\(\int \frac {a+b \text {arcsinh}(c x)}{x (d+c^2 d x^2)^{5/2}} \, dx\) [172]

Optimal result

Integrand size = 26, antiderivative size = 262 \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^{5/2}} \, dx=-\frac {b c x}{6 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {a+b \text {arcsinh}(c x)}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {a+b \text {arcsinh}(c x)}{d^2 \sqrt {d+c^2 d x^2}}-\frac {7 b \sqrt {1+c^2 x^2} \arctan (c x)}{6 d^2 \sqrt {d+c^2 d x^2}}-\frac {2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {b \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {b \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}} \]

[Out]

1/3*(a+b*arcsinh(c*x))/d/(c^2*d*x^2+d)^(3/2)+(a+b*arcsinh(c*x))/d^2/(c^2*d*x^2+d)^(1/2)-1/6*b*c*x/d^2/(c^2*x^2
+1)^(1/2)/(c^2*d*x^2+d)^(1/2)-7/6*b*arctan(c*x)*(c^2*x^2+1)^(1/2)/d^2/(c^2*d*x^2+d)^(1/2)-2*(a+b*arcsinh(c*x))
*arctanh(c*x+(c^2*x^2+1)^(1/2))*(c^2*x^2+1)^(1/2)/d^2/(c^2*d*x^2+d)^(1/2)-b*polylog(2,-c*x-(c^2*x^2+1)^(1/2))*
(c^2*x^2+1)^(1/2)/d^2/(c^2*d*x^2+d)^(1/2)+b*polylog(2,c*x+(c^2*x^2+1)^(1/2))*(c^2*x^2+1)^(1/2)/d^2/(c^2*d*x^2+
d)^(1/2)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {5811, 5816, 4267, 2317, 2438, 209, 205} \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^{5/2}} \, dx=-\frac {2 \sqrt {c^2 x^2+1} \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{d^2 \sqrt {c^2 d x^2+d}}+\frac {a+b \text {arcsinh}(c x)}{d^2 \sqrt {c^2 d x^2+d}}+\frac {a+b \text {arcsinh}(c x)}{3 d \left (c^2 d x^2+d\right )^{3/2}}-\frac {b \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {c^2 d x^2+d}}+\frac {b \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {7 b \sqrt {c^2 x^2+1} \arctan (c x)}{6 d^2 \sqrt {c^2 d x^2+d}}-\frac {b c x}{6 d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}} \]

[In]

Int[(a + b*ArcSinh[c*x])/(x*(d + c^2*d*x^2)^(5/2)),x]

[Out]

-1/6*(b*c*x)/(d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]) + (a + b*ArcSinh[c*x])/(3*d*(d + c^2*d*x^2)^(3/2)) +
(a + b*ArcSinh[c*x])/(d^2*Sqrt[d + c^2*d*x^2]) - (7*b*Sqrt[1 + c^2*x^2]*ArcTan[c*x])/(6*d^2*Sqrt[d + c^2*d*x^2
]) - (2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/(d^2*Sqrt[d + c^2*d*x^2]) - (b*Sqrt[1
+ c^2*x^2]*PolyLog[2, -E^ArcSinh[c*x]])/(d^2*Sqrt[d + c^2*d*x^2]) + (b*Sqrt[1 + c^2*x^2]*PolyLog[2, E^ArcSinh[
c*x]])/(d^2*Sqrt[d + c^2*d*x^2])

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 5811

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/(2*d*f*(p + 1))), x] + (Dist[(m + 2*p + 3)/(2*d*(
p + 1)), Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] + Dist[b*c*(n/(2*f*(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, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] &&  !GtQ[m, 1] && (IntegerQ[m] ||
IntegerQ[p] || EqQ[n, 1])

Rule 5816

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[(1/c^(m
+ 1))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]], Subst[Int[(a + b*x)^n*Sinh[x]^m, x], x, ArcSinh[c*x]], x] /; F
reeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \frac {a+b \text {arcsinh}(c x)}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {\int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^{3/2}} \, dx}{d}-\frac {\left (b c \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\left (1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b c x}{6 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {a+b \text {arcsinh}(c x)}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {a+b \text {arcsinh}(c x)}{d^2 \sqrt {d+c^2 d x^2}}+\frac {\int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {d+c^2 d x^2}} \, dx}{d^2}-\frac {\left (b c \sqrt {1+c^2 x^2}\right ) \int \frac {1}{1+c^2 x^2} \, dx}{6 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (b c \sqrt {1+c^2 x^2}\right ) \int \frac {1}{1+c^2 x^2} \, dx}{d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b c x}{6 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {a+b \text {arcsinh}(c x)}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {a+b \text {arcsinh}(c x)}{d^2 \sqrt {d+c^2 d x^2}}-\frac {7 b \sqrt {1+c^2 x^2} \arctan (c x)}{6 d^2 \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arcsinh}(c x))}{d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b c x}{6 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {a+b \text {arcsinh}(c x)}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {a+b \text {arcsinh}(c x)}{d^2 \sqrt {d+c^2 d x^2}}-\frac {7 b \sqrt {1+c^2 x^2} \arctan (c x)}{6 d^2 \sqrt {d+c^2 d x^2}}-\frac {2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b c x}{6 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {a+b \text {arcsinh}(c x)}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {a+b \text {arcsinh}(c x)}{d^2 \sqrt {d+c^2 d x^2}}-\frac {7 b \sqrt {1+c^2 x^2} \arctan (c x)}{6 d^2 \sqrt {d+c^2 d x^2}}-\frac {2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b c x}{6 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {a+b \text {arcsinh}(c x)}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {a+b \text {arcsinh}(c x)}{d^2 \sqrt {d+c^2 d x^2}}-\frac {7 b \sqrt {1+c^2 x^2} \arctan (c x)}{6 d^2 \sqrt {d+c^2 d x^2}}-\frac {2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {b \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {b \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.00 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.94 \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {\frac {2 a \left (4+3 c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{\left (1+c^2 x^2\right )^2}+6 a \sqrt {d} \log (x)-6 a \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+\frac {b d^2 \left (1+c^2 x^2\right )^{3/2} \left (-\frac {c x}{1+c^2 x^2}+\frac {2 \text {arcsinh}(c x)}{\left (1+c^2 x^2\right )^{3/2}}+\frac {6 \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}-14 \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+6 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-6 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+6 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-6 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right )}{\left (d+c^2 d x^2\right )^{3/2}}}{6 d^3} \]

