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

Optimal. Leaf size=339 \[ \frac{b d \cos \left (a+\frac{b \sqrt{c f-d e}}{\sqrt{f}}\right ) \text{CosIntegral}\left (\frac{b \sqrt{c f-d e}}{\sqrt{f}}-b \sqrt{c+d x}\right )}{2 f^{3/2} \sqrt{c f-d e}}-\frac{b d \cos \left (a-\frac{b \sqrt{c f-d e}}{\sqrt{f}}\right ) \text{CosIntegral}\left (\frac{b \sqrt{c f-d e}}{\sqrt{f}}+b \sqrt{c+d x}\right )}{2 f^{3/2} \sqrt{c f-d e}}+\frac{b d \sin \left (a+\frac{b \sqrt{c f-d e}}{\sqrt{f}}\right ) \text{Si}\left (\frac{b \sqrt{c f-d e}}{\sqrt{f}}-b \sqrt{c+d x}\right )}{2 f^{3/2} \sqrt{c f-d e}}+\frac{b d \sin \left (a-\frac{b \sqrt{c f-d e}}{\sqrt{f}}\right ) \text{Si}\left (\frac{\sqrt{c f-d e} b}{\sqrt{f}}+\sqrt{c+d x} b\right )}{2 f^{3/2} \sqrt{c f-d e}}-\frac{\sin \left (a+b \sqrt{c+d x}\right )}{f (e+f x)} \]

[Out]

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

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Rubi [A]  time = 0.982629, antiderivative size = 339, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3431, 3341, 3334, 3303, 3299, 3302} \[ \frac{b d \cos \left (a+\frac{b \sqrt{c f-d e}}{\sqrt{f}}\right ) \text{CosIntegral}\left (\frac{b \sqrt{c f-d e}}{\sqrt{f}}-b \sqrt{c+d x}\right )}{2 f^{3/2} \sqrt{c f-d e}}-\frac{b d \cos \left (a-\frac{b \sqrt{c f-d e}}{\sqrt{f}}\right ) \text{CosIntegral}\left (\frac{b \sqrt{c f-d e}}{\sqrt{f}}+b \sqrt{c+d x}\right )}{2 f^{3/2} \sqrt{c f-d e}}+\frac{b d \sin \left (a+\frac{b \sqrt{c f-d e}}{\sqrt{f}}\right ) \text{Si}\left (\frac{b \sqrt{c f-d e}}{\sqrt{f}}-b \sqrt{c+d x}\right )}{2 f^{3/2} \sqrt{c f-d e}}+\frac{b d \sin \left (a-\frac{b \sqrt{c f-d e}}{\sqrt{f}}\right ) \text{Si}\left (\frac{\sqrt{c f-d e} b}{\sqrt{f}}+\sqrt{c+d x} b\right )}{2 f^{3/2} \sqrt{c f-d e}}-\frac{\sin \left (a+b \sqrt{c+d x}\right )}{f (e+f x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 3341

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(e^m*(a + b*x^
n)^(p + 1)*Sin[c + d*x])/(b*n*(p + 1)), x] - Dist[(d*e^m)/(b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*Cos[c + d*x],
x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, -1] && EqQ[m, n - 1] && (IntegerQ[n] || GtQ[e, 0])

