\(\int (d+e x^2) (a+b \text {arcsinh}(c x))^{3/2} \, dx\) [188]

Optimal result

Integrand size = 20, antiderivative size = 427 \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^{3/2} \, dx=-\frac {3 b d \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{2 c}+\frac {b e \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^3}-\frac {b e x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{6 c}+d x (a+b \text {arcsinh}(c x))^{3/2}+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))^{3/2}+\frac {3 b^{3/2} d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {3 b^{3/2} e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {b^{3/2} e e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{96 c^3}+\frac {3 b^{3/2} d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {3 b^{3/2} e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {b^{3/2} e e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{96 c^3} \] Output:

-3/2*b*d*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^(1/2)/c+1/3*b*e*(c^2*x^2+1)^ 
(1/2)*(a+b*arcsinh(c*x))^(1/2)/c^3-1/6*b*e*x^2*(c^2*x^2+1)^(1/2)*(a+b*arcs 
inh(c*x))^(1/2)/c+d*x*(a+b*arcsinh(c*x))^(3/2)+1/3*e*x^3*(a+b*arcsinh(c*x) 
)^(3/2)+3/8*b^(3/2)*d*exp(a/b)*Pi^(1/2)*erf((a+b*arcsinh(c*x))^(1/2)/b^(1/ 
2))/c-3/32*b^(3/2)*e*exp(a/b)*Pi^(1/2)*erf((a+b*arcsinh(c*x))^(1/2)/b^(1/2 
))/c^3+1/288*b^(3/2)*e*exp(3*a/b)*3^(1/2)*Pi^(1/2)*erf(3^(1/2)*(a+b*arcsin 
h(c*x))^(1/2)/b^(1/2))/c^3+3/8*b^(3/2)*d*Pi^(1/2)*erfi((a+b*arcsinh(c*x))^ 
(1/2)/b^(1/2))/c/exp(a/b)-3/32*b^(3/2)*e*Pi^(1/2)*erfi((a+b*arcsinh(c*x))^ 
(1/2)/b^(1/2))/c^3/exp(a/b)+1/288*b^(3/2)*e*3^(1/2)*Pi^(1/2)*erfi(3^(1/2)* 
(a+b*arcsinh(c*x))^(1/2)/b^(1/2))/c^3/exp(3*a/b)
 

Mathematica [A] (warning: unable to verify)

Time = 1.73 (sec) , antiderivative size = 770, normalized size of antiderivative = 1.80 \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

(a*d*Sqrt[a + b*ArcSinh[c*x]]*(-((E^((2*a)/b)*Gamma[3/2, a/b + ArcSinh[c*x 
]])/Sqrt[a/b + ArcSinh[c*x]]) + Gamma[3/2, -((a + b*ArcSinh[c*x])/b)]/Sqrt 
[-((a + b*ArcSinh[c*x])/b)]))/(2*c*E^(a/b)) + (a*e*Sqrt[a + b*ArcSinh[c*x] 
]*(9*E^((4*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[3/2, a/b + ArcSinh[ 
c*x]] + Sqrt[3]*Sqrt[a/b + ArcSinh[c*x]]*Gamma[3/2, (-3*(a + b*ArcSinh[c*x 
]))/b] - 9*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[3/2, -((a + b*ArcSin 
h[c*x])/b)] - Sqrt[3]*E^((6*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[3/ 
2, (3*(a + b*ArcSinh[c*x]))/b]))/(72*c^3*E^((3*a)/b)*Sqrt[-((a + b*ArcSinh 
[c*x])^2/b^2)]) + (Sqrt[b]*d*(4*Sqrt[b]*Sqrt[a + b*ArcSinh[c*x]]*(-3*Sqrt[ 
1 + c^2*x^2] + 2*c*x*ArcSinh[c*x]) + (2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b* 
ArcSinh[c*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (-2*a + 3*b)*Sqrt[Pi]*Erf 
[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])))/(8*c) + (Sqrt 
[b]*e*(-9*(4*Sqrt[b]*Sqrt[a + b*ArcSinh[c*x]]*(-3*Sqrt[1 + c^2*x^2] + 2*c* 
x*ArcSinh[c*x]) + (2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[ 
b]]*(Cosh[a/b] - Sinh[a/b]) + (-2*a + 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh 
[c*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])) + (2*a + b)*Sqrt[3*Pi]*Erfi[(Sqrt 
[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]]*(Cosh[(3*a)/b] - Sinh[(3*a)/b]) + ( 
-2*a + b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]]*(Cosh 
[(3*a)/b] + Sinh[(3*a)/b]) + 12*Sqrt[b]*Sqrt[a + b*ArcSinh[c*x]]*(-Cosh[3* 
ArcSinh[c*x]] + 2*ArcSinh[c*x]*Sinh[3*ArcSinh[c*x]])))/(288*c^3)
 

Rubi [A] (verified)

Time = 1.65 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6208, 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.

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

\(\Big \downarrow \) 6208

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 \sqrt {\pi } b^{3/2} e e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {\sqrt {\frac {\pi }{3}} b^{3/2} e e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{96 c^3}-\frac {3 \sqrt {\pi } b^{3/2} e e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {\sqrt {\frac {\pi }{3}} b^{3/2} e e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{96 c^3}+\frac {3 \sqrt {\pi } b^{3/2} d e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 c}+\frac {3 \sqrt {\pi } b^{3/2} d e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {3 b d \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{2 c}-\frac {b e x^2 \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{6 c}+\frac {b e \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^3}+d x (a+b \text {arcsinh}(c x))^{3/2}+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))^{3/2}\)

Input:

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

Output:

(-3*b*d*Sqrt[1 + c^2*x^2]*Sqrt[a + b*ArcSinh[c*x]])/(2*c) + (b*e*Sqrt[1 + 
c^2*x^2]*Sqrt[a + b*ArcSinh[c*x]])/(3*c^3) - (b*e*x^2*Sqrt[1 + c^2*x^2]*Sq 
rt[a + b*ArcSinh[c*x]])/(6*c) + d*x*(a + b*ArcSinh[c*x])^(3/2) + (e*x^3*(a 
 + b*ArcSinh[c*x])^(3/2))/3 + (3*b^(3/2)*d*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b 
*ArcSinh[c*x]]/Sqrt[b]])/(8*c) - (3*b^(3/2)*e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a 
+ b*ArcSinh[c*x]]/Sqrt[b]])/(32*c^3) + (b^(3/2)*e*E^((3*a)/b)*Sqrt[Pi/3]*E 
rf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(96*c^3) + (3*b^(3/2)*d*Sq 
rt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*c*E^(a/b)) - (3*b^(3/2)* 
e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(32*c^3*E^(a/b)) + (b^( 
3/2)*e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(96*c^ 
3*E^((3*a)/b))
 

Defintions of rubi rules used

rule 2009

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

rule 6208

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n, (d + e*x^2)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[e, c^2*d] && IntegerQ[p] && (p > 
 0 || IGtQ[n, 0])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

integrate((e*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]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(asinh(c*x)*b + a),x)*a*d + int(sqrt(asinh(c*x)*b + a)*asinh(c*x)* 
x**2,x)*b*e + int(sqrt(asinh(c*x)*b + a)*asinh(c*x),x)*b*d + int(sqrt(asin 
h(c*x)*b + a)*x**2,x)*a*e