3.49 \(\int x^2 (a+b x^2)^2 \sin (c+d x) \, dx\)

Optimal. Leaf size=236 \[ \frac {2 a^2 \cos (c+d x)}{d^3}+\frac {2 a^2 x \sin (c+d x)}{d^2}-\frac {a^2 x^2 \cos (c+d x)}{d}-\frac {48 a b \cos (c+d x)}{d^5}-\frac {48 a b x \sin (c+d x)}{d^4}+\frac {24 a b x^2 \cos (c+d x)}{d^3}+\frac {8 a b x^3 \sin (c+d x)}{d^2}-\frac {2 a b x^4 \cos (c+d x)}{d}+\frac {720 b^2 \cos (c+d x)}{d^7}+\frac {720 b^2 x \sin (c+d x)}{d^6}-\frac {360 b^2 x^2 \cos (c+d x)}{d^5}-\frac {120 b^2 x^3 \sin (c+d x)}{d^4}+\frac {30 b^2 x^4 \cos (c+d x)}{d^3}+\frac {6 b^2 x^5 \sin (c+d x)}{d^2}-\frac {b^2 x^6 \cos (c+d x)}{d} \]

[Out]

720*b^2*cos(d*x+c)/d^7-48*a*b*cos(d*x+c)/d^5+2*a^2*cos(d*x+c)/d^3-360*b^2*x^2*cos(d*x+c)/d^5+24*a*b*x^2*cos(d*
x+c)/d^3-a^2*x^2*cos(d*x+c)/d+30*b^2*x^4*cos(d*x+c)/d^3-2*a*b*x^4*cos(d*x+c)/d-b^2*x^6*cos(d*x+c)/d+720*b^2*x*
sin(d*x+c)/d^6-48*a*b*x*sin(d*x+c)/d^4+2*a^2*x*sin(d*x+c)/d^2-120*b^2*x^3*sin(d*x+c)/d^4+8*a*b*x^3*sin(d*x+c)/
d^2+6*b^2*x^5*sin(d*x+c)/d^2

________________________________________________________________________________________

Rubi [A]  time = 0.33, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3339, 3296, 2638} \[ \frac {2 a^2 x \sin (c+d x)}{d^2}+\frac {2 a^2 \cos (c+d x)}{d^3}-\frac {a^2 x^2 \cos (c+d x)}{d}+\frac {8 a b x^3 \sin (c+d x)}{d^2}+\frac {24 a b x^2 \cos (c+d x)}{d^3}-\frac {48 a b x \sin (c+d x)}{d^4}-\frac {48 a b \cos (c+d x)}{d^5}-\frac {2 a b x^4 \cos (c+d x)}{d}+\frac {6 b^2 x^5 \sin (c+d x)}{d^2}-\frac {120 b^2 x^3 \sin (c+d x)}{d^4}+\frac {30 b^2 x^4 \cos (c+d x)}{d^3}-\frac {360 b^2 x^2 \cos (c+d x)}{d^5}+\frac {720 b^2 x \sin (c+d x)}{d^6}+\frac {720 b^2 \cos (c+d x)}{d^7}-\frac {b^2 x^6 \cos (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^2)^2*Sin[c + d*x],x]

[Out]

(720*b^2*Cos[c + d*x])/d^7 - (48*a*b*Cos[c + d*x])/d^5 + (2*a^2*Cos[c + d*x])/d^3 - (360*b^2*x^2*Cos[c + d*x])
/d^5 + (24*a*b*x^2*Cos[c + d*x])/d^3 - (a^2*x^2*Cos[c + d*x])/d + (30*b^2*x^4*Cos[c + d*x])/d^3 - (2*a*b*x^4*C
os[c + d*x])/d - (b^2*x^6*Cos[c + d*x])/d + (720*b^2*x*Sin[c + d*x])/d^6 - (48*a*b*x*Sin[c + d*x])/d^4 + (2*a^
2*x*Sin[c + d*x])/d^2 - (120*b^2*x^3*Sin[c + d*x])/d^4 + (8*a*b*x^3*Sin[c + d*x])/d^2 + (6*b^2*x^5*Sin[c + d*x
])/d^2