Rule 3334

Int[Cos[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Cos[c + d*x], (a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -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 \frac{\sin \left (a+b \sqrt{c+d x}\right )}{(e+f x)^2} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{x \sin (a+b x)}{\left (e-\frac{c f}{d}+\frac{f x^2}{d}\right )^2} \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=-\frac{\sin \left (a+b \sqrt{c+d x}\right )}{f (e+f x)}+\frac{b \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{e-\frac{c f}{d}+\frac{f x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{f}\\ &=-\frac{\sin \left (a+b \sqrt{c+d x}\right )}{f (e+f x)}+\frac{b \operatorname{Subst}\left (\int \left (\frac{\sqrt{-d e+c f} \cos (a+b x)}{2 \left (e-\frac{c f}{d}\right ) \left (\sqrt{-d e+c f}-\sqrt{f} x\right )}+\frac{\sqrt{-d e+c f} \cos (a+b x)}{2 \left (e-\frac{c f}{d}\right ) \left (\sqrt{-d e+c f}+\sqrt{f} x\right )}\right ) \, dx,x,\sqrt{c+d x}\right )}{f}\\ &=-\frac{\sin \left (a+b \sqrt{c+d x}\right )}{f (e+f x)}-\frac{(b d) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{\sqrt{-d e+c f}-\sqrt{f} x} \, dx,x,\sqrt{c+d x}\right )}{2 f \sqrt{-d e+c f}}-\frac{(b d) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{\sqrt{-d e+c f}+\sqrt{f} x} \, dx,x,\sqrt{c+d x}\right )}{2 f \sqrt{-d e+c f}}\\ &=-\frac{\sin \left (a+b \sqrt{c+d x}\right )}{f (e+f x)}-\frac{\left (b d \cos \left (a-\frac{b \sqrt{-d e+c f}}{\sqrt{f}}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{b \sqrt{-d e+c f}}{\sqrt{f}}+b x\right )}{\sqrt{-d e+c f}+\sqrt{f} x} \, dx,x,\sqrt{c+d x}\right )}{2 f \sqrt{-d e+c f}}-\frac{\left (b d \cos \left (a+\frac{b \sqrt{-d e+c f}}{\sqrt{f}}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{b \sqrt{-d e+c f}}{\sqrt{f}}-b x\right )}{\sqrt{-d e+c f}-\sqrt{f} x} \, dx,x,\sqrt{c+d x}\right )}{2 f \sqrt{-d e+c f}}+\frac{\left (b d \sin \left (a-\frac{b \sqrt{-d e+c f}}{\sqrt{f}}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{b \sqrt{-d e+c f}}{\sqrt{f}}+b x\right )}{\sqrt{-d e+c f}+\sqrt{f} x} \, dx,x,\sqrt{c+d x}\right )}{2 f \sqrt{-d e+c f}}-\frac{\left (b d \sin \left (a+\frac{b \sqrt{-d e+c f}}{\sqrt{f}}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{b \sqrt{-d e+c f}}{\sqrt{f}}-b x\right )}{\sqrt{-d e+c f}-\sqrt{f} x} \, dx,x,\sqrt{c+d x}\right )}{2 f \sqrt{-d e+c f}}\\ &=\frac{b d \cos \left (a+\frac{b \sqrt{-d e+c f}}{\sqrt{f}}\right ) \text{Ci}\left (\frac{b \sqrt{-d e+c f}}{\sqrt{f}}-b \sqrt{c+d x}\right )}{2 f^{3/2} \sqrt{-d e+c f}}-\frac{b d \cos \left (a-\frac{b \sqrt{-d e+c f}}{\sqrt{f}}\right ) \text{Ci}\left (\frac{b \sqrt{-d e+c f}}{\sqrt{f}}+b \sqrt{c+d x}\right )}{2 f^{3/2} \sqrt{-d e+c f}}-\frac{\sin \left (a+b \sqrt{c+d x}\right )}{f (e+f x)}+\frac{b d \sin \left (a+\frac{b \sqrt{-d e+c f}}{\sqrt{f}}\right ) \text{Si}\left (\frac{b \sqrt{-d e+c f}}{\sqrt{f}}-b \sqrt{c+d x}\right )}{2 f^{3/2} \sqrt{-d e+c f}}+\frac{b d \sin \left (a-\frac{b \sqrt{-d e+c f}}{\sqrt{f}}\right ) \text{Si}\left (\frac{b \sqrt{-d e+c f}}{\sqrt{f}}+b \sqrt{c+d x}\right )}{2 f^{3/2} \sqrt{-d e+c f}}\\ \end{align*}

