\(\int x^m (d+c^2 d x^2) (a+b \text {arcsinh}(c x)) \, dx\) [199]

Optimal result

Integrand size = 22, antiderivative size = 128 \[ \int x^m \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=-\frac {b c d x^{2+m} \sqrt {1+c^2 x^2}}{(3+m)^2}+\frac {d x^{1+m} (a+b \text {arcsinh}(c x))}{1+m}+\frac {c^2 d x^{3+m} (a+b \text {arcsinh}(c x))}{3+m}-\frac {b c d (7+3 m) x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},-c^2 x^2\right )}{(1+m) (2+m) (3+m)^2} \] Output:

-b*c*d*x^(2+m)*(c^2*x^2+1)^(1/2)/(3+m)^2+d*x^(1+m)*(a+b*arcsinh(c*x))/(1+m 
)+c^2*d*x^(3+m)*(a+b*arcsinh(c*x))/(3+m)-b*c*d*(7+3*m)*x^(2+m)*hypergeom([ 
1/2, 1+1/2*m],[2+1/2*m],-c^2*x^2)/(1+m)/(2+m)/(3+m)^2
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.92 \[ \int x^m \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {d x^{1+m} \left ((2+m) \left (3+m+c^2 x^2+c^2 m x^2\right ) (a+b \text {arcsinh}(c x))-b c (1+m) x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1+\frac {m}{2},2+\frac {m}{2},-c^2 x^2\right )-2 b c x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1+\frac {m}{2},2+\frac {m}{2},-c^2 x^2\right )\right )}{(1+m) (2+m) (3+m)} \] Input:

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

Output:

(d*x^(1 + m)*((2 + m)*(3 + m + c^2*x^2 + c^2*m*x^2)*(a + b*ArcSinh[c*x]) - 
 b*c*(1 + m)*x*Hypergeometric2F1[-1/2, 1 + m/2, 2 + m/2, -(c^2*x^2)] - 2*b 
*c*x*Hypergeometric2F1[1/2, 1 + m/2, 2 + m/2, -(c^2*x^2)]))/((1 + m)*(2 + 
m)*(3 + m))
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6218, 27, 363, 278}

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.

\(\displaystyle \int x^m \left (c^2 d x^2+d\right ) (a+b \text {arcsinh}(c x)) \, dx\)

\(\Big \downarrow \) 6218

\(\displaystyle -b c \int \frac {d x^{m+1} \left (\frac {c^2 x^2}{m+3}+\frac {1}{m+1}\right )}{\sqrt {c^2 x^2+1}}dx+\frac {c^2 d x^{m+3} (a+b \text {arcsinh}(c x))}{m+3}+\frac {d x^{m+1} (a+b \text {arcsinh}(c x))}{m+1}\)

\(\Big \downarrow \) 27

\(\displaystyle -b c d \int \frac {x^{m+1} \left (\frac {c^2 x^2}{m+3}+\frac {1}{m+1}\right )}{\sqrt {c^2 x^2+1}}dx+\frac {c^2 d x^{m+3} (a+b \text {arcsinh}(c x))}{m+3}+\frac {d x^{m+1} (a+b \text {arcsinh}(c x))}{m+1}\)

\(\Big \downarrow \) 363

\(\displaystyle -b c d \left (\frac {(3 m+7) \int \frac {x^{m+1}}{\sqrt {c^2 x^2+1}}dx}{(m+1) (m+3)^2}+\frac {\sqrt {c^2 x^2+1} x^{m+2}}{(m+3)^2}\right )+\frac {c^2 d x^{m+3} (a+b \text {arcsinh}(c x))}{m+3}+\frac {d x^{m+1} (a+b \text {arcsinh}(c x))}{m+1}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {c^2 d x^{m+3} (a+b \text {arcsinh}(c x))}{m+3}+\frac {d x^{m+1} (a+b \text {arcsinh}(c x))}{m+1}-b c d \left (\frac {(3 m+7) x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},-c^2 x^2\right )}{(m+1) (m+2) (m+3)^2}+\frac {\sqrt {c^2 x^2+1} x^{m+2}}{(m+3)^2}\right )\)

Input:

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

Output:

(d*x^(1 + m)*(a + b*ArcSinh[c*x]))/(1 + m) + (c^2*d*x^(3 + m)*(a + b*ArcSi 
nh[c*x]))/(3 + m) - b*c*d*((x^(2 + m)*Sqrt[1 + c^2*x^2])/(3 + m)^2 + ((7 + 
 3*m)*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, -(c^2*x^2)])/ 
((1 + m)*(2 + m)*(3 + m)^2))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 363

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), 
 x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3))   Int[(e*x)^ 
m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d 
, 0] && NeQ[m + 2*p + 3, 0]
 

rule 6218

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ 
)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Simp 
[(a + b*ArcSinh[c*x])   u, x] - Simp[b*c   Int[SimplifyIntegrand[u/Sqrt[1 + 
 c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] 
&& IGtQ[p, 0]
 

Maple [F]

\[\int x^{m} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int x^m \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (c^{2} d x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{m} \,d x } \] Input:

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

Output:

integral((a*c^2*d*x^2 + a*d + (b*c^2*d*x^2 + b*d)*arcsinh(c*x))*x^m, x)
 

Sympy [F]

\[ \int x^m \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=d \left (\int a x^{m}\, dx + \int b x^{m} \operatorname {asinh}{\left (c x \right )}\, dx + \int a c^{2} x^{2} x^{m}\, dx + \int b c^{2} x^{2} x^{m} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \] Input:

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

Output:

d*(Integral(a*x**m, x) + Integral(b*x**m*asinh(c*x), x) + Integral(a*c**2* 
x**2*x**m, x) + Integral(b*c**2*x**2*x**m*asinh(c*x), x))
 

Maxima [F]

\[ \int x^m \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (c^{2} d x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{m} \,d x } \] Input:

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

Output:

a*c^2*d*x^(m + 3)/(m + 3) + a*d*x^(m + 1)/(m + 1) + (b*c^2*d*(m + 1)*x^3 + 
 b*d*(m + 3)*x)*x^m*log(c*x + sqrt(c^2*x^2 + 1))/(m^2 + 4*m + 3) - integra 
te((b*c^3*d*(m + 1)*x^3 + b*c*d*(m + 3)*x)*x^m/((m^2 + 4*m + 3)*c^3*x^3 + 
(m^2 + 4*m + 3)*c*x + ((m^2 + 4*m + 3)*c^2*x^2 + m^2 + 4*m + 3)*sqrt(c^2*x 
^2 + 1)), x) - integrate((b*c^4*d*(m + 1)*x^4 + b*c^2*d*(m + 3)*x^2)*x^m/( 
(m^2 + 4*m + 3)*c^2*x^2 + m^2 + 4*m + 3), x)
 

Giac [F(-2)]

Exception generated. \[ \int x^m \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int x^m \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\int x^m\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\left (d\,c^2\,x^2+d\right ) \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int x^m \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {d \left (x^{m} a \,c^{2} m \,x^{3}+x^{m} a \,c^{2} x^{3}+x^{m} a m x +3 x^{m} a x +\left (\int x^{m} \mathit {asinh} \left (c x \right ) x^{2}d x \right ) b \,c^{2} m^{2}+4 \left (\int x^{m} \mathit {asinh} \left (c x \right ) x^{2}d x \right ) b \,c^{2} m +3 \left (\int x^{m} \mathit {asinh} \left (c x \right ) x^{2}d x \right ) b \,c^{2}+\left (\int x^{m} \mathit {asinh} \left (c x \right )d x \right ) b \,m^{2}+4 \left (\int x^{m} \mathit {asinh} \left (c x \right )d x \right ) b m +3 \left (\int x^{m} \mathit {asinh} \left (c x \right )d x \right ) b \right )}{m^{2}+4 m +3} \] Input:

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

Output:

(d*(x**m*a*c**2*m*x**3 + x**m*a*c**2*x**3 + x**m*a*m*x + 3*x**m*a*x + int( 
x**m*asinh(c*x)*x**2,x)*b*c**2*m**2 + 4*int(x**m*asinh(c*x)*x**2,x)*b*c**2 
*m + 3*int(x**m*asinh(c*x)*x**2,x)*b*c**2 + int(x**m*asinh(c*x),x)*b*m**2 
+ 4*int(x**m*asinh(c*x),x)*b*m + 3*int(x**m*asinh(c*x),x)*b))/(m**2 + 4*m 
+ 3)