\(\int \frac {a+b \text {arcsinh}(c x)}{(d+c^2 d x^2)^{7/2}} \, dx\) [56]

Optimal result

Integrand size = 23, antiderivative size = 215 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^{7/2}} \, dx=\frac {b}{20 c d^3 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2}}+\frac {2 b}{15 c d^3 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {x (a+b \text {arcsinh}(c x))}{5 d \left (d+c^2 d x^2\right )^{5/2}}+\frac {4 x (a+b \text {arcsinh}(c x))}{15 d^2 \left (d+c^2 d x^2\right )^{3/2}}+\frac {8 x (a+b \text {arcsinh}(c x))}{15 d^3 \sqrt {d+c^2 d x^2}}-\frac {4 b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{15 c d^3 \sqrt {d+c^2 d x^2}} \] Output:

1/20*b/c/d^3/(c^2*x^2+1)^(3/2)/(c^2*d*x^2+d)^(1/2)+2/15*b/c/d^3/(c^2*x^2+1 
)^(1/2)/(c^2*d*x^2+d)^(1/2)+1/5*x*(a+b*arcsinh(c*x))/d/(c^2*d*x^2+d)^(5/2) 
+4/15*x*(a+b*arcsinh(c*x))/d^2/(c^2*d*x^2+d)^(3/2)+8/15*x*(a+b*arcsinh(c*x 
))/d^3/(c^2*d*x^2+d)^(1/2)-4/15*b*(c^2*x^2+1)^(1/2)*ln(c^2*x^2+1)/c/d^3/(c 
^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.86 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^{7/2}} \, dx=\frac {\sqrt {d+c^2 d x^2} \left (11 b+19 b c^2 x^2+8 b c^4 x^4+60 a c x \sqrt {1+c^2 x^2}+80 a c^3 x^3 \sqrt {1+c^2 x^2}+32 a c^5 x^5 \sqrt {1+c^2 x^2}+4 b c x \sqrt {1+c^2 x^2} \left (15+20 c^2 x^2+8 c^4 x^4\right ) \text {arcsinh}(c x)-16 b \left (1+c^2 x^2\right )^3 \log \left (1+c^2 x^2\right )\right )}{60 c d^4 \left (1+c^2 x^2\right )^{7/2}} \] Input:

Integrate[(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^(7/2),x]
 

Output:

(Sqrt[d + c^2*d*x^2]*(11*b + 19*b*c^2*x^2 + 8*b*c^4*x^4 + 60*a*c*x*Sqrt[1 
+ c^2*x^2] + 80*a*c^3*x^3*Sqrt[1 + c^2*x^2] + 32*a*c^5*x^5*Sqrt[1 + c^2*x^ 
2] + 4*b*c*x*Sqrt[1 + c^2*x^2]*(15 + 20*c^2*x^2 + 8*c^4*x^4)*ArcSinh[c*x] 
- 16*b*(1 + c^2*x^2)^3*Log[1 + c^2*x^2]))/(60*c*d^4*(1 + c^2*x^2)^(7/2))
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6203, 241, 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.

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

\(\Big \downarrow \) 6203

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

\(\Big \downarrow \) 241

\(\displaystyle \frac {4 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 d x^2+d\right )^{5/2}}dx}{5 d}+\frac {x (a+b \text {arcsinh}(c x))}{5 d \left (c^2 d x^2+d\right )^{5/2}}+\frac {b}{20 c d^3 \left (c^2 x^2+1\right )^{3/2} \sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 6203

\(\displaystyle \frac {4 \left (\frac {2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 d x^2+d\right )^{3/2}}dx}{3 d}-\frac {b c \sqrt {c^2 x^2+1} \int \frac {x}{\left (c^2 x^2+1\right )^2}dx}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {x (a+b \text {arcsinh}(c x))}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )}{5 d}+\frac {x (a+b \text {arcsinh}(c x))}{5 d \left (c^2 d x^2+d\right )^{5/2}}+\frac {b}{20 c d^3 \left (c^2 x^2+1\right )^{3/2} \sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 241

\(\displaystyle \frac {4 \left (\frac {2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 d x^2+d\right )^{3/2}}dx}{3 d}+\frac {x (a+b \text {arcsinh}(c x))}{3 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {b}{6 c d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}}\right )}{5 d}+\frac {x (a+b \text {arcsinh}(c x))}{5 d \left (c^2 d x^2+d\right )^{5/2}}+\frac {b}{20 c d^3 \left (c^2 x^2+1\right )^{3/2} \sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 6202

