Integrand size = 26, antiderivative size = 94 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^8} \, dx=-\frac {b c \pi ^{5/2}}{42 x^6}-\frac {3 b c^3 \pi ^{5/2}}{28 x^4}-\frac {3 b c^5 \pi ^{5/2}}{14 x^2}-\frac {\left (\pi +c^2 \pi x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 \pi x^7}+\frac {1}{7} b c^7 \pi ^{5/2} \log (x) \] Output:
-1/42*b*c*Pi^(5/2)/x^6-3/28*b*c^3*Pi^(5/2)/x^4-3/14*b*c^5*Pi^(5/2)/x^2-1/7 *(Pi*c^2*x^2+Pi)^(7/2)*(a+b*arcsinh(c*x))/Pi/x^7+1/7*b*c^7*Pi^(5/2)*ln(x)
Time = 0.31 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.68 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^8} \, dx=\frac {\pi ^{5/2} \left (-10 b c x-45 b c^3 x^3-90 b c^5 x^5-147 b c^7 x^7-60 a \sqrt {1+c^2 x^2}-180 a c^2 x^2 \sqrt {1+c^2 x^2}-180 a c^4 x^4 \sqrt {1+c^2 x^2}-60 a c^6 x^6 \sqrt {1+c^2 x^2}-60 b \left (1+c^2 x^2\right )^{7/2} \text {arcsinh}(c x)+60 b c^7 x^7 \log (x)\right )}{420 x^7} \] Input:
Integrate[((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^8,x]
Output:
(Pi^(5/2)*(-10*b*c*x - 45*b*c^3*x^3 - 90*b*c^5*x^5 - 147*b*c^7*x^7 - 60*a* Sqrt[1 + c^2*x^2] - 180*a*c^2*x^2*Sqrt[1 + c^2*x^2] - 180*a*c^4*x^4*Sqrt[1 + c^2*x^2] - 60*a*c^6*x^6*Sqrt[1 + c^2*x^2] - 60*b*(1 + c^2*x^2)^(7/2)*Ar cSinh[c*x] + 60*b*c^7*x^7*Log[x]))/(420*x^7)
Time = 0.34 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.83, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6215, 243, 49, 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.
\(\displaystyle \int \frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{x^8} \, dx\) |
\(\Big \downarrow \) 6215 |
\(\displaystyle \frac {1}{7} \pi ^{5/2} b c \int \frac {\left (c^2 x^2+1\right )^3}{x^7}dx-\frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{7 \pi x^7}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{14} \pi ^{5/2} b c \int \frac {\left (c^2 x^2+1\right )^3}{x^8}dx^2-\frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{7 \pi x^7}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{14} \pi ^{5/2} b c \int \left (\frac {c^6}{x^2}+\frac {3 c^4}{x^4}+\frac {3 c^2}{x^6}+\frac {1}{x^8}\right )dx^2-\frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{7 \pi x^7}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{14} \pi ^{5/2} b c \left (c^6 \log \left (x^2\right )-\frac {3 c^4}{x^2}-\frac {3 c^2}{2 x^4}-\frac {1}{3 x^6}\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{7 \pi x^7}\) |
Input:
Int[((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^8,x]
Output:
-1/7*((Pi + c^2*Pi*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(Pi*x^7) + (b*c*Pi^(5/ 2)*(-1/3*1/x^6 - (3*c^2)/(2*x^4) - (3*c^4)/x^2 + c^6*Log[x^2]))/14
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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 + 1)*((a + b*ArcSinh[c*x])^n/(d*f*(m + 1))), 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, m, p}, x] && EqQ [e, c^2*d] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(2164\) vs. \(2(74)=148\).
