\(\int x^2 (a+b x^2)^2 \sin (c+d x) \, dx\) [49]

Optimal result

Integrand size = 19, antiderivative size = 236 \[ \int x^2 \left (a+b x^2\right )^2 \sin (c+d x) \, dx=\frac {720 b^2 \cos (c+d x)}{d^7}-\frac {48 a b \cos (c+d x)}{d^5}+\frac {2 a^2 \cos (c+d x)}{d^3}-\frac {360 b^2 x^2 \cos (c+d x)}{d^5}+\frac {24 a b x^2 \cos (c+d x)}{d^3}-\frac {a^2 x^2 \cos (c+d x)}{d}+\frac {30 b^2 x^4 \cos (c+d x)}{d^3}-\frac {2 a b x^4 \cos (c+d x)}{d}-\frac {b^2 x^6 \cos (c+d x)}{d}+\frac {720 b^2 x \sin (c+d x)}{d^6}-\frac {48 a b x \sin (c+d x)}{d^4}+\frac {2 a^2 x \sin (c+d x)}{d^2}-\frac {120 b^2 x^3 \sin (c+d x)}{d^4}+\frac {8 a b x^3 \sin (c+d x)}{d^2}+\frac {6 b^2 x^5 \sin (c+d x)}{d^2} \] Output:

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*si 
n(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*si 
n(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
 

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.59 \[ \int x^2 \left (a+b x^2\right )^2 \sin (c+d x) \, dx=\frac {-\left (\left (a^2 d^4 \left (-2+d^2 x^2\right )+2 a b d^2 \left (24-12 d^2 x^2+d^4 x^4\right )+b^2 \left (-720+360 d^2 x^2-30 d^4 x^4+d^6 x^6\right )\right ) \cos (c+d x)\right )+2 d x \left (a^2 d^4+4 a b d^2 \left (-6+d^2 x^2\right )+3 b^2 \left (120-20 d^2 x^2+d^4 x^4\right )\right ) \sin (c+d x)}{d^7} \] Input:

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

Output:

(-((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
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3820, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^2 \left (a+b x^2\right )^2 \sin (c+d x) \, dx\)

\(\Big \downarrow \) 3820

\(\displaystyle \int \left (a^2 x^2 \sin (c+d x)+2 a b x^4 \sin (c+d x)+b^2 x^6 \sin (c+d x)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 a^2 \cos (c+d x)}{d^3}+\frac {2 a^2 x \sin (c+d x)}{d^2}-\frac {a^2 x^2 \cos (c+d x)}{d}-\frac {48 a b \cos (c+d x)}{d^5}-\frac {48 a b x \sin (c+d x)}{d^4}+\frac {24 a b x^2 \cos (c+d x)}{d^3}+\frac {8 a b x^3 \sin (c+d x)}{d^2}-\frac {2 a b x^4 \cos (c+d x)}{d}+\frac {720 b^2 \cos (c+d x)}{d^7}+\frac {720 b^2 x \sin (c+d x)}{d^6}-\frac {360 b^2 x^2 \cos (c+d x)}{d^5}-\frac {120 b^2 x^3 \sin (c+d x)}{d^4}+\frac {30 b^2 x^4 \cos (c+d x)}{d^3}+\frac {6 b^2 x^5 \sin (c+d x)}{d^2}-\frac {b^2 x^6 \cos (c+d x)}{d}\)

Input:

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

Output:

(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*Cos 
[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
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3820

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*Sin[(c_.) + (d_.)*(x_ 
)], x_Symbol] :> Int[ExpandIntegrand[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]
 

Maple [A] (verified)

