\(\int (d+e x^2)^2 \sqrt {a+b \text {arcsinh}(c x)} \, dx\) [183]

Optimal result

Integrand size = 22, antiderivative size = 672 \[ \int \left (d+e x^2\right )^2 \sqrt {a+b \text {arcsinh}(c x)} \, dx=d^2 x \sqrt {a+b \text {arcsinh}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \text {arcsinh}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \text {arcsinh}(c x)}+\frac {\sqrt {b} d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} d e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {\sqrt {b} e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 c^5}+\frac {\sqrt {b} d e e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {\sqrt {b} e^2 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 c^5}+\frac {\sqrt {b} e^2 e^{\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{320 c^5}-\frac {\sqrt {b} d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 c}+\frac {\sqrt {b} d e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 c^3}-\frac {\sqrt {b} e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 c^5}-\frac {\sqrt {b} d e e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{24 c^3}+\frac {\sqrt {b} e^2 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 c^5}-\frac {\sqrt {b} e^2 e^{-\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{320 c^5} \] Output:

d^2*x*(a+b*arcsinh(c*x))^(1/2)+2/3*d*e*x^3*(a+b*arcsinh(c*x))^(1/2)+1/5*e^ 
2*x^5*(a+b*arcsinh(c*x))^(1/2)+1/4*b^(1/2)*d^2*exp(a/b)*Pi^(1/2)*erf((a+b* 
arcsinh(c*x))^(1/2)/b^(1/2))/c-1/8*b^(1/2)*d*e*exp(a/b)*Pi^(1/2)*erf((a+b* 
arcsinh(c*x))^(1/2)/b^(1/2))/c^3+1/32*b^(1/2)*e^2*exp(a/b)*Pi^(1/2)*erf((a 
+b*arcsinh(c*x))^(1/2)/b^(1/2))/c^5+1/72*b^(1/2)*d*e*exp(3*a/b)*3^(1/2)*Pi 
^(1/2)*erf(3^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))/c^3-1/192*b^(1/2)*e^2 
*exp(3*a/b)*3^(1/2)*Pi^(1/2)*erf(3^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2)) 
/c^5+1/1600*b^(1/2)*e^2*exp(5*a/b)*5^(1/2)*Pi^(1/2)*erf(5^(1/2)*(a+b*arcsi 
nh(c*x))^(1/2)/b^(1/2))/c^5-1/4*b^(1/2)*d^2*Pi^(1/2)*erfi((a+b*arcsinh(c*x 
))^(1/2)/b^(1/2))/c/exp(a/b)+1/8*b^(1/2)*d*e*Pi^(1/2)*erfi((a+b*arcsinh(c* 
x))^(1/2)/b^(1/2))/c^3/exp(a/b)-1/32*b^(1/2)*e^2*Pi^(1/2)*erfi((a+b*arcsin 
h(c*x))^(1/2)/b^(1/2))/c^5/exp(a/b)-1/72*b^(1/2)*d*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)+1/192*b^(1/2)*e^ 
2*3^(1/2)*Pi^(1/2)*erfi(3^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))/c^5/exp( 
3*a/b)-1/1600*b^(1/2)*e^2*5^(1/2)*Pi^(1/2)*erfi(5^(1/2)*(a+b*arcsinh(c*x)) 
^(1/2)/b^(1/2))/c^5/exp(5*a/b)
 

Mathematica [A] (warning: unable to verify)

Time = 4.23 (sec) , antiderivative size = 535, normalized size of antiderivative = 0.80 \[ \int \left (d+e x^2\right )^2 \sqrt {a+b \text {arcsinh}(c x)} \, dx=-\frac {b e^{-\frac {5 a}{b}} \left (450 e^{\frac {6 a}{b}} \left (8 a c^4 d^2 \sqrt {\frac {a}{b}+\text {arcsinh}(c x)}+8 b c^4 d^2 \text {arcsinh}(c x) \sqrt {\frac {a}{b}+\text {arcsinh}(c x)}+b \left (4 c^2 d-e\right ) e \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \sqrt {-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}}\right ) \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )+9 \sqrt {5} b e^2 \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \sqrt {-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}} \Gamma \left (\frac {3}{2},-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )+25 \sqrt {3} b \left (8 c^2 d-3 e\right ) e e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \sqrt {-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}} \Gamma \left (\frac {3}{2},-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+450 e^{\frac {4 a}{b}} \left (8 a c^4 d^2 \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}}+8 b c^4 d^2 \text {arcsinh}(c x) \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}}+b e \left (-4 c^2 d+e\right ) \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \sqrt {-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}}\right ) \Gamma \left (\frac {3}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )-b e e^{\frac {8 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \sqrt {-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}} \left (25 \sqrt {3} \left (8 c^2 d-3 e\right ) \Gamma \left (\frac {3}{2},\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+9 \sqrt {5} e e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )\right )\right )}{7200 c^5 (a+b \text {arcsinh}(c x))^{3/2}} \] Input:

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

Output:

