3.57 \(\int x \sqrt{\pi +c^2 \pi x^2} (a+b \sinh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=61 \[ \frac{\left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi c^2}-\frac{1}{9} \sqrt{\pi } b c x^3-\frac{\sqrt{\pi } b x}{3 c} \]

[Out]

-(b*Sqrt[Pi]*x)/(3*c) - (b*c*Sqrt[Pi]*x^3)/9 + ((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*c^2*Pi)

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Rubi [A]  time = 0.0678192, antiderivative size = 105, normalized size of antiderivative = 1.72, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {5717} \[ \frac{\left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi c^2}-\frac{b c x^3 \sqrt{\pi c^2 x^2+\pi }}{9 \sqrt{c^2 x^2+1}}-\frac{b x \sqrt{\pi c^2 x^2+\pi }}{3 c \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*x*Sqrt[Pi + c^2*Pi*x^2])/(3*c*Sqrt[1 + c^2*x^2]) - (b*c*x^3*Sqrt[Pi + c^2*Pi*x^2])/(9*Sqrt[1 + c^2*x^2]) +
((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*c^2*Pi)

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]

Rubi steps

\begin{align*} \int x \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac{\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \pi }-\frac{\left (b \sqrt{\pi +c^2 \pi x^2}\right ) \int \left (1+c^2 x^2\right ) \, dx}{3 c \sqrt{1+c^2 x^2}}\\ &=-\frac{b x \sqrt{\pi +c^2 \pi x^2}}{3 c \sqrt{1+c^2 x^2}}-\frac{b c x^3 \sqrt{\pi +c^2 \pi x^2}}{9 \sqrt{1+c^2 x^2}}+\frac{\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \pi }\\ \end{align*}

Mathematica [A]  time = 0.116606, size = 63, normalized size = 1.03 \[ \frac{\sqrt{\pi } \left (3 a \left (c^2 x^2+1\right )^{3/2}-b c x \left (c^2 x^2+3\right )+3 b \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x)\right )}{9 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[Pi]*(3*a*(1 + c^2*x^2)^(3/2) - b*c*x*(3 + c^2*x^2) + 3*b*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]))/(9*c^2)

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Maple [B]  time = 0.047, size = 108, normalized size = 1.8 \begin{align*}{\frac{a}{3\,\pi \,{c}^{2}} \left ( \pi \,{c}^{2}{x}^{2}+\pi \right ) ^{{\frac{3}{2}}}}+{\frac{b\sqrt{\pi }}{9\,{c}^{2}} \left ( 3\,{\it Arcsinh} \left ( cx \right ){c}^{4}{x}^{4}+6\,{\it Arcsinh} \left ( cx \right ){c}^{2}{x}^{2}-{c}^{3}{x}^{3}\sqrt{{c}^{2}{x}^{2}+1}+3\,{\it Arcsinh} \left ( cx \right ) -3\,cx\sqrt{{c}^{2}{x}^{2}+1} \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))*(Pi*c^2*x^2+Pi)^(1/2),x)

[Out]

1/3*a/Pi/c^2*(Pi*c^2*x^2+Pi)^(3/2)+1/9*b/c^2*Pi^(1/2)/(c^2*x^2+1)^(1/2)*(3*arcsinh(c*x)*c^4*x^4+6*arcsinh(c*x)
*c^2*x^2-c^3*x^3*(c^2*x^2+1)^(1/2)+3*arcsinh(c*x)-3*c*x*(c^2*x^2+1)^(1/2))

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Maxima [A]  time = 1.22158, size = 99, normalized size = 1.62 \begin{align*} \frac{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac{3}{2}} b \operatorname{arsinh}\left (c x\right )}{3 \, \pi c^{2}} - \frac{{\left (\pi ^{\frac{3}{2}} c^{2} x^{3} + 3 \, \pi ^{\frac{3}{2}} x\right )} b}{9 \, \pi c} + \frac{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac{3}{2}} a}{3 \, \pi c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(pi + pi*c^2*x^2)^(3/2)*b*arcsinh(c*x)/(pi*c^2) - 1/9*(pi^(3/2)*c^2*x^3 + 3*pi^(3/2)*x)*b/(pi*c) + 1/3*(pi
+ pi*c^2*x^2)^(3/2)*a/(pi*c^2)

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Fricas [B]  time = 2.54178, size = 278, normalized size = 4.56 \begin{align*} \frac{3 \, \sqrt{\pi + \pi c^{2} x^{2}}{\left (b c^{4} x^{4} + 2 \, b c^{2} x^{2} + b\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + \sqrt{\pi + \pi c^{2} x^{2}}{\left (3 \, a c^{4} x^{4} + 6 \, a c^{2} x^{2} -{\left (b c^{3} x^{3} + 3 \, b c x\right )} \sqrt{c^{2} x^{2} + 1} + 3 \, a\right )}}{9 \,{\left (c^{4} x^{2} + c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(3*sqrt(pi + pi*c^2*x^2)*(b*c^4*x^4 + 2*b*c^2*x^2 + b)*log(c*x + sqrt(c^2*x^2 + 1)) + sqrt(pi + pi*c^2*x^2
)*(3*a*c^4*x^4 + 6*a*c^2*x^2 - (b*c^3*x^3 + 3*b*c*x)*sqrt(c^2*x^2 + 1) + 3*a))/(c^4*x^2 + c^2)

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Sympy [A]  time = 2.65681, size = 141, normalized size = 2.31 \begin{align*} \begin{cases} \frac{\sqrt{\pi } a x^{2} \sqrt{c^{2} x^{2} + 1}}{3} + \frac{\sqrt{\pi } a \sqrt{c^{2} x^{2} + 1}}{3 c^{2}} - \frac{\sqrt{\pi } b c x^{3}}{9} + \frac{\sqrt{\pi } b x^{2} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{3} - \frac{\sqrt{\pi } b x}{3 c} + \frac{\sqrt{\pi } b \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{3 c^{2}} & \text{for}\: c \neq 0 \\\frac{\sqrt{\pi } a x^{2}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((sqrt(pi)*a*x**2*sqrt(c**2*x**2 + 1)/3 + sqrt(pi)*a*sqrt(c**2*x**2 + 1)/(3*c**2) - sqrt(pi)*b*c*x**3
/9 + sqrt(pi)*b*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/3 - sqrt(pi)*b*x/(3*c) + sqrt(pi)*b*sqrt(c**2*x**2 + 1)*as
inh(c*x)/(3*c**2), Ne(c, 0)), (sqrt(pi)*a*x**2/2, True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError