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

Optimal. Leaf size=410 \[ \frac{12 f (c+d x) (d e-c f) \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}+\frac{2 (d e-c f)^2 \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{24 f (d e-c f) \sin \left (a+b \sqrt{c+d x}\right )}{b^4 d^3}+\frac{24 f \sqrt{c+d x} (d e-c f) \cos \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}+\frac{10 f^2 (c+d x)^2 \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{120 f^2 (c+d x) \sin \left (a+b \sqrt{c+d x}\right )}{b^4 d^3}+\frac{240 f^2 \sin \left (a+b \sqrt{c+d x}\right )}{b^6 d^3}+\frac{40 f^2 (c+d x)^{3/2} \cos \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}-\frac{240 f^2 \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b^5 d^3}-\frac{4 f (c+d x)^{3/2} (d e-c f) \cos \left (a+b \sqrt{c+d x}\right )}{b d^3}-\frac{2 \sqrt{c+d x} (d e-c f)^2 \cos \left (a+b \sqrt{c+d x}\right )}{b d^3}-\frac{2 f^2 (c+d x)^{5/2} \cos \left (a+b \sqrt{c+d x}\right )}{b d^3} \]

[Out]

(-240*f^2*Sqrt[c + d*x]*Cos[a + b*Sqrt[c + d*x]])/(b^5*d^3) + (24*f*(d*e - c*f)*Sqrt[c + d*x]*Cos[a + b*Sqrt[c
+ d*x]])/(b^3*d^3) - (2*(d*e - c*f)^2*Sqrt[c + d*x]*Cos[a + b*Sqrt[c + d*x]])/(b*d^3) + (40*f^2*(c + d*x)^(3/
2)*Cos[a + b*Sqrt[c + d*x]])/(b^3*d^3) - (4*f*(d*e - c*f)*(c + d*x)^(3/2)*Cos[a + b*Sqrt[c + d*x]])/(b*d^3) -
(2*f^2*(c + d*x)^(5/2)*Cos[a + b*Sqrt[c + d*x]])/(b*d^3) + (240*f^2*Sin[a + b*Sqrt[c + d*x]])/(b^6*d^3) - (24*
f*(d*e - c*f)*Sin[a + b*Sqrt[c + d*x]])/(b^4*d^3) + (2*(d*e - c*f)^2*Sin[a + b*Sqrt[c + d*x]])/(b^2*d^3) - (12
0*f^2*(c + d*x)*Sin[a + b*Sqrt[c + d*x]])/(b^4*d^3) + (12*f*(d*e - c*f)*(c + d*x)*Sin[a + b*Sqrt[c + d*x]])/(b
^2*d^3) + (10*f^2*(c + d*x)^2*Sin[a + b*Sqrt[c + d*x]])/(b^2*d^3)