Time = 1.09 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.68

method	result	size
risch	\(-\frac {\left (b^{2} x^{6} d^{6}+2 a b \,d^{6} x^{4}+a^{2} d^{6} x^{2}-30 b^{2} x^{4} d^{4}-24 a b \,d^{4} x^{2}-2 a^{2} d^{4}+360 x^{2} d^{2} b^{2}+48 b \,d^{2} a -720 b^{2}\right ) \cos \left (d x +c \right )}{d^{7}}+\frac {2 x \left (3 b^{2} x^{4} d^{4}+4 a b \,d^{4} x^{2}+a^{2} d^{4}-60 x^{2} d^{2} b^{2}-24 b \,d^{2} a +360 b^{2}\right ) \sin \left (d x +c \right )}{d^{6}}\)	\(160\)
parallelrisch	\(\frac {\left (x^{2} \left (b \,x^{2}+a \right )^{2} d^{6}+\left (-30 b^{2} x^{4}-24 a b \,x^{2}-4 a^{2}\right ) d^{4}+\left (360 x^{2} b^{2}+96 a b \right ) d^{2}-1440 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+4 d x \left (\left (3 b \,x^{2}+a \right ) \left (b \,x^{2}+a \right ) d^{4}+\left (-60 x^{2} b^{2}-24 a b \right ) d^{2}+360 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\left (\left (b \,x^{2}+a \right )^{2} d^{4}+\left (-30 x^{2} b^{2}-24 a b \right ) d^{2}+360 b^{2}\right ) d^{2} x^{2}}{d^{7} \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}\)	\(196\)
norman	\(\frac {\frac {b^{2} x^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}+\frac {\left (a^{2} d^{4}-24 b \,d^{2} a +360 b^{2}\right ) x^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d^{5}}-\frac {b^{2} x^{6}}{d}-\frac {\left (a^{2} d^{4}-24 b \,d^{2} a +360 b^{2}\right ) x^{2}}{d^{5}}-\frac {\left (4 a^{2} d^{4}-96 b \,d^{2} a +1440 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d^{7}}+\frac {12 b^{2} x^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d^{2}}-\frac {2 b \left (d^{2} a -15 b \right ) x^{4}}{d^{3}}+\frac {4 \left (a^{2} d^{4}-24 b \,d^{2} a +360 b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d^{6}}+\frac {16 b \left (d^{2} a -15 b \right ) x^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d^{4}}+\frac {2 b \left (d^{2} a -15 b \right ) x^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d^{3}}}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\)	\(282\)
orering	\(\frac {4 \left (3 b^{3} d^{6} x^{8}+7 a \,b^{2} d^{6} x^{6}+5 a^{2} b \,d^{6} x^{4}-75 b^{3} d^{4} x^{6}+a^{3} d^{6} x^{2}-93 a \,b^{2} d^{4} x^{4}-27 a^{2} b \,d^{4} x^{2}+720 b^{3} d^{2} x^{4}-a^{3} d^{4}+432 a \,b^{2} d^{2} x^{2}+24 a^{2} b \,d^{2}-1080 b^{3} x^{2}-360 b^{2} a \right ) \sin \left (d x +c \right )}{d^{8} x \left (b \,x^{2}+a \right )}-\frac {\left (b^{2} x^{6} d^{6}+2 a b \,d^{6} x^{4}+a^{2} d^{6} x^{2}-30 b^{2} x^{4} d^{4}-24 a b \,d^{4} x^{2}-2 a^{2} d^{4}+360 x^{2} d^{2} b^{2}+48 b \,d^{2} a -720 b^{2}\right ) \left (2 x \left (b \,x^{2}+a \right )^{2} \sin \left (d x +c \right )+4 x^{3} \left (b \,x^{2}+a \right ) \sin \left (d x +c \right ) b +x^{2} \left (b \,x^{2}+a \right )^{2} d \cos \left (d x +c \right )\right )}{d^{8} x^{2} \left (b \,x^{2}+a \right )^{2}}\)	\(318\)
meijerg	\(\frac {64 b^{2} \sin \left (c \right ) \sqrt {\pi }\, \left (\frac {x \left (d^{2}\right )^{\frac {7}{2}} \left (\frac {21}{8} x^{4} d^{4}-\frac {105}{2} x^{2} d^{2}+315\right ) \cos \left (d x \right )}{28 \sqrt {\pi }\, d^{6}}-\frac {\left (d^{2}\right )^{\frac {7}{2}} \left (-\frac {7}{16} x^{6} d^{6}+\frac {105}{8} x^{4} d^{4}-\frac {315}{2} x^{2} d^{2}+315\right ) \sin \left (d x \right )}{28 \sqrt {\pi }\, d^{7}}\right )}{d^{6} \sqrt {d^{2}}}+\frac {64 b^{2} \cos \left (c \right ) \sqrt {\pi }\, \left (-\frac {45}{4 \sqrt {\pi }}+\frac {\left (-\frac {1}{16} x^{6} d^{6}+\frac {15}{8} x^{4} d^{4}-\frac {45}{2} x^{2} d^{2}+45\right ) \cos \left (d x \right )}{4 \sqrt {\pi }}+\frac {x d \left (\frac {3}{8} x^{4} d^{4}-\frac {15}{2} x^{2} d^{2}+45\right ) \sin \left (d x \right )}{4 \sqrt {\pi }}\right )}{d^{7}}+\frac {32 a b \sin \left (c \right ) \sqrt {\pi }\, \left (-\frac {x \left (d^{2}\right )^{\frac {5}{2}} \left (-\frac {5 x^{2} d^{2}}{2}+15\right ) \cos \left (d x \right )}{10 \sqrt {\pi }\, d^{4}}+\frac {\left (d^{2}\right )^{\frac {5}{2}} \left (\frac {5}{8} x^{4} d^{4}-\frac {15}{2} x^{2} d^{2}+15\right ) \sin \left (d x \right )}{10 \sqrt {\pi }\, d^{5}}\right )}{d^{4} \sqrt {d^{2}}}+\frac {32 a b \cos \left (c \right ) \sqrt {\pi }\, \left (\frac {3}{2 \sqrt {\pi }}-\frac {\left (\frac {3}{8} x^{4} d^{4}-\frac {9}{2} x^{2} d^{2}+9\right ) \cos \left (d x \right )}{6 \sqrt {\pi }}-\frac {x d \left (-\frac {3 x^{2} d^{2}}{2}+9\right ) \sin \left (d x \right )}{6 \sqrt {\pi }}\right )}{d^{5}}+\frac {4 a^{2} \sin \left (c \right ) \sqrt {\pi }\, \left (\frac {x \left (d^{2}\right )^{\frac {3}{2}} \cos \left (d x \right )}{2 \sqrt {\pi }\, d^{2}}-\frac {\left (d^{2}\right )^{\frac {3}{2}} \left (-\frac {3 x^{2} d^{2}}{2}+3\right ) \sin \left (d x \right )}{6 \sqrt {\pi }\, d^{3}}\right )}{d^{2} \sqrt {d^{2}}}+\frac {4 a^{2} \cos \left (c \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (-\frac {x^{2} d^{2}}{2}+1\right ) \cos \left (d x \right )}{2 \sqrt {\pi }}+\frac {x d \sin \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{3}}\)	\(443\)
parts	\(-\frac {b^{2} x^{6} \cos \left (d x +c \right )}{d}-\frac {2 a b \,x^{4} \cos \left (d x +c \right )}{d}-\frac {a^{2} x^{2} \cos \left (d x +c \right )}{d}+\frac {-\frac {2 a^{2} c \sin \left (d x +c \right )}{d}+\frac {2 a^{2} \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )}{d}-\frac {8 a b \,c^{3} \sin \left (d x +c \right )}{d^{3}}+\frac {24 a b \,c^{2} \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}-\frac {24 a b c \left (\left (d x +c \right )^{2} \sin \left (d x +c \right )-2 \sin \left (d x +c \right )+2 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{3}}+\frac {8 a b \left (\left (d x +c \right )^{3} \sin \left (d x +c \right )+3 \left (d x +c \right )^{2} \cos \left (d x +c \right )-6 \cos \left (d x +c \right )-6 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}-\frac {6 b^{2} c^{5} \sin \left (d x +c \right )}{d^{5}}+\frac {30 b^{2} c^{4} \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{5}}-\frac {60 b^{2} c^{3} \left (\left (d x +c \right )^{2} \sin \left (d x +c \right )-2 \sin \left (d x +c \right )+2 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{5}}+\frac {60 b^{2} c^{2} \left (\left (d x +c \right )^{3} \sin \left (d x +c \right )+3 \left (d x +c \right )^{2} \cos \left (d x +c \right )-6 \cos \left (d x +c \right )-6 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{5}}-\frac {30 b^{2} c \left (\left (d x +c \right )^{4} \sin \left (d x +c \right )+4 \left (d x +c \right )^{3} \cos \left (d x +c \right )-12 \left (d x +c \right )^{2} \sin \left (d x +c \right )+24 \sin \left (d x +c \right )-24 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{5}}+\frac {6 b^{2} \left (\left (d x +c \right )^{5} \sin \left (d x +c \right )+5 \left (d x +c \right )^{4} \cos \left (d x +c \right )-20 \left (d x +c \right )^{3} \sin \left (d x +c \right )-60 \left (d x +c \right )^{2} \cos \left (d x +c \right )+120 \cos \left (d x +c \right )+120 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{5}}}{d^{2}}\)	\(566\)
derivativedivides	\(\frac {-a^{2} c^{2} \cos \left (d x +c \right )-2 a^{2} c \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )+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 {2 a b \,c^{4} \cos \left (d x +c \right )}{d^{2}}-\frac {8 a b \,c^{3} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{2}}+\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 {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 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{2}}+\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 {b^{2} c^{6} \cos \left (d x +c \right )}{d^{4}}-\frac {6 b^{2} c^{5} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{4}}+\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}}-\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 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{4}}+\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 {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 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{4}}+\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}}}{d^{3}}\)	\(746\)
default	\(\frac {-a^{2} c^{2} \cos \left (d x +c \right )-2 a^{2} c \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )+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 {2 a b \,c^{4} \cos \left (d x +c \right )}{d^{2}}-\frac {8 a b \,c^{3} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{2}}+\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 {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 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{2}}+\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 {b^{2} c^{6} \cos \left (d x +c \right )}{d^{4}}-\frac {6 b^{2} c^{5} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{4}}+\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}}-\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 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{4}}+\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 {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 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{4}}+\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}}}{d^{3}}\)	\(746\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.65 \[ \int x^2 \left (a+b x^2\right )^2 \sin (c+d x) \, dx=-\frac {{\left (b^{2} d^{6} x^{6} - 2 \, a^{2} d^{4} + 2 \, {\left (a b d^{6} - 15 \, b^{2} d^{4}\right )} x^{4} + 48 \, a b d^{2} + {\left (a^{2} d^{6} - 24 \, a b d^{4} + 360 \, b^{2} d^{2}\right )} x^{2} - 720 \, b^{2}\right )} \cos \left (d x + c\right ) - 2 \, {\left (3 \, b^{2} d^{5} x^{5} + 4 \, {\left (a b d^{5} - 15 \, b^{2} d^{3}\right )} x^{3} + {\left (a^{2} d^{5} - 24 \, a b d^{3} + 360 \, b^{2} d\right )} x\right )} \sin \left (d x + c\right )}{d^{7}} \] Input:

