3.7.38 \(\int \frac {(d+e x^2)^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx\) [638]

3.7.38.1 Optimal result

Integrand size = 22, antiderivative size = 608 \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\frac {d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}-\frac {d e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^3}+\frac {e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 \sqrt {b} c^5}+\frac {d e e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^3}-\frac {e^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5}+\frac {e^2 e^{\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5}+\frac {d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}-\frac {d e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^3}+\frac {e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 \sqrt {b} c^5}+\frac {d e e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^3}-\frac {e^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5}+\frac {e^2 e^{-\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5} \]

output

1/160*e^2*exp(5*a/b)*erf(5^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))*5^(1/2) 
*Pi^(1/2)/c^5/b^(1/2)+1/160*e^2*erfi(5^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1 
/2))*5^(1/2)*Pi^(1/2)/c^5/exp(5*a/b)/b^(1/2)+1/12*d*e*exp(3*a/b)*erf(3^(1/ 
2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/c^3/b^(1/2)+1/12*d*e 
*erfi(3^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/c^3/exp(3 
*a/b)/b^(1/2)+1/2*d^2*exp(a/b)*erf((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*Pi^(1 
/2)/c/b^(1/2)-1/4*d*e*exp(a/b)*erf((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*Pi^(1 
/2)/c^3/b^(1/2)+1/16*e^2*exp(a/b)*erf((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*Pi 
^(1/2)/c^5/b^(1/2)+1/2*d^2*erfi((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*Pi^(1/2) 
/c/exp(a/b)/b^(1/2)-1/4*d*e*erfi((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*Pi^(1/2 
)/c^3/exp(a/b)/b^(1/2)+1/16*e^2*erfi((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*Pi^ 
(1/2)/c^5/exp(a/b)/b^(1/2)-1/32*e^2*exp(3*a/b)*erf(3^(1/2)*(a+b*arcsinh(c* 
x))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/c^5/b^(1/2)-1/32*e^2*erfi(3^(1/2)*(a+b 
*arcsinh(c*x))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/c^5/exp(3*a/b)/b^(1/2)
 

3.7.38.2 Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 530, normalized size of antiderivative = 0.87 \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\frac {e^{-\frac {5 a}{b}} \left (-30 \left (8 c^4 d^2-4 c^2 d e+e^2\right ) e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )+3 \sqrt {5} e^2 \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )+40 \sqrt {3} c^2 d e e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-15 \sqrt {3} e^2 e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+240 c^4 d^2 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )-120 c^2 d e e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )+30 e^2 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )-40 \sqrt {3} c^2 d e e^{\frac {8 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+15 \sqrt {3} e^2 e^{\frac {8 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-3 \sqrt {5} e^2 e^{\frac {10 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{480 c^5 \sqrt {a+b \text {arcsinh}(c x)}} \]

input

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

output

(-30*(8*c^4*d^2 - 4*c^2*d*e + e^2)*E^((6*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Ga 
mma[1/2, a/b + ArcSinh[c*x]] + 3*Sqrt[5]*e^2*Sqrt[-((a + b*ArcSinh[c*x])/b 
)]*Gamma[1/2, (-5*(a + b*ArcSinh[c*x]))/b] + 40*Sqrt[3]*c^2*d*e*E^((2*a)/b 
)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcSinh[c*x]))/b] 
- 15*Sqrt[3]*e^2*E^((2*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (- 
3*(a + b*ArcSinh[c*x]))/b] + 240*c^4*d^2*E^((4*a)/b)*Sqrt[-((a + b*ArcSinh 
[c*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c*x])/b)] - 120*c^2*d*e*E^((4*a)/b) 
*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c*x])/b)] + 3 
0*e^2*E^((4*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, -((a + b*ArcS 
inh[c*x])/b)] - 40*Sqrt[3]*c^2*d*e*E^((8*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Ga 
mma[1/2, (3*(a + b*ArcSinh[c*x]))/b] + 15*Sqrt[3]*e^2*E^((8*a)/b)*Sqrt[a/b 
 + ArcSinh[c*x]]*Gamma[1/2, (3*(a + b*ArcSinh[c*x]))/b] - 3*Sqrt[5]*e^2*E^ 
((10*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (5*(a + b*ArcSinh[c*x]))/b] 
)/(480*c^5*E^((5*a)/b)*Sqrt[a + b*ArcSinh[c*x]])
 

3.7.38.3 Rubi [A] (verified)

Time = 1.40 (sec) , antiderivative size = 608, 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.

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

\(\Big \downarrow \) 6208

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {\pi } e^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 \sqrt {b} c^5}-\frac {\sqrt {3 \pi } e^2 e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5}+\frac {\sqrt {\frac {\pi }{5}} e^2 e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5}+\frac {\sqrt {\pi } e^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 \sqrt {b} c^5}-\frac {\sqrt {3 \pi } e^2 e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5}+\frac {\sqrt {\frac {\pi }{5}} e^2 e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5}-\frac {\sqrt {\pi } d e e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^3}+\frac {\sqrt {\frac {\pi }{3}} d e e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^3}-\frac {\sqrt {\pi } d e e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^3}+\frac {\sqrt {\frac {\pi }{3}} d e e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^3}+\frac {\sqrt {\pi } d^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}+\frac {\sqrt {\pi } d^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}\)

input

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

output

(d^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(2*Sqrt[b]*c) 
 - (d*e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*Sqrt[b] 
*c^3) + (e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(16*S 
qrt[b]*c^5) + (d*e*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[ 
c*x]])/Sqrt[b]])/(4*Sqrt[b]*c^3) - (e^2*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3 
]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(32*Sqrt[b]*c^5) + (e^2*E^((5*a)/b)* 
Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(32*Sqrt[b]*c^ 
5) + (d^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(2*Sqrt[b]*c*E^ 
(a/b)) - (d*e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*Sqrt[b]* 
c^3*E^(a/b)) + (e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(16*S 
qrt[b]*c^5*E^(a/b)) + (d*e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x 
]])/Sqrt[b]])/(4*Sqrt[b]*c^3*E^((3*a)/b)) - (e^2*Sqrt[3*Pi]*Erfi[(Sqrt[3]* 
Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(32*Sqrt[b]*c^5*E^((3*a)/b)) + (e^2*Sq 
rt[Pi/5]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(32*Sqrt[b]*c^5 
*E^((5*a)/b))
 

3.7.38.3.1 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])
 

3.7.38.4 Maple [F]

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

input

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

output

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

3.7.38.5 Fricas [F(-2)]

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

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)
 

3.7.38.6 Sympy [F]

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

input

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

output

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

3.7.38.7 Maxima [F]

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

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)
 

3.7.38.8 Giac [F]

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

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)
 

3.7.38.9 Mupad [F(-1)]

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

input

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

output

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