\(\int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^4} \, dx\) [127]

Optimal result

Integrand size = 26, antiderivative size = 106 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=-\frac {b c \sqrt {d+c^2 d x^2}}{6 x^2 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 d x^3}+\frac {b c^3 \sqrt {d+c^2 d x^2} \log (x)}{3 \sqrt {1+c^2 x^2}} \]

[Out]

-1/3*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))/d/x^3-1/6*b*c*(c^2*d*x^2+d)^(1/2)/x^2/(c^2*x^2+1)^(1/2)+1/3*b*c^3*
ln(x)*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {5800, 14} \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 d x^3}-\frac {b c \sqrt {c^2 d x^2+d}}{6 x^2 \sqrt {c^2 x^2+1}}+\frac {b c^3 \log (x) \sqrt {c^2 d x^2+d}}{3 \sqrt {c^2 x^2+1}} \]

[In]

Int[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/x^4,x]

[Out]

-1/6*(b*c*Sqrt[d + c^2*d*x^2])/(x^2*Sqrt[1 + c^2*x^2]) - ((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*d*x^3
) + (b*c^3*Sqrt[d + c^2*d*x^2]*Log[x])/(3*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 5800

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/(d*f*(m + 1))), 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, m, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[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))}{3 d x^3}+\frac {\left (b c \sqrt {d+c^2 d x^2}\right ) \int \frac {1+c^2 x^2}{x^3} \, dx}{3 \sqrt {1+c^2 x^2}} \\ & = -\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 d x^3}+\frac {\left (b c \sqrt {d+c^2 d x^2}\right ) \int \left (\frac {1}{x^3}+\frac {c^2}{x}\right ) \, dx}{3 \sqrt {1+c^2 x^2}} \\ & = -\frac {b c \sqrt {d+c^2 d x^2}}{6 x^2 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 d x^3}+\frac {b c^3 \sqrt {d+c^2 d x^2} \log (x)}{3 \sqrt {1+c^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=-\frac {\sqrt {d+c^2 d x^2} \left (b c x+3 b c^3 x^3+2 a \sqrt {1+c^2 x^2}+2 a c^2 x^2 \sqrt {1+c^2 x^2}+2 b \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)-2 b c^3 x^3 \log (x)\right )}{6 x^3 \sqrt {1+c^2 x^2}} \]

[In]

Integrate[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/x^4,x]

[Out]

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

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.29

method	result	size
default	\(-\frac {a \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 d \,x^{3}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 \,\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}-2 \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) x^{3} c^{3}+2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+c x \right )}{6 \sqrt {c^{2} x^{2}+1}\, x^{3}}\)	\(137\)
parts	\(-\frac {a \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 d \,x^{3}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 \,\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}-2 \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) x^{3} c^{3}+2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+c x \right )}{6 \sqrt {c^{2} x^{2}+1}\, x^{3}}\)	\(137\)

[In]

int((a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*a/d/x^3*(c^2*d*x^2+d)^(3/2)-1/6*b*(d*(c^2*x^2+1))^(1/2)*(2*arcsinh(c*x)*c^3*x^3-2*ln((c*x+(c^2*x^2+1)^(1/
2))^2-1)*x^3*c^3+2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x^2*c^2+2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+c*x)/(c^2*x^2+1)^(1
/2)/x^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (90) = 180\).

Time = 0.28 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.05 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=-\frac {2 \, {\left (b c^{4} x^{4} + 2 \, b c^{2} x^{2} + b\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (b c^{5} x^{5} + b c^{3} x^{3}\right )} \sqrt {d} \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} + d x^{4} + \sqrt {c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} + 1} {\left (x^{4} - 1\right )} \sqrt {d} + d}{c^{2} x^{4} + x^{2}}\right ) + {\left (2 \, a c^{4} x^{4} + 4 \, a c^{2} x^{2} - {\left (b c x^{3} - b c x\right )} \sqrt {c^{2} x^{2} + 1} + 2 \, a\right )} \sqrt {c^{2} d x^{2} + d}}{6 \, {\left (c^{2} x^{5} + x^{3}\right )}} \]

[In]

integrate((a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/x^4,x, algorithm="fricas")

[Out]

-1/6*(2*(b*c^4*x^4 + 2*b*c^2*x^2 + b)*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1)) - (b*c^5*x^5 + b*c^3*x^
3)*sqrt(d)*log((c^2*d*x^6 + c^2*d*x^2 + d*x^4 + sqrt(c^2*d*x^2 + d)*sqrt(c^2*x^2 + 1)*(x^4 - 1)*sqrt(d) + d)/(
c^2*x^4 + x^2)) + (2*a*c^4*x^4 + 4*a*c^2*x^2 - (b*c*x^3 - b*c*x)*sqrt(c^2*x^2 + 1) + 2*a)*sqrt(c^2*d*x^2 + d))
/(c^2*x^5 + x^3)

Sympy [F]

\[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\int \frac {\sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{x^{4}}\, dx \]

[In]

integrate((a+b*asinh(c*x))*(c**2*d*x**2+d)**(1/2)/x**4,x)

[Out]

Integral(sqrt(d*(c**2*x**2 + 1))*(a + b*asinh(c*x))/x**4, x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.25 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=-\frac {{\left (\left (-1\right )^{2 \, c^{2} d x^{2} + 2 \, d} c^{2} d^{\frac {3}{2}} \log \left (2 \, c^{2} d + \frac {2 \, d}{x^{2}}\right ) - c^{2} d^{\frac {3}{2}} \log \left (x^{2} + \frac {1}{c^{2}}\right ) + \frac {\sqrt {c^{4} d x^{4} + 2 \, c^{2} d x^{2} + d} d}{x^{2}}\right )} b c}{6 \, d} - \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} b \operatorname {arsinh}\left (c x\right )}{3 \, d x^{3}} - \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a}{3 \, d x^{3}} \]

[In]

integrate((a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/x^4,x, algorithm="maxima")

[Out]

-1/6*((-1)^(2*c^2*d*x^2 + 2*d)*c^2*d^(3/2)*log(2*c^2*d + 2*d/x^2) - c^2*d^(3/2)*log(x^2 + 1/c^2) + sqrt(c^4*d*
x^4 + 2*c^2*d*x^2 + d)*d/x^2)*b*c/d - 1/3*(c^2*d*x^2 + d)^(3/2)*b*arcsinh(c*x)/(d*x^3) - 1/3*(c^2*d*x^2 + d)^(
3/2)*a/(d*x^3)

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/x^4,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. \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {d\,c^2\,x^2+d}}{x^4} \,d x \]

[In]

int(((a + b*asinh(c*x))*(d + c^2*d*x^2)^(1/2))/x^4,x)

[Out]

int(((a + b*asinh(c*x))*(d + c^2*d*x^2)^(1/2))/x^4, x)