Mathematica [C]  time = 3.57704, size = 397, normalized size = 1.17 \[ \frac{i e^{-i a} d \left (e^{2 i a} \left (-\frac{i b e^{\frac{i b \sqrt{c f-d e}}{\sqrt{f}}} \text{Ei}\left (i b \left (\sqrt{c+d x}-\frac{\sqrt{c f-d e}}{\sqrt{f}}\right )\right )}{\sqrt{c f-d e}}+\frac{i b e^{-\frac{i b \sqrt{c f-d e}}{\sqrt{f}}} \text{Ei}\left (i b \left (\frac{\sqrt{c f-d e}}{\sqrt{f}}+\sqrt{c+d x}\right )\right )}{\sqrt{c f-d e}}+\frac{2 \sqrt{f} e^{i b \sqrt{c+d x}}}{d e+d f x}\right )-\frac{i b e^{-\frac{i b \sqrt{c f-d e}}{\sqrt{f}}} \text{Ei}\left (-i b \left (\sqrt{c+d x}-\frac{\sqrt{c f-d e}}{\sqrt{f}}\right )\right )}{\sqrt{c f-d e}}+\frac{i b e^{\frac{i b \sqrt{c f-d e}}{\sqrt{f}}} \text{Ei}\left (-i b \left (\frac{\sqrt{c f-d e}}{\sqrt{f}}+\sqrt{c+d x}\right )\right )}{\sqrt{c f-d e}}-\frac{2 \sqrt{f} e^{-i b \sqrt{c+d x}}}{d e+d f x}\right )}{4 f^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((I/4)*d*((-2*Sqrt[f])/(E^(I*b*Sqrt[c + d*x])*(d*e + d*f*x)) - (I*b*ExpIntegralEi[(-I)*b*(-(Sqrt[-(d*e) + c*f]
/Sqrt[f]) + Sqrt[c + d*x])])/(E^((I*b*Sqrt[-(d*e) + c*f])/Sqrt[f])*Sqrt[-(d*e) + c*f]) + (I*b*E^((I*b*Sqrt[-(d
*e) + c*f])/Sqrt[f])*ExpIntegralEi[(-I)*b*(Sqrt[-(d*e) + c*f]/Sqrt[f] + Sqrt[c + d*x])])/Sqrt[-(d*e) + c*f] +
E^((2*I)*a)*((2*E^(I*b*Sqrt[c + d*x])*Sqrt[f])/(d*e + d*f*x) - (I*b*E^((I*b*Sqrt[-(d*e) + c*f])/Sqrt[f])*ExpIn
tegralEi[I*b*(-(Sqrt[-(d*e) + c*f]/Sqrt[f]) + Sqrt[c + d*x])])/Sqrt[-(d*e) + c*f] + (I*b*ExpIntegralEi[I*b*(Sq
rt[-(d*e) + c*f]/Sqrt[f] + Sqrt[c + d*x])])/(E^((I*b*Sqrt[-(d*e) + c*f])/Sqrt[f])*Sqrt[-(d*e) + c*f]))))/(E^(I
*a)*f^(3/2))