________________________________________________________________________________________

Rubi [A]  time = 0.39884, antiderivative size = 410, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {3431, 3296, 2637} \[ \frac{12 f (c+d x) (d e-c f) \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}+\frac{2 (d e-c f)^2 \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{24 f (d e-c f) \sin \left (a+b \sqrt{c+d x}\right )}{b^4 d^3}+\frac{24 f \sqrt{c+d x} (d e-c f) \cos \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}+\frac{10 f^2 (c+d x)^2 \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{120 f^2 (c+d x) \sin \left (a+b \sqrt{c+d x}\right )}{b^4 d^3}+\frac{240 f^2 \sin \left (a+b \sqrt{c+d x}\right )}{b^6 d^3}+\frac{40 f^2 (c+d x)^{3/2} \cos \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}-\frac{240 f^2 \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b^5 d^3}-\frac{4 f (c+d x)^{3/2} (d e-c f) \cos \left (a+b \sqrt{c+d x}\right )}{b d^3}-\frac{2 \sqrt{c+d x} (d e-c f)^2 \cos \left (a+b \sqrt{c+d x}\right )}{b d^3}-\frac{2 f^2 (c+d x)^{5/2} \cos \left (a+b \sqrt{c+d x}\right )}{b d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-240*f^2*Sqrt[c + d*x]*Cos[a + b*Sqrt[c + d*x]])/(b^5*d^3) + (24*f*(d*e - c*f)*Sqrt[c + d*x]*Cos[a + b*Sqrt[c
+ d*x]])/(b^3*d^3) - (2*(d*e - c*f)^2*Sqrt[c + d*x]*Cos[a + b*Sqrt[c + d*x]])/(b*d^3) + (40*f^2*(c + d*x)^(3/
2)*Cos[a + b*Sqrt[c + d*x]])/(b^3*d^3) - (4*f*(d*e - c*f)*(c + d*x)^(3/2)*Cos[a + b*Sqrt[c + d*x]])/(b*d^3) -
(2*f^2*(c + d*x)^(5/2)*Cos[a + b*Sqrt[c + d*x]])/(b*d^3) + (240*f^2*Sin[a + b*Sqrt[c + d*x]])/(b^6*d^3) - (24*
f*(d*e - c*f)*Sin[a + b*Sqrt[c + d*x]])/(b^4*d^3) + (2*(d*e - c*f)^2*Sin[a + b*Sqrt[c + d*x]])/(b^2*d^3) - (12
0*f^2*(c + d*x)*Sin[a + b*Sqrt[c + d*x]])/(b^4*d^3) + (12*f*(d*e - c*f)*(c + d*x)*Sin[a + b*Sqrt[c + d*x]])/(b
^2*d^3) + (10*f^2*(c + d*x)^2*Sin[a + b*Sqrt[c + d*x]])/(b^2*d^3)

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 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int (e+f x)^2 \sin \left (a+b \sqrt{c+d x}\right ) \, dx &=\frac{2 \operatorname{Subst}\left (\int \left (\frac{(d e-c f)^2 x \sin (a+b x)}{d^2}+\frac{2 f (d e-c f) x^3 \sin (a+b x)}{d^2}+\frac{f^2 x^5 \sin (a+b x)}{d^2}\right ) \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=\frac{\left (2 f^2\right ) \operatorname{Subst}\left (\int x^5 \sin (a+b x) \, dx,x,\sqrt{c+d x}\right )}{d^3}+\frac{(4 f (d e-c f)) \operatorname{Subst}\left (\int x^3 \sin (a+b x) \, dx,x,\sqrt{c+d x}\right )}{d^3}+\frac{\left (2 (d e-c f)^2\right ) \operatorname{Subst}\left (\int x \sin (a+b x) \, dx,x,\sqrt{c+d x}\right )}{d^3}\\ &=-\frac{2 (d e-c f)^2 \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b d^3}-\frac{4 f (d e-c f) (c+d x)^{3/2} \cos \left (a+b \sqrt{c+d x}\right )}{b d^3}-\frac{2 f^2 (c+d x)^{5/2} \cos \left (a+b \sqrt{c+d x}\right )}{b d^3}+\frac{\left (10 f^2\right ) \operatorname{Subst}\left (\int x^4 \cos (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b d^3}+\frac{(12 f (d e-c f)) \operatorname{Subst}\left (\int x^2 \cos (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b d^3}+\frac{\left (2 (d e-c f)^2\right ) \operatorname{Subst}\left (\int \cos (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b d^3}\\ &=-\frac{2 (d e-c f)^2 \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b d^3}-\frac{4 f (d e-c f) (c+d x)^{3/2} \cos \left (a+b \sqrt{c+d x}\right )}{b d^3}-\frac{2 f^2 (c+d x)^{5/2} \cos \left (a+b \sqrt{c+d x}\right )}{b d^3}+\frac{2 (d e-c f)^2 \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}+\frac{12 f (d e-c f) (c+d x) \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}+\frac{10 f^2 (c+d x)^2 \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{\left (40 f^2\right ) \operatorname{Subst}\left (\int x^3 \sin (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b^2 d^3}-\frac{(24 f (d e-c f)) \operatorname{Subst}\left (\int x \sin (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b^2 d^3}\\ &=\frac{24 f (d e-c f) \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}-\frac{2 (d e-c f)^2 \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b d^3}+\frac{40 f^2 (c+d x)^{3/2} \cos \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}-\frac{4 f (d e-c f) (c+d x)^{3/2} \cos \left (a+b \sqrt{c+d x}\right )}{b d^3}-\frac{2 f^2 (c+d x)^{5/2} \cos \left (a+b \sqrt{c+d x}\right )}{b d^3}+\frac{2 (d e-c f)^2 \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}+\frac{12 f (d e-c f) (c+d x) \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}+\frac{10 f^2 (c+d x)^2 \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{\left (120 f^2\right ) \operatorname{Subst}\left (\int x^2 \cos (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b^3 d^3}-\frac{(24 f (d e-c f)) \operatorname{Subst}\left (\int \cos (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b^3 d^3}\\ &=\frac{24 f (d e-c f) \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}-\frac{2 (d e-c f)^2 \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b d^3}+\frac{40 f^2 (c+d x)^{3/2} \cos \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}-\frac{4 f (d e-c f) (c+d x)^{3/2} \cos \left (a+b \sqrt{c+d x}\right )}{b d^3}-\frac{2 f^2 (c+d x)^{5/2} \cos \left (a+b \sqrt{c+d x}\right )}{b d^3}-\frac{24 f (d e-c f) \sin \left (a+b \sqrt{c+d x}\right )}{b^4 d^3}+\frac{2 (d e-c f)^2 \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{120 f^2 (c+d x) \sin \left (a+b \sqrt{c+d x}\right )}{b^4 d^3}+\frac{12 f (d e-c f) (c+d x) \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}+\frac{10 f^2 (c+d x)^2 \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}+\frac{\left (240 f^2\right ) \operatorname{Subst}\left (\int x \sin (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b^4 d^3}\\ &=-\frac{240 f^2 \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b^5 d^3}+\frac{24 f (d e-c f) \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}-\frac{2 (d e-c f)^2 \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b d^3}+\frac{40 f^2 (c+d x)^{3/2} \cos \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}-\frac{4 f (d e-c f) (c+d x)^{3/2} \cos \left (a+b \sqrt{c+d x}\right )}{b d^3}-\frac{2 f^2 (c+d x)^{5/2} \cos \left (a+b \sqrt{c+d x}\right )}{b d^3}-\frac{24 f (d e-c f) \sin \left (a+b \sqrt{c+d x}\right )}{b^4 d^3}+\frac{2 (d e-c f)^2 \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{120 f^2 (c+d x) \sin \left (a+b \sqrt{c+d x}\right )}{b^4 d^3}+\frac{12 f (d e-c f) (c+d x) \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}+\frac{10 f^2 (c+d x)^2 \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}+\frac{\left (240 f^2\right ) \operatorname{Subst}\left (\int \cos (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b^5 d^3}\\ &=-\frac{240 f^2 \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b^5 d^3}+\frac{24 f (d e-c f) \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}-\frac{2 (d e-c f)^2 \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b d^3}+\frac{40 f^2 (c+d x)^{3/2} \cos \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}-\frac{4 f (d e-c f) (c+d x)^{3/2} \cos \left (a+b \sqrt{c+d x}\right )}{b d^3}-\frac{2 f^2 (c+d x)^{5/2} \cos \left (a+b \sqrt{c+d x}\right )}{b d^3}+\frac{240 f^2 \sin \left (a+b \sqrt{c+d x}\right )}{b^6 d^3}-\frac{24 f (d e-c f) \sin \left (a+b \sqrt{c+d x}\right )}{b^4 d^3}+\frac{2 (d e-c f)^2 \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{120 f^2 (c+d x) \sin \left (a+b \sqrt{c+d x}\right )}{b^4 d^3}+\frac{12 f (d e-c f) (c+d x) \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}+\frac{10 f^2 (c+d x)^2 \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}\\ \end{align*}

Mathematica [A]  time = 1.76851, size = 138, normalized size = 0.34 \[ \frac{2 \sin \left (a+b \sqrt{c+d x}\right ) \left (b^4 d (e+f x) (4 c f+d (e+5 f x))-12 b^2 f (4 c f+d (e+5 f x))+120 f^2\right )-2 b \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right ) \left (-4 b^2 f (2 c f+3 d e+5 d f x)+b^4 d^2 (e+f x)^2+120 f^2\right )}{b^6 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*Sqrt[c + d*x]*(120*f^2 + b^4*d^2*(e + f*x)^2 - 4*b^2*f*(3*d*e + 2*c*f + 5*d*f*x))*Cos[a + b*Sqrt[c + d*x
]] + 2*(120*f^2 - 12*b^2*f*(4*c*f + d*(e + 5*f*x)) + b^4*d*(e + f*x)*(4*c*f + d*(e + 5*f*x)))*Sin[a + b*Sqrt[c
+ d*x]])/(b^6*d^3)

