Integrand size = 26, antiderivative size = 228 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {b c \sqrt {d+c^2 d x^2}}{6 d^2 x^2 \sqrt {1+c^2 x^2}}-\frac {a+b \text {arcsinh}(c x)}{3 d x^3 \sqrt {d+c^2 d x^2}}+\frac {4 c^2 (a+b \text {arcsinh}(c x))}{3 d x \sqrt {d+c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arcsinh}(c x))}{3 d \sqrt {d+c^2 d x^2}}-\frac {5 b c^3 \sqrt {d+c^2 d x^2} \log (x)}{3 d^2 \sqrt {1+c^2 x^2}}-\frac {b c^3 \sqrt {d+c^2 d x^2} \log \left (1+c^2 x^2\right )}{2 d^2 \sqrt {1+c^2 x^2}} \]
1/3*(-a-b*arcsinh(c*x))/d/x^3/(c^2*d*x^2+d)^(1/2)+4/3*c^2*(a+b*arcsinh(c*x ))/d/x/(c^2*d*x^2+d)^(1/2)+8/3*c^4*x*(a+b*arcsinh(c*x))/d/(c^2*d*x^2+d)^(1 /2)-1/6*b*c*(c^2*d*x^2+d)^(1/2)/d^2/x^2/(c^2*x^2+1)^(1/2)-5/3*b*c^3*ln(x)* (c^2*d*x^2+d)^(1/2)/d^2/(c^2*x^2+1)^(1/2)-1/2*b*c^3*ln(c^2*x^2+1)*(c^2*d*x ^2+d)^(1/2)/d^2/(c^2*x^2+1)^(1/2)
Time = 0.32 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.95 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {\sqrt {d+c^2 d x^2} \left (-b c x-b c^3 x^3-2 a \sqrt {1+c^2 x^2}+8 a c^2 x^2 \sqrt {1+c^2 x^2}+16 a c^4 x^4 \sqrt {1+c^2 x^2}+2 b \sqrt {1+c^2 x^2} \left (-1+4 c^2 x^2+8 c^4 x^4\right ) \text {arcsinh}(c x)+5 b c^3 x^3 \left (1+c^2 x^2\right ) \log \left (1+\frac {1}{c^2 x^2}\right )-8 b c^3 x^3 \log \left (1+c^2 x^2\right )-8 b c^5 x^5 \log \left (1+c^2 x^2\right )\right )}{6 d^2 x^3 \left (1+c^2 x^2\right )^{3/2}} \]
(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 = 0.48 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.73, 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 |
\(\displaystyle \frac {4 c^2 (a+b \text {arcsinh}(c x))}{3 d x \sqrt {c^2 d x^2+d}}-\frac {a+b \text {arcsinh}(c x)}{3 d x^3 \sqrt {c^2 d x^2+d}}+\frac {8 c^4 x (a+b \text {arcsinh}(c x))}{3 d \sqrt {c^2 d x^2+d}}+\frac {b c \sqrt {c^2 d x^2+d} \left (-5 c^2 \log \left (x^2\right )-3 c^2 \log \left (c^2 x^2+1\right )-\frac {1}{x^2}\right )}{6 d^2 \sqrt {c^2 x^2+1}}\) |
-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])
3.2.64.3.1 Defintions of rubi rules used
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. \(2(201)=402\).
Time = 0.22 (sec) , antiderivative size = 968, normalized size of antiderivative = 4.25
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 (c x \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 (c x \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 (c x \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 (c x \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 (c x \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 )}\, c^{3} \sqrt {c^{2} x^{2}+1}}{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 (c x \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 )}\, c \sqrt {c^{2} x^{2}+1}}{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 (c x \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 (c x +\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 (c x +\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 (c x \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 (c x \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 (c x \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 (c x \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 (c x \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 )}\, c^{3} \sqrt {c^{2} x^{2}+1}}{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 (c x \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 )}\, c \sqrt {c^{2} x^{2}+1}}{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 (c x \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 (c x +\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 (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) c^{3}}{3 \sqrt {c^{2} x^{2}+1}\, d^{2}}\) | \(968\) |
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(c*x)*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(c*x)*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(c*x)*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(c*x)*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(c*x)*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^3*(c^2*x^2+1)^(1/2)-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(c*x)*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*(c^2*x^2+1)^(1/2)+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(c*x)-b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/ 2)/d^2*ln(1+(c*x+(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((c*x+(c^2*x^2+1)^(1/2))^2-1)*c^3
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
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)
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{x^{4} \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
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)
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^4\,{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \]