\(\int \frac {(d+c^2 d x^2)^2}{x (a+b \text {arcsinh}(c x))^{3/2}} \, dx\) [471]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {\left (d+c^2 d x^2\right )^2}{x (a+b \text {arcsinh}(c x))^{3/2}} \, dx=-\frac {2 d^2 \left (1+c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \text {arcsinh}(c x)}}+\frac {d^2 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2}}-\frac {d^2 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2}}+\frac {d^2 e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b^{3/2}}+\frac {d^2 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2}}-\frac {d^2 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2}}+\frac {d^2 e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b^{3/2}}-\frac {2 d^2 \text {Int}\left (\frac {1}{x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}},x\right )}{b c} \]

[Out]

3/4*d^2*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)+3/4*d^2*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)+1/4*d^2*exp(4*a/b)*erf(2*(a+b*arcsinh(c
*x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(3/2)+1/4*d^2*erfi(2*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(3/2)/exp(4*a
/b)-2*d^2*(c^2*x^2+1)^(5/2)/b/c/x/(a+b*arcsinh(c*x))^(1/2)-2*d^2*Unintegrable(1/x^2/(c^2*x^2+1)^(1/2)/(a+b*arc
sinh(c*x))^(1/2),x)/b/c

Rubi [N/A]

Not integrable

Time = 0.97 (sec) , antiderivative size = 28, 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 = {} \[ \int \frac {\left (d+c^2 d x^2\right )^2}{x (a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int \frac {\left (d+c^2 d x^2\right )^2}{x (a+b \text {arcsinh}(c x))^{3/2}} \, dx \]

[In]

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

[Out]

(-2*d^2*(1 + c^2*x^2)^(5/2))/(b*c*x*Sqrt[a + b*ArcSinh[c*x]]) + (d^2*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*Ar
cSinh[c*x]])/Sqrt[b]])/(4*b^(3/2)) - (d^2*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b
]])/(2*b^(3/2)) + (d^2*E^((2*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/b^(3/2) + (d^2*
Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(4*b^(3/2)*E^((4*a)/b)) - (d^2*Sqrt[Pi/2]*Erfi[(Sqrt[2]*S
qrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(2*b^(3/2)*E^((2*a)/b)) + (d^2*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[
c*x]])/Sqrt[b]])/(b^(3/2)*E^((2*a)/b)) - (2*d^2*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^2 \left (1+c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \text {arcsinh}(c x)}}-\frac {\left (2 d^2\right ) \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x^2 \sqrt {a+b \text {arcsinh}(c x)}} \, dx}{b c}+\frac {\left (8 c d^2\right ) \int \frac {\left (1+c^2 x^2\right )^{3/2}}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx}{b} \\ & = -\frac {2 d^2 \left (1+c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \text {arcsinh}(c x)}}+\frac {\left (8 d^2\right ) \text {Subst}\left (\int \frac {\cosh ^4\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2}-\frac {\left (2 d^2\right ) \int \left (\frac {2 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)}}+\frac {c^4 x^2}{\sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}\right ) \, dx}{b c} \\ & = -\frac {2 d^2 \left (1+c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \text {arcsinh}(c x)}}+\frac {\left (8 d^2\right ) \text {Subst}\left (\int \left (\frac {3}{8 \sqrt {x}}+\frac {\cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{8 \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 {\left (2 d^2\right ) \int \frac {1}{x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}} \, dx}{b c}-\frac {\left (4 c d^2\right ) \int \frac {1}{\sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}} \, dx}{b}-\frac {\left (2 c^3 d^2\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}} \, dx}{b} \\ & = -\frac {2 d^2 \left (1+c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \text {arcsinh}(c x)}}-\frac {2 d^2 \sqrt {a+b \text {arcsinh}(c x)}}{b^2}+\frac {d^2 \text {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2}-\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {\sinh ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2}+\frac {\left (4 d^2\right ) \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 {\left (2 d^2\right ) \int \frac {1}{x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}} \, dx}{b c} \\ & = -\frac {2 d^2 \left (1+c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \text {arcsinh}(c x)}}-\frac {2 d^2 \sqrt {a+b \text {arcsinh}(c x)}}{b^2}+\frac {d^2 \text {Subst}\left (\int \frac {e^{-i \left (\frac {4 i a}{b}-\frac {4 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{2 b^2}+\frac {d^2 \text {Subst}\left (\int \frac {e^{i \left (\frac {4 i a}{b}-\frac {4 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{2 b^2}+\frac {\left (2 d^2\right ) \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 {\left (2 d^2\right ) \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 {\left (2 d^2\right ) \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 {\left (2 d^2\right ) \int \frac {1}{x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}} \, dx}{b c} \\ & = -\frac {2 d^2 \left (1+c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \text {arcsinh}(c x)}}+\frac {d^2 \text {Subst}\left (\int e^{\frac {4 a}{b}-\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2}+\frac {d^2 \text {Subst}\left (\int e^{-\frac {4 a}{b}+\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2}-\frac {d^2 \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 {\left (4 d^2\right ) \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 {\left (4 d^2\right ) \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 {\left (2 d^2\right ) \int \frac {1}{x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}} \, dx}{b c} \\ & = -\frac {2 d^2 \left (1+c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \text {arcsinh}(c x)}}+\frac {d^2 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2}}+\frac {d^2 e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b^{3/2}}+\frac {d^2 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2}}+\frac {d^2 e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b^{3/2}}-\frac {d^2 \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 )}{2 b^2}-\frac {d^2 \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 )}{2 b^2}-\frac {\left (2 d^2\right ) \int \frac {1}{x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}} \, dx}{b c} \\ & = -\frac {2 d^2 \left (1+c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \text {arcsinh}(c x)}}+\frac {d^2 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2}}+\frac {d^2 e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b^{3/2}}+\frac {d^2 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2}}+\frac {d^2 e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b^{3/2}}-\frac {d^2 \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 {d^2 \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 {\left (2 d^2\right ) \int \frac {1}{x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}} \, dx}{b c} \\ & = -\frac {2 d^2 \left (1+c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \text {arcsinh}(c x)}}+\frac {d^2 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2}}-\frac {d^2 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 )}{2 b^{3/2}}+\frac {d^2 e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b^{3/2}}+\frac {d^2 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2}}-\frac {d^2 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 )}{2 b^{3/2}}+\frac {d^2 e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{b^{3/2}}-\frac {\left (2 d^2\right ) \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 = 1.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\left (d+c^2 d x^2\right )^2}{x (a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int \frac {\left (d+c^2 d x^2\right )^2}{x (a+b \text {arcsinh}(c x))^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate((c^2*d*x^2+d)^2/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 = 5.97 (sec) , antiderivative size = 131, normalized size of antiderivative = 4.68 \[ \int \frac {\left (d+c^2 d x^2\right )^2}{x (a+b \text {arcsinh}(c x))^{3/2}} \, dx=d^{2} \left (\int \frac {2 c^{2} x^{2}}{a x \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b x \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx + \int \frac {c^{4} x^{4}}{a x \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b x \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx + \int \frac {1}{a x \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b x \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx\right ) \]

[In]

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

[Out]

d**2*(Integral(2*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
(c**4*x**4/(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 = 0.74 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+c^2 d x^2\right )^2}{x (a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{2}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}} x} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

integrate((c^2*d*x^2+d)^2/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 = 2.83 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+c^2 d x^2\right )^2}{x (a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int \frac {{\left (d\,c^2\,x^2+d\right )}^2}{x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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