Rule 2638

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

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 3339

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegran
d[Sin[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int x^2 \left (a+b x^2\right )^2 \sin (c+d x) \, dx &=\int \left (a^2 x^2 \sin (c+d x)+2 a b x^4 \sin (c+d x)+b^2 x^6 \sin (c+d x)\right ) \, dx\\ &=a^2 \int x^2 \sin (c+d x) \, dx+(2 a b) \int x^4 \sin (c+d x) \, dx+b^2 \int x^6 \sin (c+d x) \, dx\\ &=-\frac {a^2 x^2 \cos (c+d x)}{d}-\frac {2 a b x^4 \cos (c+d x)}{d}-\frac {b^2 x^6 \cos (c+d x)}{d}+\frac {\left (2 a^2\right ) \int x \cos (c+d x) \, dx}{d}+\frac {(8 a b) \int x^3 \cos (c+d x) \, dx}{d}+\frac {\left (6 b^2\right ) \int x^5 \cos (c+d x) \, dx}{d}\\ &=-\frac {a^2 x^2 \cos (c+d x)}{d}-\frac {2 a b x^4 \cos (c+d x)}{d}-\frac {b^2 x^6 \cos (c+d x)}{d}+\frac {2 a^2 x \sin (c+d x)}{d^2}+\frac {8 a b x^3 \sin (c+d x)}{d^2}+\frac {6 b^2 x^5 \sin (c+d x)}{d^2}-\frac {\left (2 a^2\right ) \int \sin (c+d x) \, dx}{d^2}-\frac {(24 a b) \int x^2 \sin (c+d x) \, dx}{d^2}-\frac {\left (30 b^2\right ) \int x^4 \sin (c+d x) \, dx}{d^2}\\ &=\frac {2 a^2 \cos (c+d x)}{d^3}+\frac {24 a b x^2 \cos (c+d x)}{d^3}-\frac {a^2 x^2 \cos (c+d x)}{d}+\frac {30 b^2 x^4 \cos (c+d x)}{d^3}-\frac {2 a b x^4 \cos (c+d x)}{d}-\frac {b^2 x^6 \cos (c+d x)}{d}+\frac {2 a^2 x \sin (c+d x)}{d^2}+\frac {8 a b x^3 \sin (c+d x)}{d^2}+\frac {6 b^2 x^5 \sin (c+d x)}{d^2}-\frac {(48 a b) \int x \cos (c+d x) \, dx}{d^3}-\frac {\left (120 b^2\right ) \int x^3 \cos (c+d x) \, dx}{d^3}\\ &=\frac {2 a^2 \cos (c+d x)}{d^3}+\frac {24 a b x^2 \cos (c+d x)}{d^3}-\frac {a^2 x^2 \cos (c+d x)}{d}+\frac {30 b^2 x^4 \cos (c+d x)}{d^3}-\frac {2 a b x^4 \cos (c+d x)}{d}-\frac {b^2 x^6 \cos (c+d x)}{d}-\frac {48 a b x \sin (c+d x)}{d^4}+\frac {2 a^2 x \sin (c+d x)}{d^2}-\frac {120 b^2 x^3 \sin (c+d x)}{d^4}+\frac {8 a b x^3 \sin (c+d x)}{d^2}+\frac {6 b^2 x^5 \sin (c+d x)}{d^2}+\frac {(48 a b) \int \sin (c+d x) \, dx}{d^4}+\frac {\left (360 b^2\right ) \int x^2 \sin (c+d x) \, dx}{d^4}\\ &=-\frac {48 a b \cos (c+d x)}{d^5}+\frac {2 a^2 \cos (c+d x)}{d^3}-\frac {360 b^2 x^2 \cos (c+d x)}{d^5}+\frac {24 a b x^2 \cos (c+d x)}{d^3}-\frac {a^2 x^2 \cos (c+d x)}{d}+\frac {30 b^2 x^4 \cos (c+d x)}{d^3}-\frac {2 a b x^4 \cos (c+d x)}{d}-\frac {b^2 x^6 \cos (c+d x)}{d}-\frac {48 a b x \sin (c+d x)}{d^4}+\frac {2 a^2 x \sin (c+d x)}{d^2}-\frac {120 b^2 x^3 \sin (c+d x)}{d^4}+\frac {8 a b x^3 \sin (c+d x)}{d^2}+\frac {6 b^2 x^5 \sin (c+d x)}{d^2}+\frac {\left (720 b^2\right ) \int x \cos (c+d x) \, dx}{d^5}\\ &=-\frac {48 a b \cos (c+d x)}{d^5}+\frac {2 a^2 \cos (c+d x)}{d^3}-\frac {360 b^2 x^2 \cos (c+d x)}{d^5}+\frac {24 a b x^2 \cos (c+d x)}{d^3}-\frac {a^2 x^2 \cos (c+d x)}{d}+\frac {30 b^2 x^4 \cos (c+d x)}{d^3}-\frac {2 a b x^4 \cos (c+d x)}{d}-\frac {b^2 x^6 \cos (c+d x)}{d}+\frac {720 b^2 x \sin (c+d x)}{d^6}-\frac {48 a b x \sin (c+d x)}{d^4}+\frac {2 a^2 x \sin (c+d x)}{d^2}-\frac {120 b^2 x^3 \sin (c+d x)}{d^4}+\frac {8 a b x^3 \sin (c+d x)}{d^2}+\frac {6 b^2 x^5 \sin (c+d x)}{d^2}-\frac {\left (720 b^2\right ) \int \sin (c+d x) \, dx}{d^6}\\ &=\frac {720 b^2 \cos (c+d x)}{d^7}-\frac {48 a b \cos (c+d x)}{d^5}+\frac {2 a^2 \cos (c+d x)}{d^3}-\frac {360 b^2 x^2 \cos (c+d x)}{d^5}+\frac {24 a b x^2 \cos (c+d x)}{d^3}-\frac {a^2 x^2 \cos (c+d x)}{d}+\frac {30 b^2 x^4 \cos (c+d x)}{d^3}-\frac {2 a b x^4 \cos (c+d x)}{d}-\frac {b^2 x^6 \cos (c+d x)}{d}+\frac {720 b^2 x \sin (c+d x)}{d^6}-\frac {48 a b x \sin (c+d x)}{d^4}+\frac {2 a^2 x \sin (c+d x)}{d^2}-\frac {120 b^2 x^3 \sin (c+d x)}{d^4}+\frac {8 a b x^3 \sin (c+d x)}{d^2}+\frac {6 b^2 x^5 \sin (c+d x)}{d^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.42, size = 139, normalized size = 0.59 \[ \frac {2 d x \left (a^2 d^4+4 a b d^2 \left (d^2 x^2-6\right )+3 b^2 \left (d^4 x^4-20 d^2 x^2+120\right )\right ) \sin (c+d x)-\left (a^2 d^4 \left (d^2 x^2-2\right )+2 a b d^2 \left (d^4 x^4-12 d^2 x^2+24\right )+b^2 \left (d^6 x^6-30 d^4 x^4+360 d^2 x^2-720\right )\right ) \cos (c+d x)}{d^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((a^2*d^4*(-2 + d^2*x^2) + 2*a*b*d^2*(24 - 12*d^2*x^2 + d^4*x^4) + b^2*(-720 + 360*d^2*x^2 - 30*d^4*x^4 + d^
6*x^6))*Cos[c + d*x]) + 2*d*x*(a^2*d^4 + 4*a*b*d^2*(-6 + d^2*x^2) + 3*b^2*(120 - 20*d^2*x^2 + d^4*x^4))*Sin[c
+ d*x])/d^7

