\(\int x^2 (a+b x)^2 \cosh (c+d x) \, dx\) [10]

Optimal result

Integrand size = 17, antiderivative size = 184 \[ \int x^2 (a+b x)^2 \cosh (c+d x) \, dx=-\frac {12 a b \cosh (c+d x)}{d^4}-\frac {24 b^2 x \cosh (c+d x)}{d^4}-\frac {2 a^2 x \cosh (c+d x)}{d^2}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}-\frac {4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac {24 b^2 \sinh (c+d x)}{d^5}+\frac {2 a^2 \sinh (c+d x)}{d^3}+\frac {12 a b x \sinh (c+d x)}{d^3}+\frac {12 b^2 x^2 \sinh (c+d x)}{d^3}+\frac {a^2 x^2 \sinh (c+d x)}{d}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {b^2 x^4 \sinh (c+d x)}{d} \]

[Out]

-12*a*b*cosh(d*x+c)/d^4-24*b^2*x*cosh(d*x+c)/d^4-2*a^2*x*cosh(d*x+c)/d^2-6*a*b*x^2*cosh(d*x+c)/d^2-4*b^2*x^3*c
osh(d*x+c)/d^2+24*b^2*sinh(d*x+c)/d^5+2*a^2*sinh(d*x+c)/d^3+12*a*b*x*sinh(d*x+c)/d^3+12*b^2*x^2*sinh(d*x+c)/d^
3+a^2*x^2*sinh(d*x+c)/d+2*a*b*x^3*sinh(d*x+c)/d+b^2*x^4*sinh(d*x+c)/d

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6874, 3377, 2717, 2718} \[ \int x^2 (a+b x)^2 \cosh (c+d x) \, dx=\frac {2 a^2 \sinh (c+d x)}{d^3}-\frac {2 a^2 x \cosh (c+d x)}{d^2}+\frac {a^2 x^2 \sinh (c+d x)}{d}-\frac {12 a b \cosh (c+d x)}{d^4}+\frac {12 a b x \sinh (c+d x)}{d^3}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {24 b^2 \sinh (c+d x)}{d^5}-\frac {24 b^2 x \cosh (c+d x)}{d^4}+\frac {12 b^2 x^2 \sinh (c+d x)}{d^3}-\frac {4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac {b^2 x^4 \sinh (c+d x)}{d} \]

[In]

Int[x^2*(a + b*x)^2*Cosh[c + d*x],x]

[Out]

