Integrand size = 26, antiderivative size = 119 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=-\frac {b}{6 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {x^3 (a+b \text {arcsinh}(c x))}{3 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{6 c^3 d^2 \sqrt {d+c^2 d x^2}} \]
1/3*x^3*(a+b*arcsinh(c*x))/d/(c^2*d*x^2+d)^(3/2)-1/6*b/c^3/d^2/(c^2*x^2+1) ^(1/2)/(c^2*d*x^2+d)^(1/2)-1/6*b*ln(c^2*x^2+1)*(c^2*x^2+1)^(1/2)/c^3/d^2/( c^2*d*x^2+d)^(1/2)
Time = 0.28 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.99 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=-\frac {\sqrt {d+c^2 d x^2} \left (b+b c^2 x^2-2 a c^3 x^3 \sqrt {1+c^2 x^2}-2 b c^3 x^3 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+b \left (1+c^2 x^2\right )^2 \log \left (1+c^2 x^2\right )\right )}{6 c^3 d^3 \left (1+c^2 x^2\right )^{5/2}} \]
-1/6*(Sqrt[d + c^2*d*x^2]*(b + b*c^2*x^2 - 2*a*c^3*x^3*Sqrt[1 + c^2*x^2] - 2*b*c^3*x^3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + b*(1 + c^2*x^2)^2*Log[1 + c^ 2*x^2]))/(c^3*d^3*(1 + c^2*x^2)^(5/2))
Time = 0.36 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.83, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6215, 243, 49, 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.
| \(\displaystyle \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (c^2 d x^2+d\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6215 |
| \(\displaystyle \frac {x^3 (a+b \text {arcsinh}(c x))}{3 d \left (c^2 d x^2+d\right )^{3/2}}-\frac {b c \sqrt {c^2 x^2+1} \int \frac {x^3}{\left (c^2 x^2+1\right )^2}dx}{3 d^2 \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {x^3 (a+b \text {arcsinh}(c x))}{3 d \left (c^2 d x^2+d\right )^{3/2}}-\frac {b c \sqrt {c^2 x^2+1} \int \frac {x^2}{\left (c^2 x^2+1\right )^2}dx^2}{6 d^2 \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {x^3 (a+b \text {arcsinh}(c x))}{3 d \left (c^2 d x^2+d\right )^{3/2}}-\frac {b c \sqrt {c^2 x^2+1} \int \left (\frac {1}{c^2 \left (c^2 x^2+1\right )}-\frac {1}{c^2 \left (c^2 x^2+1\right )^2}\right )dx^2}{6 d^2 \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^3 (a+b \text {arcsinh}(c x))}{3 d \left (c^2 d x^2+d\right )^{3/2}}-\frac {b c \sqrt {c^2 x^2+1} \left (\frac {1}{c^4 \left (c^2 x^2+1\right )}+\frac {\log \left (c^2 x^2+1\right )}{c^4}\right )}{6 d^2 \sqrt {c^2 d x^2+d}}\) |
(x^3*(a + b*ArcSinh[c*x]))/(3*d*(d + c^2*d*x^2)^(3/2)) - (b*c*Sqrt[1 + c^2 *x^2]*(1/(c^4*(1 + c^2*x^2)) + Log[1 + c^2*x^2]/c^4))/(6*d^2*Sqrt[d + c^2* d*x^2])
3.2.69.3.1 Defintions of rubi rules used
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_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] - Simp[b*c*(n/(f*(m + 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, m, p}, x] && EqQ [e, c^2*d] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(412\) vs. \(2(103)=206\).
Time = 0.26 (sec) , antiderivative size = 413, normalized size of antiderivative = 3.47
| method | result | size |
| default | \(a \left (-\frac {x}{2 c^{2} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {\frac {x}{3 d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {c^{2} d \,x^{2}+d}}}{2 c^{2}}\right )+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{3} x^{3}+c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+\sqrt {c^{2} x^{2}+1}\right ) \left (-2 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{6} c^{6}+2 \sqrt {c^{2} x^{2}+1}\, \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{5} c^{5}+6 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}-6 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{4} c^{4}+2 \sqrt {c^{2} x^{2}+1}\, \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{3} c^{3}-c^{4} x^{4}+c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+6 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}-6 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-2 c^{2} x^{2}+2 \,\operatorname {arcsinh}\left (c x \right )-2 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-1\right )}{6 \left (3 c^{8} x^{8}+9 c^{6} x^{6}+10 c^{4} x^{4}+5 c^{2} x^{2}+1\right ) c^{3} d^{3}}\) | \(413\) |
| parts | \(a \left (-\frac {x}{2 c^{2} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {\frac {x}{3 d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {c^{2} d \,x^{2}+d}}}{2 c^{2}}\right )+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{3} x^{3}+c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+\sqrt {c^{2} x^{2}+1}\right ) \left (-2 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{6} c^{6}+2 \sqrt {c^{2} x^{2}+1}\, \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{5} c^{5}+6 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}-6 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{4} c^{4}+2 \sqrt {c^{2} x^{2}+1}\, \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{3} c^{3}-c^{4} x^{4}+c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+6 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}-6 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-2 c^{2} x^{2}+2 \,\operatorname {arcsinh}\left (c x \right )-2 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-1\right )}{6 \left (3 c^{8} x^{8}+9 c^{6} x^{6}+10 c^{4} x^{4}+5 c^{2} x^{2}+1\right ) c^{3} d^{3}}\) | \(413\) |
a*(-1/2*x/c^2/d/(c^2*d*x^2+d)^(3/2)+1/2/c^2*(1/3/d*x/(c^2*d*x^2+d)^(3/2)+2 /3/d^2*x/(c^2*d*x^2+d)^(1/2)))+1/6*b*(d*(c^2*x^2+1))^(1/2)*(c^3*x^3+c^2*x^ 2*(c^2*x^2+1)^(1/2)+(c^2*x^2+1)^(1/2))*(-2*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2) *x^6*c^6+2*(c^2*x^2+1)^(1/2)*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)*x^5*c^5+6*arc sinh(c*x)*c^4*x^4-6*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)*x^4*c^4+2*(c^2*x^2+1)^ (1/2)*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)*x^3*c^3-c^4*x^4+c^3*x^3*(c^2*x^2+1)^ (1/2)+6*arcsinh(c*x)*c^2*x^2-6*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)*x^2*c^2-2*c ^2*x^2+2*arcsinh(c*x)-2*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)-1)/(3*c^8*x^8+9*c^ 6*x^6+10*c^4*x^4+5*c^2*x^2+1)/c^3/d^3
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
integral(sqrt(c^2*d*x^2 + d)*(b*x^2*arcsinh(c*x) + a*x^2)/(c^6*d^3*x^6 + 3 *c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x)
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^{2} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Time = 0.20 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.15 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=-\frac {1}{6} \, b c {\left (\frac {1}{c^{6} d^{\frac {5}{2}} x^{2} + c^{4} d^{\frac {5}{2}}} + \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4} d^{\frac {5}{2}}}\right )} + \frac {1}{3} \, b {\left (\frac {x}{\sqrt {c^{2} d x^{2} + d} c^{2} d^{2}} - \frac {x}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d}\right )} \operatorname {arsinh}\left (c x\right ) + \frac {1}{3} \, a {\left (\frac {x}{\sqrt {c^{2} d x^{2} + d} c^{2} d^{2}} - \frac {x}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d}\right )} \]
-1/6*b*c*(1/(c^6*d^(5/2)*x^2 + c^4*d^(5/2)) + log(c^2*x^2 + 1)/(c^4*d^(5/2 ))) + 1/3*b*(x/(sqrt(c^2*d*x^2 + d)*c^2*d^2) - x/((c^2*d*x^2 + d)^(3/2)*c^ 2*d))*arcsinh(c*x) + 1/3*a*(x/(sqrt(c^2*d*x^2 + d)*c^2*d^2) - x/((c^2*d*x^ 2 + d)^(3/2)*c^2*d))
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \]