Integrand size = 26, antiderivative size = 281 \[ \int \frac {x^6 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=-\frac {b}{6 c^7 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2}}{4 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {x^5 (a+b \text {arcsinh}(c x))}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {5 x^3 (a+b \text {arcsinh}(c x))}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {5 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 c^6 d^3}-\frac {5 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{4 b c^7 d^2 \sqrt {d+c^2 d x^2}}-\frac {7 b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{6 c^7 d^2 \sqrt {d+c^2 d x^2}} \]
-1/3*x^5*(a+b*arcsinh(c*x))/c^2/d/(c^2*d*x^2+d)^(3/2)-5/3*x^3*(a+b*arcsinh (c*x))/c^4/d^2/(c^2*d*x^2+d)^(1/2)-1/6*b/c^7/d^2/(c^2*x^2+1)^(1/2)/(c^2*d* x^2+d)^(1/2)-1/4*b*x^2*(c^2*x^2+1)^(1/2)/c^5/d^2/(c^2*d*x^2+d)^(1/2)-5/4*( a+b*arcsinh(c*x))^2*(c^2*x^2+1)^(1/2)/b/c^7/d^2/(c^2*d*x^2+d)^(1/2)-7/6*b* ln(c^2*x^2+1)*(c^2*x^2+1)^(1/2)/c^7/d^2/(c^2*d*x^2+d)^(1/2)+5/2*x*(a+b*arc sinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^6/d^3
Time = 0.94 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.79 \[ \int \frac {x^6 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {4 a c d x \left (15+20 c^2 x^2+3 c^4 x^4\right )+b d \left (4 c x \left (15+20 c^2 x^2+3 c^4 x^4\right ) \text {arcsinh}(c x)-30 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)^2-\sqrt {1+c^2 x^2} \left (7+9 c^2 x^2+6 c^4 x^4+28 \left (1+c^2 x^2\right ) \log \left (1+c^2 x^2\right )\right )\right )-60 a \sqrt {d} \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )}{24 c^7 d^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}} \]
(4*a*c*d*x*(15 + 20*c^2*x^2 + 3*c^4*x^4) + b*d*(4*c*x*(15 + 20*c^2*x^2 + 3 *c^4*x^4)*ArcSinh[c*x] - 30*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]^2 - Sqrt[1 + c^2*x^2]*(7 + 9*c^2*x^2 + 6*c^4*x^4 + 28*(1 + c^2*x^2)*Log[1 + c^2*x^2])) - 60*a*Sqrt[d]*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[ d + c^2*d*x^2]])/(24*c^7*d^3*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])
Time = 1.20 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.23, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {6225, 243, 49, 2009, 6225, 243, 49, 2009, 6227, 15, 6198}
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.
| \(\displaystyle \int \frac {x^6 (a+b \text {arcsinh}(c x))}{\left (c^2 d x^2+d\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6225 |
| \(\displaystyle \frac {5 \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (c^2 d x^2+d\right )^{3/2}}dx}{3 c^2 d}+\frac {b \sqrt {c^2 x^2+1} \int \frac {x^5}{\left (c^2 x^2+1\right )^2}dx}{3 c d^2 \sqrt {c^2 d x^2+d}}-\frac {x^5 (a+b \text {arcsinh}(c x))}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {5 \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (c^2 d x^2+d\right )^{3/2}}dx}{3 c^2 d}+\frac {b \sqrt {c^2 x^2+1} \int \frac {x^4}{\left (c^2 x^2+1\right )^2}dx^2}{6 c d^2 \sqrt {c^2 d x^2+d}}-\frac {x^5 (a+b \text {arcsinh}(c x))}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {5 \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (c^2 d x^2+d\right )^{3/2}}dx}{3 c^2 d}+\frac {b \sqrt {c^2 x^2+1} \int \left (\frac {1}{c^4}-\frac {2}{c^4 \left (c^2 x^2+1\right )}+\frac {1}{c^4 \left (c^2 x^2+1\right )^2}\right )dx^2}{6 c d^2 \sqrt {c^2 d x^2+d}}-\frac {x^5 (a+b \text {arcsinh}(c x))}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5 \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (c^2 d x^2+d\right )^{3/2}}dx}{3 c^2 d}-\frac {x^5 (a+b \text {arcsinh}(c x))}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {b \sqrt {c^2 x^2+1} \left (\frac {x^2}{c^4}-\frac {1}{c^6 \left (c^2 x^2+1\right )}-\frac {2 \log \left (c^2 x^2+1\right )}{c^6}\right )}{6 c d^2 \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 6225 |
| \(\displaystyle \frac {5 \left (\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}}dx}{c^2 d}+\frac {b \sqrt {c^2 x^2+1} \int \frac {x^3}{c^2 x^2+1}dx}{c d \sqrt {c^2 d x^2+d}}-\frac {x^3 (a+b \text {arcsinh}(c x))}{c^2 d \sqrt {c^2 d x^2+d}}\right )}{3 c^2 d}-\frac {x^5 (a+b \text {arcsinh}(c x))}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {b \sqrt {c^2 x^2+1} \left (\frac {x^2}{c^4}-\frac {1}{c^6 \left (c^2 x^2+1\right )}-\frac {2 \log \left (c^2 x^2+1\right )}{c^6}\right )}{6 c d^2 \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {5 \left (\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}}dx}{c^2 d}+\frac {b \sqrt {c^2 x^2+1} \int \frac {x^2}{c^2 x^2+1}dx^2}{2 c d \sqrt {c^2 d x^2+d}}-\frac {x^3 (a+b \text {arcsinh}(c x))}{c^2 d \sqrt {c^2 d x^2+d}}\right )}{3 c^2 d}-\frac {x^5 (a+b \text {arcsinh}(c x))}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {b \sqrt {c^2 x^2+1} \left (\frac {x^2}{c^4}-\frac {1}{c^6 \left (c^2 x^2+1\right )}-\frac {2 \log \left (c^2 x^2+1\right )}{c^6}\right )}{6 c d^2 \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {5 \left (\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}}dx}{c^2 d}+\frac {b \sqrt {c^2 x^2+1} \int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (c^2 x^2+1\right )}\right )dx^2}{2 c d \sqrt {c^2 d x^2+d}}-\frac {x^3 (a+b \text {arcsinh}(c x))}{c^2 d \sqrt {c^2 d x^2+d}}\right )}{3 c^2 d}-\frac {x^5 (a+b \text {arcsinh}(c x))}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {b \sqrt {c^2 x^2+1} \left (\frac {x^2}{c^4}-\frac {1}{c^6 \left (c^2 x^2+1\right )}-\frac {2 \log \left (c^2 x^2+1\right )}{c^6}\right )}{6 c d^2 \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5 \left (\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}}dx}{c^2 d}-\frac {x^3 (a+b \text {arcsinh}(c x))}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {b \sqrt {c^2 x^2+1} \left (\frac {x^2}{c^2}-\frac {\log \left (c^2 x^2+1\right )}{c^4}\right )}{2 c d \sqrt {c^2 d x^2+d}}\right )}{3 c^2 d}-\frac {x^5 (a+b \text {arcsinh}(c x))}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {b \sqrt {c^2 x^2+1} \left (\frac {x^2}{c^4}-\frac {1}{c^6 \left (c^2 x^2+1\right )}-\frac {2 \log \left (c^2 x^2+1\right )}{c^6}\right )}{6 c d^2 \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 d x^2+d}}dx}{2 c^2}-\frac {b \sqrt {c^2 x^2+1} \int xdx}{2 c \sqrt {c^2 d x^2+d}}+\frac {x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{2 c^2 d}\right )}{c^2 d}-\frac {x^3 (a+b \text {arcsinh}(c x))}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {b \sqrt {c^2 x^2+1} \left (\frac {x^2}{c^2}-\frac {\log \left (c^2 x^2+1\right )}{c^4}\right )}{2 c d \sqrt {c^2 d x^2+d}}\right )}{3 c^2 d}-\frac {x^5 (a+b \text {arcsinh}(c x))}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {b \sqrt {c^2 x^2+1} \left (\frac {x^2}{c^4}-\frac {1}{c^6 \left (c^2 x^2+1\right )}-\frac {2 \log \left (c^2 x^2+1\right )}{c^6}\right )}{6 c d^2 \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 d x^2+d}}dx}{2 c^2}+\frac {x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{2 c^2 d}-\frac {b x^2 \sqrt {c^2 x^2+1}}{4 c \sqrt {c^2 d x^2+d}}\right )}{c^2 d}-\frac {x^3 (a+b \text {arcsinh}(c x))}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {b \sqrt {c^2 x^2+1} \left (\frac {x^2}{c^2}-\frac {\log \left (c^2 x^2+1\right )}{c^4}\right )}{2 c d \sqrt {c^2 d x^2+d}}\right )}{3 c^2 d}-\frac {x^5 (a+b \text {arcsinh}(c x))}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {b \sqrt {c^2 x^2+1} \left (\frac {x^2}{c^4}-\frac {1}{c^6 \left (c^2 x^2+1\right )}-\frac {2 \log \left (c^2 x^2+1\right )}{c^6}\right )}{6 c d^2 \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle -\frac {x^5 (a+b \text {arcsinh}(c x))}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {5 \left (-\frac {x^3 (a+b \text {arcsinh}(c x))}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {3 \left (\frac {x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{2 c^2 d}-\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{4 b c^3 \sqrt {c^2 d x^2+d}}-\frac {b x^2 \sqrt {c^2 x^2+1}}{4 c \sqrt {c^2 d x^2+d}}\right )}{c^2 d}+\frac {b \sqrt {c^2 x^2+1} \left (\frac {x^2}{c^2}-\frac {\log \left (c^2 x^2+1\right )}{c^4}\right )}{2 c d \sqrt {c^2 d x^2+d}}\right )}{3 c^2 d}+\frac {b \sqrt {c^2 x^2+1} \left (\frac {x^2}{c^4}-\frac {1}{c^6 \left (c^2 x^2+1\right )}-\frac {2 \log \left (c^2 x^2+1\right )}{c^6}\right )}{6 c d^2 \sqrt {c^2 d x^2+d}}\) |
-1/3*(x^5*(a + b*ArcSinh[c*x]))/(c^2*d*(d + c^2*d*x^2)^(3/2)) + (b*Sqrt[1 + c^2*x^2]*(x^2/c^4 - 1/(c^6*(1 + c^2*x^2)) - (2*Log[1 + c^2*x^2])/c^6))/( 6*c*d^2*Sqrt[d + c^2*d*x^2]) + (5*(-((x^3*(a + b*ArcSinh[c*x]))/(c^2*d*Sqr t[d + c^2*d*x^2])) + (3*(-1/4*(b*x^2*Sqrt[1 + c^2*x^2])/(c*Sqrt[d + c^2*d* x^2]) + (x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(2*c^2*d) - (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*c^3*Sqrt[d + c^2*d*x^2])))/(c^2*d) + (b*Sqrt[1 + c^2*x^2]*(x^2/c^2 - Log[1 + c^2*x^2]/c^4))/(2*c*d*Sqrt[d + c ^2*d*x^2])))/(3*c^2*d)
3.2.65.3.1 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_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c ^2*d] && NeQ[n, -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/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] - S imp[b*f*(n/(2*c*(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]) /; Fre eQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && IG tQ[m, 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]
Time = 0.25 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.46
| method | result | size |
| default | \(\frac {a \,x^{5}}{2 c^{2} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a \,x^{3}}{6 c^{4} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a x}{2 c^{6} d^{2} \sqrt {c^{2} d \,x^{2}+d}}-\frac {5 a \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 c^{6} d^{2} \sqrt {c^{2} d}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} x^{2}+1}\, \left (-12 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+6 c^{6} x^{6}+30 \operatorname {arcsinh}\left (c x \right )^{2} x^{4} c^{4}-56 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}+56 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{4} c^{4}-80 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+15 c^{4} x^{4}+60 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-112 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+112 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-60 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+16 c^{2} x^{2}+30 \operatorname {arcsinh}\left (c x \right )^{2}-56 \,\operatorname {arcsinh}\left (c x \right )+56 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+7\right )}{24 \left (c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) c^{7} d^{3}}\) | \(410\) |
| parts | \(\frac {a \,x^{5}}{2 c^{2} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a \,x^{3}}{6 c^{4} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a x}{2 c^{6} d^{2} \sqrt {c^{2} d \,x^{2}+d}}-\frac {5 a \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 c^{6} d^{2} \sqrt {c^{2} d}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} x^{2}+1}\, \left (-12 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+6 c^{6} x^{6}+30 \operatorname {arcsinh}\left (c x \right )^{2} x^{4} c^{4}-56 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}+56 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{4} c^{4}-80 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+15 c^{4} x^{4}+60 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-112 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+112 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-60 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+16 c^{2} x^{2}+30 \operatorname {arcsinh}\left (c x \right )^{2}-56 \,\operatorname {arcsinh}\left (c x \right )+56 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+7\right )}{24 \left (c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) c^{7} d^{3}}\) | \(410\) |
1/2*a*x^5/c^2/d/(c^2*d*x^2+d)^(3/2)+5/6*a/c^4*x^3/d/(c^2*d*x^2+d)^(3/2)+5/ 2*a/c^6/d^2*x/(c^2*d*x^2+d)^(1/2)-5/2*a/c^6/d^2*ln(c^2*d*x/(c^2*d)^(1/2)+( c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)-1/24*b*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+1) ^(1/2)*(-12*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x^5*c^5+6*c^6*x^6+30*arcsinh(c* x)^2*x^4*c^4-56*arcsinh(c*x)*c^4*x^4+56*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)*x^ 4*c^4-80*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x^3*c^3+15*c^4*x^4+60*arcsinh(c*x) ^2*x^2*c^2-112*arcsinh(c*x)*c^2*x^2+112*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)*x^ 2*c^2-60*arcsinh(c*x)*c*x*(c^2*x^2+1)^(1/2)+16*c^2*x^2+30*arcsinh(c*x)^2-5 6*arcsinh(c*x)+56*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)+7)/(c^6*x^6+3*c^4*x^4+3* c^2*x^2+1)/c^7/d^3
\[ \int \frac {x^6 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{6}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
integral((b*x^6*arcsinh(c*x) + a*x^6)*sqrt(c^2*d*x^2 + d)/(c^6*d^3*x^6 + 3 *c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x)
\[ \int \frac {x^6 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^{6} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {x^6 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{6}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
1/6*a*(3*x^5/((c^2*d*x^2 + d)^(3/2)*c^2*d) + 5*x*(3*x^2/((c^2*d*x^2 + d)^( 3/2)*c^2*d) + 2/((c^2*d*x^2 + d)^(3/2)*c^4*d))/c^2 + 5*x/(sqrt(c^2*d*x^2 + d)*c^6*d^2) - 15*arcsinh(c*x)/(c^7*d^(5/2))) + b*integrate(x^6*log(c*x + sqrt(c^2*x^2 + 1))/(c^2*d*x^2 + d)^(5/2), x)
Exception generated. \[ \int \frac {x^6 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {x^6 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^6\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \]