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

Optimal. Leaf size=186 \[ \frac{2 a^2 x \sin (c+d x)}{d^2}+\frac{2 a^2 \cos (c+d x)}{d^3}-\frac{a^2 x^2 \cos (c+d x)}{d}+\frac{6 a b x^2 \sin (c+d x)}{d^2}-\frac{12 a b \sin (c+d x)}{d^4}+\frac{12 a b x \cos (c+d x)}{d^3}-\frac{2 a b x^3 \cos (c+d x)}{d}+\frac{4 b^2 x^3 \sin (c+d x)}{d^2}+\frac{12 b^2 x^2 \cos (c+d x)}{d^3}-\frac{24 b^2 x \sin (c+d x)}{d^4}-\frac{24 b^2 \cos (c+d x)}{d^5}-\frac{b^2 x^4 \cos (c+d x)}{d} \]

[Out]

(-24*b^2*Cos[c + d*x])/d^5 + (2*a^2*Cos[c + d*x])/d^3 + (12*a*b*x*Cos[c + d*x])/d^3 + (12*b^2*x^2*Cos[c + d*x]
)/d^3 - (a^2*x^2*Cos[c + d*x])/d - (2*a*b*x^3*Cos[c + d*x])/d - (b^2*x^4*Cos[c + d*x])/d - (12*a*b*Sin[c + d*x
])/d^4 - (24*b^2*x*Sin[c + d*x])/d^4 + (2*a^2*x*Sin[c + d*x])/d^2 + (6*a*b*x^2*Sin[c + d*x])/d^2 + (4*b^2*x^3*
Sin[c + d*x])/d^2

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Rubi [A]  time = 0.320068, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {6742, 3296, 2638, 2637} \[ \frac{2 a^2 x \sin (c+d x)}{d^2}+\frac{2 a^2 \cos (c+d x)}{d^3}-\frac{a^2 x^2 \cos (c+d x)}{d}+\frac{6 a b x^2 \sin (c+d x)}{d^2}-\frac{12 a b \sin (c+d x)}{d^4}+\frac{12 a b x \cos (c+d x)}{d^3}-\frac{2 a b x^3 \cos (c+d x)}{d}+\frac{4 b^2 x^3 \sin (c+d x)}{d^2}+\frac{12 b^2 x^2 \cos (c+d x)}{d^3}-\frac{24 b^2 x \sin (c+d x)}{d^4}-\frac{24 b^2 \cos (c+d x)}{d^5}-\frac{b^2 x^4 \cos (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)^2*Sin[c + d*x],x]

[Out]

(-24*b^2*Cos[c + d*x])/d^5 + (2*a^2*Cos[c + d*x])/d^3 + (12*a*b*x*Cos[c + d*x])/d^3 + (12*b^2*x^2*Cos[c + d*x]
)/d^3 - (a^2*x^2*Cos[c + d*x])/d - (2*a*b*x^3*Cos[c + d*x])/d - (b^2*x^4*Cos[c + d*x])/d - (12*a*b*Sin[c + d*x
])/d^4 - (24*b^2*x*Sin[c + d*x])/d^4 + (2*a^2*x*Sin[c + d*x])/d^2 + (6*a*b*x^2*Sin[c + d*x])/d^2 + (4*b^2*x^3*
Sin[c + d*x])/d^2

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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]

