3.2.91 \(\int \frac {\sin (a+b \sqrt {c+d x})}{(e+f x)^2} \, dx\) [191]

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

[Out]

-sin(a+b*(d*x+c)^(1/2))/f/(f*x+e)-1/2*b*d*Ci(b*(c*f-d*e)^(1/2)/f^(1/2)+b*(d*x+c)^(1/2))*cos(a-b*(c*f-d*e)^(1/2
)/f^(1/2))/f^(3/2)/(c*f-d*e)^(1/2)+1/2*b*d*Ci(b*(c*f-d*e)^(1/2)/f^(1/2)-b*(d*x+c)^(1/2))*cos(a+b*(c*f-d*e)^(1/
2)/f^(1/2))/f^(3/2)/(c*f-d*e)^(1/2)+1/2*b*d*Si(b*(c*f-d*e)^(1/2)/f^(1/2)+b*(d*x+c)^(1/2))*sin(a-b*(c*f-d*e)^(1
/2)/f^(1/2))/f^(3/2)/(c*f-d*e)^(1/2)-1/2*b*d*Si(-b*(c*f-d*e)^(1/2)/f^(1/2)+b*(d*x+c)^(1/2))*sin(a+b*(c*f-d*e)^
(1/2)/f^(1/2))/f^(3/2)/(c*f-d*e)^(1/2)

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Rubi [A]
time = 0.68, antiderivative size = 339, normalized size of antiderivative = 1.00, 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 = {3512, 3422, 3415, 3384, 3380, 3383} \begin {gather*} \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)} \end {gather*}

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 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 3415

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 3422

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 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 \frac {\sin \left (a+b \sqrt {c+d x}\right )}{(e+f x)^2} \, dx &=\frac {2 \text {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 \text {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 \text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 2.24, size = 397, normalized size = 1.17 \begin {gather*} \frac {i d e^{-i a} \left (-\frac {2 e^{-i b \sqrt {c+d x}} \sqrt {f}}{d e+d f x}-\frac {i b e^{-\frac {i b \sqrt {-d e+c f}}{\sqrt {f}}} \text {Ei}\left (-i b \left (-\frac {\sqrt {-d e+c f}}{\sqrt {f}}+\sqrt {c+d x}\right )\right )}{\sqrt {-d e+c f}}+\frac {i b e^{\frac {i b \sqrt {-d e+c f}}{\sqrt {f}}} \text {Ei}\left (-i b \left (\frac {\sqrt {-d e+c f}}{\sqrt {f}}+\sqrt {c+d x}\right )\right )}{\sqrt {-d e+c f}}+e^{2 i a} \left (\frac {2 e^{i b \sqrt {c+d x}} \sqrt {f}}{d e+d f x}-\frac {i b e^{\frac {i b \sqrt {-d e+c f}}{\sqrt {f}}} \text {Ei}\left (i b \left (-\frac {\sqrt {-d e+c f}}{\sqrt {f}}+\sqrt {c+d x}\right )\right )}{\sqrt {-d e+c f}}+\frac {i b e^{-\frac {i b \sqrt {-d e+c f}}{\sqrt {f}}} \text {Ei}\left (i b \left (\frac {\sqrt {-d e+c f}}{\sqrt {f}}+\sqrt {c+d x}\right )\right )}{\sqrt {-d e+c f}}\right )\right )}{4 f^{3/2}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(1830\) vs. \(2(273)=546\).
time = 0.07, size = 1831, normalized size = 5.40

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1831\)
default	\(\text {Expression too large to display}\)	\(1831\)

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,method=_RETURNVERBOSE)

[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^2*f-2*a*f*(a+b*(d*x+c)^(1/2))+f*(a+b*(d*x+c)^(1/2))^2)+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^2*f-a*(a*f+(b
^2*c*f^2-b^2*d*e*f)^(1/2)))/(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^2*f+a*(-a*f+(b^2*c
*f^2-b^2*d*e*f)^(1/2)))/(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^2*f-2*a*f*(a+b*(d*x+c)^(1/2))+f*(a+b*(d*x+c)
^(1/2))^2)+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)-C
i(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Result contains complex when optimal does not.
time = 0.47, size = 446, normalized size = 1.32 \begin {gather*} -\frac {{\left (i \, d f x + i \, d e\right )} \sqrt {-\frac {b^{2} c f - b^{2} d e}{f}} {\rm Ei}\left (i \, \sqrt {d x + c} b - \sqrt {-\frac {b^{2} c f - b^{2} d e}{f}}\right ) e^{\left (i \, a + \sqrt {-\frac {b^{2} c f - b^{2} d e}{f}}\right )} + {\left (-i \, d f x - i \, d e\right )} \sqrt {-\frac {b^{2} c f - b^{2} d e}{f}} {\rm Ei}\left (i \, \sqrt {d x + c} b + \sqrt {-\frac {b^{2} c f - b^{2} d e}{f}}\right ) e^{\left (i \, a - \sqrt {-\frac {b^{2} c f - b^{2} d e}{f}}\right )} + {\left (-i \, d f x - i \, d e\right )} \sqrt {-\frac {b^{2} c f - b^{2} d e}{f}} {\rm Ei}\left (-i \, \sqrt {d x + c} b - \sqrt {-\frac {b^{2} c f - b^{2} d e}{f}}\right ) e^{\left (-i \, a + \sqrt {-\frac {b^{2} c f - b^{2} d e}{f}}\right )} + {\left (i \, d f x + i \, d e\right )} \sqrt {-\frac {b^{2} c f - b^{2} d e}{f}} {\rm Ei}\left (-i \, \sqrt {d x + c} b + \sqrt {-\frac {b^{2} c f - b^{2} d e}{f}}\right ) e^{\left (-i \, a - \sqrt {-\frac {b^{2} c f - b^{2} d e}{f}}\right )} + 4 \, {\left (c f - d e\right )} \sin \left (\sqrt {d x + c} b + a\right )}{4 \, {\left (c f^{3} x - d f e^{2} - {\left (d f^{2} x - c f^{2}\right )} e\right )}} \end {gather*}

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

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

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sin \left (a+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)

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