\(\displaystyle \frac {4 \left (\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))}{d \sqrt {c^2 d x^2+d}}-\frac {b c \sqrt {c^2 x^2+1} \int \frac {x}{c^2 x^2+1}dx}{d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))}{3 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {b}{6 c d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}}\right )}{5 d}+\frac {x (a+b \text {arcsinh}(c x))}{5 d \left (c^2 d x^2+d\right )^{5/2}}+\frac {b}{20 c d^3 \left (c^2 x^2+1\right )^{3/2} \sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 240

\(\displaystyle \frac {4 \left (\frac {x (a+b \text {arcsinh}(c x))}{3 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))}{d \sqrt {c^2 d x^2+d}}-\frac {b \sqrt {c^2 x^2+1} \log \left (c^2 x^2+1\right )}{2 c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {b}{6 c d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}}\right )}{5 d}+\frac {x (a+b \text {arcsinh}(c x))}{5 d \left (c^2 d x^2+d\right )^{5/2}}+\frac {b}{20 c d^3 \left (c^2 x^2+1\right )^{3/2} \sqrt {c^2 d x^2+d}}\)

Input:

Int[(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^(7/2),x]
 

Output:

b/(20*c*d^3*(1 + c^2*x^2)^(3/2)*Sqrt[d + c^2*d*x^2]) + (x*(a + b*ArcSinh[c 
*x]))/(5*d*(d + c^2*d*x^2)^(5/2)) + (4*(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)))/(5*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. \(2185\) vs. \(2(185)=370\).

Time = 1.20 (sec) , antiderivative size = 2186, normalized size of antiderivative = 10.17

method	result	size
default	\(\text {Expression too large to display}\)	\(2186\)
parts	\(\text {Expression too large to display}\)	\(2186\)

Input:

int((a+b*arcsinh(x*c))/(c^2*d*x^2+d)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-3526/15*b*(d*(c^2*x^2+1))^(1/2)/(40*c^10*x^10+215*c^8*x^8+469*c^6*x^6+517 
*c^4*x^4+287*c^2*x^2+64)/d^4*c^4*x^5-334/3*b*(d*(c^2*x^2+1))^(1/2)/(40*c^1 
0*x^10+215*c^8*x^8+469*c^6*x^6+517*c^4*x^4+287*c^2*x^2+64)/d^4*c^2*x^3+176 
/15*b*(d*(c^2*x^2+1))^(1/2)/(40*c^10*x^10+215*c^8*x^8+469*c^6*x^6+517*c^4* 
x^4+287*c^2*x^2+64)/d^4/c*(c^2*x^2+1)^(1/2)-8/15*b*(d*(c^2*x^2+1))^(1/2)/( 
c^2*x^2+1)^(1/2)/d^4/c*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)+16/15*b*(d*(c^2*x^2 
+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^4/c*arcsinh(x*c)+22*b*(d*(c^2*x^2+1))^(1/2) 
/(40*c^10*x^10+215*c^8*x^8+469*c^6*x^6+517*c^4*x^4+287*c^2*x^2+64)/d^4*(c^ 
2*x^2+1)*x+64*b*(d*(c^2*x^2+1))^(1/2)/(40*c^10*x^10+215*c^8*x^8+469*c^6*x^ 
6+517*c^4*x^4+287*c^2*x^2+64)/d^4*arcsinh(x*c)*x-128/15*b*(d*(c^2*x^2+1))^ 
(1/2)/(40*c^10*x^10+215*c^8*x^8+469*c^6*x^6+517*c^4*x^4+287*c^2*x^2+64)/d^ 
4*c^12*x^13-176/3*b*(d*(c^2*x^2+1))^(1/2)/(40*c^10*x^10+215*c^8*x^8+469*c^ 
6*x^6+517*c^4*x^4+287*c^2*x^2+64)/d^4*c^10*x^11-2552/15*b*(d*(c^2*x^2+1))^ 
(1/2)/(40*c^10*x^10+215*c^8*x^8+469*c^6*x^6+517*c^4*x^4+287*c^2*x^2+64)/d^ 
4*c^8*x^9-3986/15*b*(d*(c^2*x^2+1))^(1/2)/(40*c^10*x^10+215*c^8*x^8+469*c^ 
6*x^6+517*c^4*x^4+287*c^2*x^2+64)/d^4*c^6*x^7-22*b*(d*(c^2*x^2+1))^(1/2)/( 
40*c^10*x^10+215*c^8*x^8+469*c^6*x^6+517*c^4*x^4+287*c^2*x^2+64)/d^4*x+541 
/3*b*(d*(c^2*x^2+1))^(1/2)/(40*c^10*x^10+215*c^8*x^8+469*c^6*x^6+517*c^4*x 
^4+287*c^2*x^2+64)/d^4*c^2*arcsinh(x*c)*x^3+519/20*b*(d*(c^2*x^2+1))^(1/2) 
/(40*c^10*x^10+215*c^8*x^8+469*c^6*x^6+517*c^4*x^4+287*c^2*x^2+64)/d^4*...
 

