3.2.98 \(\int (e+f x) \sin (a+\frac {b}{\sqrt {c+d x}}) \, dx\) [198]

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

[Out]

1/6*b*f*(d*x+c)^(3/2)*cos(a+b/(d*x+c)^(1/2))/d^2-1/12*b^4*f*cos(a)*Si(b/(d*x+c)^(1/2))/d^2+b^2*(-c*f+d*e)*cos(
a)*Si(b/(d*x+c)^(1/2))/d^2-1/12*b^4*f*Ci(b/(d*x+c)^(1/2))*sin(a)/d^2+b^2*(-c*f+d*e)*Ci(b/(d*x+c)^(1/2))*sin(a)
/d^2-1/12*b^2*f*(d*x+c)*sin(a+b/(d*x+c)^(1/2))/d^2+(-c*f+d*e)*(d*x+c)*sin(a+b/(d*x+c)^(1/2))/d^2+1/2*f*(d*x+c)
^2*sin(a+b/(d*x+c)^(1/2))/d^2-1/12*b^3*f*cos(a+b/(d*x+c)^(1/2))*(d*x+c)^(1/2)/d^2+b*(-c*f+d*e)*cos(a+b/(d*x+c)
^(1/2))*(d*x+c)^(1/2)/d^2

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Rubi [A]
time = 0.27, antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3512, 3378, 3384, 3380, 3383} \begin {gather*} -\frac {b^4 f \sin (a) \text {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right )}{12 d^2}-\frac {b^4 f \cos (a) \text {Si}\left (\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 {b^2 \sin (a) (d e-c f) \text {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right )}{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^2 f (c+d x) \sin \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*(b^3*f*Sqrt[c + d*x]*Cos[a + b/Sqrt[c + d*x]])/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 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) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx &=-\frac {2 \text {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) \text {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)) \text {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) \text {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)) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.41, size = 367, normalized size = 1.22 \begin {gather*} \frac {e \sqrt {c+d x} \cos \left (\frac {b}{\sqrt {c+d x}}\right ) \left (b \cos (a)+\sqrt {c+d x} \sin (a)\right )}{d}+\frac {f \sqrt {c+d x} \cos \left (\frac {b}{\sqrt {c+d x}}\right ) \left (-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)\right )}{12 d^2}+\frac {e \sqrt {c+d x} \left (\sqrt {c+d x} \cos (a)-b \sin (a)\right ) \sin \left (\frac {b}{\sqrt {c+d x}}\right )}{d}+\frac {f \sqrt {c+d x} \left (-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)\right ) \sin \left (\frac {b}{\sqrt {c+d x}}\right )}{12 d^2}+\frac {b^2 e \left (\text {Ci}\left (\frac {b}{\sqrt {c+d x}}\right ) \sin (a)+\cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )\right )}{d}-\frac {b^2 \left (b^2+12 c\right ) f \left (\text {Ci}\left (\frac {b}{\sqrt {c+d x}}\right ) \sin (a)+\cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )\right )}{12 d^2} \end {gather*}

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 = 0.03, size = 295, normalized size = 0.98

method	result	size
derivativedivides	\(-\frac {2 b^{2} \left (-c f \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{2 b^{2}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{2 b}-\frac {\sinIntegral \left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{2}-\frac {\cosineIntegral \left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{2}\right )+d e \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{2 b^{2}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{2 b}-\frac {\sinIntegral \left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{2}-\frac {\cosineIntegral \left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{2}\right )+f \,b^{2} \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{2}}{4 b^{4}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{\frac {3}{2}}}{12 b^{3}}+\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{24 b^{2}}+\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{24 b}+\frac {\sinIntegral \left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{24}+\frac {\cosineIntegral \left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{24}\right )\right )}{d^{2}}\)	\(295\)
default	\(-\frac {2 b^{2} \left (-c f \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{2 b^{2}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{2 b}-\frac {\sinIntegral \left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{2}-\frac {\cosineIntegral \left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{2}\right )+d e \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{2 b^{2}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{2 b}-\frac {\sinIntegral \left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{2}-\frac {\cosineIntegral \left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{2}\right )+f \,b^{2} \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{2}}{4 b^{4}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{\frac {3}{2}}}{12 b^{3}}+\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{24 b^{2}}+\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{24 b}+\frac {\sinIntegral \left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{24}+\frac {\cosineIntegral \left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{24}\right )\right )}{d^{2}}\)	\(295\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/d^2*b^2*(-c*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))+d*e*(-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))+f*b^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*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)+1/24*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] Result contains complex when optimal does not.
time = 0.53, size = 409, normalized size = 1.36 \begin {gather*} -\frac {\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} - 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 {{\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 {gather*}