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

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2718

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

Rule 3377

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 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 x^2 \cosh (c+d x)+2 a b x^3 \cosh (c+d x)+b^2 x^4 \cosh (c+d x)\right ) \, dx \\ & = a^2 \int x^2 \cosh (c+d x) \, dx+(2 a b) \int x^3 \cosh (c+d x) \, dx+b^2 \int x^4 \cosh (c+d x) \, dx \\ & = \frac {a^2 x^2 \sinh (c+d x)}{d}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {b^2 x^4 \sinh (c+d x)}{d}-\frac {\left (2 a^2\right ) \int x \sinh (c+d x) \, dx}{d}-\frac {(6 a b) \int x^2 \sinh (c+d x) \, dx}{d}-\frac {\left (4 b^2\right ) \int x^3 \sinh (c+d x) \, dx}{d} \\ & = -\frac {2 a^2 x \cosh (c+d x)}{d^2}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}-\frac {4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac {a^2 x^2 \sinh (c+d x)}{d}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {b^2 x^4 \sinh (c+d x)}{d}+\frac {\left (2 a^2\right ) \int \cosh (c+d x) \, dx}{d^2}+\frac {(12 a b) \int x \cosh (c+d x) \, dx}{d^2}+\frac {\left (12 b^2\right ) \int x^2 \cosh (c+d x) \, dx}{d^2} \\ & = -\frac {2 a^2 x \cosh (c+d x)}{d^2}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}-\frac {4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac {2 a^2 \sinh (c+d x)}{d^3}+\frac {12 a b x \sinh (c+d x)}{d^3}+\frac {12 b^2 x^2 \sinh (c+d x)}{d^3}+\frac {a^2 x^2 \sinh (c+d x)}{d}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {b^2 x^4 \sinh (c+d x)}{d}-\frac {(12 a b) \int \sinh (c+d x) \, dx}{d^3}-\frac {\left (24 b^2\right ) \int x \sinh (c+d x) \, dx}{d^3} \\ & = -\frac {12 a b \cosh (c+d x)}{d^4}-\frac {24 b^2 x \cosh (c+d x)}{d^4}-\frac {2 a^2 x \cosh (c+d x)}{d^2}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}-\frac {4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac {2 a^2 \sinh (c+d x)}{d^3}+\frac {12 a b x \sinh (c+d x)}{d^3}+\frac {12 b^2 x^2 \sinh (c+d x)}{d^3}+\frac {a^2 x^2 \sinh (c+d x)}{d}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {b^2 x^4 \sinh (c+d x)}{d}+\frac {\left (24 b^2\right ) \int \cosh (c+d x) \, dx}{d^4} \\ & = -\frac {12 a b \cosh (c+d x)}{d^4}-\frac {24 b^2 x \cosh (c+d x)}{d^4}-\frac {2 a^2 x \cosh (c+d x)}{d^2}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}-\frac {4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac {24 b^2 \sinh (c+d x)}{d^5}+\frac {2 a^2 \sinh (c+d x)}{d^3}+\frac {12 a b x \sinh (c+d x)}{d^3}+\frac {12 b^2 x^2 \sinh (c+d x)}{d^3}+\frac {a^2 x^2 \sinh (c+d x)}{d}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {b^2 x^4 \sinh (c+d x)}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.54 \[ \int x^2 (a+b x)^2 \cosh (c+d x) \, dx=\frac {-2 d (a+2 b x) \left (a d^2 x+b \left (6+d^2 x^2\right )\right ) \cosh (c+d x)+\left (a^2 d^2 \left (2+d^2 x^2\right )+2 a b d^2 x \left (6+d^2 x^2\right )+b^2 \left (24+12 d^2 x^2+d^4 x^4\right )\right ) \sinh (c+d x)}{d^5} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.73

method	result	size
parallelrisch	\(\frac {2 d x \left (\left (2 b x +a \right ) \left (b x +a \right ) d^{2}+12 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \left (-x^{2} \left (b x +a \right )^{2} d^{4}+2 \left (-6 x^{2} b^{2}-6 a b x -a^{2}\right ) d^{2}-24 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \left (b x +a \right ) d \left (x \left (2 b x +a \right ) d^{2}+12 b \right )}{d^{5} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\)	\(134\)
risch	\(\frac {\left (b^{2} x^{4} d^{4}+2 a b \,d^{4} x^{3}+a^{2} d^{4} x^{2}-4 b^{2} d^{3} x^{3}-6 a b \,d^{3} x^{2}-2 a^{2} d^{3} x +12 x^{2} d^{2} b^{2}+12 a b \,d^{2} x +2 a^{2} d^{2}-24 b^{2} d x -12 d a b +24 b^{2}\right ) {\mathrm e}^{d x +c}}{2 d^{5}}-\frac {\left (b^{2} x^{4} d^{4}+2 a b \,d^{4} x^{3}+a^{2} d^{4} x^{2}+4 b^{2} d^{3} x^{3}+6 a b \,d^{3} x^{2}+2 a^{2} d^{3} x +12 x^{2} d^{2} b^{2}+12 a b \,d^{2} x +2 a^{2} d^{2}+24 b^{2} d x +12 d a b +24 b^{2}\right ) {\mathrm e}^{-d x -c}}{2 d^{5}}\)	\(237\)
parts	\(\frac {b^{2} x^{4} \sinh \left (d x +c \right )}{d}+\frac {2 a b \,x^{3} \sinh \left (d x +c \right )}{d}+\frac {a^{2} x^{2} \sinh \left (d x +c \right )}{d}-\frac {2 \left (\frac {2 b^{2} \left (\left (d x +c \right )^{3} \cosh \left (d x +c \right )-3 \left (d x +c \right )^{2} \sinh \left (d x +c \right )+6 \left (d x +c \right ) \cosh \left (d x +c \right )-6 \sinh \left (d x +c \right )\right )}{d^{3}}-\frac {6 b^{2} c \left (\left (d x +c \right )^{2} \cosh \left (d x +c \right )-2 \left (d x +c \right ) \sinh \left (d x +c \right )+2 \cosh \left (d x +c \right )\right )}{d^{3}}+\frac {3 b a \left (\left (d x +c \right )^{2} \cosh \left (d x +c \right )-2 \left (d x +c \right ) \sinh \left (d x +c \right )+2 \cosh \left (d x +c \right )\right )}{d^{2}}+\frac {6 b^{2} c^{2} \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )}{d^{3}}-\frac {6 b c a \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )}{d^{2}}+\frac {a^{2} \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )}{d}-\frac {2 b^{2} c^{3} \cosh \left (d x +c \right )}{d^{3}}+\frac {3 b a \,c^{2} \cosh \left (d x +c \right )}{d^{2}}-\frac {a^{2} c \cosh \left (d x +c \right )}{d}\right )}{d^{2}}\)	\(340\)
meijerg	\(-\frac {16 i b^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (-\frac {i x d \left (\frac {5 x^{2} d^{2}}{2}+15\right ) \cosh \left (d x \right )}{10 \sqrt {\pi }}+\frac {i \left (\frac {5}{8} d^{4} x^{4}+\frac {15}{2} x^{2} d^{2}+15\right ) \sinh \left (d x \right )}{10 \sqrt {\pi }}\right )}{d^{5}}-\frac {16 b^{2} \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {3}{2 \sqrt {\pi }}-\frac {\left (\frac {3}{8} d^{4} x^{4}+\frac {9}{2} x^{2} d^{2}+9\right ) \cosh \left (d x \right )}{6 \sqrt {\pi }}+\frac {x d \left (\frac {3 x^{2} d^{2}}{2}+9\right ) \sinh \left (d x \right )}{6 \sqrt {\pi }}\right )}{d^{5}}+\frac {16 a b \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {3}{4 \sqrt {\pi }}-\frac {\left (\frac {3 x^{2} d^{2}}{2}+3\right ) \cosh \left (d x \right )}{4 \sqrt {\pi }}+\frac {d x \left (\frac {x^{2} d^{2}}{2}+3\right ) \sinh \left (d x \right )}{4 \sqrt {\pi }}\right )}{d^{4}}-\frac {16 i b a \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {i x d \left (\frac {5 x^{2} d^{2}}{2}+15\right ) \cosh \left (d x \right )}{20 \sqrt {\pi }}-\frac {i \left (\frac {15 x^{2} d^{2}}{2}+15\right ) \sinh \left (d x \right )}{20 \sqrt {\pi }}\right )}{d^{4}}+\frac {4 i a^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {i x d \cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {i \left (\frac {3 x^{2} d^{2}}{2}+3\right ) \sinh \left (d x \right )}{6 \sqrt {\pi }}\right )}{d^{3}}+\frac {4 a^{2} \sinh \left (c \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (\frac {x^{2} d^{2}}{2}+1\right ) \cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {d x \sinh \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{3}}\)	\(344\)
derivativedivides	\(\frac {\frac {6 b^{2} c^{2} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{2}}-\frac {4 b^{2} c \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{2}}+\frac {b^{2} \left (\left (d x +c \right )^{4} \sinh \left (d x +c \right )-4 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+12 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-24 \left (d x +c \right ) \cosh \left (d x +c \right )+24 \sinh \left (d x +c \right )\right )}{d^{2}}-\frac {4 b^{2} c^{3} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{2}}+\frac {6 b a \,c^{2} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d}-\frac {6 b c a \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d}+\frac {2 b a \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d}+\frac {b^{2} c^{4} \sinh \left (d x +c \right )}{d^{2}}-\frac {2 b \,c^{3} a \sinh \left (d x +c \right )}{d}+a^{2} c^{2} \sinh \left (d x +c \right )-2 a^{2} c \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )+a^{2} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{3}}\)	\(463\)
default	\(\frac {\frac {6 b^{2} c^{2} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{2}}-\frac {4 b^{2} c \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{2}}+\frac {b^{2} \left (\left (d x +c \right )^{4} \sinh \left (d x +c \right )-4 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+12 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-24 \left (d x +c \right ) \cosh \left (d x +c \right )+24 \sinh \left (d x +c \right )\right )}{d^{2}}-\frac {4 b^{2} c^{3} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{2}}+\frac {6 b a \,c^{2} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d}-\frac {6 b c a \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d}+\frac {2 b a \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d}+\frac {b^{2} c^{4} \sinh \left (d x +c \right )}{d^{2}}-\frac {2 b \,c^{3} a \sinh \left (d x +c \right )}{d}+a^{2} c^{2} \sinh \left (d x +c \right )-2 a^{2} c \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )+a^{2} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{3}}\)	\(463\)