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 x^2 (a+b x)^2 \sin (c+d x) \, dx &=\int \left (a^2 x^2 \sin (c+d x)+2 a b x^3 \sin (c+d x)+b^2 x^4 \sin (c+d x)\right ) \, dx\\ &=a^2 \int x^2 \sin (c+d x) \, dx+(2 a b) \int x^3 \sin (c+d x) \, dx+b^2 \int x^4 \sin (c+d x) \, dx\\ &=-\frac{a^2 x^2 \cos (c+d x)}{d}-\frac{2 a b x^3 \cos (c+d x)}{d}-\frac{b^2 x^4 \cos (c+d x)}{d}+\frac{\left (2 a^2\right ) \int x \cos (c+d x) \, dx}{d}+\frac{(6 a b) \int x^2 \cos (c+d x) \, dx}{d}+\frac{\left (4 b^2\right ) \int x^3 \cos (c+d x) \, dx}{d}\\ &=-\frac{a^2 x^2 \cos (c+d x)}{d}-\frac{2 a b x^3 \cos (c+d x)}{d}-\frac{b^2 x^4 \cos (c+d x)}{d}+\frac{2 a^2 x \sin (c+d x)}{d^2}+\frac{6 a b x^2 \sin (c+d x)}{d^2}+\frac{4 b^2 x^3 \sin (c+d x)}{d^2}-\frac{\left (2 a^2\right ) \int \sin (c+d x) \, dx}{d^2}-\frac{(12 a b) \int x \sin (c+d x) \, dx}{d^2}-\frac{\left (12 b^2\right ) \int x^2 \sin (c+d x) \, dx}{d^2}\\ &=\frac{2 a^2 \cos (c+d x)}{d^3}+\frac{12 a b x \cos (c+d x)}{d^3}+\frac{12 b^2 x^2 \cos (c+d x)}{d^3}-\frac{a^2 x^2 \cos (c+d x)}{d}-\frac{2 a b x^3 \cos (c+d x)}{d}-\frac{b^2 x^4 \cos (c+d x)}{d}+\frac{2 a^2 x \sin (c+d x)}{d^2}+\frac{6 a b x^2 \sin (c+d x)}{d^2}+\frac{4 b^2 x^3 \sin (c+d x)}{d^2}-\frac{(12 a b) \int \cos (c+d x) \, dx}{d^3}-\frac{\left (24 b^2\right ) \int x \cos (c+d x) \, dx}{d^3}\\ &=\frac{2 a^2 \cos (c+d x)}{d^3}+\frac{12 a b x \cos (c+d x)}{d^3}+\frac{12 b^2 x^2 \cos (c+d x)}{d^3}-\frac{a^2 x^2 \cos (c+d x)}{d}-\frac{2 a b x^3 \cos (c+d x)}{d}-\frac{b^2 x^4 \cos (c+d x)}{d}-\frac{12 a b \sin (c+d x)}{d^4}-\frac{24 b^2 x \sin (c+d x)}{d^4}+\frac{2 a^2 x \sin (c+d x)}{d^2}+\frac{6 a b x^2 \sin (c+d x)}{d^2}+\frac{4 b^2 x^3 \sin (c+d x)}{d^2}+\frac{\left (24 b^2\right ) \int \sin (c+d x) \, dx}{d^4}\\ &=-\frac{24 b^2 \cos (c+d x)}{d^5}+\frac{2 a^2 \cos (c+d x)}{d^3}+\frac{12 a b x \cos (c+d x)}{d^3}+\frac{12 b^2 x^2 \cos (c+d x)}{d^3}-\frac{a^2 x^2 \cos (c+d x)}{d}-\frac{2 a b x^3 \cos (c+d x)}{d}-\frac{b^2 x^4 \cos (c+d x)}{d}-\frac{12 a b \sin (c+d x)}{d^4}-\frac{24 b^2 x \sin (c+d x)}{d^4}+\frac{2 a^2 x \sin (c+d x)}{d^2}+\frac{6 a b x^2 \sin (c+d x)}{d^2}+\frac{4 b^2 x^3 \sin (c+d x)}{d^2}\\ \end{align*}

Mathematica [A]  time = 0.266428, size = 101, normalized size = 0.54 \[ \frac{2 d (a+2 b x) \left (a d^2 x+b \left (d^2 x^2-6\right )\right ) \sin (c+d x)-\left (a^2 d^2 \left (d^2 x^2-2\right )+2 a b d^2 x \left (d^2 x^2-6\right )+b^2 \left (d^4 x^4-12 d^2 x^2+24\right )\right ) \cos (c+d x)}{d^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((2*a*b*d^2*x*(-6 + d^2*x^2) + a^2*d^2*(-2 + d^2*x^2) + b^2*(24 - 12*d^2*x^2 + d^4*x^4))*Cos[c + d*x]) + 2*d
*(a + 2*b*x)*(a*d^2*x + b*(-6 + d^2*x^2))*Sin[c + d*x])/d^5

