Integrand size = 26, antiderivative size = 130 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {x (a+b \text {arcsinh}(c x))}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{2 b c^3 d \sqrt {d+c^2 d x^2}}+\frac {b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{2 c^3 d \sqrt {d+c^2 d x^2}} \]
-x*(a+b*arcsinh(c*x))/c^2/d/(c^2*d*x^2+d)^(1/2)+1/2*(a+b*arcsinh(c*x))^2*( c^2*x^2+1)^(1/2)/b/c^3/d/(c^2*d*x^2+d)^(1/2)+1/2*b*ln(c^2*x^2+1)*(c^2*x^2+ 1)^(1/2)/c^3/d/(c^2*d*x^2+d)^(1/2)
Time = 0.33 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.12 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {a x \sqrt {d \left (1+c^2 x^2\right )}}{c^2 d^2 \left (1+c^2 x^2\right )}+\frac {b \left (-2 c x \text {arcsinh}(c x)+\sqrt {1+c^2 x^2} \left (\text {arcsinh}(c x)^2+2 \log \left (\sqrt {1+c^2 x^2}\right )\right )\right )}{2 c^3 d \sqrt {d \left (1+c^2 x^2\right )}}+\frac {a \log \left (c d x+\sqrt {d} \sqrt {d \left (1+c^2 x^2\right )}\right )}{c^3 d^{3/2}} \]
-((a*x*Sqrt[d*(1 + c^2*x^2)])/(c^2*d^2*(1 + c^2*x^2))) + (b*(-2*c*x*ArcSin h[c*x] + Sqrt[1 + c^2*x^2]*(ArcSinh[c*x]^2 + 2*Log[Sqrt[1 + c^2*x^2]])))/( 2*c^3*d*Sqrt[d*(1 + c^2*x^2)]) + (a*Log[c*d*x + Sqrt[d]*Sqrt[d*(1 + c^2*x^ 2)]])/(c^3*d^(3/2))
Time = 0.44 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {6225, 240, 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.
| \(\displaystyle \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (c^2 d x^2+d\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6225 |
| \(\displaystyle \frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 d x^2+d}}dx}{c^2 d}+\frac {b \sqrt {c^2 x^2+1} \int \frac {x}{c^2 x^2+1}dx}{c d \sqrt {c^2 d x^2+d}}-\frac {x (a+b \text {arcsinh}(c x))}{c^2 d \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 d x^2+d}}dx}{c^2 d}-\frac {x (a+b \text {arcsinh}(c x))}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {b \sqrt {c^2 x^2+1} \log \left (c^2 x^2+1\right )}{2 c^3 d \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle -\frac {x (a+b \text {arcsinh}(c x))}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{2 b c^3 d \sqrt {c^2 d x^2+d}}+\frac {b \sqrt {c^2 x^2+1} \log \left (c^2 x^2+1\right )}{2 c^3 d \sqrt {c^2 d x^2+d}}\) |
-((x*(a + b*ArcSinh[c*x]))/(c^2*d*Sqrt[d + c^2*d*x^2])) + (Sqrt[1 + c^2*x^ 2]*(a + b*ArcSinh[c*x])^2)/(2*b*c^3*d*Sqrt[d + c^2*d*x^2]) + (b*Sqrt[1 + c ^2*x^2]*Log[1 + c^2*x^2])/(2*c^3*d*Sqrt[d + c^2*d*x^2])
3.2.58.3.1 Defintions of rubi rules used
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
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]
Time = 0.20 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.78
| method | result | size |
| default | \(-\frac {a x}{c^{2} d \sqrt {c^{2} d \,x^{2}+d}}+\frac {a \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{c^{2} d \sqrt {c^{2} d}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2}}{2 \sqrt {c^{2} x^{2}+1}\, c^{3} d^{2}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{3} d^{2}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) x}{c^{2} d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, c^{3} d^{2}}\) | \(232\) |
| parts | \(-\frac {a x}{c^{2} d \sqrt {c^{2} d \,x^{2}+d}}+\frac {a \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{c^{2} d \sqrt {c^{2} d}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2}}{2 \sqrt {c^{2} x^{2}+1}\, c^{3} d^{2}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{3} d^{2}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) x}{c^{2} d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, c^{3} d^{2}}\) | \(232\) |
-a*x/c^2/d/(c^2*d*x^2+d)^(1/2)+a/c^2/d*ln(c^2*d*x/(c^2*d)^(1/2)+(c^2*d*x^2 +d)^(1/2))/(c^2*d)^(1/2)+1/2*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^3 /d^2*arcsinh(c*x)^2-b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^3/d^2*arcs inh(c*x)-b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)/c^2/d^2/(c^2*x^2+1)*x+b*(d*( c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^3/d^2*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2 )
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
integral(sqrt(c^2*d*x^2 + d)*(b*x^2*arcsinh(c*x) + a*x^2)/(c^4*d^2*x^4 + 2 *c^2*d^2*x^2 + d^2), x)
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^{2} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
-a*(x/(sqrt(c^2*d*x^2 + d)*c^2*d) - arcsinh(c*x)/(c^3*d^(3/2))) + b*integr ate(x^2*log(c*x + sqrt(c^2*x^2 + 1))/(c^2*d*x^2 + d)^(3/2), x)
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \]