\(\int \frac {(a+b x)^2 \cosh (c+d x)}{x^5} \, dx\) [17]

Optimal result

Integrand size = 17, antiderivative size = 248 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^5} \, dx=-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {2 a b \cosh (c+d x)}{3 x^3}-\frac {b^2 \cosh (c+d x)}{2 x^2}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac {a b d^2 \cosh (c+d x)}{3 x}+\frac {1}{2} b^2 d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{24} a^2 d^4 \cosh (c) \text {Chi}(d x)+\frac {1}{3} a b d^3 \text {Chi}(d x) \sinh (c)-\frac {a^2 d \sinh (c+d x)}{12 x^3}-\frac {a b d \sinh (c+d x)}{3 x^2}-\frac {b^2 d \sinh (c+d x)}{2 x}-\frac {a^2 d^3 \sinh (c+d x)}{24 x}+\frac {1}{3} a b d^3 \cosh (c) \text {Shi}(d x)+\frac {1}{2} b^2 d^2 \sinh (c) \text {Shi}(d x)+\frac {1}{24} a^2 d^4 \sinh (c) \text {Shi}(d x) \]

[Out]

1/2*b^2*d^2*Chi(d*x)*cosh(c)+1/24*a^2*d^4*Chi(d*x)*cosh(c)-1/4*a^2*cosh(d*x+c)/x^4-2/3*a*b*cosh(d*x+c)/x^3-1/2
*b^2*cosh(d*x+c)/x^2-1/24*a^2*d^2*cosh(d*x+c)/x^2-1/3*a*b*d^2*cosh(d*x+c)/x+1/3*a*b*d^3*cosh(c)*Shi(d*x)+1/3*a
*b*d^3*Chi(d*x)*sinh(c)+1/2*b^2*d^2*Shi(d*x)*sinh(c)+1/24*a^2*d^4*Shi(d*x)*sinh(c)-1/12*a^2*d*sinh(d*x+c)/x^3-
1/3*a*b*d*sinh(d*x+c)/x^2-1/2*b^2*d*sinh(d*x+c)/x-1/24*a^2*d^3*sinh(d*x+c)/x

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6874, 3378, 3384, 3379, 3382} \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^5} \, dx=\frac {1}{24} a^2 d^4 \cosh (c) \text {Chi}(d x)+\frac {1}{24} a^2 d^4 \sinh (c) \text {Shi}(d x)-\frac {a^2 d^3 \sinh (c+d x)}{24 x}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a^2 d \sinh (c+d x)}{12 x^3}+\frac {1}{3} a b d^3 \sinh (c) \text {Chi}(d x)+\frac {1}{3} a b d^3 \cosh (c) \text {Shi}(d x)-\frac {a b d^2 \cosh (c+d x)}{3 x}-\frac {2 a b \cosh (c+d x)}{3 x^3}-\frac {a b d \sinh (c+d x)}{3 x^2}+\frac {1}{2} b^2 d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{2} b^2 d^2 \sinh (c) \text {Shi}(d x)-\frac {b^2 \cosh (c+d x)}{2 x^2}-\frac {b^2 d \sinh (c+d x)}{2 x} \]

[In]

Int[((a + b*x)^2*Cosh[c + d*x])/x^5,x]

[Out]

-1/4*(a^2*Cosh[c + d*x])/x^4 - (2*a*b*Cosh[c + d*x])/(3*x^3) - (b^2*Cosh[c + d*x])/(2*x^2) - (a^2*d^2*Cosh[c +
d*x])/(24*x^2) - (a*b*d^2*Cosh[c + d*x])/(3*x) + (b^2*d^2*Cosh[c]*CoshIntegral[d*x])/2 + (a^2*d^4*Cosh[c]*Cos
hIntegral[d*x])/24 + (a*b*d^3*CoshIntegral[d*x]*Sinh[c])/3 - (a^2*d*Sinh[c + d*x])/(12*x^3) - (a*b*d*Sinh[c +
d*x])/(3*x^2) - (b^2*d*Sinh[c + d*x])/(2*x) - (a^2*d^3*Sinh[c + d*x])/(24*x) + (a*b*d^3*Cosh[c]*SinhIntegral[d
*x])/3 + (b^2*d^2*Sinh[c]*SinhIntegral[d*x])/2 + (a^2*d^4*Sinh[c]*SinhIntegral[d*x])/24