[In]

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

[Out]

2*(d*x*((2*b*x+a)*(b*x+a)*d^2+12*b^2)*tanh(1/2*d*x+1/2*c)^2+(-x^2*(b*x+a)^2*d^4+2*(-6*b^2*x^2-6*a*b*x-a^2)*d^2
-24*b^2)*tanh(1/2*d*x+1/2*c)+(b*x+a)*d*(x*(2*b*x+a)*d^2+12*b))/d^5/(tanh(1/2*d*x+1/2*c)^2-1)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.69 \[ \int x^2 (a+b x)^2 \cosh (c+d x) \, dx=-\frac {2 \, {\left (2 \, b^{2} d^{3} x^{3} + 3 \, a b d^{3} x^{2} + 6 \, a b d + {\left (a^{2} d^{3} + 12 \, b^{2} d\right )} x\right )} \cosh \left (d x + c\right ) - {\left (b^{2} d^{4} x^{4} + 2 \, a b d^{4} x^{3} + 12 \, a b d^{2} x + 2 \, a^{2} d^{2} + {\left (a^{2} d^{4} + 12 \, b^{2} d^{2}\right )} x^{2} + 24 \, b^{2}\right )} \sinh \left (d x + c\right )}{d^{5}} \]

[In]

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

[Out]

-(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)*cosh(d*x + c) - (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)*sinh(d*x + c))/d^5