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Maple [B]  time = 0.013, size = 1246, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

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)

[Out]

2/d^3/b^2*(c^2*f^2*(sin(a+b*(d*x+c)^(1/2))-(a+b*(d*x+c)^(1/2))*cos(a+b*(d*x+c)^(1/2)))-2*c*d*e*f*(sin(a+b*(d*x
+c)^(1/2))-(a+b*(d*x+c)^(1/2))*cos(a+b*(d*x+c)^(1/2)))+d^2*e^2*(sin(a+b*(d*x+c)^(1/2))-(a+b*(d*x+c)^(1/2))*cos
(a+b*(d*x+c)^(1/2)))+a*c^2*f^2*cos(a+b*(d*x+c)^(1/2))-2*a*c*d*e*f*cos(a+b*(d*x+c)^(1/2))+a*d^2*e^2*cos(a+b*(d*
x+c)^(1/2))-2/b^2*c*f^2*(-(a+b*(d*x+c)^(1/2))^3*cos(a+b*(d*x+c)^(1/2))+3*(a+b*(d*x+c)^(1/2))^2*sin(a+b*(d*x+c)
^(1/2))-6*sin(a+b*(d*x+c)^(1/2))+6*(a+b*(d*x+c)^(1/2))*cos(a+b*(d*x+c)^(1/2)))+2/b^2*d*e*f*(-(a+b*(d*x+c)^(1/2
))^3*cos(a+b*(d*x+c)^(1/2))+3*(a+b*(d*x+c)^(1/2))^2*sin(a+b*(d*x+c)^(1/2))-6*sin(a+b*(d*x+c)^(1/2))+6*(a+b*(d*
x+c)^(1/2))*cos(a+b*(d*x+c)^(1/2)))+6/b^2*a*c*f^2*(-(a+b*(d*x+c)^(1/2))^2*cos(a+b*(d*x+c)^(1/2))+2*cos(a+b*(d*
x+c)^(1/2))+2*(a+b*(d*x+c)^(1/2))*sin(a+b*(d*x+c)^(1/2)))-6/b^2*a*d*e*f*(-(a+b*(d*x+c)^(1/2))^2*cos(a+b*(d*x+c
)^(1/2))+2*cos(a+b*(d*x+c)^(1/2))+2*(a+b*(d*x+c)^(1/2))*sin(a+b*(d*x+c)^(1/2)))-6/b^2*a^2*c*f^2*(sin(a+b*(d*x+
c)^(1/2))-(a+b*(d*x+c)^(1/2))*cos(a+b*(d*x+c)^(1/2)))+6/b^2*a^2*d*e*f*(sin(a+b*(d*x+c)^(1/2))-(a+b*(d*x+c)^(1/
2))*cos(a+b*(d*x+c)^(1/2)))-2/b^2*a^3*c*f^2*cos(a+b*(d*x+c)^(1/2))+2/b^2*a^3*d*e*f*cos(a+b*(d*x+c)^(1/2))+1/b^
4*f^2*(-(a+b*(d*x+c)^(1/2))^5*cos(a+b*(d*x+c)^(1/2))+5*(a+b*(d*x+c)^(1/2))^4*sin(a+b*(d*x+c)^(1/2))+20*(a+b*(d
*x+c)^(1/2))^3*cos(a+b*(d*x+c)^(1/2))-60*(a+b*(d*x+c)^(1/2))^2*sin(a+b*(d*x+c)^(1/2))+120*sin(a+b*(d*x+c)^(1/2
))-120*(a+b*(d*x+c)^(1/2))*cos(a+b*(d*x+c)^(1/2)))-5/b^4*a*f^2*(-(a+b*(d*x+c)^(1/2))^4*cos(a+b*(d*x+c)^(1/2))+
4*(a+b*(d*x+c)^(1/2))^3*sin(a+b*(d*x+c)^(1/2))+12*(a+b*(d*x+c)^(1/2))^2*cos(a+b*(d*x+c)^(1/2))-24*cos(a+b*(d*x
+c)^(1/2))-24*(a+b*(d*x+c)^(1/2))*sin(a+b*(d*x+c)^(1/2)))+10/b^4*a^2*f^2*(-(a+b*(d*x+c)^(1/2))^3*cos(a+b*(d*x+
c)^(1/2))+3*(a+b*(d*x+c)^(1/2))^2*sin(a+b*(d*x+c)^(1/2))-6*sin(a+b*(d*x+c)^(1/2))+6*(a+b*(d*x+c)^(1/2))*cos(a+
b*(d*x+c)^(1/2)))-10/b^4*a^3*f^2*(-(a+b*(d*x+c)^(1/2))^2*cos(a+b*(d*x+c)^(1/2))+2*cos(a+b*(d*x+c)^(1/2))+2*(a+
b*(d*x+c)^(1/2))*sin(a+b*(d*x+c)^(1/2)))+5/b^4*a^4*f^2*(sin(a+b*(d*x+c)^(1/2))-(a+b*(d*x+c)^(1/2))*cos(a+b*(d*
x+c)^(1/2)))+1/b^4*a^5*f^2*cos(a+b*(d*x+c)^(1/2)))