[In]

Integrate[(a + b*ArcSinh[c*x])/(x*(d + c^2*d*x^2)^(5/2)),x]

[Out]

((2*a*(4 + 3*c^2*x^2)*Sqrt[d + c^2*d*x^2])/(1 + c^2*x^2)^2 + 6*a*Sqrt[d]*Log[x] - 6*a*Sqrt[d]*Log[d + Sqrt[d]*
Sqrt[d + c^2*d*x^2]] + (b*d^2*(1 + c^2*x^2)^(3/2)*(-((c*x)/(1 + c^2*x^2)) + (2*ArcSinh[c*x])/(1 + c^2*x^2)^(3/
2) + (6*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] - 14*ArcTan[Tanh[ArcSinh[c*x]/2]] + 6*ArcSinh[c*x]*Log[1 - E^(-ArcSinh
[c*x])] - 6*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 6*PolyLog[2, -E^(-ArcSinh[c*x])] - 6*PolyLog[2, E^(-ArcS
inh[c*x])]))/(d + c^2*d*x^2)^(3/2))/(6*d^3)

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.39

method	result	size
default	\(\frac {a}{3 d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {a}{d^{2} \sqrt {c^{2} d \,x^{2}+d}}-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {5}{2}}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) x^{2} c^{2}}{\left (c^{2} x^{2}+1\right )^{2} d^{3}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c x}{6 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} d^{3}}+\frac {4 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )}{3 \left (c^{2} x^{2}+1\right )^{2} d^{3}}-\frac {7 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arctan \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{3 \sqrt {c^{2} x^{2}+1}\, d^{3}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {dilog}\left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d^{3}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d^{3}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {dilog}\left (c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d^{3}}\)	\(364\)
parts	\(\frac {a}{3 d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {a}{d^{2} \sqrt {c^{2} d \,x^{2}+d}}-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {5}{2}}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) x^{2} c^{2}}{\left (c^{2} x^{2}+1\right )^{2} d^{3}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c x}{6 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} d^{3}}+\frac {4 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )}{3 \left (c^{2} x^{2}+1\right )^{2} d^{3}}-\frac {7 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arctan \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{3 \sqrt {c^{2} x^{2}+1}\, d^{3}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {dilog}\left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d^{3}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d^{3}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {dilog}\left (c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d^{3}}\)	\(364\)

[In]

int((a+b*arcsinh(c*x))/x/(c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/3*a/d/(c^2*d*x^2+d)^(3/2)+a/d^2/(c^2*d*x^2+d)^(1/2)-a/d^(5/2)*ln((2*d+2*d^(1/2)*(c^2*d*x^2+d)^(1/2))/x)+b*(d
*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^2/d^3*arcsinh(c*x)*x^2*c^2-1/6*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(3/2)/d^3*c
*x+4/3*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^2/d^3*arcsinh(c*x)-7/3*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^
3*arctan(c*x+(c^2*x^2+1)^(1/2))-b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^3*dilog(1+c*x+(c^2*x^2+1)^(1/2))-b
*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^3*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))-b*(d*(c^2*x^2+1))^(1/2)/
(c^2*x^2+1)^(1/2)/d^3*dilog(c*x+(c^2*x^2+1)^(1/2))

Fricas [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x} \,d x } \]

[In]

integrate((a+b*arcsinh(c*x))/x/(c^2*d*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)/(c^6*d^3*x^7 + 3*c^4*d^3*x^5 + 3*c^2*d^3*x^3 + d^3*x), x)

Sympy [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{x \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate((a+b*asinh(c*x))/x/(c**2*d*x**2+d)**(5/2),x)

[Out]

Integral((a + b*asinh(c*x))/(x*(d*(c**2*x**2 + 1))**(5/2)), x)

Maxima [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x} \,d x } \]

[In]

integrate((a+b*arcsinh(c*x))/x/(c^2*d*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

-1/3*a*(3*arcsinh(1/(c*abs(x)))/d^(5/2) - 3/(sqrt(c^2*d*x^2 + d)*d^2) - 1/((c^2*d*x^2 + d)^(3/2)*d)) + b*integ
rate(log(c*x + sqrt(c^2*x^2 + 1))/((c^2*d*x^2 + d)^(5/2)*x), x)

Giac [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x} \,d x } \]

[In]

integrate((a+b*arcsinh(c*x))/x/(c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arcsinh(c*x) + a)/((c^2*d*x^2 + d)^(5/2)*x), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x\,{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \]

[In]

int((a + b*asinh(c*x))/(x*(d + c^2*d*x^2)^(5/2)),x)

[Out]

int((a + b*asinh(c*x))/(x*(d + c^2*d*x^2)^(5/2)), x)