Integrand size = 26, antiderivative size = 217 \[ \int x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {2 b d x \sqrt {d+c^2 d x^2}}{35 c^3 \sqrt {1+c^2 x^2}}-\frac {b d x^3 \sqrt {d+c^2 d x^2}}{105 c \sqrt {1+c^2 x^2}}-\frac {8 b c d x^5 \sqrt {d+c^2 d x^2}}{175 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4 d}+\frac {\left (d+c^2 d x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4 d^2} \]
-1/5*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))/c^4/d+1/7*(c^2*d*x^2+d)^(7/2)* (a+b*arcsinh(c*x))/c^4/d^2+2/35*b*d*x*(c^2*d*x^2+d)^(1/2)/c^3/(c^2*x^2+1)^ (1/2)-1/105*b*d*x^3*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-8/175*b*c*d*x^ 5*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/49*b*c^3*d*x^7*(c^2*d*x^2+d)^(1/ 2)/(c^2*x^2+1)^(1/2)
Time = 0.19 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.60 \[ \int x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {d \sqrt {d+c^2 d x^2} \left (105 a \left (1+c^2 x^2\right )^3 \left (-2+5 c^2 x^2\right )-b c x \sqrt {1+c^2 x^2} \left (-210+35 c^2 x^2+168 c^4 x^4+75 c^6 x^6\right )+105 b \left (1+c^2 x^2\right )^3 \left (-2+5 c^2 x^2\right ) \text {arcsinh}(c x)\right )}{3675 c^4 \left (1+c^2 x^2\right )} \]
(d*Sqrt[d + c^2*d*x^2]*(105*a*(1 + c^2*x^2)^3*(-2 + 5*c^2*x^2) - b*c*x*Sqr t[1 + c^2*x^2]*(-210 + 35*c^2*x^2 + 168*c^4*x^4 + 75*c^6*x^6) + 105*b*(1 + c^2*x^2)^3*(-2 + 5*c^2*x^2)*ArcSinh[c*x]))/(3675*c^4*(1 + c^2*x^2))
Time = 0.41 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.62, 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.
| \(\displaystyle \int x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6219 |
| \(\displaystyle -\frac {b c \sqrt {c^2 d x^2+d} \int -\frac {d \left (2-5 c^2 x^2\right ) \left (c^2 x^2+1\right )^2}{35 c^4}dx}{\sqrt {c^2 x^2+1}}+\frac {\left (c^2 d x^2+d\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4 d^2}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b d \sqrt {c^2 d x^2+d} \int \left (2-5 c^2 x^2\right ) \left (c^2 x^2+1\right )^2dx}{35 c^3 \sqrt {c^2 x^2+1}}+\frac {\left (c^2 d x^2+d\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4 d^2}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4 d}\) |
| \(\Big \downarrow \) 290 |
| \(\displaystyle \frac {b d \sqrt {c^2 d x^2+d} \int \left (-5 c^6 x^6-8 c^4 x^4-c^2 x^2+2\right )dx}{35 c^3 \sqrt {c^2 x^2+1}}+\frac {\left (c^2 d x^2+d\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4 d^2}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (c^2 d x^2+d\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4 d^2}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4 d}+\frac {b d \left (-\frac {5}{7} c^6 x^7-\frac {8 c^4 x^5}{5}-\frac {c^2 x^3}{3}+2 x\right ) \sqrt {c^2 d x^2+d}}{35 c^3 \sqrt {c^2 x^2+1}}\) |
(b*d*Sqrt[d + c^2*d*x^2]*(2*x - (c^2*x^3)/3 - (8*c^4*x^5)/5 - (5*c^6*x^7)/ 7))/(35*c^3*Sqrt[1 + c^2*x^2]) - ((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x ]))/(5*c^4*d) + ((d + c^2*d*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(7*c^4*d^2)
3.2.28.3.1 Defintions of rubi rules used
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])
Leaf count of result is larger than twice the leaf count of optimal. \(871\) vs. \(2(185)=370\).
Time = 0.22 (sec) , antiderivative size = 872, normalized size of antiderivative = 4.02
| method | result | size |
| default | \(a \left (\frac {x^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{7 c^{2} d}-\frac {2 \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{35 d \,c^{4}}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (64 c^{8} x^{8}+64 c^{7} x^{7} \sqrt {c^{2} x^{2}+1}+144 c^{6} x^{6}+112 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+104 c^{4} x^{4}+56 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+25 c^{2} x^{2}+7 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+7 \,\operatorname {arcsinh}\left (c x \right )\right ) d}{6272 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}+16 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+28 c^{4} x^{4}+20 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+13 c^{2} x^{2}+5 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+5 \,\operatorname {arcsinh}\left (c x \right )\right ) d}{3200 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 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}+3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (c x \right )\right ) d}{384 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}+c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+\operatorname {arcsinh}\left (c x \right )\right ) d}{128 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}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (\operatorname {arcsinh}\left (c x \right )+1\right ) d}{128 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 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}-3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (3 \,\operatorname {arcsinh}\left (c x \right )+1\right ) d}{384 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}-16 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+28 c^{4} x^{4}-20 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+13 c^{2} x^{2}-5 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (1+5 \,\operatorname {arcsinh}\left (c x \right )\right ) d}{3200 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (64 c^{8} x^{8}-64 c^{7} x^{7} \sqrt {c^{2} x^{2}+1}+144 c^{6} x^{6}-112 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+104 c^{4} x^{4}-56 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+25 c^{2} x^{2}-7 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (1+7 \,\operatorname {arcsinh}\left (c x \right )\right ) d}{6272 c^{4} \left (c^{2} x^{2}+1\right )}\right )\) | \(872\) |
| parts | \(a \left (\frac {x^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{7 c^{2} d}-\frac {2 \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{35 d \,c^{4}}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (64 c^{8} x^{8}+64 c^{7} x^{7} \sqrt {c^{2} x^{2}+1}+144 c^{6} x^{6}+112 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+104 c^{4} x^{4}+56 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+25 c^{2} x^{2}+7 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+7 \,\operatorname {arcsinh}\left (c x \right )\right ) d}{6272 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}+16 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+28 c^{4} x^{4}+20 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+13 c^{2} x^{2}+5 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+5 \,\operatorname {arcsinh}\left (c x \right )\right ) d}{3200 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 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}+3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (c x \right )\right ) d}{384 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}+c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+\operatorname {arcsinh}\left (c x \right )\right ) d}{128 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}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (\operatorname {arcsinh}\left (c x \right )+1\right ) d}{128 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 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}-3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (3 \,\operatorname {arcsinh}\left (c x \right )+1\right ) d}{384 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}-16 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+28 