Integrand size = 26, antiderivative size = 266 \[ \int x^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {2 b d^2 x \sqrt {d+c^2 d x^2}}{63 c^3 \sqrt {1+c^2 x^2}}-\frac {b d^2 x^3 \sqrt {d+c^2 d x^2}}{189 c \sqrt {1+c^2 x^2}}-\frac {b c d^2 x^5 \sqrt {d+c^2 d x^2}}{21 \sqrt {1+c^2 x^2}}-\frac {19 b c^3 d^2 x^7 \sqrt {d+c^2 d x^2}}{441 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^9 \sqrt {d+c^2 d x^2}}{81 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4 d}+\frac {\left (d+c^2 d x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4 d^2} \] Output:
2/63*b*d^2*x*(c^2*d*x^2+d)^(1/2)/c^3/(c^2*x^2+1)^(1/2)-1/189*b*d^2*x^3*(c^ 2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-1/21*b*c*d^2*x^5*(c^2*d*x^2+d)^(1/2)/ (c^2*x^2+1)^(1/2)-19/441*b*c^3*d^2*x^7*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/ 2)-1/81*b*c^5*d^2*x^9*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/7*(c^2*d*x^2 +d)^(7/2)*(a+b*arcsinh(c*x))/c^4/d+1/9*(c^2*d*x^2+d)^(9/2)*(a+b*arcsinh(c* x))/c^4/d^2
Time = 0.22 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.53 \[ \int x^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {d^2 \sqrt {d+c^2 d x^2} \left (63 a \left (1+c^2 x^2\right )^4 \left (-2+7 c^2 x^2\right )-b c x \sqrt {1+c^2 x^2} \left (-126+21 c^2 x^2+189 c^4 x^4+171 c^6 x^6+49 c^8 x^8\right )+63 b \left (1+c^2 x^2\right )^4 \left (-2+7 c^2 x^2\right ) \text {arcsinh}(c x)\right )}{3969 c^4 \left (1+c^2 x^2\right )} \] Input:
Integrate[x^3*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]
Output:
(d^2*Sqrt[d + c^2*d*x^2]*(63*a*(1 + c^2*x^2)^4*(-2 + 7*c^2*x^2) - b*c*x*Sq rt[1 + c^2*x^2]*(-126 + 21*c^2*x^2 + 189*c^4*x^4 + 171*c^6*x^6 + 49*c^8*x^ 8) + 63*b*(1 + c^2*x^2)^4*(-2 + 7*c^2*x^2)*ArcSinh[c*x]))/(3969*c^4*(1 + c ^2*x^2))
Time = 0.65 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.55, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6219, 27, 290, 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.
| \(\displaystyle \int x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6219 |
| \(\displaystyle -\frac {b c \sqrt {c^2 d x^2+d} \int -\frac {d^2 \left (2-7 c^2 x^2\right ) \left (c^2 x^2+1\right )^3}{63 c^4}dx}{\sqrt {c^2 x^2+1}}+\frac {\left (c^2 d x^2+d\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4 d^2}-\frac {\left (c^2 d x^2+d\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b d^2 \sqrt {c^2 d x^2+d} \int \left (2-7 c^2 x^2\right ) \left (c^2 x^2+1\right )^3dx}{63 c^3 \sqrt {c^2 x^2+1}}+\frac {\left (c^2 d x^2+d\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4 d^2}-\frac {\left (c^2 d x^2+d\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4 d}\) |
| \(\Big \downarrow \) 290 |
| \(\displaystyle \frac {b d^2 \sqrt {c^2 d x^2+d} \int \left (-7 c^8 x^8-19 c^6 x^6-15 c^4 x^4-c^2 x^2+2\right )dx}{63 c^3 \sqrt {c^2 x^2+1}}+\frac {\left (c^2 d x^2+d\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4 d^2}-\frac {\left (c^2 d x^2+d\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (c^2 d x^2+d\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4 d^2}-\frac {\left (c^2 d x^2+d\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4 d}+\frac {b d^2 \left (-\frac {7}{9} c^8 x^9-\frac {19 c^6 x^7}{7}-3 c^4 x^5-\frac {c^2 x^3}{3}+2 x\right ) \sqrt {c^2 d x^2+d}}{63 c^3 \sqrt {c^2 x^2+1}}\) |
Input:
Int[x^3*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]
Output:
(b*d^2*Sqrt[d + c^2*d*x^2]*(2*x - (c^2*x^3)/3 - 3*c^4*x^5 - (19*c^6*x^7)/7 - (7*c^8*x^9)/9))/(63*c^3*Sqrt[1 + c^2*x^2]) - ((d + c^2*d*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(7*c^4*d) + ((d + c^2*d*x^2)^(9/2)*(a + b*ArcSinh[c*x]) )/(9*c^4*d^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> I nt[ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d }, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_ ), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSi nh[c*x]) u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] Int[S implifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x ] && EqQ[e, c^2*d] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1) /2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
Time = 0.98 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.82
| method | result | size |
