3.466 \(\int \frac{d+c^2 d x^2}{x (a+b \sinh ^{-1}(c x))^{3/2}} \, dx\)

Optimal. Leaf size=174 \[ -\frac{2 d \text{Unintegrable}\left (\frac{1}{x^2 \sqrt{c^2 x^2+1} \sqrt{a+b \sinh ^{-1}(c x)}},x\right )}{b c}+\frac{\sqrt{\frac{\pi }{2}} d e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2}}+\frac{\sqrt{\frac{\pi }{2}} d e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2}}-\frac{2 d \left (c^2 x^2+1\right )^{3/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}} \]

[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*Unintegrable[1/(x^2*Sqrt[1 + c^2*x^2]*Sqrt[a + b*ArcSinh[c*x]]), x])/(b*c)

________________________________________________________________________________________

Rubi [A]  time = 0.796411, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{d+c^2 d x^2}{x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[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*} \int \frac{d+c^2 d x^2}{x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac{2 d \left (1+c^2 x^2\right )^{3/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{(2 d) \int \frac{\sqrt{1+c^2 x^2}}{x^2 \sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{b c}+\frac{(4 c d) \int \frac{\sqrt{1+c^2 x^2}}{\sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{b}\\ &=-\frac{2 d \left (1+c^2 x^2\right )^{3/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{(4 d) \operatorname{Subst}\left (\int \frac{\cosh ^2(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b}-\frac{(2 d) \int \left (\frac{c^2}{\sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{1}{x^2 \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}\right ) \, dx}{b c}\\ &=-\frac{2 d \left (1+c^2 x^2\right )^{3/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{(4 d) \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{a+b x}}+\frac{\cosh (2 x)}{2 \sqrt{a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b}-\frac{(2 d) \int \frac{1}{x^2 \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{b c}-\frac{(2 c d) \int \frac{1}{\sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{b}\\ &=-\frac{2 d \left (1+c^2 x^2\right )^{3/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{(2 d) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b}-\frac{(2 d) \int \frac{1}{x^2 \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{b c}\\ &=-\frac{2 d \left (1+c^2 x^2\right )^{3/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{d \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b}+\frac{d \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b}-\frac{(2 d) \int \frac{1}{x^2 \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{b c}\\ &=-\frac{2 d \left (1+c^2 x^2\right )^{3/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{(2 d) \operatorname{Subst}\left (\int e^{\frac{2 a}{b}-\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{b^2}+\frac{(2 d) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b}+\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{b^2}-\frac{(2 d) \int \frac{1}{x^2 \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{b c}\\ &=-\frac{2 d \left (1+c^2 x^2\right )^{3/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{d e^{\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(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 \sinh ^{-1}(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 \sinh ^{-1}(c x)}} \, dx}{b c}\\ \end{align*}

Mathematica [A]  time = 4.07211, size = 0, normalized size = 0. \[ \int \frac{d+c^2 d x^2}{x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[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 [A]  time = 0.184, size = 0, normalized size = 0. \begin{align*} \int{\frac{{c}^{2}d{x}^{2}+d}{x} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{c^{2} d x^{2} + d}{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} 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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{c^{2} d x^{2} + d}{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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