\(\int x (d+c^2 d x^2)^{5/2} (a+b \text {arcsinh}(c x)) \, dx\) [147]

Optimal result

Integrand size = 24, antiderivative size = 193 \[ \int x \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {b d^2 x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}-\frac {b c d^2 x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d^2 x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}+\frac {\left (d+c^2 d x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^2 d} \] Output:

-1/7*b*d^2*x*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-1/7*b*c*d^2*x^3*(c^2* 
d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-3/35*b*c^3*d^2*x^5*(c^2*d*x^2+d)^(1/2)/(c 
^2*x^2+1)^(1/2)-1/49*b*c^5*d^2*x^7*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+1 
/7*(c^2*d*x^2+d)^(7/2)*(a+b*arcsinh(c*x))/c^2/d
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.58 \[ \int x \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {d^2 \sqrt {d+c^2 d x^2} \left (35 a \left (1+c^2 x^2\right )^4-b c x \sqrt {1+c^2 x^2} \left (35+35 c^2 x^2+21 c^4 x^4+5 c^6 x^6\right )+35 b \left (1+c^2 x^2\right )^4 \text {arcsinh}(c x)\right )}{245 c^2 \left (1+c^2 x^2\right )} \] Input:

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

Output:

(d^2*Sqrt[d + c^2*d*x^2]*(35*a*(1 + c^2*x^2)^4 - b*c*x*Sqrt[1 + c^2*x^2]*( 
35 + 35*c^2*x^2 + 21*c^4*x^4 + 5*c^6*x^6) + 35*b*(1 + c^2*x^2)^4*ArcSinh[c 
*x]))/(245*c^2*(1 + c^2*x^2))
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.52, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6213, 210, 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.

\(\displaystyle \int x \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx\)

\(\Big \downarrow \) 6213

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

\(\Big \downarrow \) 210

\(\displaystyle \frac {\left (c^2 d x^2+d\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^2 d}-\frac {b d^2 \sqrt {c^2 d x^2+d} \int \left (c^6 x^6+3 c^4 x^4+3 c^2 x^2+1\right )dx}{7 c \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (c^2 d x^2+d\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^2 d}-\frac {b d^2 \left (\frac {c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+c^2 x^3+x\right ) \sqrt {c^2 d x^2+d}}{7 c \sqrt {c^2 x^2+1}}\)

Input:

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

Output:

-1/7*(b*d^2*Sqrt[d + c^2*d*x^2]*(x + c^2*x^3 + (3*c^4*x^5)/5 + (c^6*x^7)/7 
))/(c*Sqrt[1 + c^2*x^2]) + ((d + c^2*d*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(7 
*c^2*d)
 

Defintions of rubi rules used

rule 210

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 
)^p, x], x] /; FreeQ[{a, b}, x] && IGtQ[p, 0]
 

rule 2009

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

rule 6213

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p 
+ 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] 
 Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ 
{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
 

Maple [A] (verified)

Time = 0.94 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.99

method	result	size
orering	\(\frac {\left (65 c^{8} x^{8}+271 c^{6} x^{6}+441 c^{4} x^{4}+385 c^{2} x^{2}+70\right ) \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{245 c^{2} \left (c^{2} x^{2}+1\right )^{3}}-\frac {\left (5 c^{6} x^{6}+21 c^{4} x^{4}+35 c^{2} x^{2}+35\right ) \left (\left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )+5 x^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{2} d +\frac {x \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} b c}{\sqrt {c^{2} x^{2}+1}}\right )}{245 c^{2} \left (c^{2} x^{2}+1\right )^{2}}\)	\(192\)
default	\(\frac {a \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{7 c^{2} d}+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (64 c^{8} x^{8}+64 x^{7} c^{7} \sqrt {c^{2} x^{2}+1}+144 c^{6} x^{6}+112 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+104 c^{4} x^{4}+56 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+25 c^{2} x^{2}+7 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+7 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{6272 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}+16 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+28 c^{4} x^{4}+20 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+13 c^{2} x^{2}+5 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+5 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{640 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{128 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {5 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )-1\right ) d^{2}}{128 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {5 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )+1\right ) d^{2}}{128 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+3 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{128 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}-16 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+28 c^{4} x^{4}-20 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+13 c^{2} x^{2}-5 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+5 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{640 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (64 c^{8} x^{8}-64 x^{7} c^{7} \sqrt {c^{2} x^{2}+1}+144 c^{6} x^{6}-112 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+104 c^{4} x^{4}-56 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+25 c^{2} x^{2}-7 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+7 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{6272 c^{2} \left (c^{2} x^{2}+1\right )}\right )\)	\(863\)
parts	\(\frac {a \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{7 c^{2} d}+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (64 c^{8} x^{8}+64 x^{7} c^{7} \sqrt {c^{2} x^{2}+1}+144 c^{6} x^{6}+112 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+104 c^{4} x^{4}+56 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+25 c^{2} x^{2}+7 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+7 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{6272 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}+16 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+28 c^{4} x^{4}+20 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+13 c^{2} x^{2}+5 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+5 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{640 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{128 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {5 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )-1\right ) d^{2}}{128 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {5 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )+1\right ) d^{2}}{128 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+3 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{128 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}-16 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+28 c^{4} x^{4}-20 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+13 c^{2} x^{2}-5 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+5 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{640 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (64 c^{8} x^{8}-64 x^{7} c^{7} \sqrt {c^{2} x^{2}+1}+144 c^{6} x^{6}-112 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+104 c^{4} x^{4}-56 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+25 c^{2} x^{2}-7 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+7 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{6272 c^{2} \left (c^{2} x^{2}+1\right )}\right )\)	\(863\)

