\(\int (d+e x^2)^3 (a+b \text {arcsinh}(c x)) \, dx\) [607]

Optimal result

Integrand size = 18, antiderivative size = 221 \[ \int \left (d+e x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=-\frac {b \left (35 c^6 d^3-35 c^4 d^2 e+21 c^2 d e^2-5 e^3\right ) \sqrt {1+c^2 x^2}}{35 c^7}-\frac {b e \left (35 c^4 d^2-42 c^2 d e+15 e^2\right ) \left (1+c^2 x^2\right )^{3/2}}{105 c^7}-\frac {3 b \left (7 c^2 d-5 e\right ) e^2 \left (1+c^2 x^2\right )^{5/2}}{175 c^7}-\frac {b e^3 \left (1+c^2 x^2\right )^{7/2}}{49 c^7}+d^3 x (a+b \text {arcsinh}(c x))+d^2 e x^3 (a+b \text {arcsinh}(c x))+\frac {3}{5} d e^2 x^5 (a+b \text {arcsinh}(c x))+\frac {1}{7} e^3 x^7 (a+b \text {arcsinh}(c x)) \]

[Out]

-1/105*b*e*(35*c^4*d^2-42*c^2*d*e+15*e^2)*(c^2*x^2+1)^(3/2)/c^7-3/175*b*(7*c^2*d-5*e)*e^2*(c^2*x^2+1)^(5/2)/c^
7-1/49*b*e^3*(c^2*x^2+1)^(7/2)/c^7+d^3*x*(a+b*arcsinh(c*x))+d^2*e*x^3*(a+b*arcsinh(c*x))+3/5*d*e^2*x^5*(a+b*ar
csinh(c*x))+1/7*e^3*x^7*(a+b*arcsinh(c*x))-1/35*b*(35*c^6*d^3-35*c^4*d^2*e+21*c^2*d*e^2-5*e^3)*(c^2*x^2+1)^(1/
2)/c^7

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {200, 5792, 12, 1813, 1864} \[ \int \left (d+e x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=d^3 x (a+b \text {arcsinh}(c x))+d^2 e x^3 (a+b \text {arcsinh}(c x))+\frac {3}{5} d e^2 x^5 (a+b \text {arcsinh}(c x))+\frac {1}{7} e^3 x^7 (a+b \text {arcsinh}(c x))-\frac {3 b e^2 \left (c^2 x^2+1\right )^{5/2} \left (7 c^2 d-5 e\right )}{175 c^7}-\frac {b e^3 \left (c^2 x^2+1\right )^{7/2}}{49 c^7}-\frac {b e \left (c^2 x^2+1\right )^{3/2} \left (35 c^4 d^2-42 c^2 d e+15 e^2\right )}{105 c^7}-\frac {b \sqrt {c^2 x^2+1} \left (35 c^6 d^3-35 c^4 d^2 e+21 c^2 d e^2-5 e^3\right )}{35 c^7} \]

[In]

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

[Out]

-1/35*(b*(35*c^6*d^3 - 35*c^4*d^2*e + 21*c^2*d*e^2 - 5*e^3)*Sqrt[1 + c^2*x^2])/c^7 - (b*e*(35*c^4*d^2 - 42*c^2
*d*e + 15*e^2)*(1 + c^2*x^2)^(3/2))/(105*c^7) - (3*b*(7*c^2*d - 5*e)*e^2*(1 + c^2*x^2)^(5/2))/(175*c^7) - (b*e
^3*(1 + c^2*x^2)^(7/2))/(49*c^7) + d^3*x*(a + b*ArcSinh[c*x]) + d^2*e*x^3*(a + b*ArcSinh[c*x]) + (3*d*e^2*x^5*
(a + b*ArcSinh[c*x]))/5 + (e^3*x^7*(a + b*ArcSinh[c*x]))/7

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 1813

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1864

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rule 5792

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /;
FreeQ[{a, b, c, d, e}, x] && NeQ[e, c^2*d] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])