________________________________________________________________________________________

fricas [A]  time = 0.75, size = 154, normalized size = 0.65 \[ -\frac {{\left (b^{2} d^{6} x^{6} - 2 \, a^{2} d^{4} + 2 \, {\left (a b d^{6} - 15 \, b^{2} d^{4}\right )} x^{4} + 48 \, a b d^{2} + {\left (a^{2} d^{6} - 24 \, a b d^{4} + 360 \, b^{2} d^{2}\right )} x^{2} - 720 \, b^{2}\right )} \cos \left (d x + c\right ) - 2 \, {\left (3 \, b^{2} d^{5} x^{5} + 4 \, {\left (a b d^{5} - 15 \, b^{2} d^{3}\right )} x^{3} + {\left (a^{2} d^{5} - 24 \, a b d^{3} + 360 \, b^{2} d\right )} x\right )} \sin \left (d x + c\right )}{d^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^2+a)^2*sin(d*x+c),x, algorithm="fricas")

[Out]

-((b^2*d^6*x^6 - 2*a^2*d^4 + 2*(a*b*d^6 - 15*b^2*d^4)*x^4 + 48*a*b*d^2 + (a^2*d^6 - 24*a*b*d^4 + 360*b^2*d^2)*
x^2 - 720*b^2)*cos(d*x + c) - 2*(3*b^2*d^5*x^5 + 4*(a*b*d^5 - 15*b^2*d^3)*x^3 + (a^2*d^5 - 24*a*b*d^3 + 360*b^
2*d)*x)*sin(d*x + c))/d^7

________________________________________________________________________________________

giac [A]  time = 0.56, size = 162, normalized size = 0.69 \[ -\frac {{\left (b^{2} d^{6} x^{6} + 2 \, a b d^{6} x^{4} + a^{2} d^{6} x^{2} - 30 \, b^{2} d^{4} x^{4} - 24 \, a b d^{4} x^{2} - 2 \, a^{2} d^{4} + 360 \, b^{2} d^{2} x^{2} + 48 \, a b d^{2} - 720 \, b^{2}\right )} \cos \left (d x + c\right )}{d^{7}} + \frac {2 \, {\left (3 \, b^{2} d^{5} x^{5} + 4 \, a b d^{5} x^{3} + a^{2} d^{5} x - 60 \, b^{2} d^{3} x^{3} - 24 \, a b d^{3} x + 360 \, b^{2} d x\right )} \sin \left (d x + c\right )}{d^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b^2*d^6*x^6 + 2*a*b*d^6*x^4 + a^2*d^6*x^2 - 30*b^2*d^4*x^4 - 24*a*b*d^4*x^2 - 2*a^2*d^4 + 360*b^2*d^2*x^2 +
48*a*b*d^2 - 720*b^2)*cos(d*x + c)/d^7 + 2*(3*b^2*d^5*x^5 + 4*a*b*d^5*x^3 + a^2*d^5*x - 60*b^2*d^3*x^3 - 24*a*
b*d^3*x + 360*b^2*d*x)*sin(d*x + c)/d^7

________________________________________________________________________________________

maple [B]  time = 0.02, size = 746, normalized size = 3.16 \[ \frac {\frac {b^{2} \left (-\left (d x +c \right )^{6} \cos \left (d x +c \right )+6 \left (d x +c \right )^{5} \sin \left (d x +c \right )+30 \left (d x +c \right )^{4} \cos \left (d x +c \right )-120 \left (d x +c \right )^{3} \sin \left (d x +c \right )-360 \left (d x +c \right )^{2} \cos \left (d x +c \right )+720 \cos \left (d x +c \right )+720 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{4}}-\frac {6 b^{2} c \left (-\left (d x +c \right )^{5} \cos \left (d x +c \right )+5 \left (d x +c \right )^{4} \sin \left (d x +c \right )+20 \left (d x +c \right )^{3} \cos \left (d x +c \right )-60 \left (d x +c \right )^{2} \sin \left (d x +c \right )+120 \sin \left (d x +c \right )-120 \left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{4}}+\frac {2 a b \left (-\left (d x +c \right )^{4} \cos \left (d x +c \right )+4 \left (d x +c \right )^{3} \sin \left (d x +c \right )+12 \left (d x +c \right )^{2} \cos \left (d x +c \right )-24 \cos \left (d x +c \right )-24 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{2}}+\frac {15 b^{2} c^{2} \left (-\left (d x +c \right )^{4} \cos \left (d x +c \right )+4 \left (d x +c \right )^{3} \sin \left (d x +c \right )+12 \left (d x +c \right )^{2} \cos \left (d x +c \right )-24 \cos \left (d x +c \right )-24 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{4}}-\frac {8 a b c \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{2}}-\frac {20 b^{2} c^{3} \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{4}}+a^{2} \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )+\frac {12 a b \,c^{2} \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{2}}+\frac {15 b^{2} c^{4} \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{4}}-2 a^{2} c \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )-\frac {8 a b \,c^{3} \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{2}}-\frac {6 b^{2} c^{5} \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{4}}-a^{2} c^{2} \cos \left (d x +c \right )-\frac {2 a b \,c^{4} \cos \left (d x +c \right )}{d^{2}}-\frac {b^{2} c^{6} \cos \left (d x +c \right )}{d^{4}}}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x^2+a)^2*sin(d*x+c),x)

