\(\int \frac {(a+b \text {arcsinh}(c x))^2}{(d+c^2 d x^2)^{5/2}} \, dx\) [315]

Optimal result

Integrand size = 25, antiderivative size = 292 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=-\frac {b^2 x}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {b (a+b \text {arcsinh}(c x))}{3 c d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {2 x (a+b \text {arcsinh}(c x))^2}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 c d^2 \sqrt {d+c^2 d x^2}}-\frac {4 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 c d^2 \sqrt {d+c^2 d x^2}}-\frac {2 b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 c d^2 \sqrt {d+c^2 d x^2}} \]

[Out]

1/3*x*(a+b*arcsinh(c*x))^2/d/(c^2*d*x^2+d)^(3/2)-1/3*b^2*x/d^2/(c^2*d*x^2+d)^(1/2)+2/3*x*(a+b*arcsinh(c*x))^2/
d^2/(c^2*d*x^2+d)^(1/2)+1/3*b*(a+b*arcsinh(c*x))/c/d^2/(c^2*x^2+1)^(1/2)/(c^2*d*x^2+d)^(1/2)+2/3*(a+b*arcsinh(
c*x))^2*(c^2*x^2+1)^(1/2)/c/d^2/(c^2*d*x^2+d)^(1/2)-4/3*b*(a+b*arcsinh(c*x))*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)*(
c^2*x^2+1)^(1/2)/c/d^2/(c^2*d*x^2+d)^(1/2)-2/3*b^2*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2)*(c^2*x^2+1)^(1/2)/c/d
^2/(c^2*d*x^2+d)^(1/2)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {5788, 5787, 5797, 3799, 2221, 2317, 2438, 5798, 197} \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {b (a+b \text {arcsinh}(c x))}{3 c d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}}+\frac {2 x (a+b \text {arcsinh}(c x))^2}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{3 c d^2 \sqrt {c^2 d x^2+d}}-\frac {4 b \sqrt {c^2 x^2+1} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))}{3 c d^2 \sqrt {c^2 d x^2+d}}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}-\frac {2 b^2 \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 c d^2 \sqrt {c^2 d x^2+d}}-\frac {b^2 x}{3 d^2 \sqrt {c^2 d x^2+d}} \]

[In]

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

[Out]

-1/3*(b^2*x)/(d^2*Sqrt[d + c^2*d*x^2]) + (b*(a + b*ArcSinh[c*x]))/(3*c*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^
2]) + (x*(a + b*ArcSinh[c*x])^2)/(3*d*(d + c^2*d*x^2)^(3/2)) + (2*x*(a + b*ArcSinh[c*x])^2)/(3*d^2*Sqrt[d + c^
2*d*x^2]) + (2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(3*c*d^2*Sqrt[d + c^2*d*x^2]) - (4*b*Sqrt[1 + c^2*x^2
]*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(3*c*d^2*Sqrt[d + c^2*d*x^2]) - (2*b^2*Sqrt[1 + c^2*x^2]*P
olyLog[2, -E^(2*ArcSinh[c*x])])/(3*c*d^2*Sqrt[d + c^2*d*x^2])

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 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 3799

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

Rule 5787

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[x*((a + b*ArcSinh
[c*x])^n/(d*Sqrt[d + e*x^2])), x] - Dist[b*c*(n/d)*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]], Int[x*((a + b*ArcS
inh[c*x])^(n - 1)/(1 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0]

Rule 5788

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*(d + e*x^2)^(
p + 1)*((a + b*ArcSinh[c*x])^n/(2*d*(p + 1))), x] + (Dist[(2*p + 3)/(2*d*(p + 1)), Int[(d + e*x^2)^(p + 1)*(a
+ b*ArcSinh[c*x])^n, x], x] + Dist[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p], Int[x*(1 + c^2*x^2
)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] &
& LtQ[p, -1] && NeQ[p, -3/2]

