Integrand size = 26, antiderivative size = 224 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {b c \sqrt {1+c^2 x^2}}{6 d x^2 \sqrt {d+c^2 d x^2}}+\frac {a+b \text {arcsinh}(c x)}{d x^3 \sqrt {d+c^2 d x^2}}-\frac {4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 d^2 x^3}+\frac {8 c^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 d^2 x}-\frac {5 b c^3 \sqrt {1+c^2 x^2} \log (x)}{3 d \sqrt {d+c^2 d x^2}}-\frac {b c^3 \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{2 d \sqrt {d+c^2 d x^2}} \] Output:
-1/6*b*c*(c^2*x^2+1)^(1/2)/d/x^2/(c^2*d*x^2+d)^(1/2)+(a+b*arcsinh(c*x))/d/ x^3/(c^2*d*x^2+d)^(1/2)-4/3*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/d^2/x^3 +8/3*c^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/d^2/x-5/3*b*c^3*(c^2*x^2+1 )^(1/2)*ln(x)/d/(c^2*d*x^2+d)^(1/2)-1/2*b*c^3*(c^2*x^2+1)^(1/2)*ln(c^2*x^2 +1)/d/(c^2*d*x^2+d)^(1/2)
Time = 0.30 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.96 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {\sqrt {d+c^2 d x^2} \left (-b c x-b c^3 x^3-2 a \sqrt {1+c^2 x^2}+8 a c^2 x^2 \sqrt {1+c^2 x^2}+16 a c^4 x^4 \sqrt {1+c^2 x^2}+2 b \sqrt {1+c^2 x^2} \left (-1+4 c^2 x^2+8 c^4 x^4\right ) \text {arcsinh}(c x)+5 b c^3 x^3 \left (1+c^2 x^2\right ) \log \left (1+\frac {1}{c^2 x^2}\right )-8 b c^3 x^3 \log \left (1+c^2 x^2\right )-8 b c^5 x^5 \log \left (1+c^2 x^2\right )\right )}{6 d^2 x^3 \left (1+c^2 x^2\right )^{3/2}} \] Input:
Integrate[(a + b*ArcSinh[c*x])/(x^4*(d + c^2*d*x^2)^(3/2)),x]
Output:
(Sqrt[d + c^2*d*x^2]*(-(b*c*x) - b*c^3*x^3 - 2*a*Sqrt[1 + c^2*x^2] + 8*a*c ^2*x^2*Sqrt[1 + c^2*x^2] + 16*a*c^4*x^4*Sqrt[1 + c^2*x^2] + 2*b*Sqrt[1 + c ^2*x^2]*(-1 + 4*c^2*x^2 + 8*c^4*x^4)*ArcSinh[c*x] + 5*b*c^3*x^3*(1 + c^2*x ^2)*Log[1 + 1/(c^2*x^2)] - 8*b*c^3*x^3*Log[1 + c^2*x^2] - 8*b*c^5*x^5*Log[ 1 + c^2*x^2]))/(6*d^2*x^3*(1 + c^2*x^2)^(3/2))
Time = 0.61 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.75, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {6219, 27, 1578, 1195, 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.
| \(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (c^2 d x^2+d\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6219 |
| \(\displaystyle -\frac {b c \sqrt {c^2 d x^2+d} \int -\frac {-8 c^4 x^4-4 c^2 x^2+1}{3 d^2 x^3 \left (c^2 x^2+1\right )}dx}{\sqrt {c^2 x^2+1}}+\frac {4 c^2 (a+b \text {arcsinh}(c x))}{3 d x \sqrt {c^2 d x^2+d}}-\frac {a+b \text {arcsinh}(c x)}{3 d x^3 \sqrt {c^2 d x^2+d}}+\frac {8 c^4 x (a+b \text {arcsinh}(c x))}{3 d \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c \sqrt {c^2 d x^2+d} \int \frac {-8 c^4 x^4-4 c^2 x^2+1}{x^3 \left (c^2 x^2+1\right )}dx}{3 d^2 \sqrt {c^2 x^2+1}}+\frac {4 c^2 (a+b \text {arcsinh}(c x))}{3 d x \sqrt {c^2 d x^2+d}}-\frac {a+b \text {arcsinh}(c x)}{3 d x^3 \sqrt {c^2 d x^2+d}}+\frac {8 c^4 x (a+b \text {arcsinh}(c x))}{3 d \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {b c \sqrt {c^2 d x^2+d} \int \frac {-8 c^4 x^4-4 c^2 x^2+1}{x^4 \left (c^2 x^2+1\right )}dx^2}{6 d^2 \sqrt {c^2 x^2+1}}+\frac {4 c^2 (a+b \text {arcsinh}(c x))}{3 d x \sqrt {c^2 d x^2+d}}-\frac {a+b \text {arcsinh}(c x)}{3 d x^3 \sqrt {c^2 d x^2+d}}+\frac {8 c^4 x (a+b \text {arcsinh}(c x))}{3 d \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \frac {b c \sqrt {c^2 d x^2+d} \int \left (-\frac {3 c^4}{c^2 x^2+1}-\frac {5 c^2}{x^2}+\frac {1}{x^4}\right )dx^2}{6 d^2 \sqrt {c^2 x^2+1}}+\frac {4 c^2 (a+b \text {arcsinh}(c x))}{3 d x \sqrt {c^2 d x^2+d}}-\frac {a+b \text {arcsinh}(c x)}{3 d x^3 \sqrt {c^2 d x^2+d}}+\frac {8 c^4 x (a+b \text {arcsinh}(c x))}{3 d \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 c^2 (a+b \text {arcsinh}(c x))}{3 d x \sqrt {c^2 d x^2+d}}-\frac {a+b \text {arcsinh}(c x)}{3 d x^3 \sqrt {c^2 d x^2+d}}+\frac {8 c^4 x (a+b \text {arcsinh}(c x))}{3 d \sqrt {c^2 d x^2+d}}+\frac {b c \sqrt {c^2 d x^2+d} \left (-5 c^2 \log \left (x^2\right )-3 c^2 \log \left (c^2 x^2+1\right )-\frac {1}{x^2}\right )}{6 d^2 \sqrt {c^2 x^2+1}}\) |
Input:
Int[(a + b*ArcSinh[c*x])/(x^4*(d + c^2*d*x^2)^(3/2)),x]
Output:
-1/3*(a + b*ArcSinh[c*x])/(d*x^3*Sqrt[d + c^2*d*x^2]) + (4*c^2*(a + b*ArcS inh[c*x]))/(3*d*x*Sqrt[d + c^2*d*x^2]) + (8*c^4*x*(a + b*ArcSinh[c*x]))/(3 *d*Sqrt[d + c^2*d*x^2]) + (b*c*Sqrt[d + c^2*d*x^2]*(-x^(-2) - 5*c^2*Log[x^ 2] - 3*c^2*Log[1 + c^2*x^2]))/(6*d^2*Sqrt[1 + c^2*x^2])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/2]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_ ), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSi nh[c*x]) u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] Int[S implifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x ] && EqQ[e, c^2*d] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1) /2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(967\) vs. \(2(196)=392\).
Time = 0.98 (sec) , antiderivative size = 968, normalized size of antiderivative = 4.32
| method | result | size |
| default | \(a \left (-\frac {1}{3 d \,x^{3} \sqrt {c^{2} d \,x^{2}+d}}-\frac {4 c^{2} \left (-\frac {1}{d x \sqrt {c^{2} d \,x^{2}+d}}-\frac {2 c^{2} x}{d \sqrt {c^{2} d \,x^{2}+d}}\right )}{3}\right )+\frac {16 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) c^{3}}{3 \sqrt {c^{2} x^{2}+1}\, d^{2}}+\frac {32 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{7} c^{10}}{3 \left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2}}-\frac {32 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{5} \left (c^{2} x^{2}+1\right ) c^{8}}{3 \left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2}}+\frac {16 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{5} c^{8}}{\left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2}}-\frac {16 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{3} \left (c^{2} x^{2}+1\right ) c^{6}}{3 \left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2}}+\frac {64 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{3} \operatorname {arcsinh}\left (x c \right ) c^{6}}{3 \left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2}}-\frac {64 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{2} \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) c^{5}}{3 \left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2}}+\frac {4 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{3} c^{6}}{\left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2}}+\frac {4 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x \left (c^{2} x^{2}+1\right ) c^{4}}{3 \left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2}}+\frac {8 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x \,\operatorname {arcsinh}\left (x c \right ) c^{4}}{\left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2}}+\frac {8 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) c^{3}}{3 \left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2}}-\frac {4 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x \,c^{4}}{3 \left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2}}-\frac {4 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} x^{2}+1}\, c^{3}}{3 \left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2}}-\frac {4 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) c^{2}}{\left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2} x}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} x^{2}+1}\, c}{6 \left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2} x^{2}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )}{3 \left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2} x^{3}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) c^{3}}{\sqrt {c^{2} x^{2}+1}\, d^{2}}-\frac {5 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) c^{3}}{3 \sqrt {c^{2} x^{2}+1}\, d^{2}}\) | \(968\) |
| parts | \(a \left (-\frac {1}{3 d \,x^{3} \sqrt {c^{2} d \,x^{2}+d}}-\frac {4 c^{2} \left (-\frac {1}{d x \sqrt {c^{2} d \,x^{2}+d}}-\frac {2 c^{2} x}{d \sqrt {c^{2} d \,x^{2}+d}}\right )}{3}\right )+\frac {16 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) c^{3}}{3 \sqrt {c^{2} x^{2}+1}\, d^{2}}+\frac {32 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{7} c^{10}}{3 \left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2}}-\frac {32 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{5} \left (c^{2} x^{2}+1\right ) c^{8}}{3 \left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2}}+\frac {16 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{5} c^{8}}{\left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2}}-\frac {16 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{3} \left (c^{2} x^{2}+1\right ) c^{6}}{3 \left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2}}+\frac {64 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{3} \operatorname {arcsinh}\left (x c \right ) c^{6}}{3 \left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2}}-\frac {64 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{2} \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) c^{5}}{3 \left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2}}+\frac {4 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{3} c^{6}}{\left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2}}+\frac {4 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x \left (c^{2} x^{2}+1\right ) c^{4}}{3 \left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2}}+\frac {8 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x \,\operatorname {arcsinh}\left (x c \right ) c^{4}}{\left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2}}+\frac {8 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) c^{3}}{3 \left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2}}-\frac {4 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x \,c^{4}}{3 \left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2}}-\frac {4 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} x^{2}+1}\, c^{3}}{3 \left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2}}-\frac {4 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) c^{2}}{\left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2} x}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} x^{2}+1}\, c}{6 \left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2} x^{2}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )}{3 \left (8 c^{4} x^{4}+7 c^{2} x^{2}-1\right ) d^{2} x^{3}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) c^{3}}{\sqrt {c^{2} x^{2}+1}\, d^{2}}-\frac {5 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) c^{3}}{3 \sqrt {c^{2} x^{2}+1}\, d^{2}}\) | \(968\) |
Input:
int((a+b*arcsinh(x*c))/x^4/(c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)
Output:
a*(-1/3/d/x^3/(c^2*d*x^2+d)^(1/2)-4/3*c^2*(-1/d/x/(c^2*d*x^2+d)^(1/2)-2*c^ 2/d*x/(c^2*d*x^2+d)^(1/2)))+16/3*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2) /d^2*arcsinh(x*c)*c^3+32/3*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1) /d^2*x^7*c^10-32/3*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x^5 *(c^2*x^2+1)*c^8+16*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x^ 5*c^8-16/3*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x^3*(c^2*x^ 2+1)*c^6+64/3*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x^3*arcs inh(x*c)*c^6-64/3*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x^2* (c^2*x^2+1)^(1/2)*arcsinh(x*c)*c^5+4*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7* c^2*x^2-1)/d^2*x^3*c^6+4/3*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1) /d^2*x*(c^2*x^2+1)*c^4+8*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d ^2*x*arcsinh(x*c)*c^4+8/3*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/ d^2*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*c^3-4/3*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4* x^4+7*c^2*x^2-1)/d^2*x*c^4-4/3*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^ 2-1)/d^2*(c^2*x^2+1)^(1/2)*c^3-4*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2* x^2-1)/d^2/x*arcsinh(x*c)*c^2+1/6*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2 *x^2-1)/d^2/x^2*(c^2*x^2+1)^(1/2)*c+1/3*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4 +7*c^2*x^2-1)/d^2/x^3*arcsinh(x*c)-b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/ 2)/d^2*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)*c^3-5/3*b*(d*(c^2*x^2+1))^(1/2)/(c^ 2*x^2+1)^(1/2)/d^2*ln((x*c+(c^2*x^2+1)^(1/2))^2-1)*c^3
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))/x^4/(c^2*d*x^2+d)^(3/2),x, algorithm="fricas" )
Output:
integral(sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)/(c^4*d^2*x^8 + 2*c^2*d^2 *x^6 + d^2*x^4), x)
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{x^{4} \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a+b*asinh(c*x))/x**4/(c**2*d*x**2+d)**(3/2),x)
Output:
Integral((a + b*asinh(c*x))/(x**4*(d*(c**2*x**2 + 1))**(3/2)), x)
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))/x^4/(c^2*d*x^2+d)^(3/2),x, algorithm="maxima" )
Output:
1/3*(8*c^4*x/(sqrt(c^2*d*x^2 + d)*d) + 4*c^2/(sqrt(c^2*d*x^2 + d)*d*x) - 1 /(sqrt(c^2*d*x^2 + d)*d*x^3))*a + b*integrate(log(c*x + sqrt(c^2*x^2 + 1)) /((c^2*d*x^2 + d)^(3/2)*x^4), x)
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))/x^4/(c^2*d*x^2+d)^(3/2),x, algorithm="giac")
Output:
integrate((b*arcsinh(c*x) + a)/((c^2*d*x^2 + d)^(3/2)*x^4), x)
Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^4\,{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \] Input:
int((a + b*asinh(c*x))/(x^4*(d + c^2*d*x^2)^(3/2)),x)
Output:
int((a + b*asinh(c*x))/(x^4*(d + c^2*d*x^2)^(3/2)), x)
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {8 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, a +3 \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{6}+\sqrt {c^{2} x^{2}+1}\, x^{4}}d x \right ) b \,c^{2} x^{5}+3 \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{6}+\sqrt {c^{2} x^{2}+1}\, x^{4}}d x \right ) b \,x^{3}-8 a \,c^{5} x^{5}-8 a \,c^{3} x^{3}}{3 \sqrt {d}\, d \,x^{3} \left (c^{2} x^{2}+1\right )} \] Input:
int((a+b*asinh(c*x))/x^4/(c^2*d*x^2+d)^(3/2),x)
Output:
(8*sqrt(c**2*x**2 + 1)*a*c**4*x**4 + 4*sqrt(c**2*x**2 + 1)*a*c**2*x**2 - s qrt(c**2*x**2 + 1)*a + 3*int(asinh(c*x)/(sqrt(c**2*x**2 + 1)*c**2*x**6 + s qrt(c**2*x**2 + 1)*x**4),x)*b*c**2*x**5 + 3*int(asinh(c*x)/(sqrt(c**2*x**2 + 1)*c**2*x**6 + sqrt(c**2*x**2 + 1)*x**4),x)*b*x**3 - 8*a*c**5*x**5 - 8* a*c**3*x**3)/(3*sqrt(d)*d*x**3*(c**2*x**2 + 1))