| orering | \(\frac {\left (833 c^{10} x^{10}+3153 c^{8} x^{8}+4167 c^{6} x^{6}+1743 c^{4} x^{4}-1008 c^{2} x^{2}-504\right ) \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{3969 c^{4} \left (c^{2} x^{2}+1\right )^{3}}-\frac {\left (49 c^{8} x^{8}+171 c^{6} x^{6}+189 c^{4} x^{4}+21 c^{2} x^{2}-126\right ) \left (3 x^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )+5 x^{4} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{2} d +\frac {x^{3} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} b c}{\sqrt {c^{2} x^{2}+1}}\right )}{3969 x^{2} c^{4} \left (c^{2} x^{2}+1\right )^{2}}\) | \(217\) |
| default | \(a \left (\frac {x^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{9 c^{2} d}-\frac {2 \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{63 d \,c^{4}}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (256 c^{10} x^{10}+256 \sqrt {c^{2} x^{2}+1}\, x^{9} c^{9}+704 c^{8} x^{8}+576 x^{7} c^{7} \sqrt {c^{2} x^{2}+1}+688 c^{6} x^{6}+432 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+280 c^{4} x^{4}+120 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+41 c^{2} x^{2}+9 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+9 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{41472 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (64 c^{8} x^{8}+64 x^{7} c^{7} \sqrt {c^{2} x^{2}+1}+144 c^{6} x^{6}+112 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+104 c^{4} x^{4}+56 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+25 c^{2} x^{2}+7 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+7 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{25088 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{576 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )-1\right ) d^{2}}{256 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )+1\right ) d^{2}}{256 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+3 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{576 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (64 c^{8} x^{8}-64 x^{7} c^{7} \sqrt {c^{2} x^{2}+1}+144 c^{6} x^{6}-112 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+104 c^{4} x^{4}-56 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+25 c^{2} x^{2}-7 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+7 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{25088 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (256 c^{10} x^{10}-256 \sqrt {c^{2} x^{2}+1}\, x^{9} c^{9}+704 c^{8} x^{8}-576 x^{7} c^{7} \sqrt {c^{2} x^{2}+1}+688 c^{6} x^{6}-432 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+280 c^{4} x^{4}-120 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+41 c^{2} x^{2}-9 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+9 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{41472 c^{4} \left (c^{2} x^{2}+1\right )}\right )\) | \(996\) |
| parts | \(a \left (\frac {x^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{9 c^{2} d}-\frac {2 \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{63 d \,c^{4}}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (256 c^{10} x^{10}+256 \sqrt {c^{2} x^{2}+1}\, x^{9} c^{9}+704 c^{8} x^{8}+576 x^{7} c^{7} \sqrt {c^{2} x^{2}+1}+688 c^{6} x^{6}+432 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+280 c^{4} x^{4}+120 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+41 c^{2} x^{2}+9 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+9 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{41472 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (64 c^{8} x^{8}+64 x^{7} c^{7} \sqrt {c^{2} x^{2}+1}+144 c^{6} x^{6}+112 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+104 c^{4} x^{4}+56 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+25 c^{2} x^{2}+7 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+7 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{25088 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{576 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )-1\right ) d^{2}}{256 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )+1\right ) d^{2}}{256 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+3 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{576 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (64 c^{8} x^{8}-64 x^{7} c^{7} \sqrt {c^{2} x^{2}+1}+144 c^{6} x^{6}-112 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+104 c^{4} x^{4}-56 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+25 c^{2} x^{2}-7 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+7 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{25088 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (256 c^{10} x^{10}-256 \sqrt {c^{2} x^{2}+1}\, x^{9} c^{9}+704 c^{8} x^{8}-576 x^{7} c^{7} \sqrt {c^{2} x^{2}+1}+688 c^{6} x^{6}-432 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+280 c^{4} x^{4}-120 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+41 c^{2} x^{2}-9 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+9 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{41472 c^{4} \left (c^{2} x^{2}+1\right )}\right )\) | \(996\) |