Rule 5797

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

Rule 5798

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^
(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)
^p], Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e
, c^2*d] && GtQ[n, 0] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {2 \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx}{3 d}-\frac {\left (2 b c \sqrt {1+c^2 x^2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{\left (1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d+c^2 d x^2}} \\ & = \frac {b (a+b \text {arcsinh}(c x))}{3 c d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {2 x (a+b \text {arcsinh}(c x))^2}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (4 b c \sqrt {1+c^2 x^2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{1+c^2 x^2} \, dx}{3 d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b^2 x}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {b (a+b \text {arcsinh}(c x))}{3 c d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {2 x (a+b \text {arcsinh}(c x))^2}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (4 b \sqrt {1+c^2 x^2}\right ) \text {Subst}(\int (a+b x) \tanh (x) \, dx,x,\text {arcsinh}(c x))}{3 c d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b^2 x}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {b (a+b \text {arcsinh}(c x))}{3 c d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {2 x (a+b \text {arcsinh}(c x))^2}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 c d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (8 b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\text {arcsinh}(c x)\right )}{3 c d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b^2 x}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {b (a+b \text {arcsinh}(c x))}{3 c d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {2 x (a+b \text {arcsinh}(c x))^2}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 c d^2 \sqrt {d+c^2 d x^2}}-\frac {4 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 c d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (4 b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{3 c d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b^2 x}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {b (a+b \text {arcsinh}(c x))}{3 c d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {2 x (a+b \text {arcsinh}(c x))^2}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 c d^2 \sqrt {d+c^2 d x^2}}-\frac {4 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 c d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{3 c d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b^2 x}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {b (a+b \text {arcsinh}(c x))}{3 c d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {2 x (a+b \text {arcsinh}(c x))^2}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 c d^2 \sqrt {d+c^2 d x^2}}-\frac {4 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 c d^2 \sqrt {d+c^2 d x^2}}-\frac {2 b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 c d^2 \sqrt {d+c^2 d x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.45 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {a^2 c x \left (3+2 c^2 x^2\right )+a b \left (\left (6 c x+4 c^3 x^3\right ) \text {arcsinh}(c x)+\sqrt {1+c^2 x^2} \left (1-2 \left (1+c^2 x^2\right ) \log \left (1+c^2 x^2\right )\right )\right )-b^2 \left (c x+c^3 x^3-\sqrt {1+c^2 x^2} \text {arcsinh}(c x)-c x \text {arcsinh}(c x)^2-2 c x \left (1+c^2 x^2\right ) \text {arcsinh}(c x)^2+2 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x) \left (\text {arcsinh}(c x)+2 \log \left (1+e^{-2 \text {arcsinh}(c x)}\right )\right )-2 \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(c x)}\right )\right )}{3 d^2 \left (c+c^3 x^2\right ) \sqrt {d+c^2 d x^2}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2130\) vs. \(2(274)=548\).

Time = 0.28 (sec) , antiderivative size = 2131, normalized size of antiderivative = 7.30

method	result	size
default	\(\text {Expression too large to display}\)	\(2131\)
parts	\(\text {Expression too large to display}\)	\(2131\)

[In]

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

[Out]

