Integrand size = 26, antiderivative size = 215 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=-\frac {8 b x \sqrt {1+c^2 x^2}}{15 c^5 \sqrt {d+c^2 d x^2}}+\frac {4 b x^3 \sqrt {1+c^2 x^2}}{45 c^3 \sqrt {d+c^2 d x^2}}-\frac {b x^5 \sqrt {1+c^2 x^2}}{25 c \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{15 c^6 d}-\frac {4 x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{15 c^4 d}+\frac {x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{5 c^2 d} \]
-8/15*b*x*(c^2*x^2+1)^(1/2)/c^5/(c^2*d*x^2+d)^(1/2)+4/45*b*x^3*(c^2*x^2+1) ^(1/2)/c^3/(c^2*d*x^2+d)^(1/2)-1/25*b*x^5*(c^2*x^2+1)^(1/2)/c/(c^2*d*x^2+d )^(1/2)+8/15*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^6/d-4/15*x^2*(a+b*ar csinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^4/d+1/5*x^4*(a+b*arcsinh(c*x))*(c^2*d*x^ 2+d)^(1/2)/c^2/d
Time = 0.17 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.55 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\frac {b c x \sqrt {1+c^2 x^2} \left (-120+20 c^2 x^2-9 c^4 x^4\right )+15 a \left (8+4 c^2 x^2-c^4 x^4+3 c^6 x^6\right )+15 b \left (8+4 c^2 x^2-c^4 x^4+3 c^6 x^6\right ) \text {arcsinh}(c x)}{225 c^6 \sqrt {d+c^2 d x^2}} \]
(b*c*x*Sqrt[1 + c^2*x^2]*(-120 + 20*c^2*x^2 - 9*c^4*x^4) + 15*a*(8 + 4*c^2 *x^2 - c^4*x^4 + 3*c^6*x^6) + 15*b*(8 + 4*c^2*x^2 - c^4*x^4 + 3*c^6*x^6)*A rcSinh[c*x])/(225*c^6*Sqrt[d + c^2*d*x^2])
Time = 0.67 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {6227, 15, 6227, 15, 6213, 24}
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.
| \(\displaystyle \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}} \, dx\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle -\frac {4 \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}}dx}{5 c^2}-\frac {b \sqrt {c^2 x^2+1} \int x^4dx}{5 c \sqrt {c^2 d x^2+d}}+\frac {x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{5 c^2 d}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {4 \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}}dx}{5 c^2}+\frac {x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{5 c^2 d}-\frac {b x^5 \sqrt {c^2 x^2+1}}{25 c \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle -\frac {4 \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}}dx}{3 c^2}-\frac {b \sqrt {c^2 x^2+1} \int x^2dx}{3 c \sqrt {c^2 d x^2+d}}+\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{3 c^2 d}\right )}{5 c^2}+\frac {x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{5 c^2 d}-\frac {b x^5 \sqrt {c^2 x^2+1}}{25 c \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {4 \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}}dx}{3 c^2}+\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{3 c^2 d}-\frac {b x^3 \sqrt {c^2 x^2+1}}{9 c \sqrt {c^2 d x^2+d}}\right )}{5 c^2}+\frac {x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{5 c^2 d}-\frac {b x^5 \sqrt {c^2 x^2+1}}{25 c \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle -\frac {4 \left (-\frac {2 \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{c^2 d}-\frac {b \sqrt {c^2 x^2+1} \int 1dx}{c \sqrt {c^2 d x^2+d}}\right )}{3 c^2}+\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{3 c^2 d}-\frac {b x^3 \sqrt {c^2 x^2+1}}{9 c \sqrt {c^2 d x^2+d}}\right )}{5 c^2}+\frac {x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{5 c^2 d}-\frac {b x^5 \sqrt {c^2 x^2+1}}{25 c \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{5 c^2 d}-\frac {4 \left (\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{3 c^2 d}-\frac {2 \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{c^2 d}-\frac {b x \sqrt {c^2 x^2+1}}{c \sqrt {c^2 d x^2+d}}\right )}{3 c^2}-\frac {b x^3 \sqrt {c^2 x^2+1}}{9 c \sqrt {c^2 d x^2+d}}\right )}{5 c^2}-\frac {b x^5 \sqrt {c^2 x^2+1}}{25 c \sqrt {c^2 d x^2+d}}\) |
-1/25*(b*x^5*Sqrt[1 + c^2*x^2])/(c*Sqrt[d + c^2*d*x^2]) + (x^4*Sqrt[d + c^ 2*d*x^2]*(a + b*ArcSinh[c*x]))/(5*c^2*d) - (4*(-1/9*(b*x^3*Sqrt[1 + c^2*x^ 2])/(c*Sqrt[d + c^2*d*x^2]) + (x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x] ))/(3*c^2*d) - (2*(-((b*x*Sqrt[1 + c^2*x^2])/(c*Sqrt[d + c^2*d*x^2])) + (S qrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(c^2*d)))/(3*c^2)))/(5*c^2)
3.2.45.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ {a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int [(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] ) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ m, 1] && NeQ[m + 2*p + 1, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(624\) vs. \(2(185)=370\).
Time = 0.22 (sec) , antiderivative size = 625, normalized size of antiderivative = 2.91
| method | result | size |
| default | \(a \left (\frac {x^{4} \sqrt {c^{2} d \,x^{2}+d}}{5 c^{2} d}-\frac {4 \left (\frac {x^{2} \sqrt {c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )}{5 c^{2}}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}+16 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+28 c^{4} x^{4}+20 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+13 c^{2} x^{2}+5 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+5 \,\operatorname {arcsinh}\left (c x \right )\right )}{800 c^{6} d \left (c^{2} x^{2}+1\right )}-\frac {5 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}+3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (c x \right )\right )}{288 c^{6} d \left (c^{2} x^{2}+1\right )}+\frac {5 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+\operatorname {arcsinh}\left (c x \right )\right )}{16 c^{6} d \left (c^{2} x^{2}+1\right )}+\frac {5 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (\operatorname {arcsinh}\left (c x \right )+1\right )}{16 c^{6} d \left (c^{2} x^{2}+1\right )}-\frac {5 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}-3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (3 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{288 c^{6} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}-16 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+28 c^{4} x^{4}-20 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+13 c^{2} x^{2}-5 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (1+5 \,\operatorname {arcsinh}\left (c x \right )\right )}{800 c^{6} d \left (c^{2} x^{2}+1\right )}\right )\) | \(625\) |
| parts | \(a \left (\frac {x^{4} \sqrt {c^{2} d \,x^{2}+d}}{5 c^{2} d}-\frac {4 \left (\frac {x^{2} \sqrt {c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )}{5 c^{2}}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}+16 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+28 c^{4} x^{4}+20 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+13 c^{2} x^{2}+5 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+5 \,\operatorname {arcsinh}\left (c x \right )\right )}{800 c^{6} d \left (c^{2} x^{2}+1\right )}-\frac {5 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}+3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (c x \right )\right )}{288 c^{6} d \left (c^{2} x^{2}+1\right )}+\frac {5 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+\operatorname {arcsinh}\left (c x \right )\right )}{16 c^{6} d \left (c^{2} x^{2}+1\right )}+\frac {5 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (\operatorname {arcsinh}\left (c x \right )+1\right )}{16 c^{6} d \left (c^{2} x^{2}+1\right )}-\frac {5 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}-3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (3 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{288 c^{6} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}-16 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+28 c^{4} x^{4}-20 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+13 c^{2} x^{2}-5 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (1+5 \,\operatorname {arcsinh}\left (c x \right )\right )}{800 c^{6} d \left (c^{2} x^{2}+1\right )}\right )\) | \(625\) |
a*(1/5*x^4/c^2/d*(c^2*d*x^2+d)^(1/2)-4/5/c^2*(1/3*x^2/c^2/d*(c^2*d*x^2+d)^ (1/2)-2/3/d/c^4*(c^2*d*x^2+d)^(1/2)))+b*(1/800*(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))/c^6/d/(c^2*x^2 +1)-5/288*(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))/c^6/d/(c^2*x^2+1)+5/ 16*(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 ))/c^6/d/(c^2*x^2+1)+5/16*(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)/c^6/d/(c^2*x^2+1)-5/288*(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)/c^6/d/(c^2*x^2+1)+1/800*(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))/c^6/d/(c^2*x^2+1))
Time = 0.27 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.75 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\frac {15 \, {\left (3 \, b c^{6} x^{6} - b c^{4} x^{4} + 4 \, b c^{2} x^{2} + 8 \, b\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (45 \, a c^{6} x^{6} - 15 \, a c^{4} x^{4} + 60 \, a c^{2} x^{2} - {\left (9 \, b c^{5} x^{5} - 20 \, b c^{3} x^{3} + 120 \, b c x\right )} \sqrt {c^{2} x^{2} + 1} + 120 \, a\right )} \sqrt {c^{2} d x^{2} + d}}{225 \, {\left (c^{8} d x^{2} + c^{6} d\right )}} \]
1/225*(15*(3*b*c^6*x^6 - b*c^4*x^4 + 4*b*c^2*x^2 + 8*b)*sqrt(c^2*d*x^2 + d )*log(c*x + sqrt(c^2*x^2 + 1)) + (45*a*c^6*x^6 - 15*a*c^4*x^4 + 60*a*c^2*x ^2 - (9*b*c^5*x^5 - 20*b*c^3*x^3 + 120*b*c*x)*sqrt(c^2*x^2 + 1) + 120*a)*s qrt(c^2*d*x^2 + d))/(c^8*d*x^2 + c^6*d)
\[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\int \frac {x^{5} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \]
Time = 0.21 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.81 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\frac {1}{15} \, {\left (\frac {3 \, \sqrt {c^{2} d x^{2} + d} x^{4}}{c^{2} d} - \frac {4 \, \sqrt {c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {c^{2} d x^{2} + d}}{c^{6} d}\right )} b \operatorname {arsinh}\left (c x\right ) + \frac {1}{15} \, {\left (\frac {3 \, \sqrt {c^{2} d x^{2} + d} x^{4}}{c^{2} d} - \frac {4 \, \sqrt {c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {c^{2} d x^{2} + d}}{c^{6} d}\right )} a - \frac {{\left (9 \, c^{4} x^{5} - 20 \, c^{2} x^{3} + 120 \, x\right )} b}{225 \, c^{5} \sqrt {d}} \]
1/15*(3*sqrt(c^2*d*x^2 + d)*x^4/(c^2*d) - 4*sqrt(c^2*d*x^2 + d)*x^2/(c^4*d ) + 8*sqrt(c^2*d*x^2 + d)/(c^6*d))*b*arcsinh(c*x) + 1/15*(3*sqrt(c^2*d*x^2 + d)*x^4/(c^2*d) - 4*sqrt(c^2*d*x^2 + d)*x^2/(c^4*d) + 8*sqrt(c^2*d*x^2 + d)/(c^6*d))*a - 1/225*(9*c^4*x^5 - 20*c^2*x^3 + 120*x)*b/(c^5*sqrt(d))
Exception generated. \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{\sqrt {d\,c^2\,x^2+d}} \,d x \]