Rule 3378

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m
+ 1))), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 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 (\frac {a^2 \cosh (c+d x)}{x^5}+\frac {2 a b \cosh (c+d x)}{x^4}+\frac {b^2 \cosh (c+d x)}{x^3}\right ) \, dx \\ & = a^2 \int \frac {\cosh (c+d x)}{x^5} \, dx+(2 a b) \int \frac {\cosh (c+d x)}{x^4} \, dx+b^2 \int \frac {\cosh (c+d x)}{x^3} \, dx \\ & = -\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {2 a b \cosh (c+d x)}{3 x^3}-\frac {b^2 \cosh (c+d x)}{2 x^2}+\frac {1}{4} \left (a^2 d\right ) \int \frac {\sinh (c+d x)}{x^4} \, dx+\frac {1}{3} (2 a b d) \int \frac {\sinh (c+d x)}{x^3} \, dx+\frac {1}{2} \left (b^2 d\right ) \int \frac {\sinh (c+d x)}{x^2} \, dx \\ & = -\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {2 a b \cosh (c+d x)}{3 x^3}-\frac {b^2 \cosh (c+d x)}{2 x^2}-\frac {a^2 d \sinh (c+d x)}{12 x^3}-\frac {a b d \sinh (c+d x)}{3 x^2}-\frac {b^2 d \sinh (c+d x)}{2 x}+\frac {1}{12} \left (a^2 d^2\right ) \int \frac {\cosh (c+d x)}{x^3} \, dx+\frac {1}{3} \left (a b d^2\right ) \int \frac {\cosh (c+d x)}{x^2} \, dx+\frac {1}{2} \left (b^2 d^2\right ) \int \frac {\cosh (c+d x)}{x} \, dx \\ & = -\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {2 a b \cosh (c+d x)}{3 x^3}-\frac {b^2 \cosh (c+d x)}{2 x^2}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac {a b d^2 \cosh (c+d x)}{3 x}-\frac {a^2 d \sinh (c+d x)}{12 x^3}-\frac {a b d \sinh (c+d x)}{3 x^2}-\frac {b^2 d \sinh (c+d x)}{2 x}+\frac {1}{24} \left (a^2 d^3\right ) \int \frac {\sinh (c+d x)}{x^2} \, dx+\frac {1}{3} \left (a b d^3\right ) \int \frac {\sinh (c+d x)}{x} \, dx+\frac {1}{2} \left (b^2 d^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx+\frac {1}{2} \left (b^2 d^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx \\ & = -\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {2 a b \cosh (c+d x)}{3 x^3}-\frac {b^2 \cosh (c+d x)}{2 x^2}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac {a b d^2 \cosh (c+d x)}{3 x}+\frac {1}{2} b^2 d^2 \cosh (c) \text {Chi}(d x)-\frac {a^2 d \sinh (c+d x)}{12 x^3}-\frac {a b d \sinh (c+d x)}{3 x^2}-\frac {b^2 d \sinh (c+d x)}{2 x}-\frac {a^2 d^3 \sinh (c+d x)}{24 x}+\frac {1}{2} b^2 d^2 \sinh (c) \text {Shi}(d x)+\frac {1}{24} \left (a^2 d^4\right ) \int \frac {\cosh (c+d x)}{x} \, dx+\frac {1}{3} \left (a b d^3 \cosh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx+\frac {1}{3} \left (a b d^3 \sinh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx \\ & = -\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {2 a b \cosh (c+d x)}{3 x^3}-\frac {b^2 \cosh (c+d x)}{2 x^2}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac {a b d^2 \cosh (c+d x)}{3 x}+\frac {1}{2} b^2 d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{3} a b d^3 \text {Chi}(d x) \sinh (c)-\frac {a^2 d \sinh (c+d x)}{12 x^3}-\frac {a b d \sinh (c+d x)}{3 x^2}-\frac {b^2 d \sinh (c+d x)}{2 x}-\frac {a^2 d^3 \sinh (c+d x)}{24 x}+\frac {1}{3} a b d^3 \cosh (c) \text {Shi}(d x)+\frac {1}{2} b^2 d^2 \sinh (c) \text {Shi}(d x)+\frac {1}{24} \left (a^2 d^4 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx+\frac {1}{24} \left (a^2 d^4 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx \\ & = -\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {2 a b \cosh (c+d x)}{3 x^3}-\frac {b^2 \cosh (c+d x)}{2 x^2}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac {a b d^2 \cosh (c+d x)}{3 x}+\frac {1}{2} b^2 d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{24} a^2 d^4 \cosh (c) \text {Chi}(d x)+\frac {1}{3} a b d^3 \text {Chi}(d x) \sinh (c)-\frac {a^2 d \sinh (c+d x)}{12 x^3}-\frac {a b d \sinh (c+d x)}{3 x^2}-\frac {b^2 d \sinh (c+d x)}{2 x}-\frac {a^2 d^3 \sinh (c+d x)}{24 x}+\frac {1}{3} a b d^3 \cosh (c) \text {Shi}(d x)+\frac {1}{2} b^2 d^2 \sinh (c) \text {Shi}(d x)+\frac {1}{24} a^2 d^4 \sinh (c) \text {Shi}(d x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^5} \, dx=-\frac {6 a^2 \cosh (c+d x)+16 a b x \cosh (c+d x)+12 b^2 x^2 \cosh (c+d x)+a^2 d^2 x^2 \cosh (c+d x)+8 a b d^2 x^3 \cosh (c+d x)-d^2 x^4 \text {Chi}(d x) \left (\left (12 b^2+a^2 d^2\right ) \cosh (c)+8 a b d \sinh (c)\right )+2 a^2 d x \sinh (c+d x)+8 a b d x^2 \sinh (c+d x)+12 b^2 d x^3 \sinh (c+d x)+a^2 d^3 x^3 \sinh (c+d x)-d^2 x^4 \left (8 a b d \cosh (c)+12 b^2 \sinh (c)+a^2 d^2 \sinh (c)\right ) \text {Shi}(d x)}{24 x^4} \]

[In]

Integrate[((a + b*x)^2*Cosh[c + d*x])/x^5,x]

[Out]

-1/24*(6*a^2*Cosh[c + d*x] + 16*a*b*x*Cosh[c + d*x] + 12*b^2*x^2*Cosh[c + d*x] + a^2*d^2*x^2*Cosh[c + d*x] + 8
*a*b*d^2*x^3*Cosh[c + d*x] - d^2*x^4*CoshIntegral[d*x]*((12*b^2 + a^2*d^2)*Cosh[c] + 8*a*b*d*Sinh[c]) + 2*a^2*
d*x*Sinh[c + d*x] + 8*a*b*d*x^2*Sinh[c + d*x] + 12*b^2*d*x^3*Sinh[c + d*x] + a^2*d^3*x^3*Sinh[c + d*x] - d^2*x
^4*(8*a*b*d*Cosh[c] + 12*b^2*Sinh[c] + a^2*d^2*Sinh[c])*SinhIntegral[d*x])/x^4

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.61

method	result	size
risch	\(-\frac {{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a^{2} d^{4} x^{4}+{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a^{2} d^{4} x^{4}+8 \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a b \,d^{3} x^{4}-8 \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a b \,d^{3} x^{4}+12 \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) b^{2} d^{2} x^{4}+12 \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) b^{2} d^{2} x^{4}-{\mathrm e}^{-d x -c} a^{2} d^{3} x^{3}+{\mathrm e}^{d x +c} a^{2} d^{3} x^{3}+8 \,{\mathrm e}^{-d x -c} a b \,d^{2} x^{3}+8 \,{\mathrm e}^{d x +c} a b \,d^{2} x^{3}+{\mathrm e}^{-d x -c} a^{2} d^{2} x^{2}-12 \,{\mathrm e}^{-d x -c} b^{2} d \,x^{3}+{\mathrm e}^{d x +c} a^{2} d^{2} x^{2}+12 \,{\mathrm e}^{d x +c} b^{2} d \,x^{3}-8 \,{\mathrm e}^{-d x -c} a b d \,x^{2}+8 \,{\mathrm e}^{d x +c} a b d \,x^{2}-2 \,{\mathrm e}^{-d x -c} a^{2} d x +12 \,{\mathrm e}^{-d x -c} b^{2} x^{2}+2 \,{\mathrm e}^{d x +c} a^{2} d x +12 \,{\mathrm e}^{d x +c} b^{2} x^{2}+16 \,{\mathrm e}^{-d x -c} a b x +16 \,{\mathrm e}^{d x +c} a b x +6 \,{\mathrm e}^{-d x -c} a^{2}+6 \,{\mathrm e}^{d x +c} a^{2}}{48 x^{4}}\)	\(400\)
meijerg	\(-\frac {d^{2} b^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {4}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {2 \left (2 \gamma -3+2 \ln \left (x \right )+2 \ln \left (i d \right )\right )}{\sqrt {\pi }}-\frac {4 \left (\frac {9 x^{2} d^{2}}{2}+3\right )}{3 \sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \cosh \left (d x \right )}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \sinh \left (d x \right )}{\sqrt {\pi }\, x d}-\frac {4 \left (\operatorname {Chi}\left (d x \right )-\ln \left (d x \right )-\gamma \right )}{\sqrt {\pi }}\right )}{8}+\frac {i d^{2} b^{2} \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {4 i \cosh \left (d x \right )}{d x \sqrt {\pi }}+\frac {4 i \sinh \left (d x \right )}{x^{2} d^{2} \sqrt {\pi }}-\frac {4 i \operatorname {Shi}\left (d x \right )}{\sqrt {\pi }}\right )}{8}-\frac {i d^{3} a b \cosh \left (c \right ) \sqrt {\pi }\, \left (-\frac {8 i \left (x^{2} d^{2}+2\right ) \cosh \left (d x \right )}{3 d^{3} x^{3} \sqrt {\pi }}-\frac {8 i \sinh \left (d x \right )}{3 x^{2} d^{2} \sqrt {\pi }}+\frac {8 i \operatorname {Shi}\left (d x \right )}{3 \sqrt {\pi }}\right )}{8}-\frac {d^{3} b a \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {8}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {4 \left (2 \gamma -\frac {11}{3}+2 \ln \left (x \right )+2 \ln \left (i d \right )\right )}{3 \sqrt {\pi }}-\frac {8 \left (\frac {55 x^{2} d^{2}}{2}+45\right )}{45 \sqrt {\pi }\, x^{2} d^{2}}+\frac {8 \cosh \left (d x \right )}{3 \sqrt {\pi }\, x^{2} d^{2}}+\frac {16 \left (\frac {5 x^{2} d^{2}}{2}+5\right ) \sinh \left (d x \right )}{15 \sqrt {\pi }\, x^{3} d^{3}}-\frac {8 \left (\operatorname {Chi}\left (d x \right )-\ln \left (d x \right )-\gamma \right )}{3 \sqrt {\pi }}\right )}{8}+\frac {a^{2} \cosh \left (c \right ) \sqrt {\pi }\, d^{4} \left (-\frac {8}{\sqrt {\pi }\, x^{4} d^{4}}-\frac {8}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {\frac {4 \gamma }{3}-\frac {25}{9}+\frac {4 \ln \left (x \right )}{3}+\frac {4 \ln \left (i d \right )}{3}}{\sqrt {\pi }}+\frac {\frac {25}{9} d^{4} x^{4}+8 x^{2} d^{2}+8}{\sqrt {\pi }\, x^{4} d^{4}}-\frac {8 \left (\frac {15 x^{2} d^{2}}{2}+45\right ) \cosh \left (d x \right )}{45 \sqrt {\pi }\, x^{4} d^{4}}-\frac {8 \left (\frac {15 x^{2} d^{2}}{2}+15\right ) \sinh \left (d x \right )}{45 \sqrt {\pi }\, x^{3} d^{3}}+\frac {\frac {4 \,\operatorname {Chi}\left (d x \right )}{3}-\frac {4 \ln \left (d x \right )}{3}-\frac {4 \gamma }{3}}{\sqrt {\pi }}\right )}{32}-\frac {i a^{2} \sinh \left (c \right ) \sqrt {\pi }\, d^{4} \left (-\frac {8 i \left (\frac {x^{2} d^{2}}{2}+1\right ) \cosh \left (d x \right )}{3 d^{3} x^{3} \sqrt {\pi }}-\frac {8 i \left (\frac {x^{2} d^{2}}{2}+3\right ) \sinh \left (d x \right )}{3 d^{4} x^{4} \sqrt {\pi }}+\frac {4 i \operatorname {Shi}\left (d x \right )}{3 \sqrt {\pi }}\right )}{32}\)	\(597\)

