3.90 \(\int x^2 \cos (a+b \sqrt{c+d x}) \, dx\)

Optimal. Leaf size=346 \[ \frac{2 c^2 \cos \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{40 (c+d x)^{3/2} \sin \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}+\frac{24 c \sqrt{c+d x} \sin \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}+\frac{240 \sqrt{c+d x} \sin \left (a+b \sqrt{c+d x}\right )}{b^5 d^3}+\frac{10 (c+d x)^2 \cos \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{12 c (c+d x) \cos \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{120 (c+d x) \cos \left (a+b \sqrt{c+d x}\right )}{b^4 d^3}+\frac{24 c \cos \left (a+b \sqrt{c+d x}\right )}{b^4 d^3}+\frac{240 \cos \left (a+b \sqrt{c+d x}\right )}{b^6 d^3}+\frac{2 c^2 \sqrt{c+d x} \sin \left (a+b \sqrt{c+d x}\right )}{b d^3}+\frac{2 (c+d x)^{5/2} \sin \left (a+b \sqrt{c+d x}\right )}{b d^3}-\frac{4 c (c+d x)^{3/2} \sin \left (a+b \sqrt{c+d x}\right )}{b d^3} \]

[Out]

(240*Cos[a + b*Sqrt[c + d*x]])/(b^6*d^3) + (24*c*Cos[a + b*Sqrt[c + d*x]])/(b^4*d^3) + (2*c^2*Cos[a + b*Sqrt[c
+ d*x]])/(b^2*d^3) - (120*(c + d*x)*Cos[a + b*Sqrt[c + d*x]])/(b^4*d^3) - (12*c*(c + d*x)*Cos[a + b*Sqrt[c +
d*x]])/(b^2*d^3) + (10*(c + d*x)^2*Cos[a + b*Sqrt[c + d*x]])/(b^2*d^3) + (240*Sqrt[c + d*x]*Sin[a + b*Sqrt[c +
d*x]])/(b^5*d^3) + (24*c*Sqrt[c + d*x]*Sin[a + b*Sqrt[c + d*x]])/(b^3*d^3) + (2*c^2*Sqrt[c + d*x]*Sin[a + b*S
qrt[c + d*x]])/(b*d^3) - (40*(c + d*x)^(3/2)*Sin[a + b*Sqrt[c + d*x]])/(b^3*d^3) - (4*c*(c + d*x)^(3/2)*Sin[a
+ b*Sqrt[c + d*x]])/(b*d^3) + (2*(c + d*x)^(5/2)*Sin[a + b*Sqrt[c + d*x]])/(b*d^3)

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Rubi [A]  time = 0.304552, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3432, 3296, 2638} \[ \frac{2 c^2 \cos \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{40 (c+d x)^{3/2} \sin \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}+\frac{24 c \sqrt{c+d x} \sin \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}+\frac{240 \sqrt{c+d x} \sin \left (a+b \sqrt{c+d x}\right )}{b^5 d^3}+\frac{10 (c+d x)^2 \cos \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{12 c (c+d x) \cos \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{120 (c+d x) \cos \left (a+b \sqrt{c+d x}\right )}{b^4 d^3}+\frac{24 c \cos \left (a+b \sqrt{c+d x}\right )}{b^4 d^3}+\frac{240 \cos \left (a+b \sqrt{c+d x}\right )}{b^6 d^3}+\frac{2 c^2 \sqrt{c+d x} \sin \left (a+b \sqrt{c+d x}\right )}{b d^3}+\frac{2 (c+d x)^{5/2} \sin \left (a+b \sqrt{c+d x}\right )}{b d^3}-\frac{4 c (c+d x)^{3/2} \sin \left (a+b \sqrt{c+d x}\right )}{b d^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Cos[a + b*Sqrt[c + d*x]],x]

[Out]