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Maxima [B]  time = 1.18474, size = 1486, normalized size = 3.62 \begin{align*} \text{result too large to display} \end{align*}

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]

2*(a*e^2*cos(sqrt(d*x + c)*b + a) - 2*a*c*e*f*cos(sqrt(d*x + c)*b + a)/d + a*c^2*f^2*cos(sqrt(d*x + c)*b + a)/
d^2 - ((sqrt(d*x + c)*b + a)*cos(sqrt(d*x + c)*b + a) - sin(sqrt(d*x + c)*b + a))*e^2 + 2*((sqrt(d*x + c)*b +
a)*cos(sqrt(d*x + c)*b + a) - sin(sqrt(d*x + c)*b + a))*c*e*f/d - ((sqrt(d*x + c)*b + a)*cos(sqrt(d*x + c)*b +
a) - sin(sqrt(d*x + c)*b + a))*c^2*f^2/d^2 + 2*a^3*e*f*cos(sqrt(d*x + c)*b + a)/(b^2*d) - 2*a^3*c*f^2*cos(sqr
t(d*x + c)*b + a)/(b^2*d^2) - 6*((sqrt(d*x + c)*b + a)*cos(sqrt(d*x + c)*b + a) - sin(sqrt(d*x + c)*b + a))*a^
2*e*f/(b^2*d) + 6*((sqrt(d*x + c)*b + a)*cos(sqrt(d*x + c)*b + a) - sin(sqrt(d*x + c)*b + a))*a^2*c*f^2/(b^2*d
^2) + a^5*f^2*cos(sqrt(d*x + c)*b + a)/(b^4*d^2) + 6*(((sqrt(d*x + c)*b + a)^2 - 2)*cos(sqrt(d*x + c)*b + a) -
2*(sqrt(d*x + c)*b + a)*sin(sqrt(d*x + c)*b + a))*a*e*f/(b^2*d) - 5*((sqrt(d*x + c)*b + a)*cos(sqrt(d*x + c)*
b + a) - sin(sqrt(d*x + c)*b + a))*a^4*f^2/(b^4*d^2) - 6*(((sqrt(d*x + c)*b + a)^2 - 2)*cos(sqrt(d*x + c)*b +
a) - 2*(sqrt(d*x + c)*b + a)*sin(sqrt(d*x + c)*b + a))*a*c*f^2/(b^2*d^2) - 2*(((sqrt(d*x + c)*b + a)^3 - 6*sqr
t(d*x + c)*b - 6*a)*cos(sqrt(d*x + c)*b + a) - 3*((sqrt(d*x + c)*b + a)^2 - 2)*sin(sqrt(d*x + c)*b + a))*e*f/(
b^2*d) + 10*(((sqrt(d*x + c)*b + a)^2 - 2)*cos(sqrt(d*x + c)*b + a) - 2*(sqrt(d*x + c)*b + a)*sin(sqrt(d*x + c
)*b + a))*a^3*f^2/(b^4*d^2) + 2*(((sqrt(d*x + c)*b + a)^3 - 6*sqrt(d*x + c)*b - 6*a)*cos(sqrt(d*x + c)*b + a)
- 3*((sqrt(d*x + c)*b + a)^2 - 2)*sin(sqrt(d*x + c)*b + a))*c*f^2/(b^2*d^2) - 10*(((sqrt(d*x + c)*b + a)^3 - 6
*sqrt(d*x + c)*b - 6*a)*cos(sqrt(d*x + c)*b + a) - 3*((sqrt(d*x + c)*b + a)^2 - 2)*sin(sqrt(d*x + c)*b + a))*a
^2*f^2/(b^4*d^2) + 5*(((sqrt(d*x + c)*b + a)^4 - 12*(sqrt(d*x + c)*b + a)^2 + 24)*cos(sqrt(d*x + c)*b + a) - 4
*((sqrt(d*x + c)*b + a)^3 - 6*sqrt(d*x + c)*b - 6*a)*sin(sqrt(d*x + c)*b + a))*a*f^2/(b^4*d^2) - (((sqrt(d*x +
c)*b + a)^5 - 20*(sqrt(d*x + c)*b + a)^3 + 120*sqrt(d*x + c)*b + 120*a)*cos(sqrt(d*x + c)*b + a) - 5*((sqrt(d
*x + c)*b + a)^4 - 12*(sqrt(d*x + c)*b + a)^2 + 24)*sin(sqrt(d*x + c)*b + a))*f^2/(b^4*d^2))/(b^2*d)