7/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)*c/d^3*x^2*(c^2*x^2+1)^(1/2)-8/3*b^2*(d*(c^2*
x^2+1))^(1/2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)/c/d^3*(c^2*x^2+1)^(1/2)*arcsinh(c*x)^2+4/3*b^2*(d*(c^2*x^2+1
))^(1/2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)/c/d^3*(c^2*x^2+1)^(1/2)*arcsinh(c*x)+17/3*b^2*(d*(c^2*x^2+1))^(1/
2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)*c^2/d^3*arcsinh(c*x)^2*x^3+2*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^6*x^6+10*c^
4*x^4+11*c^2*x^2+4)/d^3*(c^2*x^2+1)*arcsinh(c*x)*x-16/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^6*x^6+10*c^4*x^4+11*c^2
*x^2+4)*c^2/d^3*arcsinh(c*x)*x^3-4/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)*c^6/d^3*arc
sinh(c*x)*x^7+2/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)*c^4/d^3*(c^2*x^2+1)*x^5+2*b^2*
(d*(c^2*x^2+1))^(1/2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)*c^4/d^3*arcsinh(c*x)^2*x^5-14/3*b^2*(d*(c^2*x^2+1))^
(1/2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)*c^4/d^3*arcsinh(c*x)*x^5+b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^6*x^6+10*c^4
*x^4+11*c^2*x^2+4)*c^3/d^3*(c^2*x^2+1)^(1/2)*x^4+4/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^
2+4)*c^2/d^3*(c^2*x^2+1)*x^3-4/3*b^2/(c^2*x^2+1)^(1/2)*(d*(c^2*x^2+1))^(1/2)/d^3/c*arcsinh(c*x)*ln(1+(c*x+(c^2
*x^2+1)^(1/2))^2)-2/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)*c^6/d^3*x^7-3*b^2*(d*(c^2*
x^2+1))^(1/2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)*c^4/d^3*x^5-13/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^6*x^6+10*c^4
*x^4+11*c^2*x^2+4)*c^2/d^3*x^3+4/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)/c/d^3*(c^2*x^
2+1)^(1/2)-2/3*b^2/(c^2*x^2+1)^(1/2)*(d*(c^2*x^2+1))^(1/2)/d^3/c*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2)+2/3*b^2
*(d*(c^2*x^2+1))^(1/2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)/d^3*(c^2*x^2+1)*x+a^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))+4/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)*c^4/d^3*(c^2*
x^2+1)*arcsinh(c*x)*x^5-2*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)*c^3/d^3*(c^2*x^2+1)^(1
/2)*arcsinh(c*x)^2*x^4+10/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)*c^2/d^3*(c^2*x^2+1)*
arcsinh(c*x)*x^3-14/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)*c/d^3*(c^2*x^2+1)^(1/2)*ar
csinh(c*x)^2*x^2+b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)*c/d^3*(c^2*x^2+1)^(1/2)*arcsinh
(c*x)*x^2+1/3*a*b*(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)+3*c*x+2*(c^2*x^2+1)^(1/2))*(-8*
ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)*x^6*c^6+8*(c^2*x^2+1)^(1/2)*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)*x^5*c^5-24*ln(1+(c
*x+(c^2*x^2+1)^(1/2))^2)*x^4*c^4+20*(c^2*x^2+1)^(1/2)*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)*x^3*c^3+2*c^4*x^4-2*c^3*
x^3*(c^2*x^2+1)^(1/2)+6*arcsinh(c*x)*c^2*x^2-24*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)*x^2*c^2+12*(c^2*x^2+1)^(1/2)*l
n(1+(c*x+(c^2*x^2+1)^(1/2))^2)*x*c+4*c^2*x^2-3*c*x*(c^2*x^2+1)^(1/2)+8*arcsinh(c*x)-8*ln(1+(c*x+(c^2*x^2+1)^(1
/2))^2)+2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)/c/d^3-2*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*
x^2+4)/d^3*x+4*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^6*x^6+10*c^4*x^4+11*c^2*x^2+4)/d^3*arcsinh(c*x)^2*x+4/3*b^2/(c^2
*x^2+1)^(1/2)*(d*(c^2*x^2+1))^(1/2)/d^3/c*arcsinh(c*x)^2-2*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^6*x^6+10*c^4*x^4+11*
c^2*x^2+4)/d^3*arcsinh(c*x)*x

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/3*a*b*c*(1/(c^4*d^(5/2)*x^2 + c^2*d^(5/2)) - 2*log(c^2*x^2 + 1)/(c^2*d^(5/2))) + 2/3*a*b*(2*x/(sqrt(c^2*d*x^
2 + d)*d^2) + x/((c^2*d*x^2 + d)^(3/2)*d))*arcsinh(c*x) + 1/3*a^2*(2*x/(sqrt(c^2*d*x^2 + d)*d^2) + x/((c^2*d*x
^2 + d)^(3/2)*d)) + b^2*integrate(log(c*x + sqrt(c^2*x^2 + 1))^2/(c^2*d*x^2 + d)^(5/2), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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