Integrand size = 24, antiderivative size = 77 \[ \int x \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {b \pi ^{3/2} x}{5 c}-\frac {2}{15} b c \pi ^{3/2} x^3-\frac {1}{25} b c^3 \pi ^{3/2} x^5+\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2 \pi } \] Output:
-1/5*b*Pi^(3/2)*x/c-2/15*b*c*Pi^(3/2)*x^3-1/25*b*c^3*Pi^(3/2)*x^5+1/5*(Pi* c^2*x^2+Pi)^(5/2)*(a+b*arcsinh(c*x))/c^2/Pi
Time = 0.13 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.94 \[ \int x \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\pi ^{3/2} \left (15 a \left (1+c^2 x^2\right )^{5/2}-b c x \left (15+10 c^2 x^2+3 c^4 x^4\right )+15 b \left (1+c^2 x^2\right )^{5/2} \text {arcsinh}(c x)\right )}{75 c^2} \] Input:
Integrate[x*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]),x]
Output:
(Pi^(3/2)*(15*a*(1 + c^2*x^2)^(5/2) - b*c*x*(15 + 10*c^2*x^2 + 3*c^4*x^4) + 15*b*(1 + c^2*x^2)^(5/2)*ArcSinh[c*x]))/(75*c^2)
Time = 0.30 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.88, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6213, 210, 2009}
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.
\(\displaystyle \int x \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx\) |
\(\Big \downarrow \) 6213 |
\(\displaystyle \frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{5 \pi c^2}-\frac {\pi ^{3/2} b \int \left (c^2 x^2+1\right )^2dx}{5 c}\) |
\(\Big \downarrow \) 210 |
\(\displaystyle \frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{5 \pi c^2}-\frac {\pi ^{3/2} b \int \left (c^4 x^4+2 c^2 x^2+1\right )dx}{5 c}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{5 \pi c^2}-\frac {\pi ^{3/2} b \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )}{5 c}\) |
Input:
Int[x*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]),x]
Output:
-1/5*(b*Pi^(3/2)*(x + (2*c^2*x^3)/3 + (c^4*x^5)/5))/c + ((Pi + c^2*Pi*x^2) ^(5/2)*(a + b*ArcSinh[c*x]))/(5*c^2*Pi)
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 )^p, x], x] /; FreeQ[{a, b}, x] && IGtQ[p, 0]
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] - Simp[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]
Leaf count of result is larger than twice the leaf count of optimal. \(138\) vs. \(2(61)=122\).
Time = 1.14 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.81
method | result | size |
default | \(\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{5 \pi \,c^{2}}+\frac {b \,\pi ^{\frac {3}{2}} \left (15 \,\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}+45 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-3 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+45 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}-10 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+15 \,\operatorname {arcsinh}\left (x c \right )-15 \sqrt {c^{2} x^{2}+1}\, x c \right )}{75 c^{2} \sqrt {c^{2} x^{2}+1}}\) | \(139\) |
parts | \(\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{5 \pi \,c^{2}}+\frac {b \,\pi ^{\frac {3}{2}} \left (15 \,\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}+45 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-3 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+45 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}-10 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+15 \,\operatorname {arcsinh}\left (x c \right )-15 \sqrt {c^{2} x^{2}+1}\, x c \right )}{75 c^{2} \sqrt {c^{2} x^{2}+1}}\) | \(139\) |
orering | \(\frac {\left (27 c^{6} x^{6}+88 c^{4} x^{4}+115 c^{2} x^{2}+30\right ) \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{75 c^{2} \left (c^{2} x^{2}+1\right )^{2}}-\frac {\left (3 c^{4} x^{4}+10 c^{2} x^{2}+15\right ) \left (\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )+3 x^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }\, \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) \pi \,c^{2}+\frac {x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}} b c}{\sqrt {c^{2} x^{2}+1}}\right )}{75 c^{2} \left (c^{2} x^{2}+1\right )}\) | \(176\) |
Input:
int(x*(Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(x*c)),x,method=_RETURNVERBOSE)
Output:
1/5*a/Pi/c^2*(Pi*c^2*x^2+Pi)^(5/2)+1/75*b*Pi^(3/2)/c^2/(c^2*x^2+1)^(1/2)*( 15*arcsinh(x*c)*x^6*c^6+45*arcsinh(x*c)*c^4*x^4-3*(c^2*x^2+1)^(1/2)*x^5*c^ 5+45*arcsinh(x*c)*c^2*x^2-10*(c^2*x^2+1)^(1/2)*c^3*x^3+15*arcsinh(x*c)-15* (c^2*x^2+1)^(1/2)*x*c)
Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (61) = 122\).
Time = 0.11 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.17 \[ \int x \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {15 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (\pi b c^{6} x^{6} + 3 \, \pi b c^{4} x^{4} + 3 \, \pi b c^{2} x^{2} + \pi b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (15 \, \pi a c^{6} x^{6} + 45 \, \pi a c^{4} x^{4} + 45 \, \pi a c^{2} x^{2} + 15 \, \pi a - {\left (3 \, \pi b c^{5} x^{5} + 10 \, \pi b c^{3} x^{3} + 15 \, \pi b c x\right )} \sqrt {c^{2} x^{2} + 1}\right )}}{75 \, {\left (c^{4} x^{2} + c^{2}\right )}} \] Input:
integrate(x*(pi*c^2*x^2+pi)^(3/2)*(a+b*arcsinh(c*x)),x, algorithm="fricas" )
Output:
1/75*(15*sqrt(pi + pi*c^2*x^2)*(pi*b*c^6*x^6 + 3*pi*b*c^4*x^4 + 3*pi*b*c^2 *x^2 + pi*b)*log(c*x + sqrt(c^2*x^2 + 1)) + sqrt(pi + pi*c^2*x^2)*(15*pi*a *c^6*x^6 + 45*pi*a*c^4*x^4 + 45*pi*a*c^2*x^2 + 15*pi*a - (3*pi*b*c^5*x^5 + 10*pi*b*c^3*x^3 + 15*pi*b*c*x)*sqrt(c^2*x^2 + 1)))/(c^4*x^2 + c^2)
Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (70) = 140\).
Time = 3.06 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.87 \[ \int x \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {\pi ^{\frac {3}{2}} a c^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{5} + \frac {2 \pi ^{\frac {3}{2}} a x^{2} \sqrt {c^{2} x^{2} + 1}}{5} + \frac {\pi ^{\frac {3}{2}} a \sqrt {c^{2} x^{2} + 1}}{5 c^{2}} - \frac {\pi ^{\frac {3}{2}} b c^{3} x^{5}}{25} + \frac {\pi ^{\frac {3}{2}} b c^{2} x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {2 \pi ^{\frac {3}{2}} b c x^{3}}{15} + \frac {2 \pi ^{\frac {3}{2}} b x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {\pi ^{\frac {3}{2}} b x}{5 c} + \frac {\pi ^{\frac {3}{2}} b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{5 c^{2}} & \text {for}\: c \neq 0 \\\frac {\pi ^{\frac {3}{2}} a x^{2}}{2} & \text {otherwise} \end {cases} \] Input:
integrate(x*(pi*c**2*x**2+pi)**(3/2)*(a+b*asinh(c*x)),x)
Output:
Piecewise((pi**(3/2)*a*c**2*x**4*sqrt(c**2*x**2 + 1)/5 + 2*pi**(3/2)*a*x** 2*sqrt(c**2*x**2 + 1)/5 + pi**(3/2)*a*sqrt(c**2*x**2 + 1)/(5*c**2) - pi**( 3/2)*b*c**3*x**5/25 + pi**(3/2)*b*c**2*x**4*sqrt(c**2*x**2 + 1)*asinh(c*x) /5 - 2*pi**(3/2)*b*c*x**3/15 + 2*pi**(3/2)*b*x**2*sqrt(c**2*x**2 + 1)*asin h(c*x)/5 - pi**(3/2)*b*x/(5*c) + pi**(3/2)*b*sqrt(c**2*x**2 + 1)*asinh(c*x )/(5*c**2), Ne(c, 0)), (pi**(3/2)*a*x**2/2, True))
Time = 0.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.10 \[ \int x \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} b \operatorname {arsinh}\left (c x\right )}{5 \, \pi c^{2}} + \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} a}{5 \, \pi c^{2}} - \frac {{\left (3 \, \pi ^{\frac {5}{2}} c^{4} x^{5} + 10 \, \pi ^{\frac {5}{2}} c^{2} x^{3} + 15 \, \pi ^{\frac {5}{2}} x\right )} b}{75 \, \pi c} \] Input:
integrate(x*(pi*c^2*x^2+pi)^(3/2)*(a+b*arcsinh(c*x)),x, algorithm="maxima" )
Output:
1/5*(pi + pi*c^2*x^2)^(5/2)*b*arcsinh(c*x)/(pi*c^2) + 1/5*(pi + pi*c^2*x^2 )^(5/2)*a/(pi*c^2) - 1/75*(3*pi^(5/2)*c^4*x^5 + 10*pi^(5/2)*c^2*x^3 + 15*p i^(5/2)*x)*b/(pi*c)
Exception generated. \[ \int x \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x*(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
Timed out. \[ \int x \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2} \,d x \] Input:
int(x*(a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(3/2),x)
Output:
int(x*(a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(3/2), x)
\[ \int x \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {\pi }\, \pi \left (\sqrt {c^{2} x^{2}+1}\, a \,c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, a +5 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{3}d x \right ) b \,c^{4}+5 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x d x \right ) b \,c^{2}\right )}{5 c^{2}} \] Input:
int(x*(Pi*c^2*x^2+Pi)^(3/2)*(a+b*asinh(c*x)),x)
Output:
(sqrt(pi)*pi*(sqrt(c**2*x**2 + 1)*a*c**4*x**4 + 2*sqrt(c**2*x**2 + 1)*a*c* *2*x**2 + sqrt(c**2*x**2 + 1)*a + 5*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x** 3,x)*b*c**4 + 5*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x,x)*b*c**2))/(5*c**2)