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Fricas [A]  time = 1.75107, size = 431, normalized size = 1.05 \begin{align*} -\frac{2 \,{\left ({\left (b^{5} d^{2} f^{2} x^{2} + b^{5} d^{2} e^{2} - 12 \, b^{3} d e f - 8 \,{\left (b^{3} c - 15 \, b\right )} f^{2} + 2 \,{\left (b^{5} d^{2} e f - 10 \, b^{3} d f^{2}\right )} x\right )} \sqrt{d x + c} \cos \left (\sqrt{d x + c} b + a\right ) -{\left (5 \, b^{4} d^{2} f^{2} x^{2} + b^{4} d^{2} e^{2} + 4 \,{\left (b^{4} c - 3 \, b^{2}\right )} d e f - 24 \,{\left (2 \, b^{2} c - 5\right )} f^{2} + 2 \,{\left (3 \, b^{4} d^{2} e f + 2 \,{\left (b^{4} c - 15 \, b^{2}\right )} d f^{2}\right )} x\right )} \sin \left (\sqrt{d x + c} b + a\right )\right )}}{b^{6} d^{3}} \end{align*}

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]

-2*((b^5*d^2*f^2*x^2 + b^5*d^2*e^2 - 12*b^3*d*e*f - 8*(b^3*c - 15*b)*f^2 + 2*(b^5*d^2*e*f - 10*b^3*d*f^2)*x)*s
qrt(d*x + c)*cos(sqrt(d*x + c)*b + a) - (5*b^4*d^2*f^2*x^2 + b^4*d^2*e^2 + 4*(b^4*c - 3*b^2)*d*e*f - 24*(2*b^2
*c - 5)*f^2 + 2*(3*b^4*d^2*e*f + 2*(b^4*c - 15*b^2)*d*f^2)*x)*sin(sqrt(d*x + c)*b + a))/(b^6*d^3)