Rubi steps \begin{align*} \text {integral}& = d^3 x (a+b \text {arcsinh}(c x))+d^2 e x^3 (a+b \text {arcsinh}(c x))+\frac {3}{5} d e^2 x^5 (a+b \text {arcsinh}(c x))+\frac {1}{7} e^3 x^7 (a+b \text {arcsinh}(c x))-(b c) \int \frac {x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )}{35 \sqrt {1+c^2 x^2}} \, dx \\ & = d^3 x (a+b \text {arcsinh}(c x))+d^2 e x^3 (a+b \text {arcsinh}(c x))+\frac {3}{5} d e^2 x^5 (a+b \text {arcsinh}(c x))+\frac {1}{7} e^3 x^7 (a+b \text {arcsinh}(c x))-\frac {1}{35} (b c) \int \frac {x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )}{\sqrt {1+c^2 x^2}} \, dx \\ & = d^3 x (a+b \text {arcsinh}(c x))+d^2 e x^3 (a+b \text {arcsinh}(c x))+\frac {3}{5} d e^2 x^5 (a+b \text {arcsinh}(c x))+\frac {1}{7} e^3 x^7 (a+b \text {arcsinh}(c x))-\frac {1}{70} (b c) \text {Subst}\left (\int \frac {35 d^3+35 d^2 e x+21 d e^2 x^2+5 e^3 x^3}{\sqrt {1+c^2 x}} \, dx,x,x^2\right ) \\ & = d^3 x (a+b \text {arcsinh}(c x))+d^2 e x^3 (a+b \text {arcsinh}(c x))+\frac {3}{5} d e^2 x^5 (a+b \text {arcsinh}(c x))+\frac {1}{7} e^3 x^7 (a+b \text {arcsinh}(c x))-\frac {1}{70} (b c) \text {Subst}\left (\int \left (\frac {35 c^6 d^3-35 c^4 d^2 e+21 c^2 d e^2-5 e^3}{c^6 \sqrt {1+c^2 x}}+\frac {e \left (35 c^4 d^2-42 c^2 d e+15 e^2\right ) \sqrt {1+c^2 x}}{c^6}+\frac {3 \left (7 c^2 d-5 e\right ) e^2 \left (1+c^2 x\right )^{3/2}}{c^6}+\frac {5 e^3 \left (1+c^2 x\right )^{5/2}}{c^6}\right ) \, dx,x,x^2\right ) \\ & = -\frac {b \left (35 c^6 d^3-35 c^4 d^2 e+21 c^2 d e^2-5 e^3\right ) \sqrt {1+c^2 x^2}}{35 c^7}-\frac {b e \left (35 c^4 d^2-42 c^2 d e+15 e^2\right ) \left (1+c^2 x^2\right )^{3/2}}{105 c^7}-\frac {3 b \left (7 c^2 d-5 e\right ) e^2 \left (1+c^2 x^2\right )^{5/2}}{175 c^7}-\frac {b e^3 \left (1+c^2 x^2\right )^{7/2}}{49 c^7}+d^3 x (a+b \text {arcsinh}(c x))+d^2 e x^3 (a+b \text {arcsinh}(c x))+\frac {3}{5} d e^2 x^5 (a+b \text {arcsinh}(c x))+\frac {1}{7} e^3 x^7 (a+b \text {arcsinh}(c x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.85 \[ \int \left (d+e x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=a \left (d^3 x+d^2 e x^3+\frac {3}{5} d e^2 x^5+\frac {e^3 x^7}{7}\right )-\frac {b \sqrt {1+c^2 x^2} \left (-240 e^3+24 c^2 e^2 \left (49 d+5 e x^2\right )-2 c^4 e \left (1225 d^2+294 d e x^2+45 e^2 x^4\right )+c^6 \left (3675 d^3+1225 d^2 e x^2+441 d e^2 x^4+75 e^3 x^6\right )\right )}{3675 c^7}+b \left (d^3 x+d^2 e x^3+\frac {3}{5} d e^2 x^5+\frac {e^3 x^7}{7}\right ) \text {arcsinh}(c x) \]