Time = 1.24 (sec) , antiderivative size = 2165, normalized size of antiderivative = 23.03
method | result | size |
default | \(\text {Expression too large to display}\) | \(2165\) |
parts | \(\text {Expression too large to display}\) | \(2165\) |
Input:
int((Pi*c^2*x^2+Pi)^(5/2)*(a+b*arcsinh(x*c))/x^8,x,method=_RETURNVERBOSE)
Output:
3*b*Pi^(5/2)/(7*c^12*x^12+21*c^10*x^10+35*c^8*x^8+35*c^6*x^6+21*c^4*x^4+7* c^2*x^2+1)*x^4*arcsinh(x*c)*c^11-40/7*b*Pi^(5/2)/(7*c^12*x^12+21*c^10*x^10 +35*c^8*x^8+35*c^6*x^6+21*c^4*x^4+7*c^2*x^2+1)*x^2*(c^2*x^2+1)*c^9+b*Pi^(5 /2)/(7*c^12*x^12+21*c^10*x^10+35*c^8*x^8+35*c^6*x^6+21*c^4*x^4+7*c^2*x^2+1 )*x^2*arcsinh(x*c)*c^9-17/14*b*Pi^(5/2)/(7*c^12*x^12+21*c^10*x^10+35*c^8*x ^8+35*c^6*x^6+21*c^4*x^4+7*c^2*x^2+1)/x^2*(c^2*x^2+1)*c^5-1/4*b*Pi^(5/2)/( 7*c^12*x^12+21*c^10*x^10+35*c^8*x^8+35*c^6*x^6+21*c^4*x^4+7*c^2*x^2+1)/x^4 *(c^2*x^2+1)*c^3-1/42*b*Pi^(5/2)/(7*c^12*x^12+21*c^10*x^10+35*c^8*x^8+35*c ^6*x^6+21*c^4*x^4+7*c^2*x^2+1)/x^6*(c^2*x^2+1)*c-1/7*b*Pi^(5/2)/(7*c^12*x^ 12+21*c^10*x^10+35*c^8*x^8+35*c^6*x^6+21*c^4*x^4+7*c^2*x^2+1)/x^7*(c^2*x^2 +1)^(1/2)*arcsinh(x*c)-39/28*b*Pi^(5/2)/(7*c^12*x^12+21*c^10*x^10+35*c^8*x ^8+35*c^6*x^6+21*c^4*x^4+7*c^2*x^2+1)*x^8*(c^2*x^2+1)*c^15+5*b*Pi^(5/2)/(7 *c^12*x^12+21*c^10*x^10+35*c^8*x^8+35*c^6*x^6+21*c^4*x^4+7*c^2*x^2+1)*x^8* arcsinh(x*c)*c^15-163/42*b*Pi^(5/2)/(7*c^12*x^12+21*c^10*x^10+35*c^8*x^8+3 5*c^6*x^6+21*c^4*x^4+7*c^2*x^2+1)*x^6*(c^2*x^2+1)*c^13+5*b*Pi^(5/2)/(7*c^1 2*x^12+21*c^10*x^10+35*c^8*x^8+35*c^6*x^6+21*c^4*x^4+7*c^2*x^2+1)*x^6*arcs inh(x*c)*c^13-169/28*b*Pi^(5/2)/(7*c^12*x^12+21*c^10*x^10+35*c^8*x^8+35*c^ 6*x^6+21*c^4*x^4+7*c^2*x^2+1)*x^4*(c^2*x^2+1)*c^11+b*Pi^(5/2)/(7*c^12*x^12 +21*c^10*x^10+35*c^8*x^8+35*c^6*x^6+21*c^4*x^4+7*c^2*x^2+1)*x^12*arcsinh(x *c)*c^19-3/14*b*Pi^(5/2)/(7*c^12*x^12+21*c^10*x^10+35*c^8*x^8+35*c^6*x^...
Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (74) = 148\).
Time = 0.16 (sec) , antiderivative size = 335, normalized size of antiderivative = 3.56 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^8} \, dx=-\frac {12 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (\pi ^{2} b c^{8} x^{8} + 4 \, \pi ^{2} b c^{6} x^{6} + 6 \, \pi ^{2} b c^{4} x^{4} + 4 \, \pi ^{2} b c^{2} x^{2} + \pi ^{2} b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 6 \, \sqrt {\pi } {\left (\pi ^{2} b c^{9} x^{9} + \pi ^{2} b c^{7} x^{7}\right )} \log \left (\frac {\pi + \pi c^{2} x^{6} + \pi c^{2} x^{2} + \pi x^{4} + \sqrt {\pi } \sqrt {\pi + \pi c^{2} x^{2}} \sqrt {c^{2} x^{2} + 1} {\left (x^{4} - 1\right )}}{c^{2} x^{4} + x^{2}}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (12 \, \pi ^{2} a c^{8} x^{8} + 48 \, \pi ^{2} a c^{6} x^{6} + 72 \, \pi ^{2} a c^{4} x^{4} + 48 \, \pi ^{2} a c^{2} x^{2} + 12 \, \pi ^{2} a + {\left (18 \, \pi ^{2} b c^{5} x^{5} - \pi ^{2} {\left (18 \, b c^{5} + 9 \, b c^{3} + 2 \, b c\right )} x^{7} + 9 \, \pi ^{2} b c^{3} x^{3} + 2 \, \pi ^{2} b c x\right )} \sqrt {c^{2} x^{2} + 1}\right )}}{84 \, {\left (c^{2} x^{9} + x^{7}\right )}} \] Input:
integrate((pi*c^2*x^2+pi)^(5/2)*(a+b*arcsinh(c*x))/x^8,x, algorithm="frica s")
Output:
-1/84*(12*sqrt(pi + pi*c^2*x^2)*(pi^2*b*c^8*x^8 + 4*pi^2*b*c^6*x^6 + 6*pi^ 2*b*c^4*x^4 + 4*pi^2*b*c^2*x^2 + pi^2*b)*log(c*x + sqrt(c^2*x^2 + 1)) - 6* sqrt(pi)*(pi^2*b*c^9*x^9 + pi^2*b*c^7*x^7)*log((pi + pi*c^2*x^6 + pi*c^2*x ^2 + pi*x^4 + sqrt(pi)*sqrt(pi + pi*c^2*x^2)*sqrt(c^2*x^2 + 1)*(x^4 - 1))/ (c^2*x^4 + x^2)) + sqrt(pi + pi*c^2*x^2)*(12*pi^2*a*c^8*x^8 + 48*pi^2*a*c^ 6*x^6 + 72*pi^2*a*c^4*x^4 + 48*pi^2*a*c^2*x^2 + 12*pi^2*a + (18*pi^2*b*c^5 *x^5 - pi^2*(18*b*c^5 + 9*b*c^3 + 2*b*c)*x^7 + 9*pi^2*b*c^3*x^3 + 2*pi^2*b *c*x)*sqrt(c^2*x^2 + 1)))/(c^2*x^9 + x^7)
Timed out. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^8} \, dx=\text {Timed out} \] Input:
integrate((pi*c**2*x**2+pi)**(5/2)*(a+b*asinh(c*x))/x**8,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (74) = 148\).
