Integrand size = 26, antiderivative size = 192 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\frac {3 b x^2 \sqrt {1+c^2 x^2}}{16 c^3 \sqrt {d+c^2 d x^2}}-\frac {b x^4 \sqrt {1+c^2 x^2}}{16 c \sqrt {d+c^2 d x^2}}-\frac {3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 c^4 d}+\frac {x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{4 c^2 d}+\frac {3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{16 b c^5 \sqrt {d+c^2 d x^2}} \]
3/16*b*x^2*(c^2*x^2+1)^(1/2)/c^3/(c^2*d*x^2+d)^(1/2)-1/16*b*x^4*(c^2*x^2+1 )^(1/2)/c/(c^2*d*x^2+d)^(1/2)+3/16*(a+b*arcsinh(c*x))^2*(c^2*x^2+1)^(1/2)/ b/c^5/(c^2*d*x^2+d)^(1/2)-3/8*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^4 /d+1/4*x^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^2/d
Time = 0.69 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.79 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\frac {\frac {16 a c x \left (-3+2 c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{d}+\frac {48 a \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )}{\sqrt {d}}+\frac {b \sqrt {1+c^2 x^2} (16 \cosh (2 \text {arcsinh}(c x))-\cosh (4 \text {arcsinh}(c x))+4 \text {arcsinh}(c x) (6 \text {arcsinh}(c x)-8 \sinh (2 \text {arcsinh}(c x))+\sinh (4 \text {arcsinh}(c x))))}{\sqrt {d+c^2 d x^2}}}{128 c^5} \]
((16*a*c*x*(-3 + 2*c^2*x^2)*Sqrt[d + c^2*d*x^2])/d + (48*a*Log[c*d*x + Sqr t[d]*Sqrt[d + c^2*d*x^2]])/Sqrt[d] + (b*Sqrt[1 + c^2*x^2]*(16*Cosh[2*ArcSi nh[c*x]] - Cosh[4*ArcSinh[c*x]] + 4*ArcSinh[c*x]*(6*ArcSinh[c*x] - 8*Sinh[ 2*ArcSinh[c*x]] + Sinh[4*ArcSinh[c*x]])))/Sqrt[d + c^2*d*x^2])/(128*c^5)
Time = 0.65 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {6227, 15, 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.
| \(\displaystyle \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}} \, dx\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle -\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}}dx}{4 c^2}-\frac {b \sqrt {c^2 x^2+1} \int x^3dx}{4 c \sqrt {c^2 d x^2+d}}+\frac {x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{4 c^2 d}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}}dx}{4 c^2}+\frac {x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{4 c^2 d}-\frac {b x^4 \sqrt {c^2 x^2+1}}{16 c \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle -\frac {3 \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 d x^2+d}}dx}{2 c^2}-\frac {b \sqrt {c^2 x^2+1} \int xdx}{2 c \sqrt {c^2 d x^2+d}}+\frac {x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{2 c^2 d}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{4 c^2 d}-\frac {b x^4 \sqrt {c^2 x^2+1}}{16 c \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {3 \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 d x^2+d}}dx}{2 c^2}+\frac {x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{2 c^2 d}-\frac {b x^2 \sqrt {c^2 x^2+1}}{4 c \sqrt {c^2 d x^2+d}}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{4 c^2 d}-\frac {b x^4 \sqrt {c^2 x^2+1}}{16 c \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{4 c^2 d}-\frac {3 \left (\frac {x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{2 c^2 d}-\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{4 b c^3 \sqrt {c^2 d x^2+d}}-\frac {b x^2 \sqrt {c^2 x^2+1}}{4 c \sqrt {c^2 d x^2+d}}\right )}{4 c^2}-\frac {b x^4 \sqrt {c^2 x^2+1}}{16 c \sqrt {c^2 d x^2+d}}\) |
-1/16*(b*x^4*Sqrt[1 + c^2*x^2])/(c*Sqrt[d + c^2*d*x^2]) + (x^3*Sqrt[d + c^ 2*d*x^2]*(a + b*ArcSinh[c*x]))/(4*c^2*d) - (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])))/(4*c^2)
3.2.46.3.1 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_.)/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/(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. \(518\) vs. \(2(166)=332\).
