3.2.87 \(\int (e+f x)^2 \sin (a+b \sqrt {c+d x}) \, dx\) [187]

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

[Out]

40*f^2*(d*x+c)^(3/2)*cos(a+b*(d*x+c)^(1/2))/b^3/d^3-4*f*(-c*f+d*e)*(d*x+c)^(3/2)*cos(a+b*(d*x+c)^(1/2))/b/d^3-
2*f^2*(d*x+c)^(5/2)*cos(a+b*(d*x+c)^(1/2))/b/d^3+240*f^2*sin(a+b*(d*x+c)^(1/2))/b^6/d^3-24*f*(-c*f+d*e)*sin(a+
b*(d*x+c)^(1/2))/b^4/d^3+2*(-c*f+d*e)^2*sin(a+b*(d*x+c)^(1/2))/b^2/d^3-120*f^2*(d*x+c)*sin(a+b*(d*x+c)^(1/2))/
b^4/d^3+12*f*(-c*f+d*e)*(d*x+c)*sin(a+b*(d*x+c)^(1/2))/b^2/d^3+10*f^2*(d*x+c)^2*sin(a+b*(d*x+c)^(1/2))/b^2/d^3
-240*f^2*cos(a+b*(d*x+c)^(1/2))*(d*x+c)^(1/2)/b^5/d^3+24*f*(-c*f+d*e)*cos(a+b*(d*x+c)^(1/2))*(d*x+c)^(1/2)/b^3
/d^3-2*(-c*f+d*e)^2*cos(a+b*(d*x+c)^(1/2))*(d*x+c)^(1/2)/b/d^3

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Rubi [A]
time = 0.28, antiderivative size = 410, normalized size of antiderivative = 1.00, 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 = {3512, 3377, 2717} \begin {gather*} \frac {240 f^2 \sin \left (a+b \sqrt {c+d x}\right )}{b^6 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) \sin \left (a+b \sqrt {c+d x}\right )}{b^4 d^3}-\frac {120 f^2 (c+d x) \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 {40 f^2 (c+d x)^{3/2} \cos \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}+\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 {10 f^2 (c+d x)^2 \sin \left (a+b \sqrt {c+d x}\right )}{b^2 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} \end {gather*}

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 2717

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

Rule 3377

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

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Mathematica [A]
time = 1.00, size = 138, normalized size = 0.34 \begin {gather*} \frac {-2 b \sqrt {c+d x} \left (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)\right ) \cos \left (a+b \sqrt {c+d x}\right )+2 \left (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))\right ) \sin \left (a+b \sqrt {c+d x}\right )}{b^6 d^3} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(1245\) vs. \(2(374)=748\).
time = 0.06, size = 1246, normalized size = 3.04

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1246\)
default	\(\text {Expression too large to display}\)	\(1246\)

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,method=_RETURNVERBOSE)

[Out]