[In]

int((b*x+a)^2*cosh(d*x+c)/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/48*(exp(c)*Ei(1,-d*x)*a^2*d^4*x^4+exp(-c)*Ei(1,d*x)*a^2*d^4*x^4+8*exp(c)*Ei(1,-d*x)*a*b*d^3*x^4-8*exp(-c)*E
i(1,d*x)*a*b*d^3*x^4+12*exp(c)*Ei(1,-d*x)*b^2*d^2*x^4+12*exp(-c)*Ei(1,d*x)*b^2*d^2*x^4-exp(-d*x-c)*a^2*d^3*x^3
+exp(d*x+c)*a^2*d^3*x^3+8*exp(-d*x-c)*a*b*d^2*x^3+8*exp(d*x+c)*a*b*d^2*x^3+exp(-d*x-c)*a^2*d^2*x^2-12*exp(-d*x
-c)*b^2*d*x^3+exp(d*x+c)*a^2*d^2*x^2+12*exp(d*x+c)*b^2*d*x^3-8*exp(-d*x-c)*a*b*d*x^2+8*exp(d*x+c)*a*b*d*x^2-2*
exp(-d*x-c)*a^2*d*x+12*exp(-d*x-c)*b^2*x^2+2*exp(d*x+c)*a^2*d*x+12*exp(d*x+c)*b^2*x^2+16*exp(-d*x-c)*a*b*x+16*
exp(d*x+c)*a*b*x+6*exp(-d*x-c)*a^2+6*exp(d*x+c)*a^2)/x^4

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^5} \, dx=-\frac {2 \, {\left (8 \, a b d^{2} x^{3} + 16 \, a b x + {\left (a^{2} d^{2} + 12 \, b^{2}\right )} x^{2} + 6 \, a^{2}\right )} \cosh \left (d x + c\right ) - {\left ({\left (a^{2} d^{4} + 8 \, a b d^{3} + 12 \, b^{2} d^{2}\right )} x^{4} {\rm Ei}\left (d x\right ) + {\left (a^{2} d^{4} - 8 \, a b d^{3} + 12 \, b^{2} d^{2}\right )} x^{4} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \, {\left (8 \, a b d x^{2} + 2 \, a^{2} d x + {\left (a^{2} d^{3} + 12 \, b^{2} d\right )} x^{3}\right )} \sinh \left (d x + c\right ) - {\left ({\left (a^{2} d^{4} + 8 \, a b d^{3} + 12 \, b^{2} d^{2}\right )} x^{4} {\rm Ei}\left (d x\right ) - {\left (a^{2} d^{4} - 8 \, a b d^{3} + 12 \, b^{2} d^{2}\right )} x^{4} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{48 \, x^{4}} \]

[In]

integrate((b*x+a)^2*cosh(d*x+c)/x^5,x, algorithm="fricas")

[Out]

-1/48*(2*(8*a*b*d^2*x^3 + 16*a*b*x + (a^2*d^2 + 12*b^2)*x^2 + 6*a^2)*cosh(d*x + c) - ((a^2*d^4 + 8*a*b*d^3 + 1
2*b^2*d^2)*x^4*Ei(d*x) + (a^2*d^4 - 8*a*b*d^3 + 12*b^2*d^2)*x^4*Ei(-d*x))*cosh(c) + 2*(8*a*b*d*x^2 + 2*a^2*d*x
+ (a^2*d^3 + 12*b^2*d)*x^3)*sinh(d*x + c) - ((a^2*d^4 + 8*a*b*d^3 + 12*b^2*d^2)*x^4*Ei(d*x) - (a^2*d^4 - 8*a*
b*d^3 + 12*b^2*d^2)*x^4*Ei(-d*x))*sinh(c))/x^4

Sympy [F]

\[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^5} \, dx=\int \frac {\left (a + b x\right )^{2} \cosh {\left (c + d x \right )}}{x^{5}}\, dx \]

[In]

integrate((b*x+a)**2*cosh(d*x+c)/x**5,x)

[Out]

Integral((a + b*x)**2*cosh(c + d*x)/x**5, x)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.52 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^5} \, dx=\frac {1}{24} \, {\left (3 \, a^{2} d^{3} e^{\left (-c\right )} \Gamma \left (-3, d x\right ) + 3 \, a^{2} d^{3} e^{c} \Gamma \left (-3, -d x\right ) + 8 \, a b d^{2} e^{\left (-c\right )} \Gamma \left (-2, d x\right ) - 8 \, a b d^{2} e^{c} \Gamma \left (-2, -d x\right ) + 6 \, b^{2} d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + 6 \, b^{2} d e^{c} \Gamma \left (-1, -d x\right )\right )} d - \frac {{\left (6 \, b^{2} x^{2} + 8 \, a b x + 3 \, a^{2}\right )} \cosh \left (d x + c\right )}{12 \, x^{4}} \]

