3.1.66 \(\int (\pi +c^2 \pi x^2)^{3/2} (a+b \text {arcsinh}(c x)) \, dx\) [66]

3.1.66.1 Optimal result

Integrand size = 23, antiderivative size = 111 \[ \int \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {5}{16} b c \pi ^{3/2} x^2-\frac {1}{16} b c^3 \pi ^{3/2} x^4+\frac {3}{8} \pi x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))+\frac {1}{4} x \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3 \pi ^{3/2} (a+b \text {arcsinh}(c x))^2}{16 b c} \]

output

-5/16*b*c*Pi^(3/2)*x^2-1/16*b*c^3*Pi^(3/2)*x^4+1/4*x*(Pi*c^2*x^2+Pi)^(3/2) 
*(a+b*arcsinh(c*x))+3/16*Pi^(3/2)*(a+b*arcsinh(c*x))^2/b/c+3/8*Pi*x*(a+b*a 
rcsinh(c*x))*(Pi*c^2*x^2+Pi)^(1/2)
 

3.1.66.2 Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00 \[ \int \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\pi ^{3/2} \left (80 a c x \sqrt {1+c^2 x^2}+32 a c^3 x^3 \sqrt {1+c^2 x^2}+24 b \text {arcsinh}(c x)^2-16 b \cosh (2 \text {arcsinh}(c x))-b \cosh (4 \text {arcsinh}(c x))+4 \text {arcsinh}(c x) (12 a+8 b \sinh (2 \text {arcsinh}(c x))+b \sinh (4 \text {arcsinh}(c x)))\right )}{128 c} \]

input

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

output

(Pi^(3/2)*(80*a*c*x*Sqrt[1 + c^2*x^2] + 32*a*c^3*x^3*Sqrt[1 + c^2*x^2] + 2 
4*b*ArcSinh[c*x]^2 - 16*b*Cosh[2*ArcSinh[c*x]] - b*Cosh[4*ArcSinh[c*x]] + 
4*ArcSinh[c*x]*(12*a + 8*b*Sinh[2*ArcSinh[c*x]] + b*Sinh[4*ArcSinh[c*x]])) 
)/(128*c)
 

3.1.66.3 Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.16, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6201, 244, 2009, 6200, 15, 6198}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx\)

\(\Big \downarrow \) 6201

\(\displaystyle \frac {3}{4} \pi \int \sqrt {c^2 \pi x^2+\pi } (a+b \text {arcsinh}(c x))dx-\frac {1}{4} \pi ^{3/2} b c \int x \left (c^2 x^2+1\right )dx+\frac {1}{4} x \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 244

\(\displaystyle \frac {3}{4} \pi \int \sqrt {c^2 \pi x^2+\pi } (a+b \text {arcsinh}(c x))dx-\frac {1}{4} \pi ^{3/2} b c \int \left (c^2 x^3+x\right )dx+\frac {1}{4} x \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3}{4} \pi \int \sqrt {c^2 \pi x^2+\pi } (a+b \text {arcsinh}(c x))dx+\frac {1}{4} x \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{4} \pi ^{3/2} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\)

\(\Big \downarrow \) 6200

\(\displaystyle \frac {3}{4} \pi \left (\frac {1}{2} \sqrt {\pi } \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx-\frac {1}{2} \sqrt {\pi } b c \int xdx+\frac {1}{2} x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))\right )+\frac {1}{4} x \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{4} \pi ^{3/2} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {3}{4} \pi \left (\frac {1}{2} \sqrt {\pi } \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))-\frac {1}{4} \sqrt {\pi } b c x^2\right )+\frac {1}{4} x \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{4} \pi ^{3/2} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\)

\(\Big \downarrow \) 6198

\(\displaystyle \frac {1}{4} x \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \pi \left (\frac {1}{2} x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))+\frac {\sqrt {\pi } (a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} \sqrt {\pi } b c x^2\right )-\frac {1}{4} \pi ^{3/2} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\)

input

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

output

-1/4*(b*c*Pi^(3/2)*(x^2/2 + (c^2*x^4)/4)) + (x*(Pi + c^2*Pi*x^2)^(3/2)*(a 
+ b*ArcSinh[c*x]))/4 + (3*Pi*(-1/4*(b*c*Sqrt[Pi]*x^2) + (x*Sqrt[Pi + c^2*P 
i*x^2]*(a + b*ArcSinh[c*x]))/2 + (Sqrt[Pi]*(a + b*ArcSinh[c*x])^2)/(4*b*c) 
))/4
 