c^{4} x^{4}-20 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+13 c^{2} x^{2}-5 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (1+5 \,\operatorname {arcsinh}\left (c x \right )\right ) d}{3200 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (64 c^{8} x^{8}-64 c^{7} x^{7} \sqrt {c^{2} x^{2}+1}+144 c^{6} x^{6}-112 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+104 c^{4} x^{4}-56 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+25 c^{2} x^{2}-7 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (1+7 \,\operatorname {arcsinh}\left (c x \right )\right ) d}{6272 c^{4} \left (c^{2} x^{2}+1\right )}\right )\) | \(872\) |
a*(1/7*x^2*(c^2*d*x^2+d)^(5/2)/c^2/d-2/35/d/c^4*(c^2*d*x^2+d)^(5/2))+b*(1/ 6272*(d*(c^2*x^2+1))^(1/2)*(64*c^8*x^8+64*c^7*x^7*(c^2*x^2+1)^(1/2)+144*c^ 6*x^6+112*c^5*x^5*(c^2*x^2+1)^(1/2)+104*c^4*x^4+56*c^3*x^3*(c^2*x^2+1)^(1/ 2)+25*c^2*x^2+7*c*x*(c^2*x^2+1)^(1/2)+1)*(-1+7*arcsinh(c*x))*d/c^4/(c^2*x^ 2+1)+1/3200*(d*(c^2*x^2+1))^(1/2)*(16*c^6*x^6+16*c^5*x^5*(c^2*x^2+1)^(1/2) +28*c^4*x^4+20*c^3*x^3*(c^2*x^2+1)^(1/2)+13*c^2*x^2+5*c*x*(c^2*x^2+1)^(1/2 )+1)*(-1+5*arcsinh(c*x))*d/c^4/(c^2*x^2+1)-1/384*(d*(c^2*x^2+1))^(1/2)*(4* c^4*x^4+4*c^3*x^3*(c^2*x^2+1)^(1/2)+5*c^2*x^2+3*c*x*(c^2*x^2+1)^(1/2)+1)*( -1+3*arcsinh(c*x))*d/c^4/(c^2*x^2+1)-3/128*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+ c*x*(c^2*x^2+1)^(1/2)+1)*(-1+arcsinh(c*x))*d/c^4/(c^2*x^2+1)-3/128*(d*(c^2 *x^2+1))^(1/2)*(c^2*x^2-c*x*(c^2*x^2+1)^(1/2)+1)*(arcsinh(c*x)+1)*d/c^4/(c ^2*x^2+1)-1/384*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4-4*c^3*x^3*(c^2*x^2+1)^(1/ 2)+5*c^2*x^2-3*c*x*(c^2*x^2+1)^(1/2)+1)*(3*arcsinh(c*x)+1)*d/c^4/(c^2*x^2+ 1)+1/3200*(d*(c^2*x^2+1))^(1/2)*(16*c^6*x^6-16*c^5*x^5*(c^2*x^2+1)^(1/2)+2 8*c^4*x^4-20*c^3*x^3*(c^2*x^2+1)^(1/2)+13*c^2*x^2-5*c*x*(c^2*x^2+1)^(1/2)+ 1)*(1+5*arcsinh(c*x))*d/c^4/(c^2*x^2+1)+1/6272*(d*(c^2*x^2+1))^(1/2)*(64*c ^8*x^8-64*c^7*x^7*(c^2*x^2+1)^(1/2)+144*c^6*x^6-112*c^5*x^5*(c^2*x^2+1)^(1 /2)+104*c^4*x^4-56*c^3*x^3*(c^2*x^2+1)^(1/2)+25*c^2*x^2-7*c*x*(c^2*x^2+1)^ (1/2)+1)*(1+7*arcsinh(c*x))*d/c^4/(c^2*x^2+1))
Time = 0.27 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.92 \[ \int x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {105 \, {\left (5 \, b c^{8} d x^{8} + 13 \, b c^{6} d x^{6} + 9 \, b c^{4} d x^{4} - b c^{2} d x^{2} - 2 \, b d\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (525 \, a c^{8} d x^{8} + 1365 \, a c^{6} d x^{6} + 945 \, a c^{4} d x^{4} - 105 \, a c^{2} d x^{2} - 210 \, a d - {\left (75 \, b c^{7} d x^{7} + 168 \, b c^{5} d x^{5} + 35 \, b c^{3} d x^{3} - 210 \, b c d x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d}}{3675 \, {\left (c^{6} x^{2} + c^{4}\right )}} \]
1/3675*(105*(5*b*c^8*d*x^8 + 13*b*c^6*d*x^6 + 9*b*c^4*d*x^4 - b*c^2*d*x^2 - 2*b*d)*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1)) + (525*a*c^8*d*x ^8 + 1365*a*c^6*d*x^6 + 945*a*c^4*d*x^4 - 105*a*c^2*d*x^2 - 210*a*d - (75* b*c^7*d*x^7 + 168*b*c^5*d*x^5 + 35*b*c^3*d*x^3 - 210*b*c*d*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d))/(c^6*x^2 + c^4)
\[ \int x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int x^{3} \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )\, dx \]
Time = 0.20 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.67 \[ \int x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{35} \, {\left (\frac {5 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} b \operatorname {arsinh}\left (c x\right ) + \frac {1}{35} \, {\left (\frac {5 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} a - \frac {{\left (75 \, c^{6} d^{\frac {3}{2}} x^{7} + 168 \, c^{4} d^{\frac {3}{2}} x^{5} + 35 \, c^{2} d^{\frac {3}{2}} x^{3} - 210 \, d^{\frac {3}{2}} x\right )} b}{3675 \, c^{3}} \]
1/35*(5*(c^2*d*x^2 + d)^(5/2)*x^2/(c^2*d) - 2*(c^2*d*x^2 + d)^(5/2)/(c^4*d ))*b*arcsinh(c*x) + 1/35*(5*(c^2*d*x^2 + d)^(5/2)*x^2/(c^2*d) - 2*(c^2*d*x ^2 + d)^(5/2)/(c^4*d))*a - 1/3675*(75*c^6*d^(3/2)*x^7 + 168*c^4*d^(3/2)*x^ 5 + 35*c^2*d^(3/2)*x^3 - 210*d^(3/2)*x)*b/c^3
Exception generated. \[ \int x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \]
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. \[ \int x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int x^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \]