\(\int \frac {(\pi +c^2 \pi x^2)^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx\) [72]

Optimal result

Integrand size = 26, antiderivative size = 108 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=-\frac {1}{4} b c^3 \pi ^{3/2} x^2+\frac {3}{2} c^2 \pi x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+\frac {3 c \pi ^{3/2} (a+b \text {arcsinh}(c x))^2}{4 b}+b c \pi ^{3/2} \log (x) \] Output:

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

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.13 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\frac {\pi ^{3/2} \left (-8 a \sqrt {1+c^2 x^2}+4 a c^2 x^2 \sqrt {1+c^2 x^2}+6 b c x \text {arcsinh}(c x)^2-b c x \cosh (2 \text {arcsinh}(c x))+8 b c x \log (c x)+2 \text {arcsinh}(c x) \left (6 a c x-4 b \sqrt {1+c^2 x^2}+b c x \sinh (2 \text {arcsinh}(c x))\right )\right )}{8 x} \] Input:

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

Output:

(Pi^(3/2)*(-8*a*Sqrt[1 + c^2*x^2] + 4*a*c^2*x^2*Sqrt[1 + c^2*x^2] + 6*b*c* 
x*ArcSinh[c*x]^2 - b*c*x*Cosh[2*ArcSinh[c*x]] + 8*b*c*x*Log[c*x] + 2*ArcSi 
nh[c*x]*(6*a*c*x - 4*b*Sqrt[1 + c^2*x^2] + b*c*x*Sinh[2*ArcSinh[c*x]])))/( 
8*x)
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {6222, 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.

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

\(\Big \downarrow \) 6222

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

\(\Big \downarrow \) 244

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6200

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6198

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

Input:

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

Output:

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

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 6222

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(207\) vs. \(2(92)=184\).

Time = 1.00 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.93

method	result	size
default	\(-\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{\pi x}+a \,c^{2} x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}+\frac {3 a \,c^{2} \pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}{2}+\frac {3 a \,c^{2} \pi ^{2} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{2 \sqrt {\pi \,c^{2}}}+\frac {b \,\pi ^{\frac {3}{2}} \left (4 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}-2 x^{3} c^{3}+6 \operatorname {arcsinh}\left (x c \right )^{2} x c -8 x c \,\operatorname {arcsinh}\left (x c \right )+8 \ln \left (\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) x c -8 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}-x c \right )}{8 x}\)	\(208\)
parts	\(-\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{\pi x}+a \,c^{2} x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}+\frac {3 a \,c^{2} \pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}{2}+\frac {3 a \,c^{2} \pi ^{2} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{2 \sqrt {\pi \,c^{2}}}+\frac {b \,\pi ^{\frac {3}{2}} \left (4 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}-2 x^{3} c^{3}+6 \operatorname {arcsinh}\left (x c \right )^{2} x c -8 x c \,\operatorname {arcsinh}\left (x c \right )+8 \ln \left (\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) x c -8 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}-x c \right )}{8 x}\)	\(208\)

Input:

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

Output:

-a/Pi/x*(Pi*c^2*x^2+Pi)^(5/2)+a*c^2*x*(Pi*c^2*x^2+Pi)^(3/2)+3/2*a*c^2*Pi*x 
*(Pi*c^2*x^2+Pi)^(1/2)+3/2*a*c^2*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)+1/8*b*Pi^(3/2)*(4*(c^2*x^2+1)^(1/2)*arcsinh(x 
*c)*x^2*c^2-2*x^3*c^3+6*arcsinh(x*c)^2*x*c-8*x*c*arcsinh(x*c)+8*ln((x*c+(c 
^2*x^2+1)^(1/2))^2-1)*x*c-8*arcsinh(x*c)*(c^2*x^2+1)^(1/2)-x*c)/x
 

Fricas [F]

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

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

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^2, x)
 

Sympy [F]

\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\pi ^{\frac {3}{2}} \left (\int a c^{2} \sqrt {c^{2} x^{2} + 1}\, dx + \int \frac {a \sqrt {c^{2} x^{2} + 1}}{x^{2}}\, dx + \int b c^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}\, dx + \int \frac {b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x^{2}}\, dx\right ) \] Input:

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

Output:

pi**(3/2)*(Integral(a*c**2*sqrt(c**2*x**2 + 1), x) + Integral(a*sqrt(c**2* 
x**2 + 1)/x**2, x) + Integral(b*c**2*sqrt(c**2*x**2 + 1)*asinh(c*x), x) + 
Integral(b*sqrt(c**2*x**2 + 1)*asinh(c*x)/x**2, x))
 

Maxima [F(-2)]

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

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

Output:

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

Giac [F(-2)]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\frac {\sqrt {\pi }\, \pi \left (4 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}-8 \sqrt {c^{2} x^{2}+1}\, a +8 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{2}}d x \right ) b x +8 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )d x \right ) b \,c^{2} x +12 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a c x -9 a c x \right )}{8 x} \] Input:

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

Output:

(sqrt(pi)*pi*(4*sqrt(c**2*x**2 + 1)*a*c**2*x**2 - 8*sqrt(c**2*x**2 + 1)*a 
+ 8*int((sqrt(c**2*x**2 + 1)*asinh(c*x))/x**2,x)*b*x + 8*int(sqrt(c**2*x** 
2 + 1)*asinh(c*x),x)*b*c**2*x + 12*log(sqrt(c**2*x**2 + 1) + c*x)*a*c*x - 
9*a*c*x))/(8*x)