3.1.66.3.1 Defintions of rubi rules used

rule 15

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

rule 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6198

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( 
a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c 
^2*d] && NeQ[n, -1]
 

rule 6200

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/2), x] + (Simp[(1 
/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   Int[(a + b*ArcSinh[c*x])^n/Sq 
rt[1 + c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2* 
x^2]]   Int[x*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[e, c^2*d] && GtQ[n, 0]
 

rule 6201

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

3.1.66.4 Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.37

method	result	size
default	\(\frac {x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}} a}{4}+\frac {3 a \pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}{8}+\frac {3 a \,\pi ^{2} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{8 \sqrt {\pi \,c^{2}}}+\frac {b \,\pi ^{\frac {3}{2}} \left (4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}-c^{4} x^{4}+10 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}-5 c^{2} x^{2}+3 \operatorname {arcsinh}\left (c x \right )^{2}-4\right )}{16 c}\)	\(152\)
parts	\(\frac {x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}} a}{4}+\frac {3 a \pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}{8}+\frac {3 a \,\pi ^{2} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{8 \sqrt {\pi \,c^{2}}}+\frac {b \,\pi ^{\frac {3}{2}} \left (4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}-c^{4} x^{4}+10 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}-5 c^{2} x^{2}+3 \operatorname {arcsinh}\left (c x \right )^{2}-4\right )}{16 c}\)	\(152\)

input

int((Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(c*x)),x,method=_RETURNVERBOSE)
 

output

1/4*x*(Pi*c^2*x^2+Pi)^(3/2)*a+3/8*a*Pi*x*(Pi*c^2*x^2+Pi)^(1/2)+3/8*a*Pi^2* 
ln(Pi*c^2*x/(Pi*c^2)^(1/2)+(Pi*c^2*x^2+Pi)^(1/2))/(Pi*c^2)^(1/2)+1/16*b*Pi 
^(3/2)*(4*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x^3*c^3-c^4*x^4+10*arcsinh(c*x)*c 
*x*(c^2*x^2+1)^(1/2)-5*c^2*x^2+3*arcsinh(c*x)^2-4)/c
 

3.1.66.5 Fricas [F]

\[ \int \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} \,d x } \]

input

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

output

integral(sqrt(pi + pi*c^2*x^2)*(pi*a*c^2*x^2 + pi*a + (pi*b*c^2*x^2 + pi*b 
)*arcsinh(c*x)), x)
 

3.1.66.6 Sympy [A] (verification not implemented)

Time = 1.58 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.67 \[ \int \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {\pi ^{\frac {3}{2}} a c^{2} x^{3} \sqrt {c^{2} x^{2} + 1}}{4} + \frac {5 \pi ^{\frac {3}{2}} a x \sqrt {c^{2} x^{2} + 1}}{8} + \frac {3 \pi ^{\frac {3}{2}} a \operatorname {asinh}{\left (c x \right )}}{8 c} - \frac {\pi ^{\frac {3}{2}} b c^{3} x^{4}}{16} + \frac {\pi ^{\frac {3}{2}} b c^{2} x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{4} - \frac {5 \pi ^{\frac {3}{2}} b c x^{2}}{16} + \frac {5 \pi ^{\frac {3}{2}} b x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{8} + \frac {3 \pi ^{\frac {3}{2}} b \operatorname {asinh}^{2}{\left (c x \right )}}{16 c} & \text {for}\: c \neq 0 \\\pi ^{\frac {3}{2}} a x & \text {otherwise} \end {cases} \]

input

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

output

Piecewise((pi**(3/2)*a*c**2*x**3*sqrt(c**2*x**2 + 1)/4 + 5*pi**(3/2)*a*x*s 
qrt(c**2*x**2 + 1)/8 + 3*pi**(3/2)*a*asinh(c*x)/(8*c) - pi**(3/2)*b*c**3*x 
**4/16 + pi**(3/2)*b*c**2*x**3*sqrt(c**2*x**2 + 1)*asinh(c*x)/4 - 5*pi**(3 
/2)*b*c*x**2/16 + 5*pi**(3/2)*b*x*sqrt(c**2*x**2 + 1)*asinh(c*x)/8 + 3*pi* 
*(3/2)*b*asinh(c*x)**2/(16*c), Ne(c, 0)), (pi**(3/2)*a*x, True))
 

3.1.66.7 Maxima [F(-2)]

Exception generated. \[ \int \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]

input

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

output

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

3.1.66.8 Giac [F(-2)]

Exception generated. \[ \int \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \]

input

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

output

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

3.1.66.9 Mupad [F(-1)]

Timed out. \[ \int \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2} \,d x \]

input

int((a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(3/2),x)
 

output

int((a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(3/2), x)