\(\int \frac {x^3 (d+c^2 d x^2)}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx\) [446]

Optimal result

Integrand size = 26, antiderivative size = 254 \[ \int \frac {x^3 \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=-\frac {2 d x^3 \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {3 d e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^4}+\frac {d e^{\frac {6 a}{b}} \sqrt {\frac {3 \pi }{2}} \text {erf}\left (\frac {\sqrt {6} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^4}-\frac {3 d e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^4}+\frac {d e^{-\frac {6 a}{b}} \sqrt {\frac {3 \pi }{2}} \text {erfi}\left (\frac {\sqrt {6} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^4} \] Output:

-2*d*x^3*(c^2*x^2+1)^(3/2)/b/c/(a+b*arcsinh(c*x))^(1/2)-3/32*d*exp(2*a/b)* 
2^(1/2)*Pi^(1/2)*erf(2^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^4 
+1/32*d*exp(6*a/b)*6^(1/2)*Pi^(1/2)*erf(6^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b 
^(1/2))/b^(3/2)/c^4-3/32*d*2^(1/2)*Pi^(1/2)*erfi(2^(1/2)*(a+b*arcsinh(c*x) 
)^(1/2)/b^(1/2))/b^(3/2)/c^4/exp(2*a/b)+1/32*d*6^(1/2)*Pi^(1/2)*erfi(6^(1/ 
2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^4/exp(6*a/b)
 

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.91 \[ \int \frac {x^3 \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\frac {d e^{-\frac {6 a}{b}} \left (\sqrt {6} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )-3 \sqrt {2} e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+3 \sqrt {2} e^{\frac {8 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )-\sqrt {6} e^{\frac {12 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )-8 e^{\frac {6 a}{b}} \sinh ^3(2 \text {arcsinh}(c x))\right )}{32 b c^4 \sqrt {a+b \text {arcsinh}(c x)}} \] Input:

Integrate[(x^3*(d + c^2*d*x^2))/(a + b*ArcSinh[c*x])^(3/2),x]
 

Output:

(d*(Sqrt[6]*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-6*(a + b*ArcSinh[ 
c*x]))/b] - 3*Sqrt[2]*E^((4*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/ 
2, (-2*(a + b*ArcSinh[c*x]))/b] + 3*Sqrt[2]*E^((8*a)/b)*Sqrt[a/b + ArcSinh 
[c*x]]*Gamma[1/2, (2*(a + b*ArcSinh[c*x]))/b] - Sqrt[6]*E^((12*a)/b)*Sqrt[ 
a/b + ArcSinh[c*x]]*Gamma[1/2, (6*(a + b*ArcSinh[c*x]))/b] - 8*E^((6*a)/b) 
*Sinh[2*ArcSinh[c*x]]^3))/(32*b*c^4*E^((6*a)/b)*Sqrt[a + b*ArcSinh[c*x]])
 

Rubi [A] (verified)

Time = 1.43 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.80, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6229, 6234, 5971, 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.

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

\(\Big \downarrow \) 6229

\(\displaystyle \frac {6 d \int \frac {x^2 \sqrt {c^2 x^2+1}}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{b c}+\frac {12 c d \int \frac {x^4 \sqrt {c^2 x^2+1}}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{b}-\frac {2 d x^3 \left (c^2 x^2+1\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\)

\(\Big \downarrow \) 6234

\(\displaystyle \frac {12 d \int \frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b^2 c^4}+\frac {6 d \int \frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b^2 c^4}-\frac {2 d x^3 \left (c^2 x^2+1\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {6 d \int \left (\frac {\cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{8 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {1}{8 \sqrt {a+b \text {arcsinh}(c x)}}\right )d(a+b \text {arcsinh}(c x))}{b^2 c^4}+\frac {12 d \int \left (\frac {\cosh \left (\frac {6 a}{b}-\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{32 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {\cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{16 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{32 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {1}{16 \sqrt {a+b \text {arcsinh}(c x)}}\right )d(a+b \text {arcsinh}(c x))}{b^2 c^4}-\frac {2 d x^3 \left (c^2 x^2+1\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {6 d \left (\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{4} \sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2 c^4}+\frac {12 d \left (-\frac {1}{64} \sqrt {\pi } \sqrt {b} e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{64} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{64} \sqrt {\frac {\pi }{6}} \sqrt {b} e^{\frac {6 a}{b}} \text {erf}\left (\frac {\sqrt {6} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{64} \sqrt {\pi } \sqrt {b} e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{64} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{64} \sqrt {\frac {\pi }{6}} \sqrt {b} e^{-\frac {6 a}{b}} \text {erfi}\left (\frac {\sqrt {6} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2 c^4}-\frac {2 d x^3 \left (c^2 x^2+1\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\)