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Sympy [A]  time = 2.70185, size = 549, normalized size = 1.34 \begin{align*} \begin{cases} \left (e^{2} x + e f x^{2} + \frac{f^{2} x^{3}}{3}\right ) \sin{\left (a \right )} & \text{for}\: b = 0 \wedge d = 0 \\\left (e^{2} x + e f x^{2} + \frac{f^{2} x^{3}}{3}\right ) \sin{\left (a + b \sqrt{c} \right )} & \text{for}\: d = 0 \\\left (e^{2} x + e f x^{2} + \frac{f^{2} x^{3}}{3}\right ) \sin{\left (a \right )} & \text{for}\: b = 0 \\- \frac{2 e^{2} \sqrt{c + d x} \cos{\left (a + b \sqrt{c + d x} \right )}}{b d} - \frac{4 e f x \sqrt{c + d x} \cos{\left (a + b \sqrt{c + d x} \right )}}{b d} - \frac{2 f^{2} x^{2} \sqrt{c + d x} \cos{\left (a + b \sqrt{c + d x} \right )}}{b d} + \frac{8 c e f \sin{\left (a + b \sqrt{c + d x} \right )}}{b^{2} d^{2}} + \frac{8 c f^{2} x \sin{\left (a + b \sqrt{c + d x} \right )}}{b^{2} d^{2}} + \frac{2 e^{2} \sin{\left (a + b \sqrt{c + d x} \right )}}{b^{2} d} + \frac{12 e f x \sin{\left (a + b \sqrt{c + d x} \right )}}{b^{2} d} + \frac{10 f^{2} x^{2} \sin{\left (a + b \sqrt{c + d x} \right )}}{b^{2} d} + \frac{16 c f^{2} \sqrt{c + d x} \cos{\left (a + b \sqrt{c + d x} \right )}}{b^{3} d^{3}} + \frac{24 e f \sqrt{c + d x} \cos{\left (a + b \sqrt{c + d x} \right )}}{b^{3} d^{2}} + \frac{40 f^{2} x \sqrt{c + d x} \cos{\left (a + b \sqrt{c + d x} \right )}}{b^{3} d^{2}} - \frac{96 c f^{2} \sin{\left (a + b \sqrt{c + d x} \right )}}{b^{4} d^{3}} - \frac{24 e f \sin{\left (a + b \sqrt{c + d x} \right )}}{b^{4} d^{2}} - \frac{120 f^{2} x \sin{\left (a + b \sqrt{c + d x} \right )}}{b^{4} d^{2}} - \frac{240 f^{2} \sqrt{c + d x} \cos{\left (a + b \sqrt{c + d x} \right )}}{b^{5} d^{3}} + \frac{240 f^{2} \sin{\left (a + b \sqrt{c + d x} \right )}}{b^{6} d^{3}} & \text{otherwise} \end{cases} \end{align*}

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]

Piecewise(((e**2*x + e*f*x**2 + f**2*x**3/3)*sin(a), Eq(b, 0) & Eq(d, 0)), ((e**2*x + e*f*x**2 + f**2*x**3/3)*
sin(a + b*sqrt(c)), Eq(d, 0)), ((e**2*x + e*f*x**2 + f**2*x**3/3)*sin(a), Eq(b, 0)), (-2*e**2*sqrt(c + d*x)*co
s(a + b*sqrt(c + d*x))/(b*d) - 4*e*f*x*sqrt(c + d*x)*cos(a + b*sqrt(c + d*x))/(b*d) - 2*f**2*x**2*sqrt(c + d*x
)*cos(a + b*sqrt(c + d*x))/(b*d) + 8*c*e*f*sin(a + b*sqrt(c + d*x))/(b**2*d**2) + 8*c*f**2*x*sin(a + b*sqrt(c
+ d*x))/(b**2*d**2) + 2*e**2*sin(a + b*sqrt(c + d*x))/(b**2*d) + 12*e*f*x*sin(a + b*sqrt(c + d*x))/(b**2*d) +
10*f**2*x**2*sin(a + b*sqrt(c + d*x))/(b**2*d) + 16*c*f**2*sqrt(c + d*x)*cos(a + b*sqrt(c + d*x))/(b**3*d**3)
+ 24*e*f*sqrt(c + d*x)*cos(a + b*sqrt(c + d*x))/(b**3*d**2) + 40*f**2*x*sqrt(c + d*x)*cos(a + b*sqrt(c + d*x))
/(b**3*d**2) - 96*c*f**2*sin(a + b*sqrt(c + d*x))/(b**4*d**3) - 24*e*f*sin(a + b*sqrt(c + d*x))/(b**4*d**2) -
120*f**2*x*sin(a + b*sqrt(c + d*x))/(b**4*d**2) - 240*f**2*sqrt(c + d*x)*cos(a + b*sqrt(c + d*x))/(b**5*d**3)
+ 240*f**2*sin(a + b*sqrt(c + d*x))/(b**6*d**3), True))