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/sq
rt(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 - 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)))*e - (((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)))*f/d)/d

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Fricas [A]
time = 0.38, size = 243, normalized size = 0.81 \begin {gather*} \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 (b^{2} d f x - 6 \, d^{2} f x^{2} + {\left (b^{2} c + 6 \, c^{2}\right )} f - 12 \, {\left (d^{2} x + c d\right )} e\right )} \sin \left (\frac {a d x + a c + \sqrt {d x + c} b}{d x + c}\right )}{24 \, d^{2}} \end {gather*}

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*(b^2*d*f*x - 6*d^2*f*x^2 + (b^2*c + 6*c^2)*f - 12*(d^2*x + c*d)*e)*sin((a*d*x + a*c + sqrt(d*x + c)*b)/
(d*x + c)))/d^2

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (e + f x\right ) \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)*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 [B] Leaf count of result is larger than twice the leaf count of optimal. 2159 vs. \(2 (271) = 542\).
time = 5.17, size = 2159, normalized size = 7.17 \begin {gather*} \text {Too large to display} \end {gather*}

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]

1/12*(12*(a^2*b^3*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(a) - a^2*b^3*cos(a)*sin_integral(
a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c)) - 2*(sqrt(d*x + c)*a + b)*a*b^3*cos_integral(-a + (sqrt(d*x + c)*a +
b)/sqrt(d*x + c))*sin(a)/sqrt(d*x + c) + 2*(sqrt(d*x + c)*a + b)*a*b^3*cos(a)*sin_integral(a - (sqrt(d*x + c)*
a + b)/sqrt(d*x + c))/sqrt(d*x + c) + (sqrt(d*x + c)*a + b)^2*b^3*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt
(d*x + c))*sin(a)/(d*x + c) - (sqrt(d*x + c)*a + b)^2*b^3*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d
*x + c))/(d*x + c) - a*b^3*cos((sqrt(d*x + c)*a + b)/sqrt(d*x + c)) + (sqrt(d*x + c)*a + b)*b^3*cos((sqrt(d*x
+ c)*a + b)/sqrt(d*x + c))/sqrt(d*x + c) + b^3*sin((sqrt(d*x + c)*a + b)/sqrt(d*x + c)))*e/((a^2 - 2*(sqrt(d*x
+ c)*a + b)*a/sqrt(d*x + c) + (sqrt(d*x + c)*a + b)^2/(d*x + c))*b) - (a^4*b^5*cos_integral(-a + (sqrt(d*x +
c)*a + b)/sqrt(d*x + c))*sin(a) - a^4*b^5*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c)) - 4*(sq
rt(d*x + c)*a + b)*a^3*b^5*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(a)/sqrt(d*x + c) + 4*(sq
rt(d*x + c)*a + b)*a^3*b^5*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c))/sqrt(d*x + c) + 6*(sqr
t(d*x + c)*a + b)^2*a^2*b^5*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(a)/(d*x + c) + 12*a^4*b
^3*c*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(a) - 6*(sqrt(d*x + c)*a + b)^2*a^2*b^5*cos(a)*
sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c))/(d*x + c) - 12*a^4*b^3*c*cos(a)*sin_integral(a - (sqrt(d
*x + c)*a + b)/sqrt(d*x + c)) - 4*(sqrt(d*x + c)*a + b)^3*a*b^5*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d