[Out]

1/d^3*(1/d^4*b^2*(-(d*x+c)^6*cos(d*x+c)+6*(d*x+c)^5*sin(d*x+c)+30*(d*x+c)^4*cos(d*x+c)-120*(d*x+c)^3*sin(d*x+c
)-360*(d*x+c)^2*cos(d*x+c)+720*cos(d*x+c)+720*(d*x+c)*sin(d*x+c))-6/d^4*b^2*c*(-(d*x+c)^5*cos(d*x+c)+5*(d*x+c)
^4*sin(d*x+c)+20*(d*x+c)^3*cos(d*x+c)-60*(d*x+c)^2*sin(d*x+c)+120*sin(d*x+c)-120*(d*x+c)*cos(d*x+c))+2/d^2*a*b
*(-(d*x+c)^4*cos(d*x+c)+4*(d*x+c)^3*sin(d*x+c)+12*(d*x+c)^2*cos(d*x+c)-24*cos(d*x+c)-24*(d*x+c)*sin(d*x+c))+15
/d^4*b^2*c^2*(-(d*x+c)^4*cos(d*x+c)+4*(d*x+c)^3*sin(d*x+c)+12*(d*x+c)^2*cos(d*x+c)-24*cos(d*x+c)-24*(d*x+c)*si
n(d*x+c))-8/d^2*a*b*c*(-(d*x+c)^3*cos(d*x+c)+3*(d*x+c)^2*sin(d*x+c)-6*sin(d*x+c)+6*(d*x+c)*cos(d*x+c))-20/d^4*
b^2*c^3*(-(d*x+c)^3*cos(d*x+c)+3*(d*x+c)^2*sin(d*x+c)-6*sin(d*x+c)+6*(d*x+c)*cos(d*x+c))+a^2*(-(d*x+c)^2*cos(d
*x+c)+2*cos(d*x+c)+2*(d*x+c)*sin(d*x+c))+12/d^2*a*b*c^2*(-(d*x+c)^2*cos(d*x+c)+2*cos(d*x+c)+2*(d*x+c)*sin(d*x+
c))+15/d^4*b^2*c^4*(-(d*x+c)^2*cos(d*x+c)+2*cos(d*x+c)+2*(d*x+c)*sin(d*x+c))-2*a^2*c*(sin(d*x+c)-(d*x+c)*cos(d
*x+c))-8/d^2*a*b*c^3*(sin(d*x+c)-(d*x+c)*cos(d*x+c))-6/d^4*b^2*c^5*(sin(d*x+c)-(d*x+c)*cos(d*x+c))-a^2*c^2*cos
(d*x+c)-2/d^2*a*b*c^4*cos(d*x+c)-1/d^4*b^2*c^6*cos(d*x+c))

