\(\int x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx\) [57]

Optimal result

Integrand size = 24, antiderivative size = 61 \[ \int x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {b \sqrt {\pi } x}{3 c}-\frac {1}{9} b c \sqrt {\pi } x^3+\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2 \pi } \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {5798} \[ \int x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 \pi c^2}-\frac {1}{9} \sqrt {\pi } b c x^3-\frac {\sqrt {\pi } b x}{3 c} \]

[In]

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

[Out]

-1/3*(b*Sqrt[Pi]*x)/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)

Rule 5798

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)
^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*} \text {integral}& = \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2 \pi }-\frac {\left (b \sqrt {\pi }\right ) \int \left (1+c^2 x^2\right ) \, dx}{3 c} \\ & = -\frac {b \sqrt {\pi } x}{3 c}-\frac {1}{9} b c \sqrt {\pi } x^3+\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2 \pi } \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.03 \[ \int x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {\pi } \left (3 a \left (1+c^2 x^2\right )^{3/2}-b c x \left (3+c^2 x^2\right )+3 b \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)\right )}{9 c^2} \]

[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)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(107\) vs. \(2(49)=98\).

Time = 0.21 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.77

method	result	size
default	\(\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3 \pi \,c^{2}}+\frac {b \sqrt {\pi }\, \left (3 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}+6 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}-c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+3 \,\operatorname {arcsinh}\left (c x \right )-3 c x \sqrt {c^{2} x^{2}+1}\right )}{9 c^{2} \sqrt {c^{2} x^{2}+1}}\)	\(108\)
parts	\(\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3 \pi \,c^{2}}+\frac {b \sqrt {\pi }\, \left (3 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}+6 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}-c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+3 \,\operatorname {arcsinh}\left (c x \right )-3 c x \sqrt {c^{2} x^{2}+1}\right )}{9 c^{2} \sqrt {c^{2} x^{2}+1}}\)	\(108\)

[In]

int(x*(a+b*arcsinh(c*x))*(Pi*c^2*x^2+Pi)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/3*a*(Pi*c^2*x^2+Pi)^(3/2)/Pi/c^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))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (49) = 98\).

Time = 0.26 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.08 \[ \int x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {3 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (b c^{4} x^{4} + 2 \, b c^{2} x^{2} + b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (3 \, a c^{4} x^{4} + 6 \, a c^{2} x^{2} - {\left (b c^{3} x^{3} + 3 \, b c x\right )} \sqrt {c^{2} x^{2} + 1} + 3 \, a\right )}}{9 \, {\left (c^{4} x^{2} + c^{2}\right )}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (53) = 106\).

Time = 0.34 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.31 \[ \int x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {\sqrt {\pi } a x^{2} \sqrt {c^{2} x^{2} + 1}}{3} + \frac {\sqrt {\pi } a \sqrt {c^{2} x^{2} + 1}}{3 c^{2}} - \frac {\sqrt {\pi } b c x^{3}}{9} + \frac {\sqrt {\pi } b x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {\sqrt {\pi } b x}{3 c} + \frac {\sqrt {\pi } b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3 c^{2}} & \text {for}\: c \neq 0 \\\frac {\sqrt {\pi } a x^{2}}{2} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.20 \[ \int x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} b \operatorname {arsinh}\left (c x\right )}{3 \, \pi c^{2}} - \frac {{\left (\pi ^{\frac {3}{2}} c^{2} x^{3} + 3 \, \pi ^{\frac {3}{2}} x\right )} b}{9 \, \pi c} + \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} a}{3 \, \pi c^{2}} \]

[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)

Giac [F(-2)]

Exception generated. \[ \int x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\int x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {\Pi \,c^2\,x^2+\Pi } \,d x \]

[In]

int(x*(a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(1/2),x)

[Out]

int(x*(a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(1/2), x)