Time = 0.20 (sec) , antiderivative size = 519, normalized size of antiderivative = 2.70
| method | result | size |
| default | \(\frac {a \,x^{3} \sqrt {c^{2} d \,x^{2}+d}}{4 c^{2} d}-\frac {3 a x \sqrt {c^{2} d \,x^{2}+d}}{8 c^{4} d}+\frac {3 a \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 c^{4} \sqrt {c^{2} d}}+b \left (\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2}}{16 \sqrt {c^{2} x^{2}+1}\, c^{5} d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}+8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}+8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+4 \,\operatorname {arcsinh}\left (c x \right )\right )}{256 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}+2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}-2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}-8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}-8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (1+4 \,\operatorname {arcsinh}\left (c x \right )\right )}{256 c^{5} d \left (c^{2} x^{2}+1\right )}\right )\) | \(519\) |
| parts | \(\frac {a \,x^{3} \sqrt {c^{2} d \,x^{2}+d}}{4 c^{2} d}-\frac {3 a x \sqrt {c^{2} d \,x^{2}+d}}{8 c^{4} d}+\frac {3 a \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 c^{4} \sqrt {c^{2} d}}+b \left (\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2}}{16 \sqrt {c^{2} x^{2}+1}\, c^{5} d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}+8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}+8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+4 \,\operatorname {arcsinh}\left (c x \right )\right )}{256 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}+2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}-2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}-8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}-8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (1+4 \,\operatorname {arcsinh}\left (c x \right )\right )}{256 c^{5} d \left (c^{2} x^{2}+1\right )}\right )\) | \(519\) |
1/4*a*x^3/c^2/d*(c^2*d*x^2+d)^(1/2)-3/8*a/c^4*x/d*(c^2*d*x^2+d)^(1/2)+3/8* a/c^4*ln(c^2*d*x/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+b*(3/16* (d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^5/d*arcsinh(c*x)^2+1/256*(d*(c^2 *x^2+1))^(1/2)*(8*c^5*x^5+8*c^4*x^4*(c^2*x^2+1)^(1/2)+12*c^3*x^3+8*c^2*x^2 *(c^2*x^2+1)^(1/2)+4*c*x+(c^2*x^2+1)^(1/2))*(-1+4*arcsinh(c*x))/c^5/d/(c^2 *x^2+1)-1/16*(d*(c^2*x^2+1))^(1/2)*(2*c^3*x^3+2*c^2*x^2*(c^2*x^2+1)^(1/2)+ 2*c*x+(c^2*x^2+1)^(1/2))*(-1+2*arcsinh(c*x))/c^5/d/(c^2*x^2+1)-1/16*(d*(c^ 2*x^2+1))^(1/2)*(2*c^3*x^3-2*c^2*x^2*(c^2*x^2+1)^(1/2)+2*c*x-(c^2*x^2+1)^( 1/2))*(1+2*arcsinh(c*x))/c^5/d/(c^2*x^2+1)+1/256*(d*(c^2*x^2+1))^(1/2)*(8* c^5*x^5-8*c^4*x^4*(c^2*x^2+1)^(1/2)+12*c^3*x^3-8*c^2*x^2*(c^2*x^2+1)^(1/2) +4*c*x-(c^2*x^2+1)^(1/2))*(1+4*arcsinh(c*x))/c^5/d/(c^2*x^2+1))
\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{\sqrt {c^{2} d x^{2} + d}} \,d x } \]
\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\int \frac {x^{4} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \]
Exception generated. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{\sqrt {c^{2} d x^{2} + d}} \,d x } \]
Timed out. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{\sqrt {d\,c^2\,x^2+d}} \,d x \]