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

Optimal. Leaf size=374 \[ -\frac{2 d^2 \text{Unintegrable}\left (\frac{1}{x^2 \sqrt{c^2 x^2+1} \sqrt{a+b \sinh ^{-1}(c x)}},x\right )}{b c}+\frac{\sqrt{\pi } d^2 e^{\frac{4 a}{b}} \text{Erf}\left (\frac{2 \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2}}+\frac{\sqrt{2 \pi } d^2 e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} d^2 e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{3/2}}+\frac{\sqrt{\pi } d^2 e^{-\frac{4 a}{b}} \text{Erfi}\left (\frac{2 \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2}}+\frac{\sqrt{2 \pi } d^2 e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} d^2 e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{3/2}}-\frac{2 d^2 \left (c^2 x^2+1\right )^{5/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}} \]

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

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Rubi [A]  time = 1.44103, 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 = {} \[ \int \frac{\left (d+c^2 d x^2\right )^2}{x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(d + c^2*d*x^2)^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*} \int \frac{\left (d+c^2 d x^2\right )^2}{x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac{2 d^2 \left (1+c^2 x^2\right )^{5/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{\left (2 d^2\right ) \int \frac{\left (1+c^2 x^2\right )^{3/2}}{x^2 \sqrt{a+b \sinh ^{-1}(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 \sinh ^{-1}(c x)}} \, dx}{b}\\ &=-\frac{2 d^2 \left (1+c^2 x^2\right )^{5/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{\left (8 d^2\right ) \operatorname{Subst}\left (\int \frac{\cosh ^4(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b}-\frac{\left (2 d^2\right ) \int \left (\frac{2 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)}}+\frac{c^4 x^2}{\sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(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 \sinh ^{-1}(c x)}}+\frac{\left (8 d^2\right ) \operatorname{Subst}\left (\int \left (\frac{3}{8 \sqrt{a+b x}}+\frac{\cosh (2 x)}{2 \sqrt{a+b x}}+\frac{\cosh (4 x)}{8 \sqrt{a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b}-\frac{\left (2 d^2\right ) \int \frac{1}{x^2 \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{b c}-\frac{\left (4 c d^2\right ) \int \frac{1}{\sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(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 \sinh ^{-1}(c x)}} \, dx}{b}\\ &=-\frac{2 d^2 \left (1+c^2 x^2\right )^{5/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{2 d^2 \sqrt{a+b \sinh ^{-1}(c x)}}{b^2}+\frac{d^2 \operatorname{Subst}\left (\int \frac{\cosh (4 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b}-\frac{\left (2 d^2\right ) \operatorname{Subst}\left (\int \frac{\sinh ^2(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b}+\frac{\left (4 d^2\right ) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b}-\frac{\left (2 d^2\right ) \int \frac{1}{x^2 \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{b c}\\ &=-\frac{2 d^2 \left (1+c^2 x^2\right )^{5/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{2 d^2 \sqrt{a+b \sinh ^{-1}(c x)}}{b^2}+\frac{d^2 \operatorname{Subst}\left (\int \frac{e^{-4 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b}+\frac{d^2 \operatorname{Subst}\left (\int \frac{e^{4 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b}+\frac{\left (2 d^2\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b}+\frac{\left (2 d^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b}+\frac{\left (2 d^2\right ) \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{\left (2 d^2\right ) \int \frac{1}{x^2 \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{b c}\\ &=-\frac{2 d^2 \left (1+c^2 x^2\right )^{5/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{d^2 \operatorname{Subst}\left (\int e^{\frac{4 a}{b}-\frac{4 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{b^2}+\frac{d^2 \operatorname{Subst}\left (\int e^{-\frac{4 a}{b}+\frac{4 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{b^2}+\frac{\left (4 d^2\right ) \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{\left (4 d^2\right ) \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{d^2 \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b}-\frac{\left (2 d^2\right ) \int \frac{1}{x^2 \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{b c}\\ &=-\frac{2 d^2 \left (1+c^2 x^2\right )^{5/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{d^2 e^{\frac{4 a}{b}} \sqrt{\pi } \text{erf}\left (\frac{2 \sqrt{a+b \sinh ^{-1}(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 \sinh ^{-1}(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 \sinh ^{-1}(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 \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2}}-\frac{d^2 \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b}-\frac{d^2 \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b}-\frac{\left (2 d^2\right ) \int \frac{1}{x^2 \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{b c}\\ &=-\frac{2 d^2 \left (1+c^2 x^2\right )^{5/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{d^2 e^{\frac{4 a}{b}} \sqrt{\pi } \text{erf}\left (\frac{2 \sqrt{a+b \sinh ^{-1}(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 \sinh ^{-1}(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 \sinh ^{-1}(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 \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2}}-\frac{d^2 \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{d^2 \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{\left (2 d^2\right ) \int \frac{1}{x^2 \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{b c}\\ &=-\frac{2 d^2 \left (1+c^2 x^2\right )^{5/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{d^2 e^{\frac{4 a}{b}} \sqrt{\pi } \text{erf}\left (\frac{2 \sqrt{a+b \sinh ^{-1}(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 \sinh ^{-1}(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 \sinh ^{-1}(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 \sinh ^{-1}(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 \sinh ^{-1}(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 \sinh ^{-1}(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 \sinh ^{-1}(c x)}} \, dx}{b c}\\ \end{align*}

Mathematica [A]  time = 3.12948, size = 0, normalized size = 0. \[ \int \frac{\left (d+c^2 d x^2\right )^2}{x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(d + c^2*d*x^2)^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]

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Maple [A]  time = 0.243, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({c}^{2}d{x}^{2}+d \right ) ^{2}}{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)^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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[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))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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