(240*Cos[a + b*Sqrt[c + d*x]])/(b^6*d^3) + (24*c*Cos[a + b*Sqrt[c + d*x]])/(b^4*d^3) + (2*c^2*Cos[a + b*Sqrt[c
+ d*x]])/(b^2*d^3) - (120*(c + d*x)*Cos[a + b*Sqrt[c + d*x]])/(b^4*d^3) - (12*c*(c + d*x)*Cos[a + b*Sqrt[c +
d*x]])/(b^2*d^3) + (10*(c + d*x)^2*Cos[a + b*Sqrt[c + d*x]])/(b^2*d^3) + (240*Sqrt[c + d*x]*Sin[a + b*Sqrt[c +
d*x]])/(b^5*d^3) + (24*c*Sqrt[c + d*x]*Sin[a + b*Sqrt[c + d*x]])/(b^3*d^3) + (2*c^2*Sqrt[c + d*x]*Sin[a + b*S
qrt[c + d*x]])/(b*d^3) - (40*(c + d*x)^(3/2)*Sin[a + b*Sqrt[c + d*x]])/(b^3*d^3) - (4*c*(c + d*x)^(3/2)*Sin[a
+ b*Sqrt[c + d*x]])/(b*d^3) + (2*(c + d*x)^(5/2)*Sin[a + b*Sqrt[c + d*x]])/(b*d^3)

Rule 3432

Int[((a_.) + Cos[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)]*(b_.))^(p_.)*((g_.) + (h_.)*(x_))^(m_.), x_Symbol] :
> Dist[1/(n*f), Subst[Int[ExpandIntegrand[(a + b*Cos[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 2638

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

Rubi steps

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

Mathematica [C]  time = 0.841162, size = 224, normalized size = 0.65 \[ \frac{e^{-i \left (a+b \sqrt{c+d x}\right )} \left (\left (-i b^5 d^2 x^2 \sqrt{c+d x}+b^4 d x (4 c+5 d x)+4 i b^3 \sqrt{c+d x} (2 c+5 d x)-12 b^2 (4 c+5 d x)-120 i b \sqrt{c+d x}+120\right ) e^{2 i \left (a+b \sqrt{c+d x}\right )}+i b^5 d^2 x^2 \sqrt{c+d x}+b^4 d x (4 c+5 d x)-4 i b^3 \sqrt{c+d x} (2 c+5 d x)-12 b^2 (4 c+5 d x)+120 i b \sqrt{c+d x}+120\right )}{b^6 d^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Cos[a + b*Sqrt[c + d*x]],x]

[Out]

(120 + (120*I)*b*Sqrt[c + d*x] + I*b^5*d^2*x^2*Sqrt[c + d*x] - (4*I)*b^3*Sqrt[c + d*x]*(2*c + 5*d*x) - 12*b^2*
(4*c + 5*d*x) + b^4*d*x*(4*c + 5*d*x) + E^((2*I)*(a + b*Sqrt[c + d*x]))*(120 - (120*I)*b*Sqrt[c + d*x] - I*b^5
*d^2*x^2*Sqrt[c + d*x] + (4*I)*b^3*Sqrt[c + d*x]*(2*c + 5*d*x) - 12*b^2*(4*c + 5*d*x) + b^4*d*x*(4*c + 5*d*x))
)/(b^6*d^3*E^(I*(a + b*Sqrt[c + d*x])))

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Maple [B]  time = 0.089, size = 825, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*cos(a+b*(d*x+c)^(1/2)),x)

[Out]