Input:

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

Output:

1/245*(65*c^8*x^8+271*c^6*x^6+441*c^4*x^4+385*c^2*x^2+70)/c^2/(c^2*x^2+1)^ 
3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(x*c))-1/245*(5*c^6*x^6+21*c^4*x^4+35*c^ 
2*x^2+35)/c^2/(c^2*x^2+1)^2*((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(x*c))+5*x^2* 
(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(x*c))*c^2*d+x*(c^2*d*x^2+d)^(5/2)*b*c/(c^ 
2*x^2+1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.17 \[ \int x \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {35 \, {\left (b c^{8} d^{2} x^{8} + 4 \, b c^{6} d^{2} x^{6} + 6 \, b c^{4} d^{2} x^{4} + 4 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (35 \, a c^{8} d^{2} x^{8} + 140 \, a c^{6} d^{2} x^{6} + 210 \, a c^{4} d^{2} x^{4} + 140 \, a c^{2} d^{2} x^{2} + 35 \, a d^{2} - {\left (5 \, b c^{7} d^{2} x^{7} + 21 \, b c^{5} d^{2} x^{5} + 35 \, b c^{3} d^{2} x^{3} + 35 \, b c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d}}{245 \, {\left (c^{4} x^{2} + c^{2}\right )}} \] Input:

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

Output:

1/245*(35*(b*c^8*d^2*x^8 + 4*b*c^6*d^2*x^6 + 6*b*c^4*d^2*x^4 + 4*b*c^2*d^2 
*x^2 + b*d^2)*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1)) + (35*a*c^8 
*d^2*x^8 + 140*a*c^6*d^2*x^6 + 210*a*c^4*d^2*x^4 + 140*a*c^2*d^2*x^2 + 35* 
a*d^2 - (5*b*c^7*d^2*x^7 + 21*b*c^5*d^2*x^5 + 35*b*c^3*d^2*x^3 + 35*b*c*d^ 
2*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d))/(c^4*x^2 + c^2)
 

Sympy [F(-1)]

Timed out. \[ \int x \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.50 \[ \int x \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}} b \operatorname {arsinh}\left (c x\right )}{7 \, c^{2} d} + \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}} a}{7 \, c^{2} d} - \frac {{\left (5 \, c^{6} d^{\frac {7}{2}} x^{7} + 21 \, c^{4} d^{\frac {7}{2}} x^{5} + 35 \, c^{2} d^{\frac {7}{2}} x^{3} + 35 \, d^{\frac {7}{2}} x\right )} b}{245 \, c d} \] Input:

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

Output:

1/7*(c^2*d*x^2 + d)^(7/2)*b*arcsinh(c*x)/(c^2*d) + 1/7*(c^2*d*x^2 + d)^(7/ 
2)*a/(c^2*d) - 1/245*(5*c^6*d^(7/2)*x^7 + 21*c^4*d^(7/2)*x^5 + 35*c^2*d^(7 
/2)*x^3 + 35*d^(7/2)*x)*b/(c*d)
 

Giac [F(-2)]

Exception generated. \[ \int x \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x*(c^2*d*x^2+d)^(5/2)*(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. \[ \int x \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{5/2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int x \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {d}\, d^{2} \left (\sqrt {c^{2} x^{2}+1}\, a \,c^{6} x^{6}+3 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} x^{4}+3 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, a +7 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{5}d x \right ) b \,c^{6}+14 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{3}d x \right ) b \,c^{4}+7 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x d x \right ) b \,c^{2}\right )}{7 c^{2}} \] Input:

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

Output:

(sqrt(d)*d**2*(sqrt(c**2*x**2 + 1)*a*c**6*x**6 + 3*sqrt(c**2*x**2 + 1)*a*c 
**4*x**4 + 3*sqrt(c**2*x**2 + 1)*a*c**2*x**2 + sqrt(c**2*x**2 + 1)*a + 7*i 
nt(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**5,x)*b*c**6 + 14*int(sqrt(c**2*x**2 + 
 1)*asinh(c*x)*x**3,x)*b*c**4 + 7*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x,x)* 
b*c**2))/(7*c**2)