________________________________________________________________________________________

maxima [B]  time = 0.38, size = 612, normalized size = 2.59 \[ -\frac {a^{2} c^{2} \cos \left (d x + c\right ) + \frac {b^{2} c^{6} \cos \left (d x + c\right )}{d^{4}} + \frac {2 \, a b c^{4} \cos \left (d x + c\right )}{d^{2}} - 2 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} a^{2} c - \frac {6 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} b^{2} c^{5}}{d^{4}} - \frac {8 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} a b c^{3}}{d^{2}} + {\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (d x + c\right )\right )} a^{2} + \frac {15 \, {\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (d x + c\right )\right )} b^{2} c^{4}}{d^{4}} + \frac {12 \, {\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (d x + c\right )\right )} a b c^{2}}{d^{2}} - \frac {20 \, {\left ({\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \cos \left (d x + c\right ) - 3 \, {\left ({\left (d x + c\right )}^{2} - 2\right )} \sin \left (d x + c\right )\right )} b^{2} c^{3}}{d^{4}} - \frac {8 \, {\left ({\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \cos \left (d x + c\right ) - 3 \, {\left ({\left (d x + c\right )}^{2} - 2\right )} \sin \left (d x + c\right )\right )} a b c}{d^{2}} + \frac {15 \, {\left ({\left ({\left (d x + c\right )}^{4} - 12 \, {\left (d x + c\right )}^{2} + 24\right )} \cos \left (d x + c\right ) - 4 \, {\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \sin \left (d x + c\right )\right )} b^{2} c^{2}}{d^{4}} + \frac {2 \, {\left ({\left ({\left (d x + c\right )}^{4} - 12 \, {\left (d x + c\right )}^{2} + 24\right )} \cos \left (d x + c\right ) - 4 \, {\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \sin \left (d x + c\right )\right )} a b}{d^{2}} - \frac {6 \, {\left ({\left ({\left (d x + c\right )}^{5} - 20 \, {\left (d x + c\right )}^{3} + 120 \, d x + 120 \, c\right )} \cos \left (d x + c\right ) - 5 \, {\left ({\left (d x + c\right )}^{4} - 12 \, {\left (d x + c\right )}^{2} + 24\right )} \sin \left (d x + c\right )\right )} b^{2} c}{d^{4}} + \frac {{\left ({\left ({\left (d x + c\right )}^{6} - 30 \, {\left (d x + c\right )}^{4} + 360 \, {\left (d x + c\right )}^{2} - 720\right )} \cos \left (d x + c\right ) - 6 \, {\left ({\left (d x + c\right )}^{5} - 20 \, {\left (d x + c\right )}^{3} + 120 \, d x + 120 \, c\right )} \sin \left (d x + c\right )\right )} b^{2}}{d^{4}}}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a^2*c^2*cos(d*x + c) + b^2*c^6*cos(d*x + c)/d^4 + 2*a*b*c^4*cos(d*x + c)/d^2 - 2*((d*x + c)*cos(d*x + c) - s
in(d*x + c))*a^2*c - 6*((d*x + c)*cos(d*x + c) - sin(d*x + c))*b^2*c^5/d^4 - 8*((d*x + c)*cos(d*x + c) - sin(d
*x + c))*a*b*c^3/d^2 + (((d*x + c)^2 - 2)*cos(d*x + c) - 2*(d*x + c)*sin(d*x + c))*a^2 + 15*(((d*x + c)^2 - 2)
*cos(d*x + c) - 2*(d*x + c)*sin(d*x + c))*b^2*c^4/d^4 + 12*(((d*x + c)^2 - 2)*cos(d*x + c) - 2*(d*x + c)*sin(d
*x + c))*a*b*c^2/d^2 - 20*(((d*x + c)^3 - 6*d*x - 6*c)*cos(d*x + c) - 3*((d*x + c)^2 - 2)*sin(d*x + c))*b^2*c^
3/d^4 - 8*(((d*x + c)^3 - 6*d*x - 6*c)*cos(d*x + c) - 3*((d*x + c)^2 - 2)*sin(d*x + c))*a*b*c/d^2 + 15*(((d*x
+ c)^4 - 12*(d*x + c)^2 + 24)*cos(d*x + c) - 4*((d*x + c)^3 - 6*d*x - 6*c)*sin(d*x + c))*b^2*c^2/d^4 + 2*(((d*
x + c)^4 - 12*(d*x + c)^2 + 24)*cos(d*x + c) - 4*((d*x + c)^3 - 6*d*x - 6*c)*sin(d*x + c))*a*b/d^2 - 6*(((d*x
+ c)^5 - 20*(d*x + c)^3 + 120*d*x + 120*c)*cos(d*x + c) - 5*((d*x + c)^4 - 12*(d*x + c)^2 + 24)*sin(d*x + c))*
b^2*c/d^4 + (((d*x + c)^6 - 30*(d*x + c)^4 + 360*(d*x + c)^2 - 720)*cos(d*x + c) - 6*((d*x + c)^5 - 20*(d*x +
c)^3 + 120*d*x + 120*c)*sin(d*x + c))*b^2/d^4)/d^3