[In]

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

[Out]

a*(d^3*x + d^2*e*x^3 + (3*d*e^2*x^5)/5 + (e^3*x^7)/7) - (b*Sqrt[1 + c^2*x^2]*(-240*e^3 + 24*c^2*e^2*(49*d + 5*
e*x^2) - 2*c^4*e*(1225*d^2 + 294*d*e*x^2 + 45*e^2*x^4) + c^6*(3675*d^3 + 1225*d^2*e*x^2 + 441*d*e^2*x^4 + 75*e
^3*x^6)))/(3675*c^7) + b*(d^3*x + d^2*e*x^3 + (3*d*e^2*x^5)/5 + (e^3*x^7)/7)*ArcSinh[c*x]

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.33

method	result	size
parts	\(a \left (\frac {1}{7} e^{3} x^{7}+\frac {3}{5} d \,e^{2} x^{5}+d^{2} e \,x^{3}+d^{3} x \right )+\frac {b \left (\frac {c \,\operatorname {arcsinh}\left (c x \right ) e^{3} x^{7}}{7}+\frac {3 c \,\operatorname {arcsinh}\left (c x \right ) d \,e^{2} x^{5}}{5}+c \,\operatorname {arcsinh}\left (c x \right ) d^{2} e \,x^{3}+\operatorname {arcsinh}\left (c x \right ) c x \,d^{3}-\frac {5 e^{3} \left (\frac {c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{35}+\frac {8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {c^{2} x^{2}+1}}{35}\right )+35 d^{3} c^{6} \sqrt {c^{2} x^{2}+1}+21 d \,c^{2} e^{2} \left (\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{15}+\frac {8 \sqrt {c^{2} x^{2}+1}}{15}\right )+35 d^{2} c^{4} e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{35 c^{6}}\right )}{c}\)	\(295\)
derivativedivides	\(\frac {\frac {a \left (d^{3} c^{7} x +d^{2} c^{7} e \,x^{3}+\frac {3}{5} d \,c^{7} e^{2} x^{5}+\frac {1}{7} e^{3} c^{7} x^{7}\right )}{c^{6}}+\frac {b \left (\operatorname {arcsinh}\left (c x \right ) d^{3} c^{7} x +\operatorname {arcsinh}\left (c x \right ) d^{2} c^{7} e \,x^{3}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) d \,c^{7} e^{2} x^{5}}{5}+\frac {\operatorname {arcsinh}\left (c x \right ) e^{3} c^{7} x^{7}}{7}-\frac {e^{3} \left (\frac {c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{35}+\frac {8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {c^{2} x^{2}+1}}{35}\right )}{7}-d^{3} c^{6} \sqrt {c^{2} x^{2}+1}-d^{2} c^{4} e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )-\frac {3 d \,c^{2} e^{2} \left (\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{15}+\frac {8 \sqrt {c^{2} x^{2}+1}}{15}\right )}{5}\right )}{c^{6}}}{c}\)	\(316\)
default	\(\frac {\frac {a \left (d^{3} c^{7} x +d^{2} c^{7} e \,x^{3}+\frac {3}{5} d \,c^{7} e^{2} x^{5}+\frac {1}{7} e^{3} c^{7} x^{7}\right )}{c^{6}}+\frac {b \left (\operatorname {arcsinh}\left (c x \right ) d^{3} c^{7} x +\operatorname {arcsinh}\left (c x \right ) d^{2} c^{7} e \,x^{3}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) d \,c^{7} e^{2} x^{5}}{5}+\frac {\operatorname {arcsinh}\left (c x \right ) e^{3} c^{7} x^{7}}{7}-\frac {e^{3} \left (\frac {c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{35}+\frac {8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {c^{2} x^{2}+1}}{35}\right )}{7}-d^{3} c^{6} \sqrt {c^{2} x^{2}+1}-d^{2} c^{4} e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )-\frac {3 d \,c^{2} e^{2} \left (\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{15}+\frac {8 \sqrt {c^{2} x^{2}+1}}{15}\right )}{5}\right )}{c^{6}}}{c}\)	\(316\)