[In]

integrate((b*x+a)^2*cosh(d*x+c)/x^5,x, algorithm="maxima")

[Out]

1/24*(3*a^2*d^3*e^(-c)*gamma(-3, d*x) + 3*a^2*d^3*e^c*gamma(-3, -d*x) + 8*a*b*d^2*e^(-c)*gamma(-2, d*x) - 8*a*
b*d^2*e^c*gamma(-2, -d*x) + 6*b^2*d*e^(-c)*gamma(-1, d*x) + 6*b^2*d*e^c*gamma(-1, -d*x))*d - 1/12*(6*b^2*x^2 +
8*a*b*x + 3*a^2)*cosh(d*x + c)/x^4

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.59 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^5} \, dx=\frac {a^{2} d^{4} x^{4} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{4} x^{4} {\rm Ei}\left (d x\right ) e^{c} - 8 \, a b d^{3} x^{4} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 8 \, a b d^{3} x^{4} {\rm Ei}\left (d x\right ) e^{c} + 12 \, b^{2} d^{2} x^{4} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 12 \, b^{2} d^{2} x^{4} {\rm Ei}\left (d x\right ) e^{c} - a^{2} d^{3} x^{3} e^{\left (d x + c\right )} + a^{2} d^{3} x^{3} e^{\left (-d x - c\right )} - 8 \, a b d^{2} x^{3} e^{\left (d x + c\right )} - 8 \, a b d^{2} x^{3} e^{\left (-d x - c\right )} - a^{2} d^{2} x^{2} e^{\left (d x + c\right )} - 12 \, b^{2} d x^{3} e^{\left (d x + c\right )} - a^{2} d^{2} x^{2} e^{\left (-d x - c\right )} + 12 \, b^{2} d x^{3} e^{\left (-d x - c\right )} - 8 \, a b d x^{2} e^{\left (d x + c\right )} + 8 \, a b d x^{2} e^{\left (-d x - c\right )} - 2 \, a^{2} d x e^{\left (d x + c\right )} - 12 \, b^{2} x^{2} e^{\left (d x + c\right )} + 2 \, a^{2} d x e^{\left (-d x - c\right )} - 12 \, b^{2} x^{2} e^{\left (-d x - c\right )} - 16 \, a b x e^{\left (d x + c\right )} - 16 \, a b x e^{\left (-d x - c\right )} - 6 \, a^{2} e^{\left (d x + c\right )} - 6 \, a^{2} e^{\left (-d x - c\right )}}{48 \, x^{4}} \]

[In]

integrate((b*x+a)^2*cosh(d*x+c)/x^5,x, algorithm="giac")

[Out]

1/48*(a^2*d^4*x^4*Ei(-d*x)*e^(-c) + a^2*d^4*x^4*Ei(d*x)*e^c - 8*a*b*d^3*x^4*Ei(-d*x)*e^(-c) + 8*a*b*d^3*x^4*Ei
(d*x)*e^c + 12*b^2*d^2*x^4*Ei(-d*x)*e^(-c) + 12*b^2*d^2*x^4*Ei(d*x)*e^c - a^2*d^3*x^3*e^(d*x + c) + a^2*d^3*x^
3*e^(-d*x - c) - 8*a*b*d^2*x^3*e^(d*x + c) - 8*a*b*d^2*x^3*e^(-d*x - c) - a^2*d^2*x^2*e^(d*x + c) - 12*b^2*d*x
^3*e^(d*x + c) - a^2*d^2*x^2*e^(-d*x - c) + 12*b^2*d*x^3*e^(-d*x - c) - 8*a*b*d*x^2*e^(d*x + c) + 8*a*b*d*x^2*
e^(-d*x - c) - 2*a^2*d*x*e^(d*x + c) - 12*b^2*x^2*e^(d*x + c) + 2*a^2*d*x*e^(-d*x - c) - 12*b^2*x^2*e^(-d*x -
c) - 16*a*b*x*e^(d*x + c) - 16*a*b*x*e^(-d*x - c) - 6*a^2*e^(d*x + c) - 6*a^2*e^(-d*x - c))/x^4

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^5} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (a+b\,x\right )}^2}{x^5} \,d x \]

[In]

int((cosh(c + d*x)*(a + b*x)^2)/x^5,x)

[Out]

int((cosh(c + d*x)*(a + b*x)^2)/x^5, x)