________________________________________________________________________________________

mupad [B]  time = 0.58, size = 186, normalized size = 0.79 \[ \frac {2\,\cos \left (c+d\,x\right )\,\left (a^2\,d^4-24\,a\,b\,d^2+360\,b^2\right )}{d^7}-\frac {b^2\,x^6\,\cos \left (c+d\,x\right )}{d}+\frac {6\,b^2\,x^5\,\sin \left (c+d\,x\right )}{d^2}+\frac {2\,x\,\sin \left (c+d\,x\right )\,\left (a^2\,d^4-24\,a\,b\,d^2+360\,b^2\right )}{d^6}-\frac {x^2\,\cos \left (c+d\,x\right )\,\left (a^2\,d^4-24\,a\,b\,d^2+360\,b^2\right )}{d^5}+\frac {2\,x^4\,\cos \left (c+d\,x\right )\,\left (15\,b^2-a\,b\,d^2\right )}{d^3}-\frac {8\,x^3\,\sin \left (c+d\,x\right )\,\left (15\,b^2-a\,b\,d^2\right )}{d^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*sin(c + d*x)*(a + b*x^2)^2,x)

[Out]

(2*cos(c + d*x)*(360*b^2 + a^2*d^4 - 24*a*b*d^2))/d^7 - (b^2*x^6*cos(c + d*x))/d + (6*b^2*x^5*sin(c + d*x))/d^
2 + (2*x*sin(c + d*x)*(360*b^2 + a^2*d^4 - 24*a*b*d^2))/d^6 - (x^2*cos(c + d*x)*(360*b^2 + a^2*d^4 - 24*a*b*d^
2))/d^5 + (2*x^4*cos(c + d*x)*(15*b^2 - a*b*d^2))/d^3 - (8*x^3*sin(c + d*x)*(15*b^2 - a*b*d^2))/d^4

________________________________________________________________________________________

sympy [A]  time = 7.53, size = 286, normalized size = 1.21 \[ \begin {cases} - \frac {a^{2} x^{2} \cos {\left (c + d x \right )}}{d} + \frac {2 a^{2} x \sin {\left (c + d x \right )}}{d^{2}} + \frac {2 a^{2} \cos {\left (c + d x \right )}}{d^{3}} - \frac {2 a b x^{4} \cos {\left (c + d x \right )}}{d} + \frac {8 a b x^{3} \sin {\left (c + d x \right )}}{d^{2}} + \frac {24 a b x^{2} \cos {\left (c + d x \right )}}{d^{3}} - \frac {48 a b x \sin {\left (c + d x \right )}}{d^{4}} - \frac {48 a b \cos {\left (c + d x \right )}}{d^{5}} - \frac {b^{2} x^{6} \cos {\left (c + d x \right )}}{d} + \frac {6 b^{2} x^{5} \sin {\left (c + d x \right )}}{d^{2}} + \frac {30 b^{2} x^{4} \cos {\left (c + d x \right )}}{d^{3}} - \frac {120 b^{2} x^{3} \sin {\left (c + d x \right )}}{d^{4}} - \frac {360 b^{2} x^{2} \cos {\left (c + d x \right )}}{d^{5}} + \frac {720 b^{2} x \sin {\left (c + d x \right )}}{d^{6}} + \frac {720 b^{2} \cos {\left (c + d x \right )}}{d^{7}} & \text {for}\: d \neq 0 \\\left (\frac {a^{2} x^{3}}{3} + \frac {2 a b x^{5}}{5} + \frac {b^{2} x^{7}}{7}\right ) \sin {\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(b*x**2+a)**2*sin(d*x+c),x)

[Out]

Piecewise((-a**2*x**2*cos(c + d*x)/d + 2*a**2*x*sin(c + d*x)/d**2 + 2*a**2*cos(c + d*x)/d**3 - 2*a*b*x**4*cos(
c + d*x)/d + 8*a*b*x**3*sin(c + d*x)/d**2 + 24*a*b*x**2*cos(c + d*x)/d**3 - 48*a*b*x*sin(c + d*x)/d**4 - 48*a*
b*cos(c + d*x)/d**5 - b**2*x**6*cos(c + d*x)/d + 6*b**2*x**5*sin(c + d*x)/d**2 + 30*b**2*x**4*cos(c + d*x)/d**
3 - 120*b**2*x**3*sin(c + d*x)/d**4 - 360*b**2*x**2*cos(c + d*x)/d**5 + 720*b**2*x*sin(c + d*x)/d**6 + 720*b**
2*cos(c + d*x)/d**7, Ne(d, 0)), ((a**2*x**3/3 + 2*a*b*x**5/5 + b**2*x**7/7)*sin(c), True))

________________________________________________________________________________________