Time = 0.06 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.14 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^8} \, dx=-\frac {{\left (6 \, \pi ^{\frac {7}{2}} \left (-1\right )^{2 \, \pi + 2 \, \pi c^{2} x^{2}} c^{6} \log \left (2 \, \pi c^{2} + \frac {2 \, \pi }{x^{2}}\right ) - 6 \, \pi ^{\frac {7}{2}} c^{6} \log \left (x^{2} + \frac {1}{c^{2}}\right ) + \frac {11 \, \pi ^{3} \sqrt {\pi + \pi c^{4} x^{4} + 2 \, \pi c^{2} x^{2}} c^{4}}{x^{2}} + \frac {7 \, \pi ^{3} \sqrt {\pi + \pi c^{4} x^{4} + 2 \, \pi c^{2} x^{2}} c^{2}}{x^{4}} + \frac {2 \, \pi ^{3} \sqrt {\pi + \pi c^{4} x^{4} + 2 \, \pi c^{2} x^{2}}}{x^{6}}\right )} b c}{84 \, \pi } - \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}} b \operatorname {arsinh}\left (c x\right )}{7 \, \pi x^{7}} - \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}} a}{7 \, \pi x^{7}} \] Input:
integrate((pi*c^2*x^2+pi)^(5/2)*(a+b*arcsinh(c*x))/x^8,x, algorithm="maxim a")
Output:
-1/84*(6*pi^(7/2)*(-1)^(2*pi + 2*pi*c^2*x^2)*c^6*log(2*pi*c^2 + 2*pi/x^2) - 6*pi^(7/2)*c^6*log(x^2 + 1/c^2) + 11*pi^3*sqrt(pi + pi*c^4*x^4 + 2*pi*c^ 2*x^2)*c^4/x^2 + 7*pi^3*sqrt(pi + pi*c^4*x^4 + 2*pi*c^2*x^2)*c^2/x^4 + 2*p i^3*sqrt(pi + pi*c^4*x^4 + 2*pi*c^2*x^2)/x^6)*b*c/pi - 1/7*(pi + pi*c^2*x^ 2)^(7/2)*b*arcsinh(c*x)/(pi*x^7) - 1/7*(pi + pi*c^2*x^2)^(7/2)*a/(pi*x^7)
Exception generated. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^8} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((pi*c^2*x^2+pi)^(5/2)*(a+b*arcsinh(c*x))/x^8,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. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^8} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2}}{x^8} \,d x \] Input:
int(((a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(5/2))/x^8,x)
Output:
int(((a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(5/2))/x^8, x)
\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^8} \, dx=\frac {\sqrt {\pi }\, \pi ^{2} \left (-\sqrt {c^{2} x^{2}+1}\, a \,c^{6} x^{6}-3 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} x^{4}-3 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, a +7 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{8}}d x \right ) b \,x^{7}+14 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{6}}d x \right ) b \,c^{2} x^{7}+7 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{4}}d x \right ) b \,c^{4} x^{7}-a \,c^{7} x^{7}\right )}{7 x^{7}} \] Input:
int((Pi*c^2*x^2+Pi)^(5/2)*(a+b*asinh(c*x))/x^8,x)
Output:
(sqrt(pi)*pi**2*( - sqrt(c**2*x**2 + 1)*a*c**6*x**6 - 3*sqrt(c**2*x**2 + 1 )*a*c**4*x**4 - 3*sqrt(c**2*x**2 + 1)*a*c**2*x**2 - sqrt(c**2*x**2 + 1)*a + 7*int((sqrt(c**2*x**2 + 1)*asinh(c*x))/x**8,x)*b*x**7 + 14*int((sqrt(c** 2*x**2 + 1)*asinh(c*x))/x**6,x)*b*c**2*x**7 + 7*int((sqrt(c**2*x**2 + 1)*a sinh(c*x))/x**4,x)*b*c**4*x**7 - a*c**7*x**7))/(7*x**7)