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Giac [B]  time = 1.6002, size = 1848, normalized size = 4.51 \begin{align*} \text{result too large to display} \end{align*}

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]

-2*(f^2*(((sqrt(d*x + c)*b + a)*b^4*c^2 - a*b^4*c^2 - 2*(sqrt(d*x + c)*b + a)^3*b^2*c + 6*(sqrt(d*x + c)*b + a
)^2*a*b^2*c - 6*(sqrt(d*x + c)*b + a)*a^2*b^2*c + 2*a^3*b^2*c + (sqrt(d*x + c)*b + a)^5 - 5*(sqrt(d*x + c)*b +
a)^4*a + 10*(sqrt(d*x + c)*b + a)^3*a^2 - 10*(sqrt(d*x + c)*b + a)^2*a^3 + 5*(sqrt(d*x + c)*b + a)*a^4 - a^5
+ 12*(sqrt(d*x + c)*b + a)*b^2*c - 12*a*b^2*c - 20*(sqrt(d*x + c)*b + a)^3 + 60*(sqrt(d*x + c)*b + a)^2*a - 60
*(sqrt(d*x + c)*b + a)*a^2 + 20*a^3 + 120*sqrt(d*x + c)*b)*cos(-(sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a
)*b - a*b) + a*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a)/(b^3*d^2) + (b^4*c^2*sgn((sqrt(d*x + c)*b + a)*b - a*b)
- 6*(sqrt(d*x + c)*b + a)^2*b^2*c*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 12*(sqrt(d*x + c)*b + a)*a*b^2*c*sgn((
sqrt(d*x + c)*b + a)*b - a*b) - 6*a^2*b^2*c*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 5*(sqrt(d*x + c)*b + a)^4*sgn
((sqrt(d*x + c)*b + a)*b - a*b) - 20*(sqrt(d*x + c)*b + a)^3*a*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 30*(sqrt(d
*x + c)*b + a)^2*a^2*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 20*(sqrt(d*x + c)*b + a)*a^3*sgn((sqrt(d*x + c)*b +
a)*b - a*b) + 5*a^4*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 12*b^2*c*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 60*(sqr
t(d*x + c)*b + a)^2*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 120*(sqrt(d*x + c)*b + a)*a*sgn((sqrt(d*x + c)*b + a)
*b - a*b) - 60*a^2*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 120*sgn((sqrt(d*x + c)*b + a)*b - a*b))*sin(-(sqrt(d*x
+ c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a*sgn((sqrt(d*x + c)*b + a)*b - a*b) - a)/(b^3*d^2))/b^2 + (
((sqrt(d*x + c)*b + a)*b - a*b)*cos(-(sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a*sgn((sqrt(d*
x + c)*b + a)*b - a*b) - a) + b*sin(-(sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a*sgn((sqrt(d*
x + c)*b + a)*b - a*b) - a)/sgn((sqrt(d*x + c)*b + a)*b - a*b))*e^2/b^2 - 2*f*(((sqrt(d*x + c)*b + a)*b^2*c -
a*b^2*c - (sqrt(d*x + c)*b + a)^3 + 3*(sqrt(d*x + c)*b + a)^2*a - 3*(sqrt(d*x + c)*b + a)*a^2 + a^3 + 6*sqrt(d
*x + c)*b)*cos(-(sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a*sgn((sqrt(d*x + c)*b + a)*b - a*b
) - a)/b + (b^2*c*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 3*(sqrt(d*x + c)*b + a)^2*sgn((sqrt(d*x + c)*b + a)*b -
a*b) + 6*(sqrt(d*x + c)*b + a)*a*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 3*a^2*sgn((sqrt(d*x + c)*b + a)*b - a*b
) + 6*sgn((sqrt(d*x + c)*b + a)*b - a*b))*sin(-(sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) + a*sg
n((sqrt(d*x + c)*b + a)*b - a*b) - a)/b)*e/(b^2*d))/(b*d)