*x + c))*sin(a)/(d*x + c)^(3/2) - 48*(sqrt(d*x + c)*a + b)*a^3*b^3*c*cos_integral(-a + (sqrt(d*x + c)*a + b)/s
qrt(d*x + c))*sin(a)/sqrt(d*x + c) + 4*(sqrt(d*x + c)*a + b)^3*a*b^5*cos(a)*sin_integral(a - (sqrt(d*x + c)*a
+ b)/sqrt(d*x + c))/(d*x + c)^(3/2) + 48*(sqrt(d*x + c)*a + b)*a^3*b^3*c*cos(a)*sin_integral(a - (sqrt(d*x + c
)*a + b)/sqrt(d*x + c))/sqrt(d*x + c) - a^3*b^5*cos((sqrt(d*x + c)*a + b)/sqrt(d*x + c)) + (sqrt(d*x + c)*a +
b)^4*b^5*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(a)/(d*x + c)^2 + 72*(sqrt(d*x + c)*a + b)^
2*a^2*b^3*c*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(a)/(d*x + c) - (sqrt(d*x + c)*a + b)^4*
b^5*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c))/(d*x + c)^2 - 72*(sqrt(d*x + c)*a + b)^2*a^2*
b^3*c*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c))/(d*x + c) + 3*(sqrt(d*x + c)*a + b)*a^2*b^5
*cos((sqrt(d*x + c)*a + b)/sqrt(d*x + c))/sqrt(d*x + c) - 48*(sqrt(d*x + c)*a + b)^3*a*b^3*c*cos_integral(-a +
(sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(a)/(d*x + c)^(3/2) + 48*(sqrt(d*x + c)*a + b)^3*a*b^3*c*cos(a)*sin_i
ntegral(a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c))/(d*x + c)^(3/2) - 3*(sqrt(d*x + c)*a + b)^2*a*b^5*cos((sqrt(d
*x + c)*a + b)/sqrt(d*x + c))/(d*x + c) - 12*a^3*b^3*c*cos((sqrt(d*x + c)*a + b)/sqrt(d*x + c)) + 12*(sqrt(d*x
+ c)*a + b)^4*b^3*c*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(a)/(d*x + c)^2 + a^2*b^5*sin((
sqrt(d*x + c)*a + b)/sqrt(d*x + c)) - 12*(sqrt(d*x + c)*a + b)^4*b^3*c*cos(a)*sin_integral(a - (sqrt(d*x + c)*
a + b)/sqrt(d*x + c))/(d*x + c)^2 + (sqrt(d*x + c)*a + b)^3*b^5*cos((sqrt(d*x + c)*a + b)/sqrt(d*x + c))/(d*x
+ c)^(3/2) + 36*(sqrt(d*x + c)*a + b)*a^2*b^3*c*cos((sqrt(d*x + c)*a + b)/sqrt(d*x + c))/sqrt(d*x + c) - 2*(sq
rt(d*x + c)*a + b)*a*b^5*sin((sqrt(d*x + c)*a + b)/sqrt(d*x + c))/sqrt(d*x + c) + 2*a*b^5*cos((sqrt(d*x + c)*a
+ b)/sqrt(d*x + c)) - 36*(sqrt(d*x + c)*a + b)^2*a*b^3*c*cos((sqrt(d*x + c)*a + b)/sqrt(d*x + c))/(d*x + c) +
(sqrt(d*x + c)*a + b)^2*b^5*sin((sqrt(d*x + c)*a + b)/sqrt(d*x + c))/(d*x + c) + 12*a^2*b^3*c*sin((sqrt(d*x +
c)*a + b)/sqrt(d*x + c)) - 2*(sqrt(d*x + c)*a + b)*b^5*cos((sqrt(d*x + c)*a + b)/sqrt(d*x + c))/sqrt(d*x + c)
+ 12*(sqrt(d*x + c)*a + b)^3*b^3*c*cos((sqrt(d*x + c)*a + b)/sqrt(d*x + c))/(d*x + c)^(3/2) - 24*(sqrt(d*x +
c)*a + b)*a*b^3*c*sin((sqrt(d*x + c)*a + b)/sqrt(d*x + c))/sqrt(d*x + c) - 6*b^5*sin((sqrt(d*x + c)*a + b)/sqr
t(d*x + c)) + 12*(sqrt(d*x + c)*a + b)^2*b^3*c*sin((sqrt(d*x + c)*a + b)/sqrt(d*x + c))/(d*x + c))*f/((a^4 - 4
*(sqrt(d*x + c)*a + b)*a^3/sqrt(d*x + c) + 6*(sqrt(d*x + c)*a + b)^2*a^2/(d*x + c) - 4*(sqrt(d*x + c)*a + b)^3
*a/(d*x + c)^(3/2) + (sqrt(d*x + c)*a + b)^4/(d*x + c)^2)*b*d))/d

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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 ) \,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),x)

[Out]

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

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