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

Optimal. Leaf size=138 \[ -\frac{2 a b x \sqrt{c^2 x^2+1}}{c \sqrt{c^2 d x^2+d}}+\frac{\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d}+\frac{2 b^2 \left (c^2 x^2+1\right )}{c^2 \sqrt{c^2 d x^2+d}}-\frac{2 b^2 x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{c \sqrt{c^2 d x^2+d}} \]

[Out]

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

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Rubi [A]  time = 0.124207, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {5717, 5653, 261} \[ -\frac{2 a b x \sqrt{c^2 x^2+1}}{c \sqrt{c^2 d x^2+d}}+\frac{\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d}+\frac{2 b^2 \left (c^2 x^2+1\right )}{c^2 \sqrt{c^2 d x^2+d}}-\frac{2 b^2 x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{c \sqrt{c^2 d x^2+d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5717

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)
^(p + 1)*(a + b*ArcSinh[c*x])^n)/(2*e*(p + 1)), x] - Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p +
1)*(1 + c^2*x^2)^FracPart[p]), Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[{a,
b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]

Rule 5653

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{d+c^2 d x^2}} \, dx &=\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d}-\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \int \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{c \sqrt{d+c^2 d x^2}}\\ &=-\frac{2 a b x \sqrt{1+c^2 x^2}}{c \sqrt{d+c^2 d x^2}}+\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d}-\frac{\left (2 b^2 \sqrt{1+c^2 x^2}\right ) \int \sinh ^{-1}(c x) \, dx}{c \sqrt{d+c^2 d x^2}}\\ &=-\frac{2 a b x \sqrt{1+c^2 x^2}}{c \sqrt{d+c^2 d x^2}}-\frac{2 b^2 x \sqrt{1+c^2 x^2} \sinh ^{-1}(c x)}{c \sqrt{d+c^2 d x^2}}+\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d}+\frac{\left (2 b^2 \sqrt{1+c^2 x^2}\right ) \int \frac{x}{\sqrt{1+c^2 x^2}} \, dx}{\sqrt{d+c^2 d x^2}}\\ &=-\frac{2 a b x \sqrt{1+c^2 x^2}}{c \sqrt{d+c^2 d x^2}}+\frac{2 b^2 \left (1+c^2 x^2\right )}{c^2 \sqrt{d+c^2 d x^2}}-\frac{2 b^2 x \sqrt{1+c^2 x^2} \sinh ^{-1}(c x)}{c \sqrt{d+c^2 d x^2}}+\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d}\\ \end{align*}

Mathematica [A]  time = 0.28431, size = 127, normalized size = 0.92 \[ \frac{\sqrt{c^2 d x^2+d} \left (a^2 \sqrt{c^2 x^2+1}-2 b \sinh ^{-1}(c x) \left (b c x-a \sqrt{c^2 x^2+1}\right )-2 a b c x+2 b^2 \sqrt{c^2 x^2+1}+b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^2\right )}{c^2 d \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.155, size = 296, normalized size = 2.1 \begin{align*}{\frac{{a}^{2}}{{c}^{2}d}\sqrt{{c}^{2}d{x}^{2}+d}}+{b}^{2} \left ({\frac{ \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}-2\,{\it Arcsinh} \left ( cx \right ) +2}{2\,{c}^{2}d \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) } \left ({c}^{2}{x}^{2}+cx\sqrt{{c}^{2}{x}^{2}+1}+1 \right ) }+{\frac{ \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}+2\,{\it Arcsinh} \left ( cx \right ) +2}{2\,{c}^{2}d \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) } \left ({c}^{2}{x}^{2}-cx\sqrt{{c}^{2}{x}^{2}+1}+1 \right ) } \right ) +2\,ab \left ( 1/2\,{\frac{\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) } \left ({c}^{2}{x}^{2}+cx\sqrt{{c}^{2}{x}^{2}+1}+1 \right ) \left ( -1+{\it Arcsinh} \left ( cx \right ) \right ) }{{c}^{2}d \left ({c}^{2}{x}^{2}+1 \right ) }}+1/2\,{\frac{\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) } \left ({c}^{2}{x}^{2}-cx\sqrt{{c}^{2}{x}^{2}+1}+1 \right ) \left ( 1+{\it Arcsinh} \left ( cx \right ) \right ) }{{c}^{2}d \left ({c}^{2}{x}^{2}+1 \right ) }} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.27355, size = 169, normalized size = 1.22 \begin{align*} -2 \, b^{2}{\left (\frac{x \operatorname{arsinh}\left (c x\right )}{c \sqrt{d}} - \frac{\sqrt{c^{2} x^{2} + 1}}{c^{2} \sqrt{d}}\right )} - \frac{2 \, a b x}{c \sqrt{d}} + \frac{\sqrt{c^{2} d x^{2} + d} b^{2} \operatorname{arsinh}\left (c x\right )^{2}}{c^{2} d} + \frac{2 \, \sqrt{c^{2} d x^{2} + d} a b \operatorname{arsinh}\left (c x\right )}{c^{2} d} + \frac{\sqrt{c^{2} d x^{2} + d} a^{2}}{c^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 3.20107, size = 385, normalized size = 2.79 \begin{align*} \frac{{\left (b^{2} c^{2} x^{2} + b^{2}\right )} \sqrt{c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 2 \,{\left (a b c^{2} x^{2} - \sqrt{c^{2} x^{2} + 1} b^{2} c x + a b\right )} \sqrt{c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) +{\left ({\left (a^{2} + 2 \, b^{2}\right )} c^{2} x^{2} - 2 \, \sqrt{c^{2} x^{2} + 1} a b c x + a^{2} + 2 \, b^{2}\right )} \sqrt{c^{2} d x^{2} + d}}{c^{4} d x^{2} + c^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{2}}{\sqrt{d \left (c^{2} x^{2} + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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