Input:
int(x^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(x*c)),x,method=_RETURNVERBOSE)
Output:
1/3969*(833*c^10*x^10+3153*c^8*x^8+4167*c^6*x^6+1743*c^4*x^4-1008*c^2*x^2- 504)/c^4/(c^2*x^2+1)^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(x*c))-1/3969/x^2*( 49*c^8*x^8+171*c^6*x^6+189*c^4*x^4+21*c^2*x^2-126)/c^4/(c^2*x^2+1)^2*(3*x^ 2*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(x*c))+5*x^4*(c^2*d*x^2+d)^(3/2)*(a+b*ar csinh(x*c))*c^2*d+x^3*(c^2*d*x^2+d)^(5/2)*b*c/(c^2*x^2+1)^(1/2))
Time = 0.10 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.99 \[ \int x^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {63 \, {\left (7 \, b c^{10} d^{2} x^{10} + 26 \, b c^{8} d^{2} x^{8} + 34 \, b c^{6} d^{2} x^{6} + 16 \, b c^{4} d^{2} x^{4} - b c^{2} d^{2} x^{2} - 2 \, b d^{2}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (441 \, a c^{10} d^{2} x^{10} + 1638 \, a c^{8} d^{2} x^{8} + 2142 \, a c^{6} d^{2} x^{6} + 1008 \, a c^{4} d^{2} x^{4} - 63 \, a c^{2} d^{2} x^{2} - 126 \, a d^{2} - {\left (49 \, b c^{9} d^{2} x^{9} + 171 \, b c^{7} d^{2} x^{7} + 189 \, b c^{5} d^{2} x^{5} + 21 \, b c^{3} d^{2} x^{3} - 126 \, b c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d}}{3969 \, {\left (c^{6} x^{2} + c^{4}\right )}} \] Input:
integrate(x^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="fricas" )
Output:
1/3969*(63*(7*b*c^10*d^2*x^10 + 26*b*c^8*d^2*x^8 + 34*b*c^6*d^2*x^6 + 16*b *c^4*d^2*x^4 - b*c^2*d^2*x^2 - 2*b*d^2)*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt (c^2*x^2 + 1)) + (441*a*c^10*d^2*x^10 + 1638*a*c^8*d^2*x^8 + 2142*a*c^6*d^ 2*x^6 + 1008*a*c^4*d^2*x^4 - 63*a*c^2*d^2*x^2 - 126*a*d^2 - (49*b*c^9*d^2* x^9 + 171*b*c^7*d^2*x^7 + 189*b*c^5*d^2*x^5 + 21*b*c^3*d^2*x^3 - 126*b*c*d ^2*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d))/(c^6*x^2 + c^4)
Timed out. \[ \int x^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Timed out} \] Input:
integrate(x**3*(c**2*d*x**2+d)**(5/2)*(a+b*asinh(c*x)),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.59 \[ \int x^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{63} \, {\left (\frac {7 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} b \operatorname {arsinh}\left (c x\right ) + \frac {1}{63} \, {\left (\frac {7 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} a - \frac {{\left (49 \, c^{8} d^{\frac {5}{2}} x^{9} + 171 \, c^{6} d^{\frac {5}{2}} x^{7} + 189 \, c^{4} d^{\frac {5}{2}} x^{5} + 21 \, c^{2} d^{\frac {5}{2}} x^{3} - 126 \, d^{\frac {5}{2}} x\right )} b}{3969 \, c^{3}} \] Input:
integrate(x^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="maxima" )
Output:
1/63*(7*(c^2*d*x^2 + d)^(7/2)*x^2/(c^2*d) - 2*(c^2*d*x^2 + d)^(7/2)/(c^4*d ))*b*arcsinh(c*x) + 1/63*(7*(c^2*d*x^2 + d)^(7/2)*x^2/(c^2*d) - 2*(c^2*d*x ^2 + d)^(7/2)/(c^4*d))*a - 1/3969*(49*c^8*d^(5/2)*x^9 + 171*c^6*d^(5/2)*x^ 7 + 189*c^4*d^(5/2)*x^5 + 21*c^2*d^(5/2)*x^3 - 126*d^(5/2)*x)*b/c^3
Exception generated. \[ \int x^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int x^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int x^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{5/2} \,d x \] Input:
int(x^3*(a + b*asinh(c*x))*(d + c^2*d*x^2)^(5/2),x)
Output:
int(x^3*(a + b*asinh(c*x))*(d + c^2*d*x^2)^(5/2), x)
\[ \int x^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {d}\, d^{2} \left (7 \sqrt {c^{2} x^{2}+1}\, a \,c^{8} x^{8}+19 \sqrt {c^{2} x^{2}+1}\, a \,c^{6} x^{6}+15 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} x^{4}+\sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}-2 \sqrt {c^{2} x^{2}+1}\, a +63 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{7}d x \right ) b \,c^{8}+126 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{5}d x \right ) b \,c^{6}+63 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{3}d x \right ) b \,c^{4}\right )}{63 c^{4}} \] Input:
int(x^3*(c^2*d*x^2+d)^(5/2)*(a+b*asinh(c*x)),x)
Output:
(sqrt(d)*d**2*(7*sqrt(c**2*x**2 + 1)*a*c**8*x**8 + 19*sqrt(c**2*x**2 + 1)* a*c**6*x**6 + 15*sqrt(c**2*x**2 + 1)*a*c**4*x**4 + sqrt(c**2*x**2 + 1)*a*c **2*x**2 - 2*sqrt(c**2*x**2 + 1)*a + 63*int(sqrt(c**2*x**2 + 1)*asinh(c*x) *x**7,x)*b*c**8 + 126*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**5,x)*b*c**6 + 63*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**3,x)*b*c**4))/(63*c**4)