-1/7200*(b*(450*E^((6*a)/b)*(8*a*c^4*d^2*Sqrt[a/b + ArcSinh[c*x]] + 8*b*c^ 
4*d^2*ArcSinh[c*x]*Sqrt[a/b + ArcSinh[c*x]] + b*(4*c^2*d - e)*e*Sqrt[-((a 
+ b*ArcSinh[c*x])/b)]*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)])*Gamma[3/2, a/b 
+ ArcSinh[c*x]] + 9*Sqrt[5]*b*e^2*Sqrt[a/b + ArcSinh[c*x]]*Sqrt[-((a + b*A 
rcSinh[c*x])^2/b^2)]*Gamma[3/2, (-5*(a + b*ArcSinh[c*x]))/b] + 25*Sqrt[3]* 
b*(8*c^2*d - 3*e)*e*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Sqrt[-((a + b*Arc 
Sinh[c*x])^2/b^2)]*Gamma[3/2, (-3*(a + b*ArcSinh[c*x]))/b] + 450*E^((4*a)/ 
b)*(8*a*c^4*d^2*Sqrt[-((a + b*ArcSinh[c*x])/b)] + 8*b*c^4*d^2*ArcSinh[c*x] 
*Sqrt[-((a + b*ArcSinh[c*x])/b)] + b*e*(-4*c^2*d + e)*Sqrt[a/b + ArcSinh[c 
*x]]*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)])*Gamma[3/2, -((a + b*ArcSinh[c*x] 
)/b)] - b*e*E^((8*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Sqrt[-((a + b*ArcS 
inh[c*x])^2/b^2)]*(25*Sqrt[3]*(8*c^2*d - 3*e)*Gamma[3/2, (3*(a + b*ArcSinh 
[c*x]))/b] + 9*Sqrt[5]*e*E^((2*a)/b)*Gamma[3/2, (5*(a + b*ArcSinh[c*x]))/b 
])))/(c^5*E^((5*a)/b)*(a + b*ArcSinh[c*x])^(3/2))
 

Rubi [A] (verified)

Time = 2.50 (sec) , antiderivative size = 672, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.

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

\(\Big \downarrow \) 6208

\(\displaystyle \int \left (d^2 \sqrt {a+b \text {arcsinh}(c x)}+2 d e x^2 \sqrt {a+b \text {arcsinh}(c x)}+e^2 x^4 \sqrt {a+b \text {arcsinh}(c x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {\pi } \sqrt {b} e^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 c^5}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e^2 e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 c^5}+\frac {\sqrt {\frac {\pi }{5}} \sqrt {b} e^2 e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{320 c^5}-\frac {\sqrt {\pi } \sqrt {b} e^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 c^5}+\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e^2 e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 c^5}-\frac {\sqrt {\frac {\pi }{5}} \sqrt {b} e^2 e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{320 c^5}-\frac {\sqrt {\pi } \sqrt {b} d e e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} d e e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{24 c^3}+\frac {\sqrt {\pi } \sqrt {b} d e e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 c^3}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} d e e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{24 c^3}+\frac {\sqrt {\pi } \sqrt {b} d^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {\pi } \sqrt {b} d^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 c}+d^2 x \sqrt {a+b \text {arcsinh}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \text {arcsinh}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \text {arcsinh}(c x)}\)

Input:

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

Output:

d^2*x*Sqrt[a + b*ArcSinh[c*x]] + (2*d*e*x^3*Sqrt[a + b*ArcSinh[c*x]])/3 + 
(e^2*x^5*Sqrt[a + b*ArcSinh[c*x]])/5 + (Sqrt[b]*d^2*E^(a/b)*Sqrt[Pi]*Erf[S 
qrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*c) - (Sqrt[b]*d*e*E^(a/b)*Sqrt[Pi]*Er 
f[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*c^3) + (Sqrt[b]*e^2*E^(a/b)*Sqrt[P 
i]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(32*c^5) + (Sqrt[b]*d*e*E^((3*a) 
/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(24*c^3) - 
 (Sqrt[b]*e^2*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]] 
)/Sqrt[b]])/(64*c^5) + (Sqrt[b]*e^2*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sq 
rt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(320*c^5) - (Sqrt[b]*d^2*Sqrt[Pi]*Erfi[S 
qrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*c*E^(a/b)) + (Sqrt[b]*d*e*Sqrt[Pi]*Er 
fi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*c^3*E^(a/b)) - (Sqrt[b]*e^2*Sqrt[ 
Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(32*c^5*E^(a/b)) - (Sqrt[b]*d* 
e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(24*c^3*E^( 
(3*a)/b)) + (Sqrt[b]*e^2*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]] 
)/Sqrt[b]])/(64*c^5*E^((3*a)/b)) - (Sqrt[b]*e^2*Sqrt[Pi/5]*Erfi[(Sqrt[5]*S 
qrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(320*c^5*E^((5*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]

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

Input:

int((e*x^2+d)^2*(a+b*arcsinh(x*c))^(1/2),x)
 

Output:

int((e*x^2+d)^2*(a+b*arcsinh(x*c))^(1/2),x)
 

Fricas [F(-2)]

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

integrate((e*x^2+d)^2*(a+b*arcsinh(c*x))^(1/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate((e*x**2+d)**2*(a+b*asinh(c*x))**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate((e*x^2+d)^2*(a+b*arcsinh(c*x))^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((e*x^2+d)^2*(a+b*arcsinh(c*x))^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((a + b*asinh(c*x))^(1/2)*(d + e*x^2)^2,x)
 

Output:

int((a + b*asinh(c*x))^(1/2)*(d + e*x^2)^2, x)
 

Reduce [F]

\[ \int \left (d+e x^2\right )^2 \sqrt {a+b \text {arcsinh}(c x)} \, dx=\left (\int \sqrt {\mathit {asinh} \left (c x \right ) b +a}d x \right ) d^{2}+\left (\int \sqrt {\mathit {asinh} \left (c x \right ) b +a}\, x^{4}d x \right ) e^{2}+2 \left (\int \sqrt {\mathit {asinh} \left (c x \right ) b +a}\, x^{2}d x \right ) d e \] Input:

int((e*x^2+d)^2*(a+b*asinh(c*x))^(1/2),x)
 

Output:

int(sqrt(asinh(c*x)*b + a),x)*d**2 + int(sqrt(asinh(c*x)*b + a)*x**4,x)*e* 
*2 + 2*int(sqrt(asinh(c*x)*b + a)*x**2,x)*d*e