\(\int \frac {d+c^2 d x^2}{x (a+b \text {arcsinh}(c x))^{3/2}} \, dx\) [466]

Optimal result

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

[Out]

1/2*d*exp(2*a/b)*erf(2^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(3/2)+1/2*d*erfi(2^(1/2)*(a+
b*arcsinh(c*x))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(3/2)/exp(2*a/b)-2*d*(c^2*x^2+1)^(3/2)/b/c/x/(a+b*arcsinh(c*
x))^(1/2)-2*d*Unintegrable(1/x^2/(c^2*x^2+1)^(1/2)/(a+b*arcsinh(c*x))^(1/2),x)/b/c

Rubi [N/A]

Not integrable

Time = 0.51 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {d+c^2 d x^2}{x (a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int \frac {d+c^2 d x^2}{x (a+b \text {arcsinh}(c x))^{3/2}} \, dx \]

[In]

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

[Out]

(-2*d*(1 + c^2*x^2)^(3/2))/(b*c*x*Sqrt[a + b*ArcSinh[c*x]]) + (d*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a +
b*ArcSinh[c*x]])/Sqrt[b]])/b^(3/2) + (d*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(b^(3/2)*
E^((2*a)/b)) - (2*d*Defer[Int][1/(x^2*Sqrt[1 + c^2*x^2]*Sqrt[a + b*ArcSinh[c*x]]), x])/(b*c)

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

Mathematica [N/A]

Not integrable

Time = 1.35 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {d+c^2 d x^2}{x (a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int \frac {d+c^2 d x^2}{x (a+b \text {arcsinh}(c x))^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [N/A]

Not integrable

Time = 4.46 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.19 \[ \int \frac {d+c^2 d x^2}{x (a+b \text {arcsinh}(c x))^{3/2}} \, dx=d \left (\int \frac {c^{2} x^{2}}{a x \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b x \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx + \int \frac {1}{a x \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b x \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.72 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {d+c^2 d x^2}{x (a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int { \frac {c^{2} d x^{2} + d}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}} x} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

integrate((c^2*d*x^2+d)/x/(a+b*arcsinh(c*x))^(3/2),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 [N/A]

Not integrable

Time = 2.85 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {d+c^2 d x^2}{x (a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int \frac {d\,c^2\,x^2+d}{x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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