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

Optimal. Leaf size=70 \[ \frac{b \sqrt{c^2 x^2+1} \tan ^{-1}(c x)}{c^2 d \sqrt{c^2 d x^2+d}}-\frac{a+b \sinh ^{-1}(c x)}{c^2 d \sqrt{c^2 d x^2+d}} \]

[Out]

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

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Rubi [A]  time = 0.0703006, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {5717, 203} \[ \frac{b \sqrt{c^2 x^2+1} \tan ^{-1}(c x)}{c^2 d \sqrt{c^2 d x^2+d}}-\frac{a+b \sinh ^{-1}(c x)}{c^2 d \sqrt{c^2 d x^2+d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\left (d+c^2 d x^2\right )^{3/2}} \, dx &=-\frac{a+b \sinh ^{-1}(c x)}{c^2 d \sqrt{d+c^2 d x^2}}+\frac{\left (b \sqrt{1+c^2 x^2}\right ) \int \frac{1}{1+c^2 x^2} \, dx}{c d \sqrt{d+c^2 d x^2}}\\ &=-\frac{a+b \sinh ^{-1}(c x)}{c^2 d \sqrt{d+c^2 d x^2}}+\frac{b \sqrt{1+c^2 x^2} \tan ^{-1}(c x)}{c^2 d \sqrt{d+c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 0.162319, size = 82, normalized size = 1.17 \[ \frac{b \sqrt{d \left (c^2 x^2+1\right )} \tan ^{-1}(c x)}{c^2 d^2 \sqrt{c^2 x^2+1}}-\frac{\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d^2 \left (c^2 x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.083, size = 164, normalized size = 2.3 \begin{align*} -{\frac{a}{{c}^{2}d}{\frac{1}{\sqrt{{c}^{2}d{x}^{2}+d}}}}-{\frac{b{\it Arcsinh} \left ( cx \right ) }{{c}^{2}{d}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}+{\frac{ib}{{c}^{2}{d}^{2}}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}+1}+i \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}-{\frac{ib}{{c}^{2}{d}^{2}}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}+1}-i \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} b{\left (\frac{-\operatorname{arsinh}\left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right )}{c^{2} d^{\frac{3}{2}}} - \frac{\log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{\sqrt{c^{2} x^{2} + 1} c^{2} d^{\frac{3}{2}}} - \int \frac{1}{c^{5} d^{\frac{3}{2}} x^{4} + c^{3} d^{\frac{3}{2}} x^{2} +{\left (c^{4} d^{\frac{3}{2}} x^{3} + c^{2} d^{\frac{3}{2}} x\right )} \sqrt{c^{2} x^{2} + 1}}\,{d x}\right )} - \frac{a}{\sqrt{c^{2} d x^{2} + d} c^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*(integrate(1/(sqrt(c^2*x^2 + 1)*x), x)/(c^2*d^(3/2)) - log(c*x + sqrt(c^2*x^2 + 1))/(sqrt(c^2*x^2 + 1)*c^2*d
^(3/2)) - integrate(1/(c^5*d^(3/2)*x^4 + c^3*d^(3/2)*x^2 + (c^4*d^(3/2)*x^3 + c^2*d^(3/2)*x)*sqrt(c^2*x^2 + 1)
), x)) - a/(sqrt(c^2*d*x^2 + d)*c^2*d)

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Fricas [B]  time = 3.18023, size = 288, normalized size = 4.11 \begin{align*} -\frac{{\left (b c^{2} x^{2} + b\right )} \sqrt{d} \arctan \left (\frac{2 \, \sqrt{c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} + 1} c \sqrt{d} x}{c^{4} d x^{4} - d}\right ) + 2 \, \sqrt{c^{2} d x^{2} + d} b \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + 2 \, \sqrt{c^{2} d x^{2} + d} a}{2 \,{\left (c^{4} d^{2} x^{2} + c^{2} d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*(a + b*asinh(c*x))/(d*(c**2*x**2 + 1))**(3/2), 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 )} x}{{\left (c^{2} d x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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