d*x)^2*Sin[a + b/Sqrt[c + d*x]]
)/(2*d^2) - (b^4*f*Cos[a]*SinIntegral[b/Sqrt[c + d*x]])/(12*d^2) + (b^2*(d*e - c*f)*Cos[a]*SinIntegral[b/Sqrt[
c + d*x]])/d^2

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) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right ) \, dx &=-\frac{2 \operatorname{Subst}\left (\int \left (\frac{f \sin (a+b x)}{d x^5}+\frac{(d e-c f) \sin (a+b x)}{d x^3}\right ) \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d}\\ &=-\frac{(2 f) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^5} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d^2}-\frac{(2 (d e-c f)) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^3} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d^2}\\ &=\frac{(d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{2 d^2}-\frac{(b f) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^4} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{2 d^2}-\frac{(b (d e-c f)) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^2} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d^2}\\ &=\frac{b (d e-c f) \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^2}+\frac{b f (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^2}+\frac{(d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{2 d^2}+\frac{\left (b^2 f\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^3} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{6 d^2}+\frac{\left (b^2 (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d^2}\\ &=\frac{b (d e-c f) \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^2}+\frac{b f (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^2}-\frac{b^2 f (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{12 d^2}+\frac{(d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{2 d^2}+\frac{\left (b^3 f\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^2} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{12 d^2}+\frac{\left (b^2 (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 )}{d^2}+\frac{\left (b^2 (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 )}{d^2}\\ &=-\frac{b^3 f \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{12 d^2}+\frac{b (d e-c f) \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^2}+\frac{b f (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^2}+\frac{b^2 (d e-c f) \text{Ci}\left (\frac{b}{\sqrt{c+d x}}\right ) \sin (a)}{d^2}-\frac{b^2 f (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{12 d^2}+\frac{(d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{2 d^2}+\frac{b^2 (d e-c f) \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{d^2}-\frac{\left (b^4 f\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{12 d^2}\\ &=-\frac{b^3 f \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{12 d^2}+\frac{b (d e-c f) \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^2}+\frac{b f (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^2}+\frac{b^2 (d e-c f) \text{Ci}\left (\frac{b}{\sqrt{c+d x}}\right ) \sin (a)}{d^2}-\frac{b^2 f (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{12 d^2}+\frac{(d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{2 d^2}+\frac{b^2 (d e-c f) \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{d^2}-\frac{\left (b^4 f \cos (a)\right ) \operatorname{Subst}\left (\int \frac{\sin (b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{12 d^2}-\frac{\left (b^4 f \sin (a)\right ) \operatorname{Subst}\left (\int \frac{\cos (b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{12 d^2}\\ &=-\frac{b^3 f \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{12 d^2}+\frac{b (d e-c f) \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^2}+\frac{b f (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^2}-\frac{b^4 f \text{Ci}\left (\frac{b}{\sqrt{c+d x}}\right ) \sin (a)}{12 d^2}+\frac{b^2 (d e-c f) \text{Ci}\left (\frac{b}{\sqrt{c+d x}}\right ) \sin (a)}{d^2}-\frac{b^2 f (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{12 d^2}+\frac{(d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{2 d^2}-\frac{b^4 f \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{12 d^2}+\frac{b^2 (d e-c f) \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{d^2}\\ \end{align*}

Mathematica [A]  time = 0.620483, size = 367, normalized size = 1.22 \[ -\frac{b^2 f \left (b^2+12 c\right ) \left (\sin (a) \text{CosIntegral}\left (\frac{b}{\sqrt{c+d x}}\right )+\cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )\right )}{12 d^2}+\frac{b^2 e \left (\sin (a) \text{CosIntegral}\left (\frac{b}{\sqrt{c+d x}}\right )+\cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )\right )}{d}+\frac{f \sqrt{c+d x} \cos \left (\frac{b}{\sqrt{c+d x}}\right ) \left (-b^2 \sin (a) \sqrt{c+d x}+b^3 (-\cos (a))+2 b \cos (a) (c+d x)-12 b c \cos (a)+6 \sin (a) (c+d x)^{3/2}-12 c \sin (a) \sqrt{c+d x}\right )}{12 d^2}+\frac{f \sqrt{c+d x} \sin \left (\frac{b}{\sqrt{c+d x}}\right ) \left (-b^2 \cos (a) \sqrt{c+d x}+b^3 \sin (a)-2 b \sin (a) (c+d x)+12 b c \sin (a)+6 \cos (a) (c+d x)^{3/2}-12 c \cos (a) \sqrt{c+d x}\right )}{12 d^2}+\frac{e \sqrt{c+d x} \cos \left (\frac{b}{\sqrt{c+d x}}\right ) \left (b \cos (a)+\sin (a) \sqrt{c+d x}\right )}{d}+\frac{e \sqrt{c+d x} \sin \left (\frac{b}{\sqrt{c+d x}}\right ) \left (\cos (a) \sqrt{c+d x}-b \sin (a)\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*Sqrt[c + d*x]*Cos[b/Sqrt[c + d*x]]*(b*Cos[a] + Sqrt[c + d*x]*Sin[a]))/d + (f*Sqrt[c + d*x]*Cos[b/Sqrt[c + d
*x]]*(-(b^3*Cos[a]) - 12*b*c*Cos[a] + 2*b*(c + d*x)*Cos[a] - b^2*Sqrt[c + d*x]*Sin[a] - 12*c*Sqrt[c + d*x]*Sin
[a] + 6*(c + d*x)^(3/2)*Sin[a]))/(12*d^2) + (e*Sqrt[c + d*x]*(Sqrt[c + d*x]*Cos[a] - b*Sin[a])*Sin[b/Sqrt[c +
d*x]])/d + (f*Sqrt[c + d*x]*(-(b^2*Sqrt[c + d*x]*Cos[a]) - 12*c*Sqrt[c + d*x]*Cos[a] + 6*(c + d*x)^(3/2)*Cos[a
] + b^3*Sin[a] + 12*b*c*Sin[a] - 2*b*(c + d*x)*Sin[a])*Sin[b/Sqrt[c + d*x]])/(12*d^2) + (b^2*e*(CosIntegral[b/
Sqrt[c + d*x]]*Sin[a] + Cos[a]*SinIntegral[b/Sqrt[c + d*x]]))/d - (b^2*(b^2 + 12*c)*f*(CosIntegral[b/Sqrt[c +
d*x]]*Sin[a] + Cos[a]*SinIntegral[b/Sqrt[c + d*x]]))/(12*d^2)

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Maple [A]  time = 0.028, size = 295, normalized size = 1. \begin{align*} -2\,{\frac{{b}^{2}}{{d}^{2}} \left ( -cf \left ( -1/2\,{\frac{dx+c}{{b}^{2}}\sin \left ( a+{\frac{b}{\sqrt{dx+c}}} \right ) }-1/2\,{\frac{\sqrt{dx+c}}{b}\cos \left ( a+{\frac{b}{\sqrt{dx+c}}} \right ) }-1/2\,{\it Si} \left ({\frac{b}{\sqrt{dx+c}}} \right ) \cos \left ( a \right ) -1/2\,{\it Ci} \left ({\frac{b}{\sqrt{dx+c}}} \right ) \sin \left ( a \right ) \right ) +de \left ( -1/2\,{\frac{dx+c}{{b}^{2}}\sin \left ( a+{\frac{b}{\sqrt{dx+c}}} \right ) }-1/2\,{\frac{\sqrt{dx+c}}{b}\cos \left ( a+{\frac{b}{\sqrt{dx+c}}} \right ) }-1/2\,{\it Si} \left ({\frac{b}{\sqrt{dx+c}}} \right ) \cos \left ( a \right ) -1/2\,{\it Ci} \left ({\frac{b}{\sqrt{dx+c}}} \right ) \sin \left ( a \right ) \right ) +{b}^{2}f \left ( -1/4\,{\frac{ \left ( dx+c \right ) ^{2}}{{b}^{4}}\sin \left ( a+{\frac{b}{\sqrt{dx+c}}} \right ) }-1/12\,{\frac{ \left ( dx+c \right ) ^{3/2}}{{b}^{3}}\cos \left ( a+{\frac{b}{\sqrt{dx+c}}} \right ) }+1/24\,{\frac{dx+c}{{b}^{2}}\sin \left ( a+{\frac{b}{\sqrt{dx+c}}} \right ) }+1/24\,{\frac{\sqrt{dx+c}}{b}\cos \left ( a+{\frac{b}{\sqrt{dx+c}}} \right ) }+1/24\,{\it Si} \left ({\frac{b}{\sqrt{dx+c}}} \right ) \cos \left ( a \right ) +1/24\,{\it Ci} \left ({\frac{b}{\sqrt{dx+c}}} \right ) \sin \left ( a \right ) \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/d^2*b^2*(-c*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*e*(-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))+b^2*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)))