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

Output:

-((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
 

Sympy [A] (verification not implemented)

Time = 0.57 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.21 \[ \int x^2 \left (a+b x^2\right )^2 \sin (c+d x) \, dx=\begin {cases} - \frac {a^{2} x^{2} \cos {\left (c + d x \right )}}{d} + \frac {2 a^{2} x \sin {\left (c + d x \right )}}{d^{2}} + \frac {2 a^{2} \cos {\left (c + d x \right )}}{d^{3}} - \frac {2 a b x^{4} \cos {\left (c + d x \right )}}{d} + \frac {8 a b x^{3} \sin {\left (c + d x \right )}}{d^{2}} + \frac {24 a b x^{2} \cos {\left (c + d x \right )}}{d^{3}} - \frac {48 a b x \sin {\left (c + d x \right )}}{d^{4}} - \frac {48 a b \cos {\left (c + d x \right )}}{d^{5}} - \frac {b^{2} x^{6} \cos {\left (c + d x \right )}}{d} + \frac {6 b^{2} x^{5} \sin {\left (c + d x \right )}}{d^{2}} + \frac {30 b^{2} x^{4} \cos {\left (c + d x \right )}}{d^{3}} - \frac {120 b^{2} x^{3} \sin {\left (c + d x \right )}}{d^{4}} - \frac {360 b^{2} x^{2} \cos {\left (c + d x \right )}}{d^{5}} + \frac {720 b^{2} x \sin {\left (c + d x \right )}}{d^{6}} + \frac {720 b^{2} \cos {\left (c + d x \right )}}{d^{7}} & \text {for}\: d \neq 0 \\\left (\frac {a^{2} x^{3}}{3} + \frac {2 a b x^{5}}{5} + \frac {b^{2} x^{7}}{7}\right ) \sin {\left (c \right )} & \text {otherwise} \end {cases} \] Input:

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

Output:

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 - 36 
0*b**2*x**2*cos(c + d*x)/d**5 + 720*b**2*x*sin(c + d*x)/d**6 + 720*b**2*co 
s(c + d*x)/d**7, Ne(d, 0)), ((a**2*x**3/3 + 2*a*b*x**5/5 + b**2*x**7/7)*si 
n(c), True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 612 vs. \(2 (236) = 472\).