Sympy [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.24 \[ \int x^2 (a+b x)^2 \cosh (c+d x) \, dx=\begin {cases} \frac {a^{2} x^{2} \sinh {\left (c + d x \right )}}{d} - \frac {2 a^{2} x \cosh {\left (c + d x \right )}}{d^{2}} + \frac {2 a^{2} \sinh {\left (c + d x \right )}}{d^{3}} + \frac {2 a b x^{3} \sinh {\left (c + d x \right )}}{d} - \frac {6 a b x^{2} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {12 a b x \sinh {\left (c + d x \right )}}{d^{3}} - \frac {12 a b \cosh {\left (c + d x \right )}}{d^{4}} + \frac {b^{2} x^{4} \sinh {\left (c + d x \right )}}{d} - \frac {4 b^{2} x^{3} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {12 b^{2} x^{2} \sinh {\left (c + d x \right )}}{d^{3}} - \frac {24 b^{2} x \cosh {\left (c + d x \right )}}{d^{4}} + \frac {24 b^{2} \sinh {\left (c + d x \right )}}{d^{5}} & \text {for}\: d \neq 0 \\\left (\frac {a^{2} x^{3}}{3} + \frac {a b x^{4}}{2} + \frac {b^{2} x^{5}}{5}\right ) \cosh {\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a**2*x**2*sinh(c + d*x)/d - 2*a**2*x*cosh(c + d*x)/d**2 + 2*a**2*sinh(c + d*x)/d**3 + 2*a*b*x**3*si
nh(c + d*x)/d - 6*a*b*x**2*cosh(c + d*x)/d**2 + 12*a*b*x*sinh(c + d*x)/d**3 - 12*a*b*cosh(c + d*x)/d**4 + b**2
*x**4*sinh(c + d*x)/d - 4*b**2*x**3*cosh(c + d*x)/d**2 + 12*b**2*x**2*sinh(c + d*x)/d**3 - 24*b**2*x*cosh(c +
d*x)/d**4 + 24*b**2*sinh(c + d*x)/d**5, Ne(d, 0)), ((a**2*x**3/3 + a*b*x**4/2 + b**2*x**5/5)*cosh(c), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.79 \[ \int x^2 (a+b x)^2 \cosh (c+d x) \, dx=-\frac {1}{60} \, d {\left (\frac {10 \, {\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} a^{2} e^{\left (d x\right )}}{d^{4}} + \frac {10 \, {\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} a^{2} e^{\left (-d x - c\right )}}{d^{4}} + \frac {15 \, {\left (d^{4} x^{4} e^{c} - 4 \, d^{3} x^{3} e^{c} + 12 \, d^{2} x^{2} e^{c} - 24 \, d x e^{c} + 24 \, e^{c}\right )} a b e^{\left (d x\right )}}{d^{5}} + \frac {15 \, {\left (d^{4} x^{4} + 4 \, d^{3} x^{3} + 12 \, d^{2} x^{2} + 24 \, d x + 24\right )} a b e^{\left (-d x - c\right )}}{d^{5}} + \frac {6 \, {\left (d^{5} x^{5} e^{c} - 5 \, d^{4} x^{4} e^{c} + 20 \, d^{3} x^{3} e^{c} - 60 \, d^{2} x^{2} e^{c} + 120 \, d x e^{c} - 120 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{d^{6}} + \frac {6 \, {\left (d^{5} x^{5} + 5 \, d^{4} x^{4} + 20 \, d^{3} x^{3} + 60 \, d^{2} x^{2} + 120 \, d x + 120\right )} b^{2} e^{\left (-d x - c\right )}}{d^{6}}\right )} + \frac {1}{30} \, {\left (6 \, b^{2} x^{5} + 15 \, a b x^{4} + 10 \, a^{2} x^{3}\right )} \cosh \left (d x + c\right ) \]

[In]

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

[Out]

-1/60*d*(10*(d^3*x^3*e^c - 3*d^2*x^2*e^c + 6*d*x*e^c - 6*e^c)*a^2*e^(d*x)/d^4 + 10*(d^3*x^3 + 3*d^2*x^2 + 6*d*
x + 6)*a^2*e^(-d*x - c)/d^4 + 15*(d^4*x^4*e^c - 4*d^3*x^3*e^c + 12*d^2*x^2*e^c - 24*d*x*e^c + 24*e^c)*a*b*e^(d
*x)/d^5 + 15*(d^4*x^4 + 4*d^3*x^3 + 12*d^2*x^2 + 24*d*x + 24)*a*b*e^(-d*x - c)/d^5 + 6*(d^5*x^5*e^c - 5*d^4*x^
4*e^c + 20*d^3*x^3*e^c - 60*d^2*x^2*e^c + 120*d*x*e^c - 120*e^c)*b^2*e^(d*x)/d^6 + 6*(d^5*x^5 + 5*d^4*x^4 + 20
*d^3*x^3 + 60*d^2*x^2 + 120*d*x + 120)*b^2*e^(-d*x - c)/d^6) + 1/30*(6*b^2*x^5 + 15*a*b*x^4 + 10*a^2*x^3)*cosh
(d*x + c)