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Maxima [C]  time = 1.55219, size = 549, normalized size = 1.82 \begin{align*} \frac{12 \,{\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 - \frac{12 \,{\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}{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^{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}{d}}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(12*(((-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/sqr
t(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 - 12*(((-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*f/d + (((I*Ei(I*b/sqrt(d*x + c)) - I*Ei(-I*b/s
qrt(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/d)/d

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Fricas [A]  time = 2.28389, size = 621, normalized size = 2.06 \begin{align*} \frac{{\left (12 \, b^{2} d e -{\left (b^{4} + 12 \, b^{2} c\right )} f\right )} \operatorname{Ci}\left (\frac{b}{\sqrt{d x + c}}\right ) \sin \left (a\right ) +{\left (12 \, b^{2} d e -{\left (b^{4} + 12 \, b^{2} c\right )} f\right )} \operatorname{Ci}\left (-\frac{b}{\sqrt{d x + c}}\right ) \sin \left (a\right ) + 2 \,{\left (12 \, b^{2} d e -{\left (b^{4} + 12 \, b^{2} c\right )} f\right )} \cos \left (a\right ) \operatorname{Si}\left (\frac{b}{\sqrt{d x + c}}\right ) + 2 \,{\left (2 \, b d f x + 12 \, b d e -{\left (b^{3} + 10 \, b c\right )} f\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 \, d^{2} f x^{2} + 12 \, c d e -{\left (b^{2} c + 6 \, c^{2}\right )} f -{\left (b^{2} d f - 12 \, d^{2} e\right )} x\right )} \sin \left (\frac{a d x + a c + \sqrt{d x + c} b}{d x + c}\right )}{24 \, d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*((12*b^2*d*e - (b^4 + 12*b^2*c)*f)*cos_integral(b/sqrt(d*x + c))*sin(a) + (12*b^2*d*e - (b^4 + 12*b^2*c)*
f)*cos_integral(-b/sqrt(d*x + c))*sin(a) + 2*(12*b^2*d*e - (b^4 + 12*b^2*c)*f)*cos(a)*sin_integral(b/sqrt(d*x
+ c)) + 2*(2*b*d*f*x + 12*b*d*e - (b^3 + 10*b*c)*f)*sqrt(d*x + c)*cos((a*d*x + a*c + sqrt(d*x + c)*b)/(d*x + c
)) + 2*(6*d^2*f*x^2 + 12*c*d*e - (b^2*c + 6*c^2)*f - (b^2*d*f - 12*d^2*e)*x)*sin((a*d*x + a*c + sqrt(d*x + c)*
b)/(d*x + c)))/d^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e + f x\right ) \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)*sin(a+b/(d*x+c)**(1/2)),x)

[Out]

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

[Out]

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