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Maple [B]  time = 0.048, size = 1817, normalized size = 5.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*d/b^2*(sin(a+b*(d*x+c)^(1/2))*(-1/2*a*b^2/(c*f-d*e)*(a+b*(d*x+c)^(1/2))+1/2*b^2*(-b^2*c*f+b^2*d*e+a^2*f)/(c*
f-d*e)/f)/(-c*f*b^2+d*e*b^2+(a+b*(d*x+c)^(1/2))^2*f-2*(a+b*(d*x+c)^(1/2))*a*f+a^2*f)-1/4*a*b^2/(c*f-d*e)/f/((a
*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f-a)*(Si(b*(d*x+c)^(1/2)+a-(a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)*cos((a*f+(b^2*
c*f^2-b^2*d*e*f)^(1/2))/f)+Ci(b*(d*x+c)^(1/2)+a-(a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)*sin((a*f+(b^2*c*f^2-b^2*d
*e*f)^(1/2))/f))-1/4*a*b^2/(c*f-d*e)/f/(-(-a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f-a)*(Si(b*(d*x+c)^(1/2)+a+(-a*f+(
b^2*c*f^2-b^2*d*e*f)^(1/2))/f)*cos((-a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)-Ci(b*(d*x+c)^(1/2)+a+(-a*f+(b^2*c*f^2
-b^2*d*e*f)^(1/2))/f)*sin((-a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f))+1/4*b^2*(c*f*b^2-d*e*b^2+(a*f+(b^2*c*f^2-b^2*
d*e*f)^(1/2))*a-a^2*f)/(c*f-d*e)/f^2/((a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f-a)*(-Si(b*(d*x+c)^(1/2)+a-(a*f+(b^2*
c*f^2-b^2*d*e*f)^(1/2))/f)*sin((a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)+Ci(b*(d*x+c)^(1/2)+a-(a*f+(b^2*c*f^2-b^2*d
*e*f)^(1/2))/f)*cos((a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f))+1/4*b^2*(c*f*b^2-d*e*b^2-(-a*f+(b^2*c*f^2-b^2*d*e*f)
^(1/2))*a-a^2*f)/(c*f-d*e)/f^2/(-(-a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f-a)*(Si(b*(d*x+c)^(1/2)+a+(-a*f+(b^2*c*f^
2-b^2*d*e*f)^(1/2))/f)*sin((-a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)+Ci(b*(d*x+c)^(1/2)+a+(-a*f+(b^2*c*f^2-b^2*d*e
*f)^(1/2))/f)*cos((-a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f))-a*b^4*(sin(a+b*(d*x+c)^(1/2))*(-1/2/b^2/(c*f-d*e)*(a+
b*(d*x+c)^(1/2))+1/2*a/b^2/(c*f-d*e))/(-c*f*b^2+d*e*b^2+(a+b*(d*x+c)^(1/2))^2*f-2*(a+b*(d*x+c)^(1/2))*a*f+a^2*
f)-1/4/b^2/(c*f-d*e)/f/((a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f-a)*(Si(b*(d*x+c)^(1/2)+a-(a*f+(b^2*c*f^2-b^2*d*e*f
)^(1/2))/f)*cos((a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)+Ci(b*(d*x+c)^(1/2)+a-(a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)
*sin((a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f))-1/4/b^2/(c*f-d*e)/f/(-(-a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f-a)*(Si(b
*(d*x+c)^(1/2)+a+(-a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)*cos((-a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)-Ci(b*(d*x+c)^
(1/2)+a+(-a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)*sin((-a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f))+1/4/f/b^2/(c*f-d*e)*(
-Si(b*(d*x+c)^(1/2)+a-(a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)*sin((a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)+Ci(b*(d*x+
c)^(1/2)+a-(a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)*cos((a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f))+1/4/f/b^2/(c*f-d*e)*
(Si(b*(d*x+c)^(1/2)+a+(-a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)*sin((-a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)+Ci(b*(d*
x+c)^(1/2)+a+(-a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)*cos((-a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f))))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C]  time = 1.96785, size = 873, normalized size = 2.58 \begin{align*} -\frac{{\left (-i \, d f x - i \, d e\right )} \sqrt{\frac{b^{2} d e - b^{2} c f}{f}}{\rm Ei}\left (i \, \sqrt{d x + c} b - \sqrt{\frac{b^{2} d e - b^{2} c f}{f}}\right ) e^{\left (i \, a + \sqrt{\frac{b^{2} d e - b^{2} c f}{f}}\right )} +{\left (i \, d f x + i \, d e\right )} \sqrt{\frac{b^{2} d e - b^{2} c f}{f}}{\rm Ei}\left (i \, \sqrt{d x + c} b + \sqrt{\frac{b^{2} d e - b^{2} c f}{f}}\right ) e^{\left (i \, a - \sqrt{\frac{b^{2} d e - b^{2} c f}{f}}\right )} +{\left (i \, d f x + i \, d e\right )} \sqrt{\frac{b^{2} d e - b^{2} c f}{f}}{\rm Ei}\left (-i \, \sqrt{d x + c} b - \sqrt{\frac{b^{2} d e - b^{2} c f}{f}}\right ) e^{\left (-i \, a + \sqrt{\frac{b^{2} d e - b^{2} c f}{f}}\right )} +{\left (-i \, d f x - i \, d e\right )} \sqrt{\frac{b^{2} d e - b^{2} c f}{f}}{\rm Ei}\left (-i \, \sqrt{d x + c} b + \sqrt{\frac{b^{2} d e - b^{2} c f}{f}}\right ) e^{\left (-i \, a - \sqrt{\frac{b^{2} d e - b^{2} c f}{f}}\right )} + 4 \,{\left (d e - c f\right )} \sin \left (\sqrt{d x + c} b + a\right )}{4 \,{\left (d e^{2} f - c e f^{2} +{\left (d e f^{2} - c f^{3}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((-I*d*f*x - I*d*e)*sqrt((b^2*d*e - b^2*c*f)/f)*Ei(I*sqrt(d*x + c)*b - sqrt((b^2*d*e - b^2*c*f)/f))*e^(I*
a + sqrt((b^2*d*e - b^2*c*f)/f)) + (I*d*f*x + I*d*e)*sqrt((b^2*d*e - b^2*c*f)/f)*Ei(I*sqrt(d*x + c)*b + sqrt((
b^2*d*e - b^2*c*f)/f))*e^(I*a - sqrt((b^2*d*e - b^2*c*f)/f)) + (I*d*f*x + I*d*e)*sqrt((b^2*d*e - b^2*c*f)/f)*E
i(-I*sqrt(d*x + c)*b - sqrt((b^2*d*e - b^2*c*f)/f))*e^(-I*a + sqrt((b^2*d*e - b^2*c*f)/f)) + (-I*d*f*x - I*d*e
)*sqrt((b^2*d*e - b^2*c*f)/f)*Ei(-I*sqrt(d*x + c)*b + sqrt((b^2*d*e - b^2*c*f)/f))*e^(-I*a - sqrt((b^2*d*e - b
^2*c*f)/f)) + 4*(d*e - c*f)*sin(sqrt(d*x + c)*b + a))/(d*e^2*f - c*e*f^2 + (d*e*f^2 - c*f^3)*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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