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Maple [B]  time = 0.007, size = 468, normalized size = 2.5 \begin{align*}{\frac{1}{{d}^{3}} \left ({\frac{{b}^{2} \left ( - \left ( dx+c \right ) ^{4}\cos \left ( dx+c \right ) +4\, \left ( dx+c \right ) ^{3}\sin \left ( dx+c \right ) +12\, \left ( dx+c \right ) ^{2}\cos \left ( dx+c \right ) -24\,\cos \left ( dx+c \right ) -24\, \left ( dx+c \right ) \sin \left ( dx+c \right ) \right ) }{{d}^{2}}}+2\,{\frac{ab \left ( - \left ( dx+c \right ) ^{3}\cos \left ( dx+c \right ) +3\, \left ( dx+c \right ) ^{2}\sin \left ( dx+c \right ) -6\,\sin \left ( dx+c \right ) +6\, \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) }{d}}-4\,{\frac{{b}^{2}c \left ( - \left ( dx+c \right ) ^{3}\cos \left ( dx+c \right ) +3\, \left ( dx+c \right ) ^{2}\sin \left ( dx+c \right ) -6\,\sin \left ( dx+c \right ) +6\, \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) }{{d}^{2}}}+{a}^{2} \left ( - \left ( dx+c \right ) ^{2}\cos \left ( dx+c \right ) +2\,\cos \left ( dx+c \right ) +2\, \left ( dx+c \right ) \sin \left ( dx+c \right ) \right ) -6\,{\frac{abc \left ( - \left ( dx+c \right ) ^{2}\cos \left ( dx+c \right ) +2\,\cos \left ( dx+c \right ) +2\, \left ( dx+c \right ) \sin \left ( dx+c \right ) \right ) }{d}}+6\,{\frac{{b}^{2}{c}^{2} \left ( - \left ( dx+c \right ) ^{2}\cos \left ( dx+c \right ) +2\,\cos \left ( dx+c \right ) +2\, \left ( dx+c \right ) \sin \left ( dx+c \right ) \right ) }{{d}^{2}}}-2\,{a}^{2}c \left ( \sin \left ( dx+c \right ) - \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) +6\,{\frac{ab{c}^{2} \left ( \sin \left ( dx+c \right ) - \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) }{d}}-4\,{\frac{{b}^{2}{c}^{3} \left ( \sin \left ( dx+c \right ) - \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) }{{d}^{2}}}-{a}^{2}{c}^{2}\cos \left ( dx+c \right ) +2\,{\frac{ab{c}^{3}\cos \left ( dx+c \right ) }{d}}-{\frac{{b}^{2}{c}^{4}\cos \left ( dx+c \right ) }{{d}^{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^3*(1/d^2*b^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
)*sin(d*x+c))+2/d*a*b*(-(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))-4/d^2*b
^2*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))+a^2*(-(d*x+c)^2*cos(d*x+
c)+2*cos(d*x+c)+2*(d*x+c)*sin(d*x+c))-6/d*a*b*c*(-(d*x+c)^2*cos(d*x+c)+2*cos(d*x+c)+2*(d*x+c)*sin(d*x+c))+6/d^
2*b^2*c^2*(-(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))+6/
d*a*b*c^2*(sin(d*x+c)-(d*x+c)*cos(d*x+c))-4/d^2*b^2*c^3*(sin(d*x+c)-(d*x+c)*cos(d*x+c))-a^2*c^2*cos(d*x+c)+2/d
*a*b*c^3*cos(d*x+c)-1/d^2*b^2*c^4*cos(d*x+c))

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Maxima [B]  time = 1.07477, size = 548, normalized size = 2.95 \begin{align*} -\frac{a^{2} c^{2} \cos \left (d x + c\right ) + \frac{b^{2} c^{4} \cos \left (d x + c\right )}{d^{2}} - \frac{2 \, a b c^{3} \cos \left (d x + c\right )}{d} - 2 \,{\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} a^{2} c - \frac{4 \,{\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} b^{2} c^{3}}{d^{2}} + \frac{6 \,{\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} a b c^{2}}{d} +{\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{6 \,{\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^{2}}{d^{2}} - \frac{6 \,{\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}{d} - \frac{4 \,{\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}{d^{2}} + \frac{2 \,{\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}{d} + \frac{{\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}}{d^{2}}}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a^2*c^2*cos(d*x + c) + b^2*c^4*cos(d*x + c)/d^2 - 2*a*b*c^3*cos(d*x + c)/d - 2*((d*x + c)*cos(d*x + c) - sin
(d*x + c))*a^2*c - 4*((d*x + c)*cos(d*x + c) - sin(d*x + c))*b^2*c^3/d^2 + 6*((d*x + c)*cos(d*x + c) - sin(d*x
+ c))*a*b*c^2/d + (((d*x + c)^2 - 2)*cos(d*x + c) - 2*(d*x + c)*sin(d*x + c))*a^2 + 6*(((d*x + c)^2 - 2)*cos(
d*x + c) - 2*(d*x + c)*sin(d*x + c))*b^2*c^2/d^2 - 6*(((d*x + c)^2 - 2)*cos(d*x + c) - 2*(d*x + c)*sin(d*x + c
))*a*b*c/d - 4*(((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/d^2 + 2*(((
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/d + (((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/d^2)/d^3

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Fricas [A]  time = 1.72934, size = 270, normalized size = 1.45 \begin{align*} -\frac{{\left (b^{2} d^{4} x^{4} + 2 \, a b d^{4} x^{3} - 12 \, a b d^{2} x - 2 \, a^{2} d^{2} +{\left (a^{2} d^{4} - 12 \, b^{2} d^{2}\right )} x^{2} + 24 \, b^{2}\right )} \cos \left (d x + c\right ) - 2 \,{\left (2 \, b^{2} d^{3} x^{3} + 3 \, a b d^{3} x^{2} - 6 \, a b d +{\left (a^{2} d^{3} - 12 \, b^{2} d\right )} x\right )} \sin \left (d x + c\right )}{d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((b^2*d^4*x^4 + 2*a*b*d^4*x^3 - 12*a*b*d^2*x - 2*a^2*d^2 + (a^2*d^4 - 12*b^2*d^2)*x^2 + 24*b^2)*cos(d*x + c)
- 2*(2*b^2*d^3*x^3 + 3*a*b*d^3*x^2 - 6*a*b*d + (a^2*d^3 - 12*b^2*d)*x)*sin(d*x + c))/d^5

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Sympy [A]  time = 2.59693, size = 228, normalized size = 1.23 \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^{3} \cos{\left (c + d x \right )}}{d} + \frac{6 a b x^{2} \sin{\left (c + d x \right )}}{d^{2}} + \frac{12 a b x \cos{\left (c + d x \right )}}{d^{3}} - \frac{12 a b \sin{\left (c + d x \right )}}{d^{4}} - \frac{b^{2} x^{4} \cos{\left (c + d x \right )}}{d} + \frac{4 b^{2} x^{3} \sin{\left (c + d x \right )}}{d^{2}} + \frac{12 b^{2} x^{2} \cos{\left (c + d x \right )}}{d^{3}} - \frac{24 b^{2} x \sin{\left (c + d x \right )}}{d^{4}} - \frac{24 b^{2} \cos{\left (c + d x \right )}}{d^{5}} & \text{for}\: d \neq 0 \\\left (\frac{a^{2} x^{3}}{3} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{5}}{5}\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+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**3*cos(
c + d*x)/d + 6*a*b*x**2*sin(c + d*x)/d**2 + 12*a*b*x*cos(c + d*x)/d**3 - 12*a*b*sin(c + d*x)/d**4 - b**2*x**4*
cos(c + d*x)/d + 4*b**2*x**3*sin(c + d*x)/d**2 + 12*b**2*x**2*cos(c + d*x)/d**3 - 24*b**2*x*sin(c + d*x)/d**4
- 24*b**2*cos(c + d*x)/d**5, Ne(d, 0)), ((a**2*x**3/3 + a*b*x**4/2 + b**2*x**5/5)*sin(c), True))

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Giac [A]  time = 1.11363, size = 173, normalized size = 0.93 \begin{align*} -\frac{{\left (b^{2} d^{4} x^{4} + 2 \, a b d^{4} x^{3} + a^{2} d^{4} x^{2} - 12 \, b^{2} d^{2} x^{2} - 12 \, a b d^{2} x - 2 \, a^{2} d^{2} + 24 \, b^{2}\right )} \cos \left (d x + c\right )}{d^{5}} + \frac{2 \,{\left (2 \, b^{2} d^{3} x^{3} + 3 \, a b d^{3} x^{2} + a^{2} d^{3} x - 12 \, b^{2} d x - 6 \, a b d\right )} \sin \left (d x + c\right )}{d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b^2*d^4*x^4 + 2*a*b*d^4*x^3 + a^2*d^4*x^2 - 12*b^2*d^2*x^2 - 12*a*b*d^2*x - 2*a^2*d^2 + 24*b^2)*cos(d*x + c)
/d^5 + 2*(2*b^2*d^3*x^3 + 3*a*b*d^3*x^2 + a^2*d^3*x - 12*b^2*d*x - 6*a*b*d)*sin(d*x + c)/d^5