[In]

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

[Out]

a*(1/7*e^3*x^7+3/5*d*e^2*x^5+d^2*e*x^3+d^3*x)+b/c*(1/7*c*arcsinh(c*x)*e^3*x^7+3/5*c*arcsinh(c*x)*d*e^2*x^5+c*a
rcsinh(c*x)*d^2*e*x^3+arcsinh(c*x)*c*x*d^3-1/35/c^6*(5*e^3*(1/7*c^6*x^6*(c^2*x^2+1)^(1/2)-6/35*c^4*x^4*(c^2*x^
2+1)^(1/2)+8/35*c^2*x^2*(c^2*x^2+1)^(1/2)-16/35*(c^2*x^2+1)^(1/2))+35*d^3*c^6*(c^2*x^2+1)^(1/2)+21*d*c^2*e^2*(
1/5*c^4*x^4*(c^2*x^2+1)^(1/2)-4/15*c^2*x^2*(c^2*x^2+1)^(1/2)+8/15*(c^2*x^2+1)^(1/2))+35*d^2*c^4*e*(1/3*c^2*x^2
*(c^2*x^2+1)^(1/2)-2/3*(c^2*x^2+1)^(1/2))))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.09 \[ \int \left (d+e x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {525 \, a c^{7} e^{3} x^{7} + 2205 \, a c^{7} d e^{2} x^{5} + 3675 \, a c^{7} d^{2} e x^{3} + 3675 \, a c^{7} d^{3} x + 105 \, {\left (5 \, b c^{7} e^{3} x^{7} + 21 \, b c^{7} d e^{2} x^{5} + 35 \, b c^{7} d^{2} e x^{3} + 35 \, b c^{7} d^{3} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (75 \, b c^{6} e^{3} x^{6} + 3675 \, b c^{6} d^{3} - 2450 \, b c^{4} d^{2} e + 1176 \, b c^{2} d e^{2} + 9 \, {\left (49 \, b c^{6} d e^{2} - 10 \, b c^{4} e^{3}\right )} x^{4} - 240 \, b e^{3} + {\left (1225 \, b c^{6} d^{2} e - 588 \, b c^{4} d e^{2} + 120 \, b c^{2} e^{3}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{3675 \, c^{7}} \]

[In]

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

[Out]

1/3675*(525*a*c^7*e^3*x^7 + 2205*a*c^7*d*e^2*x^5 + 3675*a*c^7*d^2*e*x^3 + 3675*a*c^7*d^3*x + 105*(5*b*c^7*e^3*
x^7 + 21*b*c^7*d*e^2*x^5 + 35*b*c^7*d^2*e*x^3 + 35*b*c^7*d^3*x)*log(c*x + sqrt(c^2*x^2 + 1)) - (75*b*c^6*e^3*x
^6 + 3675*b*c^6*d^3 - 2450*b*c^4*d^2*e + 1176*b*c^2*d*e^2 + 9*(49*b*c^6*d*e^2 - 10*b*c^4*e^3)*x^4 - 240*b*e^3
+ (1225*b*c^6*d^2*e - 588*b*c^4*d*e^2 + 120*b*c^2*e^3)*x^2)*sqrt(c^2*x^2 + 1))/c^7

Sympy [A] (verification not implemented)

