\(\int \frac {(d+c^2 d x^2)^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx\) [133]

Optimal result

Integrand size = 26, antiderivative size = 177 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=-\frac {b c^3 d x^2 \sqrt {d+c^2 d x^2}}{4 \sqrt {1+c^2 x^2}}+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+\frac {3 c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b \sqrt {1+c^2 x^2}}+\frac {b c d \sqrt {d+c^2 d x^2} \log (x)}{\sqrt {1+c^2 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {5807, 5785, 5783, 30, 14} \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\frac {3}{2} c^2 d x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))+\frac {3 c d \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{4 b \sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+\frac {b c d \log (x) \sqrt {c^2 d x^2+d}}{\sqrt {c^2 x^2+1}}-\frac {b c^3 d x^2 \sqrt {c^2 d x^2+d}}{4 \sqrt {c^2 x^2+1}} \]

[In]

Int[((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x^2,x]

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 5783

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[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]

Rule 5785

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

Rule 5807

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

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

Mathematica [A] (verified)

Time = 0.83 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.13 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\frac {1}{8} \left (\frac {4 a d \left (-2+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{x}+\frac {4 b d \sqrt {d+c^2 d x^2} \left (-2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+c x \text {arcsinh}(c x)^2+2 c x \log (c x)\right )}{x \sqrt {1+c^2 x^2}}+12 a c d^{3/2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+\frac {b c d \sqrt {d+c^2 d x^2} (-\cosh (2 \text {arcsinh}(c x))+2 \text {arcsinh}(c x) (\text {arcsinh}(c x)+\sinh (2 \text {arcsinh}(c x))))}{\sqrt {1+c^2 x^2}}\right ) \]

[In]

Integrate[((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x^2,x]

[Out]

((4*a*d*(-2 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/x + (4*b*d*Sqrt[d + c^2*d*x^2]*(-2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]
+ c*x*ArcSinh[c*x]^2 + 2*c*x*Log[c*x]))/(x*Sqrt[1 + c^2*x^2]) + 12*a*c*d^(3/2)*Log[c*d*x + Sqrt[d]*Sqrt[d + c^
2*d*x^2]] + (b*c*d*Sqrt[d + c^2*d*x^2]*(-Cosh[2*ArcSinh[c*x]] + 2*ArcSinh[c*x]*(ArcSinh[c*x] + Sinh[2*ArcSinh[
c*x]])))/Sqrt[1 + c^2*x^2])/8

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.30

method	result	size
default	\(-\frac {a \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{d x}+a \,c^{2} x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}+\frac {3 \sqrt {c^{2} d \,x^{2}+d}\, a \,c^{2} d x}{2}+\frac {3 a \,c^{2} d^{2} \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-2 c^{3} x^{3}+6 \operatorname {arcsinh}\left (c x \right )^{2} x c -8 \,\operatorname {arcsinh}\left (c x \right ) c x +8 \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) x c -8 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-c x \right ) d}{8 \sqrt {c^{2} x^{2}+1}\, x}\)	\(230\)
parts	\(-\frac {a \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{d x}+a \,c^{2} x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}+\frac {3 \sqrt {c^{2} d \,x^{2}+d}\, a \,c^{2} d x}{2}+\frac {3 a \,c^{2} d^{2} \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-2 c^{3} x^{3}+6 \operatorname {arcsinh}\left (c x \right )^{2} x c -8 \,\operatorname {arcsinh}\left (c x \right ) c x +8 \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) x c -8 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-c x \right ) d}{8 \sqrt {c^{2} x^{2}+1}\, x}\)	\(230\)

[In]

int((c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))/x^2,x,method=_RETURNVERBOSE)

[Out]

-a/d/x*(c^2*d*x^2+d)^(5/2)+a*c^2*x*(c^2*d*x^2+d)^(3/2)+3/2*(c^2*d*x^2+d)^(1/2)*a*c^2*d*x+3/2*a*c^2*d^2*ln(c^2*
d*x/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+1/8*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/x*(4*arcsin
h(c*x)*(c^2*x^2+1)^(1/2)*x^2*c^2-2*c^3*x^3+6*arcsinh(c*x)^2*x*c-8*arcsinh(c*x)*c*x+8*ln((c*x+(c^2*x^2+1)^(1/2)
)^2-1)*x*c-8*arcsinh(c*x)*(c^2*x^2+1)^(1/2)-c*x)*d

Fricas [F]

\[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x^{2}} \,d x } \]

[In]

integrate((c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))/x^2,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{x^{2}}\, dx \]

[In]

integrate((c**2*d*x**2+d)**(3/2)*(a+b*asinh(c*x))/x**2,x)

[Out]

Integral((d*(c**2*x**2 + 1))**(3/2)*(a + b*asinh(c*x))/x**2, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))/x^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))/x^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{3/2}}{x^2} \,d x \]

[In]

int(((a + b*asinh(c*x))*(d + c^2*d*x^2)^(3/2))/x^2,x)

[Out]

int(((a + b*asinh(c*x))*(d + c^2*d*x^2)^(3/2))/x^2, x)