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

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Maxima [B]  time = 1.24188, size = 907, normalized size = 2.62 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(a*c^2*sin(sqrt(d*x + c)*b + a) - ((sqrt(d*x + c)*b + a)*sin(sqrt(d*x + c)*b + a) + cos(sqrt(d*x + c)*b + a
))*c^2 - 2*a^3*c*sin(sqrt(d*x + c)*b + a)/b^2 + 6*((sqrt(d*x + c)*b + a)*sin(sqrt(d*x + c)*b + a) + cos(sqrt(d
*x + c)*b + a))*a^2*c/b^2 + a^5*sin(sqrt(d*x + c)*b + a)/b^4 - 5*((sqrt(d*x + c)*b + a)*sin(sqrt(d*x + c)*b +
a) + cos(sqrt(d*x + c)*b + a))*a^4/b^4 - 6*(2*(sqrt(d*x + c)*b + a)*cos(sqrt(d*x + c)*b + a) + ((sqrt(d*x + c)
*b + a)^2 - 2)*sin(sqrt(d*x + c)*b + a))*a*c/b^2 + 10*(2*(sqrt(d*x + c)*b + a)*cos(sqrt(d*x + c)*b + a) + ((sq
rt(d*x + c)*b + a)^2 - 2)*sin(sqrt(d*x + c)*b + a))*a^3/b^4 + 2*(3*((sqrt(d*x + c)*b + a)^2 - 2)*cos(sqrt(d*x
+ c)*b + a) + ((sqrt(d*x + c)*b + a)^3 - 6*sqrt(d*x + c)*b - 6*a)*sin(sqrt(d*x + c)*b + a))*c/b^2 - 10*(3*((sq
rt(d*x + c)*b + a)^2 - 2)*cos(sqrt(d*x + c)*b + a) + ((sqrt(d*x + c)*b + a)^3 - 6*sqrt(d*x + c)*b - 6*a)*sin(s
qrt(d*x + c)*b + a))*a^2/b^4 + 5*(4*((sqrt(d*x + c)*b + a)^3 - 6*sqrt(d*x + c)*b - 6*a)*cos(sqrt(d*x + c)*b +
a) + ((sqrt(d*x + c)*b + a)^4 - 12*(sqrt(d*x + c)*b + a)^2 + 24)*sin(sqrt(d*x + c)*b + a))*a/b^4 - (5*((sqrt(d
*x + c)*b + a)^4 - 12*(sqrt(d*x + c)*b + a)^2 + 24)*cos(sqrt(d*x + c)*b + a) + ((sqrt(d*x + c)*b + a)^5 - 20*(
sqrt(d*x + c)*b + a)^3 + 120*sqrt(d*x + c)*b + 120*a)*sin(sqrt(d*x + c)*b + a))/b^4)/(b^2*d^3)

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Fricas [A]  time = 1.62633, size = 248, normalized size = 0.72 \begin{align*} \frac{2 \,{\left ({\left (b^{5} d^{2} x^{2} - 20 \, b^{3} d x - 8 \, b^{3} c + 120 \, b\right )} \sqrt{d x + c} \sin \left (\sqrt{d x + c} b + a\right ) +{\left (5 \, b^{4} d^{2} x^{2} - 48 \, b^{2} c + 4 \,{\left (b^{4} c - 15 \, b^{2}\right )} d x + 120\right )} \cos \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(x^2*cos(a+b*(d*x+c)^(1/2)),x, algorithm="fricas")

[Out]

2*((b^5*d^2*x^2 - 20*b^3*d*x - 8*b^3*c + 120*b)*sqrt(d*x + c)*sin(sqrt(d*x + c)*b + a) + (5*b^4*d^2*x^2 - 48*b
^2*c + 4*(b^4*c - 15*b^2)*d*x + 120)*cos(sqrt(d*x + c)*b + a))/(b^6*d^3)

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Sympy [A]  time = 2.36067, size = 274, normalized size = 0.79 \begin{align*} \begin{cases} \frac{x^{3} \cos{\left (a \right )}}{3} & \text{for}\: b = 0 \wedge d = 0 \\\frac{x^{3} \cos{\left (a + b \sqrt{c} \right )}}{3} & \text{for}\: d = 0 \\\frac{x^{3} \cos{\left (a \right )}}{3} & \text{for}\: b = 0 \\\frac{2 x^{2} \sqrt{c + d x} \sin{\left (a + b \sqrt{c + d x} \right )}}{b d} + \frac{8 c x \cos{\left (a + b \sqrt{c + d x} \right )}}{b^{2} d^{2}} + \frac{10 x^{2} \cos{\left (a + b \sqrt{c + d x} \right )}}{b^{2} d} - \frac{16 c \sqrt{c + d x} \sin{\left (a + b \sqrt{c + d x} \right )}}{b^{3} d^{3}} - \frac{40 x \sqrt{c + d x} \sin{\left (a + b \sqrt{c + d x} \right )}}{b^{3} d^{2}} - \frac{96 c \cos{\left (a + b \sqrt{c + d x} \right )}}{b^{4} d^{3}} - \frac{120 x \cos{\left (a + b \sqrt{c + d x} \right )}}{b^{4} d^{2}} + \frac{240 \sqrt{c + d x} \sin{\left (a + b \sqrt{c + d x} \right )}}{b^{5} d^{3}} + \frac{240 \cos{\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(x**2*cos(a+b*(d*x+c)**(1/2)),x)

[Out]

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

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Giac [B]  time = 1.39436, size = 1134, normalized size = 3.28 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*s
gn((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*(sq
rt(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*(sqrt(d*x + c)*b + a)^2*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 120*(sq
rt(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))*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) - ((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)*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^3*d)