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

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

[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

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Rubi [A]  time = 0.327105, antiderivative size = 236, normalized size of antiderivative = 1., 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 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]

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 \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 = 0.393481, 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

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Maple [B]  time = 0.007, size = 746, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

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))

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Maxima [B]  time = 1.17647, size = 826, normalized size = 3.5 \begin{align*} \text{result too large to display} \end{align*}

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

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Fricas [A]  time = 1.57149, size = 333, normalized size = 1.41 \begin{align*} -\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}} \end{align*}

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

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Sympy [A]  time = 9.83652, size = 286, normalized size = 1.21 \begin{align*} \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{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

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))

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Giac [A]  time = 1.1305, size = 219, normalized size = 0.93 \begin{align*} -\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}} \end{align*}

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