Time = 0.07 (sec) , antiderivative size = 612, normalized size of antiderivative = 2.59 \[ \int x^2 \left (a+b x^2\right )^2 \sin (c+d x) \, dx =\text {Too large to display} \] Input:

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

Output:

-(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) - sin(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 + 3 
60*(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
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.69 \[ \int x^2 \left (a+b x^2\right )^2 \sin (c+d x) \, dx=-\frac {{\left (b^{2} d^{6} x^{6} + 2 \, a b d^{6} x^{4} + a^{2} d^{6} x^{2} - 30 \, b^{2} d^{4} x^{4} - 24 \, a b d^{4} x^{2} - 2 \, a^{2} d^{4} + 360 \, b^{2} d^{2} x^{2} + 48 \, a b d^{2} - 720 \, b^{2}\right )} \cos \left (d x + c\right )}{d^{7}} + \frac {2 \, {\left (3 \, b^{2} d^{5} x^{5} + 4 \, a b d^{5} x^{3} + a^{2} d^{5} x - 60 \, b^{2} d^{3} x^{3} - 24 \, a b d^{3} x + 360 \, b^{2} d x\right )} \sin \left (d x + c\right )}{d^{7}} \] Input:

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

Output:

-(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
 

Mupad [B] (verification not implemented)

Time = 1.30 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.79 \[ \int x^2 \left (a+b x^2\right )^2 \sin (c+d x) \, dx=\frac {2\,\cos \left (c+d\,x\right )\,\left (a^2\,d^4-24\,a\,b\,d^2+360\,b^2\right )}{d^7}-\frac {b^2\,x^6\,\cos \left (c+d\,x\right )}{d}+\frac {6\,b^2\,x^5\,\sin \left (c+d\,x\right )}{d^2}+\frac {2\,x\,\sin \left (c+d\,x\right )\,\left (a^2\,d^4-24\,a\,b\,d^2+360\,b^2\right )}{d^6}-\frac {x^2\,\cos \left (c+d\,x\right )\,\left (a^2\,d^4-24\,a\,b\,d^2+360\,b^2\right )}{d^5}+\frac {2\,x^4\,\cos \left (c+d\,x\right )\,\left (15\,b^2-a\,b\,d^2\right )}{d^3}-\frac {8\,x^3\,\sin \left (c+d\,x\right )\,\left (15\,b^2-a\,b\,d^2\right )}{d^4} \] Input:

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

Output:

(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
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a+b x^2\right )^2 \sin (c+d x) \, dx=\frac {-\cos \left (d x +c \right ) a^{2} d^{6} x^{2}+2 \cos \left (d x +c \right ) a^{2} d^{4}-2 \cos \left (d x +c \right ) a b \,d^{6} x^{4}+24 \cos \left (d x +c \right ) a b \,d^{4} x^{2}-48 \cos \left (d x +c \right ) a b \,d^{2}-\cos \left (d x +c \right ) b^{2} d^{6} x^{6}+30 \cos \left (d x +c \right ) b^{2} d^{4} x^{4}-360 \cos \left (d x +c \right ) b^{2} d^{2} x^{2}+720 \cos \left (d x +c \right ) b^{2}+2 \sin \left (d x +c \right ) a^{2} d^{5} x +8 \sin \left (d x +c \right ) a b \,d^{5} x^{3}-48 \sin \left (d x +c \right ) a b \,d^{3} x +6 \sin \left (d x +c \right ) b^{2} d^{5} x^{5}-120 \sin \left (d x +c \right ) b^{2} d^{3} x^{3}+720 \sin \left (d x +c \right ) b^{2} d x}{d^{7}} \] Input:

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

Output:

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