\(\int \frac {a+b \text {arcsinh}(c x)}{(d+c^2 d x^2)^{5/2}} \, dx\) [179]

Optimal result

Integrand size = 23, antiderivative size = 147 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {b}{6 c d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {x (a+b \text {arcsinh}(c x))}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {2 x (a+b \text {arcsinh}(c x))}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{3 c d^2 \sqrt {d+c^2 d x^2}} \] Output:

1/6*b/c/d^2/(c^2*x^2+1)^(1/2)/(c^2*d*x^2+d)^(1/2)+1/3*x*(a+b*arcsinh(c*x)) 
/d/(c^2*d*x^2+d)^(3/2)+2/3*x*(a+b*arcsinh(c*x))/d^2/(c^2*d*x^2+d)^(1/2)-1/ 
3*b*(c^2*x^2+1)^(1/2)*ln(c^2*x^2+1)/c/d^2/(c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.97 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {\sqrt {d+c^2 d x^2} \left (b+b c^2 x^2+6 a c x \sqrt {1+c^2 x^2}+4 a c^3 x^3 \sqrt {1+c^2 x^2}+2 b c x \sqrt {1+c^2 x^2} \left (3+2 c^2 x^2\right ) \text {arcsinh}(c x)-2 b \left (1+c^2 x^2\right )^2 \log \left (1+c^2 x^2\right )\right )}{6 c d^3 \left (1+c^2 x^2\right )^{5/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6203, 241, 6202, 240}

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.

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

\(\Big \downarrow \) 6203

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 6202

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

\(\Big \downarrow \) 240

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
 

rule 241

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ 
(2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
 

rule 6202

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), 
x_Symbol] :> Simp[x*((a + b*ArcSinh[c*x])^n/(d*Sqrt[d + e*x^2])), x] - Simp 
[b*c*(n/d)*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]   Int[x*((a + b*ArcSinh[ 
c*x])^(n - 1)/(1 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, 
 c^2*d] && GtQ[n, 0]
 

rule 6203

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(424\) vs. \(2(127)=254\).

Time = 0.85 (sec) , antiderivative size = 425, normalized size of antiderivative = 2.89

method	result	size
default	\(a \left (\frac {x}{3 d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {c^{2} d \,x^{2}+d}}\right )+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+3 x c +2 \sqrt {c^{2} x^{2}+1}\right ) \left (-8 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{6} c^{6}+8 \sqrt {c^{2} x^{2}+1}\, \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{5} c^{5}-24 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{4} c^{4}+20 \sqrt {c^{2} x^{2}+1}\, \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{3} c^{3}+2 c^{4} x^{4}-2 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+6 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}-24 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}+12 \sqrt {c^{2} x^{2}+1}\, \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x c +4 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +8 \,\operatorname {arcsinh}\left (x c \right )-8 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+2\right )}{6 \left (3 c^{6} x^{6}+10 c^{4} x^{4}+11 c^{2} x^{2}+4\right ) d^{3} c}\)	\(425\)
parts	\(a \left (\frac {x}{3 d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {c^{2} d \,x^{2}+d}}\right )+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+3 x c +2 \sqrt {c^{2} x^{2}+1}\right ) \left (-8 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{6} c^{6}+8 \sqrt {c^{2} x^{2}+1}\, \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{5} c^{5}-24 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{4} c^{4}+20 \sqrt {c^{2} x^{2}+1}\, \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{3} c^{3}+2 c^{4} x^{4}-2 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+6 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}-24 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}+12 \sqrt {c^{2} x^{2}+1}\, \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x c +4 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +8 \,\operatorname {arcsinh}\left (x c \right )-8 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+2\right )}{6 \left (3 c^{6} x^{6}+10 c^{4} x^{4}+11 c^{2} x^{2}+4\right ) d^{3} c}\)	\(425\)

Input:

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

Output:

a*(1/3*x/d/(c^2*d*x^2+d)^(3/2)+2/3/d^2*x/(c^2*d*x^2+d)^(1/2))+1/6*b*(d*(c^ 
2*x^2+1))^(1/2)*(2*x^3*c^3+2*x^2*c^2*(c^2*x^2+1)^(1/2)+3*x*c+2*(c^2*x^2+1) 
^(1/2))*(-8*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)*x^6*c^6+8*(c^2*x^2+1)^(1/2)*ln 
(1+(x*c+(c^2*x^2+1)^(1/2))^2)*x^5*c^5-24*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)*x 
^4*c^4+20*(c^2*x^2+1)^(1/2)*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)*x^3*c^3+2*c^4* 
x^4-2*(c^2*x^2+1)^(1/2)*c^3*x^3+6*arcsinh(x*c)*c^2*x^2-24*ln(1+(x*c+(c^2*x 
^2+1)^(1/2))^2)*x^2*c^2+12*(c^2*x^2+1)^(1/2)*ln(1+(x*c+(c^2*x^2+1)^(1/2))^ 
2)*x*c+4*c^2*x^2-3*(c^2*x^2+1)^(1/2)*x*c+8*arcsinh(x*c)-8*ln(1+(x*c+(c^2*x 
^2+1)^(1/2))^2)+2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)/d^3/c
 

Fricas [F]

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

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

Output:

integral(sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)/(c^6*d^3*x^6 + 3*c^4*d^3 
*x^4 + 3*c^2*d^3*x^2 + d^3), x)
 

Sympy [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:

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

Output:

Integral((a + b*asinh(c*x))/(d*(c**2*x**2 + 1))**(5/2), x)
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.86 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {1}{6} \, b c {\left (\frac {1}{c^{4} d^{\frac {5}{2}} x^{2} + c^{2} d^{\frac {5}{2}}} - \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{2} d^{\frac {5}{2}}}\right )} + \frac {1}{3} \, b {\left (\frac {2 \, x}{\sqrt {c^{2} d x^{2} + d} d^{2}} + \frac {x}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d}\right )} \operatorname {arsinh}\left (c x\right ) + \frac {1}{3} \, a {\left (\frac {2 \, x}{\sqrt {c^{2} d x^{2} + d} d^{2}} + \frac {x}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d}\right )} \] Input:

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

Output:

1/6*b*c*(1/(c^4*d^(5/2)*x^2 + c^2*d^(5/2)) - 2*log(c^2*x^2 + 1)/(c^2*d^(5/ 
2))) + 1/3*b*(2*x/(sqrt(c^2*d*x^2 + d)*d^2) + x/((c^2*d*x^2 + d)^(3/2)*d)) 
*arcsinh(c*x) + 1/3*a*(2*x/(sqrt(c^2*d*x^2 + d)*d^2) + x/((c^2*d*x^2 + d)^ 
(3/2)*d))
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*sqrt(c**2*x**2 + 1)*a*c**3*x**3 + 3*sqrt(c**2*x**2 + 1)*a*c*x + 3*int(a 
sinh(c*x)/(sqrt(c**2*x**2 + 1)*c**4*x**4 + 2*sqrt(c**2*x**2 + 1)*c**2*x**2 
 + sqrt(c**2*x**2 + 1)),x)*b*c**5*x**4 + 6*int(asinh(c*x)/(sqrt(c**2*x**2 
+ 1)*c**4*x**4 + 2*sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x) 
*b*c**3*x**2 + 3*int(asinh(c*x)/(sqrt(c**2*x**2 + 1)*c**4*x**4 + 2*sqrt(c* 
*2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b*c - 2*a*c**4*x**4 - 4*a 
*c**2*x**2 - 2*a)/(3*sqrt(d)*c*d**2*(c**4*x**4 + 2*c**2*x**2 + 1))