Fricas [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^{7/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(7/2),x, algorithm="fricas")
 

Output:

integral(sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)/(c^8*d^4*x^8 + 4*c^6*d^4 
*x^6 + 6*c^4*d^4*x^4 + 4*c^2*d^4*x^2 + d^4), x)
 

Sympy [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^{7/2}} \, dx=\int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((a+b*asinh(c*x))/(c**2*d*x**2+d)**(7/2),x)
 

Output:

Integral((a + b*asinh(c*x))/(d*(c**2*x**2 + 1))**(7/2), x)
 

Maxima [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^{7/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(7/2),x, algorithm="maxima")
 

Output:

1/15*a*(8*x/(sqrt(c^2*d*x^2 + d)*d^3) + 4*x/((c^2*d*x^2 + d)^(3/2)*d^2) + 
3*x/((c^2*d*x^2 + d)^(5/2)*d)) + b*integrate(log(c*x + sqrt(c^2*x^2 + 1))/ 
(c^2*d*x^2 + d)^(7/2), x)
 

Giac [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^{7/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(7/2),x, algorithm="giac")
 

Output:

integrate((b*arcsinh(c*x) + a)/(c^2*d*x^2 + d)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^{7/2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{{\left (d\,c^2\,x^2+d\right )}^{7/2}} \,d x \] Input:

int((a + b*asinh(c*x))/(d + c^2*d*x^2)^(7/2),x)
 

Output:

int((a + b*asinh(c*x))/(d + c^2*d*x^2)^(7/2), x)
 

Reduce [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^{7/2}} \, dx=\frac {8 \sqrt {c^{2} x^{2}+1}\, a \,c^{5} x^{5}+20 \sqrt {c^{2} x^{2}+1}\, a \,c^{3} x^{3}+15 \sqrt {c^{2} x^{2}+1}\, a c x +15 \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{6} x^{6}+3 \sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+3 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{7} x^{6}+45 \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{6} x^{6}+3 \sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+3 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{5} x^{4}+45 \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{6} x^{6}+3 \sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+3 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{3} x^{2}+15 \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{6} x^{6}+3 \sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+3 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b c -8 a \,c^{6} x^{6}-24 a \,c^{4} x^{4}-24 a \,c^{2} x^{2}-8 a}{15 \sqrt {d}\, c \,d^{3} \left (c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1\right )} \] Input:

int((a+b*asinh(c*x))/(c^2*d*x^2+d)^(7/2),x)
 

Output:

(8*sqrt(c**2*x**2 + 1)*a*c**5*x**5 + 20*sqrt(c**2*x**2 + 1)*a*c**3*x**3 + 
15*sqrt(c**2*x**2 + 1)*a*c*x + 15*int(asinh(c*x)/(sqrt(c**2*x**2 + 1)*c**6 
*x**6 + 3*sqrt(c**2*x**2 + 1)*c**4*x**4 + 3*sqrt(c**2*x**2 + 1)*c**2*x**2 
+ sqrt(c**2*x**2 + 1)),x)*b*c**7*x**6 + 45*int(asinh(c*x)/(sqrt(c**2*x**2 
+ 1)*c**6*x**6 + 3*sqrt(c**2*x**2 + 1)*c**4*x**4 + 3*sqrt(c**2*x**2 + 1)*c 
**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b*c**5*x**4 + 45*int(asinh(c*x)/(sqrt(c 
**2*x**2 + 1)*c**6*x**6 + 3*sqrt(c**2*x**2 + 1)*c**4*x**4 + 3*sqrt(c**2*x* 
*2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b*c**3*x**2 + 15*int(asinh(c*x 
)/(sqrt(c**2*x**2 + 1)*c**6*x**6 + 3*sqrt(c**2*x**2 + 1)*c**4*x**4 + 3*sqr 
t(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b*c - 8*a*c**6*x**6 - 
 24*a*c**4*x**4 - 24*a*c**2*x**2 - 8*a)/(15*sqrt(d)*c*d**3*(c**6*x**6 + 3* 
c**4*x**4 + 3*c**2*x**2 + 1))