Time = 0.63 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.76 \[ \int \left (d+e x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} a d^{3} x + a d^{2} e x^{3} + \frac {3 a d e^{2} x^{5}}{5} + \frac {a e^{3} x^{7}}{7} + b d^{3} x \operatorname {asinh}{\left (c x \right )} + b d^{2} e x^{3} \operatorname {asinh}{\left (c x \right )} + \frac {3 b d e^{2} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} + \frac {b e^{3} x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {b d^{3} \sqrt {c^{2} x^{2} + 1}}{c} - \frac {b d^{2} e x^{2} \sqrt {c^{2} x^{2} + 1}}{3 c} - \frac {3 b d e^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{25 c} - \frac {b e^{3} x^{6} \sqrt {c^{2} x^{2} + 1}}{49 c} + \frac {2 b d^{2} e \sqrt {c^{2} x^{2} + 1}}{3 c^{3}} + \frac {4 b d e^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{25 c^{3}} + \frac {6 b e^{3} x^{4} \sqrt {c^{2} x^{2} + 1}}{245 c^{3}} - \frac {8 b d e^{2} \sqrt {c^{2} x^{2} + 1}}{25 c^{5}} - \frac {8 b e^{3} x^{2} \sqrt {c^{2} x^{2} + 1}}{245 c^{5}} + \frac {16 b e^{3} \sqrt {c^{2} x^{2} + 1}}{245 c^{7}} & \text {for}\: c \neq 0 \\a \left (d^{3} x + d^{2} e x^{3} + \frac {3 d e^{2} x^{5}}{5} + \frac {e^{3} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a*d**3*x + a*d**2*e*x**3 + 3*a*d*e**2*x**5/5 + a*e**3*x**7/7 + b*d**3*x*asinh(c*x) + b*d**2*e*x**3*
asinh(c*x) + 3*b*d*e**2*x**5*asinh(c*x)/5 + b*e**3*x**7*asinh(c*x)/7 - b*d**3*sqrt(c**2*x**2 + 1)/c - b*d**2*e
*x**2*sqrt(c**2*x**2 + 1)/(3*c) - 3*b*d*e**2*x**4*sqrt(c**2*x**2 + 1)/(25*c) - b*e**3*x**6*sqrt(c**2*x**2 + 1)
/(49*c) + 2*b*d**2*e*sqrt(c**2*x**2 + 1)/(3*c**3) + 4*b*d*e**2*x**2*sqrt(c**2*x**2 + 1)/(25*c**3) + 6*b*e**3*x
**4*sqrt(c**2*x**2 + 1)/(245*c**3) - 8*b*d*e**2*sqrt(c**2*x**2 + 1)/(25*c**5) - 8*b*e**3*x**2*sqrt(c**2*x**2 +
1)/(245*c**5) + 16*b*e**3*sqrt(c**2*x**2 + 1)/(245*c**7), Ne(c, 0)), (a*(d**3*x + d**2*e*x**3 + 3*d*e**2*x**5
/5 + e**3*x**7/7), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.30 \[ \int \left (d+e x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{7} \, a e^{3} x^{7} + \frac {3}{5} \, a d e^{2} x^{5} + a d^{2} e x^{3} + \frac {1}{3} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d^{2} e + \frac {1}{25} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d e^{2} + \frac {1}{245} \, {\left (35 \, x^{7} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b e^{3} + a d^{3} x + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b d^{3}}{c} \]

[In]

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

[Out]

1/7*a*e^3*x^7 + 3/5*a*d*e^2*x^5 + a*d^2*e*x^3 + 1/3*(3*x^3*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x^2/c^2 - 2*sqr
t(c^2*x^2 + 1)/c^4))*b*d^2*e + 1/25*(15*x^5*arcsinh(c*x) - (3*sqrt(c^2*x^2 + 1)*x^4/c^2 - 4*sqrt(c^2*x^2 + 1)*
x^2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c)*b*d*e^2 + 1/245*(35*x^7*arcsinh(c*x) - (5*sqrt(c^2*x^2 + 1)*x^6/c^2 - 6*
sqrt(c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(c^2*x^2 + 1)*x^2/c^6 - 16*sqrt(c^2*x^2 + 1)/c^8)*c)*b*e^3 + a*d^3*x + (c*x*
arcsinh(c*x) - sqrt(c^2*x^2 + 1))*b*d^3/c

Giac [F(-2)]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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