Input:

Int[(x^3*(d + c^2*d*x^2))/(a + b*ArcSinh[c*x])^(3/2),x]
 

Output:

(-2*d*x^3*(1 + c^2*x^2)^(3/2))/(b*c*Sqrt[a + b*ArcSinh[c*x]]) + (6*d*(-1/4 
*Sqrt[a + b*ArcSinh[c*x]] + (Sqrt[b]*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + 
b*ArcSinh[c*x]])/Sqrt[b]])/32 + (Sqrt[b]*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcSi 
nh[c*x]])/Sqrt[b]])/(32*E^((4*a)/b))))/(b^2*c^4) + (12*d*(Sqrt[a + b*ArcSi 
nh[c*x]]/8 - (Sqrt[b]*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcSinh[c*x]] 
)/Sqrt[b]])/64 - (Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*A 
rcSinh[c*x]])/Sqrt[b]])/64 + (Sqrt[b]*E^((6*a)/b)*Sqrt[Pi/6]*Erf[(Sqrt[6]* 
Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/64 - (Sqrt[b]*Sqrt[Pi]*Erfi[(2*Sqrt[a 
+ b*ArcSinh[c*x]])/Sqrt[b]])/(64*E^((4*a)/b)) - (Sqrt[b]*Sqrt[Pi/2]*Erfi[( 
Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(64*E^((2*a)/b)) + (Sqrt[b]*Sq 
rt[Pi/6]*Erfi[(Sqrt[6]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(64*E^((6*a)/b) 
)))/(b^2*c^4)
 

Defintions of rubi rules used

rule 2009

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

rule 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 6229

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

rule 6234

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

Maple [F]

\[\int \frac {x^{3} \left (c^{2} d \,x^{2}+d \right )}{\left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{\frac {3}{2}}}d x\]

Input:

int(x^3*(c^2*d*x^2+d)/(a+b*arcsinh(x*c))^(3/2),x)
 

Output:

int(x^3*(c^2*d*x^2+d)/(a+b*arcsinh(x*c))^(3/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^3 \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^3*(c^2*d*x^2+d)/(a+b*arcsinh(c*x))^(3/2),x, algorithm="fricas" 
)
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

integrate(x**3*(c**2*d*x**2+d)/(a+b*asinh(c*x))**(3/2),x)
 

Output:

d*(Integral(x**3/(a*sqrt(a + b*asinh(c*x)) + b*sqrt(a + b*asinh(c*x))*asin 
h(c*x)), x) + Integral(c**2*x**5/(a*sqrt(a + b*asinh(c*x)) + b*sqrt(a + b* 
asinh(c*x))*asinh(c*x)), x))
 

Maxima [F]

\[ \int \frac {x^3 \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )} x^{3}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^3*(c^2*d*x^2+d)/(a+b*arcsinh(c*x))^(3/2),x, algorithm="maxima" 
)
 

Output:

integrate((c^2*d*x^2 + d)*x^3/(b*arcsinh(c*x) + a)^(3/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^3*(c^2*d*x^2+d)/(a+b*arcsinh(c*x))^(3/2),x, algorithm="giac")
 

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int \frac {x^3\,\left (d\,c^2\,x^2+d\right )}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

d*(int((sqrt(asinh(c*x)*b + a)*x**5)/(asinh(c*x)**2*b**2 + 2*asinh(c*x)*a* 
b + a**2),x)*c**2 + int((sqrt(asinh(c*x)*b + a)*x**3)/(asinh(c*x)**2*b**2 
+ 2*asinh(c*x)*a*b + a**2),x))