2/d^3/b^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)
)+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)))-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))-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)))+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)))-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)))+1/b^
4*a^5*f^2*cos(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)))-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)))+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)))-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)))+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))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1105 vs. \(2 (380) = 760\).
time = 0.37, size = 1105, normalized size = 2.70 \begin {gather*} \frac {2 \, {\left (\frac {a c^{2} f^{2} \cos \left (\sqrt {d x + c} b + a\right )}{d^{2}} - \frac {2 \, a c f \cos \left (\sqrt {d x + c} b + a\right ) e}{d} - \frac {{\left ({\left (\sqrt {d x + c} b + a\right )} \cos \left (\sqrt {d x + c} b + a\right ) - \sin \left (\sqrt {d x + c} b + a\right )\right )} c^{2} f^{2}}{d^{2}} - \frac {2 \, a^{3} c f^{2} \cos \left (\sqrt {d x + c} b + a\right )}{b^{2} d^{2}} + a \cos \left (\sqrt {d x + c} b + a\right ) e^{2} + \frac {2 \, {\left ({\left (\sqrt {d x + c} b + a\right )} \cos \left (\sqrt {d x + c} b + a\right ) - \sin \left (\sqrt {d x + c} b + a\right )\right )} c f e}{d} + \frac {2 \, a^{3} f \cos \left (\sqrt {d x + c} b + a\right ) e}{b^{2} d} + \frac {6 \, {\left ({\left (\sqrt {d x + c} b + a\right )} \cos \left (\sqrt {d x + c} b + a\right ) - \sin \left (\sqrt {d x + c} b + a\right )\right )} a^{2} c f^{2}}{b^{2} d^{2}} + \frac {a^{5} f^{2} \cos \left (\sqrt {d x + c} b + a\right )}{b^{4} d^{2}} - {\left ({\left (\sqrt {d x + c} b + a\right )} \cos \left (\sqrt {d x + c} b + a\right ) - \sin \left (\sqrt {d x + c} b + a\right )\right )} e^{2} - \frac {6 \, {\left ({\left (\sqrt {d x + c} b + a\right )} \cos \left (\sqrt {d x + c} b + a\right ) - \sin \left (\sqrt {d x + c} b + a\right )\right )} a^{2} f e}{b^{2} d} - \frac {5 \, {\left ({\left (\sqrt {d x + c} b + a\right )} \cos \left (\sqrt {d x + c} b + a\right ) - \sin \left (\sqrt {d x + c} b + a\right )\right )} a^{4} f^{2}}{b^{4} d^{2}} - \frac {6 \, {\left ({\left ({\left (\sqrt {d x + c} b + a\right )}^{2} - 2\right )} \cos \left (\sqrt {d x + c} b + a\right ) - 2 \, {\left (\sqrt {d x + c} b + a\right )} \sin \left (\sqrt {d x + c} b + a\right )\right )} a c f^{2}}{b^{2} d^{2}} + \frac {6 \, {\left ({\left ({\left (\sqrt {d x + c} b + a\right )}^{2} - 2\right )} \cos \left (\sqrt {d x + c} b + a\right ) - 2 \, {\left (\sqrt {d x + c} b + a\right )} \sin \left (\sqrt {d x + c} b + a\right )\right )} a f e}{b^{2} d} + \frac {10 \, {\left ({\left ({\left (\sqrt {d x + c} b + a\right )}^{2} - 2\right )} \cos \left (\sqrt {d x + c} b + a\right ) - 2 \, {\left (\sqrt {d x + c} b + a\right )} \sin \left (\sqrt {d x + c} b + a\right )\right )} a^{3} f^{2}}{b^{4} d^{2}} + \frac {2 \, {\left ({\left ({\left (\sqrt {d x + c} b + a\right )}^{3} - 6 \, \sqrt {d x + c} b - 6 \, a\right )} \cos \left (\sqrt {d x + c} b + a\right ) - 3 \, {\left ({\left (\sqrt {d x + c} b + a\right )}^{2} - 2\right )} \sin \left (\sqrt {d x + c} b + a\right )\right )} c f^{2}}{b^{2} d^{2}} - \frac {2 \, {\left ({\left ({\left (\sqrt {d x + c} b + a\right )}^{3} - 6 \, \sqrt {d x + c} b - 6 \, a\right )} \cos \left (\sqrt {d x + c} b + a\right ) - 3 \, {\left ({\left (\sqrt {d x + c} b + a\right )}^{2} - 2\right )} \sin \left (\sqrt {d x + c} b + a\right )\right )} f e}{b^{2} d} - \frac {10 \, {\left ({\left ({\left (\sqrt {d x + c} b + a\right )}^{3} - 6 \, \sqrt {d x + c} b - 6 \, a\right )} \cos \left (\sqrt {d x + c} b + a\right ) - 3 \, {\left ({\left (\sqrt {d x + c} b + a\right )}^{2} - 2\right )} \sin \left (\sqrt {d x + c} b + a\right )\right )} a^{2} f^{2}}{b^{4} d^{2}} + \frac {5 \, {\left ({\left ({\left (\sqrt {d x + c} b + a\right )}^{4} - 12 \, {\left (\sqrt {d x + c} b + a\right )}^{2} + 24\right )} \cos \left (\sqrt {d x + c} b + a\right ) - 4 \, {\left ({\left (\sqrt {d x + c} b + a\right )}^{3} - 6 \, \sqrt {d x + c} b - 6 \, a\right )} \sin \left (\sqrt {d x + c} b + a\right )\right )} a f^{2}}{b^{4} d^{2}} - \frac {{\left ({\left ({\left (\sqrt {d x + c} b + a\right )}^{5} - 20 \, {\left (\sqrt {d x + c} b + a\right )}^{3} + 120 \, \sqrt {d x + c} b + 120 \, a\right )} \cos \left (\sqrt {d x + c} b + a\right ) - 5 \, {\left ({\left (\sqrt {d x + c} b + a\right )}^{4} - 12 \, {\left (\sqrt {d x + c} b + a\right )}^{2} + 24\right )} \sin \left (\sqrt {d x + c} b + a\right )\right )} f^{2}}{b^{4} d^{2}}\right )}}{b^{2} d} \end {gather*}

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*c^2*f^2*cos(sqrt(d*x + c)*b + a)/d^2 - 2*a*c*f*cos(sqrt(d*x + c)*b + a)*e/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*c*f^2*cos(sqrt(d*x + c)*b + a)/(b^2*d^2)
+ a*cos(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*f*e/d + 2*a^3*f*cos(sqrt(d*x + c)*b + a)*e/(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) - ((sqrt(d*x + c)*b
+ a)*cos(sqrt(d*x + c)*b + a) - sin(sqrt(d*x + c)*b + a))*e^2 - 6*((sqrt(d*x + c)*b + a)*cos(sqrt(d*x + c)*b
+ a) - sin(sqrt(d*x + c)*b + a))*a^2*f*e/(b^2*d) - 5*((sqrt(d*x + c)*b + a)*cos(sqrt(d*x + c)*b + a) - sin(sqr
t(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) + 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*f*e/(b^2*d) + 10*(((sqrt(d*x + c)*b + a)^2 - 2)*c
os(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) - 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))*f*e/(b^2*d) - 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 = 0.35, size = 196, normalized size = 0.48 \begin {gather*} -\frac {2 \, {\left ({\left (b^{5} d^{2} f^{2} x^{2} + b^{5} d^{2} e^{2} - 20 \, b^{3} d f^{2} x - 8 \, {\left (b^{3} c - 15 \, b\right )} f^{2} + 2 \, {\left (b^{5} d^{2} f x - 6 \, b^{3} d f\right )} e\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 - 15 \, b^{2}\right )} d f^{2} x - 24 \, {\left (2 \, b^{2} c - 5\right )} f^{2} + 2 \, {\left (3 \, b^{4} d^{2} f x + 2 \, {\left (b^{4} c - 3 \, b^{2}\right )} d f\right )} e\right )} \sin \left (\sqrt {d x + c} b + a\right )\right )}}{b^{6} d^{3}} \end {gather*}

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 - 20*b^3*d*f^2*x - 8*(b^3*c - 15*b)*f^2 + 2*(b^5*d^2*f*x - 6*b^3*d*f)*e)*sq
rt(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 - 15*b^2)*d*f^2*x - 24*(2*b
^2*c - 5)*f^2 + 2*(3*b^4*d^2*f*x + 2*(b^4*c - 3*b^2)*d*f)*e)*sin(sqrt(d*x + c)*b + a))/(b^6*d^3)

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Sympy [A]
time = 0.38, size = 529, normalized size = 1.29 \begin {gather*} \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 \left (b = 0 \vee d = 0\right ) \\\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 \\- \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 {gather*}

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(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)), (-2*e**2*sqrt(c + d*x)*cos(a + b*sqrt(c + d*x))/(b*d) - 4*e*f*x*sq
rt(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*sq
rt(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 [A]
time = 2.46, size = 701, normalized size = 1.71 \begin {gather*} -\frac {2 \, {\left (\frac {f^{2} {\left (\frac {{\left ({\left (\sqrt {d x + c} b + a\right )} b^{4} c^{2} - a b^{4} c^{2} - 2 \, {\left (\sqrt {d x + c} b + a\right )}^{3} b^{2} c + 6 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a b^{2} c - 6 \, {\left (\sqrt {d x + c} b + a\right )} a^{2} b^{2} c + 2 \, a^{3} b^{2} c + {\left (\sqrt {d x + c} b + a\right )}^{5} - 5 \, {\left (\sqrt {d x + c} b + a\right )}^{4} a + 10 \, {\left (\sqrt {d x + c} b + a\right )}^{3} a^{2} - 10 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a^{3} + 5 \, {\left (\sqrt {d x + c} b + a\right )} a^{4} - a^{5} + 12 \, {\left (\sqrt {d x + c} b + a\right )} b^{2} c - 12 \, a b^{2} c - 20 \, {\left (\sqrt {d x + c} b + a\right )}^{3} + 60 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a - 60 \, {\left (\sqrt {d x + c} b + a\right )} a^{2} + 20 \, a^{3} + 120 \, \sqrt {d x + c} b\right )} \cos \left (\sqrt {d x + c} b + a\right )}{b^{4} d^{2}} - \frac {{\left (b^{4} c^{2} - 6 \, {\left (\sqrt {d x + c} b + a\right )}^{2} b^{2} c + 12 \, {\left (\sqrt {d x + c} b + a\right )} a b^{2} c - 6 \, a^{2} b^{2} c + 5 \, {\left (\sqrt {d x + c} b + a\right )}^{4} - 20 \, {\left (\sqrt {d x + c} b + a\right )}^{3} a + 30 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a^{2} - 20 \, {\left (\sqrt {d x + c} b + a\right )} a^{3} + 5 \, a^{4} + 12 \, b^{2} c - 60 \, {\left (\sqrt {d x + c} b + a\right )}^{2} + 120 \, {\left (\sqrt {d x + c} b + a\right )} a - 60 \, a^{2} + 120\right )} \sin \left (\sqrt {d x + c} b + a\right )}{b^{4} d^{2}}\right )}}{b} + \frac {{\left (\sqrt {d x + c} b \cos \left (\sqrt {d x + c} b + a\right ) - \sin \left (\sqrt {d x + c} b + a\right )\right )} e^{2}}{b} - \frac {2 \, f {\left (\frac {{\left ({\left (\sqrt {d x + c} b + a\right )} b^{2} c - a b^{2} c - {\left (\sqrt {d x + c} b + a\right )}^{3} + 3 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a - 3 \, {\left (\sqrt {d x + c} b + a\right )} a^{2} + a^{3} + 6 \, \sqrt {d x + c} b\right )} \cos \left (\sqrt {d x + c} b + a\right )}{b^{2}} - \frac {{\left (b^{2} c - 3 \, {\left (\sqrt {d x + c} b + a\right )}^{2} + 6 \, {\left (\sqrt {d x + c} b + a\right )} a - 3 \, a^{2} + 6\right )} \sin \left (\sqrt {d x + c} b + a\right )}{b^{2}}\right )} e}{b d}\right )}}{b d} \end {gather*}

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)/(b^4*d^2) - (b^4*c^2 - 6*(
sqrt(d*x + c)*b + a)^2*b^2*c + 12*(sqrt(d*x + c)*b + a)*a*b^2*c - 6*a^2*b^2*c + 5*(sqrt(d*x + c)*b + a)^4 - 20
*(sqrt(d*x + c)*b + a)^3*a + 30*(sqrt(d*x + c)*b + a)^2*a^2 - 20*(sqrt(d*x + c)*b + a)*a^3 + 5*a^4 + 12*b^2*c
- 60*(sqrt(d*x + c)*b + a)^2 + 120*(sqrt(d*x + c)*b + a)*a - 60*a^2 + 120)*sin(sqrt(d*x + c)*b + a)/(b^4*d^2))
/b + (sqrt(d*x + c)*b*cos(sqrt(d*x + c)*b + a) - sin(sqrt(d*x + c)*b + a))*e^2/b - 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)/b^2 - (b^2*c - 3*(sqrt(d*x + c)*b + a)^2 + 6*(sqrt(d*x + c)*b + a
)*a - 3*a^2 + 6)*sin(sqrt(d*x + c)*b + a)/b^2)*e/(b*d))/(b*d)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \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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