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.28 \[ \int x^2 (a+b x)^2 \cosh (c+d x) \, dx=\frac {{\left (b^{2} d^{4} x^{4} + 2 \, a b d^{4} x^{3} + a^{2} d^{4} x^{2} - 4 \, b^{2} d^{3} x^{3} - 6 \, a b d^{3} x^{2} - 2 \, a^{2} d^{3} x + 12 \, b^{2} d^{2} x^{2} + 12 \, a b d^{2} x + 2 \, a^{2} d^{2} - 24 \, b^{2} d x - 12 \, a b d + 24 \, b^{2}\right )} e^{\left (d x + c\right )}}{2 \, d^{5}} - \frac {{\left (b^{2} d^{4} x^{4} + 2 \, a b d^{4} x^{3} + a^{2} d^{4} x^{2} + 4 \, b^{2} d^{3} x^{3} + 6 \, a b d^{3} x^{2} + 2 \, a^{2} d^{3} x + 12 \, b^{2} d^{2} x^{2} + 12 \, a b d^{2} x + 2 \, a^{2} d^{2} + 24 \, b^{2} d x + 12 \, a b d + 24 \, b^{2}\right )} e^{\left (-d x - c\right )}}{2 \, d^{5}} \]

[In]

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

[Out]

1/2*(b^2*d^4*x^4 + 2*a*b*d^4*x^3 + a^2*d^4*x^2 - 4*b^2*d^3*x^3 - 6*a*b*d^3*x^2 - 2*a^2*d^3*x + 12*b^2*d^2*x^2
+ 12*a*b*d^2*x + 2*a^2*d^2 - 24*b^2*d*x - 12*a*b*d + 24*b^2)*e^(d*x + c)/d^5 - 1/2*(b^2*d^4*x^4 + 2*a*b*d^4*x^
3 + a^2*d^4*x^2 + 4*b^2*d^3*x^3 + 6*a*b*d^3*x^2 + 2*a^2*d^3*x + 12*b^2*d^2*x^2 + 12*a*b*d^2*x + 2*a^2*d^2 + 24
*b^2*d*x + 12*a*b*d + 24*b^2)*e^(-d*x - c)/d^5

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.91 \[ \int x^2 (a+b x)^2 \cosh (c+d x) \, dx=\frac {2\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a^2\,d^2+12\,b^2\right )}{d^5}-\frac {4\,b^2\,x^3\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}+\frac {b^2\,x^4\,\mathrm {sinh}\left (c+d\,x\right )}{d}-\frac {12\,a\,b\,\mathrm {cosh}\left (c+d\,x\right )}{d^4}-\frac {2\,x\,\mathrm {cosh}\left (c+d\,x\right )\,\left (a^2\,d^2+12\,b^2\right )}{d^4}+\frac {x^2\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a^2\,d^2+12\,b^2\right )}{d^3}-\frac {6\,a\,b\,x^2\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}+\frac {2\,a\,b\,x^3\,\mathrm {sinh}\left (c+d\,x\right )}{d}+\frac {12\,a\,b\,x\,\mathrm {sinh}\left (c+d\,x\right )}{d^3} \]

[In]

int(x^2*cosh(c + d*x)*(a + b*x)^2,x)

[Out]

(2*sinh(c + d*x)*(12*b^2 + a^2*d^2))/d^5 - (4*b^2*x^3*cosh(c + d*x))/d^2 + (b^2*x^4*sinh(c + d*x))/d - (12*a*b
*cosh(c + d*x))/d^4 - (2*x*cosh(c + d*x)*(12*b^2 + a^2*d^2))/d^4 + (x^2*sinh(c + d*x)*(12*b^2 + a^2*d^2))/d^3
- (6*a*b*x^2*cosh(c + d*x))/d^2 + (2*a*b*x^3*sinh(c + d*x))/d + (12*a*b*x*sinh(c + d*x))/d^3