Integrand size = 24, antiderivative size = 93 \[ \int x \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {b \pi ^{5/2} x}{7 c}-\frac {1}{7} b c \pi ^{5/2} x^3-\frac {3}{35} b c^3 \pi ^{5/2} x^5-\frac {1}{49} b c^5 \pi ^{5/2} x^7+\frac {\left (\pi +c^2 \pi x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^2 \pi } \] Output:
-1/7*b*Pi^(5/2)*x/c-1/7*b*c*Pi^(5/2)*x^3-3/35*b*c^3*Pi^(5/2)*x^5-1/49*b*c^ 5*Pi^(5/2)*x^7+1/7*(Pi*c^2*x^2+Pi)^(7/2)*(a+b*arcsinh(c*x))/c^2/Pi
Time = 0.15 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.86 \[ \int x \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\pi ^{5/2} \left (35 a \left (1+c^2 x^2\right )^{7/2}-b c x \left (35+35 c^2 x^2+21 c^4 x^4+5 c^6 x^6\right )+35 b \left (1+c^2 x^2\right )^{7/2} \text {arcsinh}(c x)\right )}{245 c^2} \] Input:
Integrate[x*(Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]
Output:
(Pi^(5/2)*(35*a*(1 + c^2*x^2)^(7/2) - b*c*x*(35 + 35*c^2*x^2 + 21*c^4*x^4 + 5*c^6*x^6) + 35*b*(1 + c^2*x^2)^(7/2)*ArcSinh[c*x]))/(245*c^2)
Time = 0.31 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.81, 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.
\(\displaystyle \int x \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx\) |
\(\Big \downarrow \) 6213 |
\(\displaystyle \frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{7 \pi c^2}-\frac {\pi ^{5/2} b \int \left (c^2 x^2+1\right )^3dx}{7 c}\) |
\(\Big \downarrow \) 210 |
\(\displaystyle \frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{7 \pi c^2}-\frac {\pi ^{5/2} b \int \left (c^6 x^6+3 c^4 x^4+3 c^2 x^2+1\right )dx}{7 c}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{7 \pi c^2}-\frac {\pi ^{5/2} b \left (\frac {c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+c^2 x^3+x\right )}{7 c}\) |
Input:
Int[x*(Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]
Output:
-1/7*(b*Pi^(5/2)*(x + c^2*x^3 + (3*c^4*x^5)/5 + (c^6*x^7)/7))/c + ((Pi + c ^2*Pi*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(7*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. \(169\) vs. \(2(73)=146\).
Time = 1.10 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.83
method | result | size |
default | \(\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{7 \pi \,c^{2}}+\frac {b \,\pi ^{\frac {5}{2}} \left (35 \,\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}+140 \,\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}-5 x^{7} c^{7} \sqrt {c^{2} x^{2}+1}+210 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-21 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+140 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}-35 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+35 \,\operatorname {arcsinh}\left (x c \right )-35 \sqrt {c^{2} x^{2}+1}\, x c \right )}{245 c^{2} \sqrt {c^{2} x^{2}+1}}\) | \(170\) |
parts | \(\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{7 \pi \,c^{2}}+\frac {b \,\pi ^{\frac {5}{2}} \left (35 \,\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}+140 \,\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}-5 x^{7} c^{7} \sqrt {c^{2} x^{2}+1}+210 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-21 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+140 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}-35 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+35 \,\operatorname {arcsinh}\left (x c \right )-35 \sqrt {c^{2} x^{2}+1}\, x c \right )}{245 c^{2} \sqrt {c^{2} x^{2}+1}}\) | \(170\) |
orering | \(\frac {\left (65 c^{8} x^{8}+271 c^{6} x^{6}+441 c^{4} x^{4}+385 c^{2} x^{2}+70\right ) \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{245 c^{2} \left (c^{2} x^{2}+1\right )^{3}}-\frac {\left (5 c^{6} x^{6}+21 c^{4} x^{4}+35 c^{2} x^{2}+35\right ) \left (\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )+5 x^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) \pi \,c^{2}+\frac {x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}} b c}{\sqrt {c^{2} x^{2}+1}}\right )}{245 c^{2} \left (c^{2} x^{2}+1\right )^{2}}\) | \(192\) |
Input:
int(x*(Pi*c^2*x^2+Pi)^(5/2)*(a+b*arcsinh(x*c)),x,method=_RETURNVERBOSE)
Output:
1/7*a/Pi/c^2*(Pi*c^2*x^2+Pi)^(7/2)+1/245*b*Pi^(5/2)/c^2/(c^2*x^2+1)^(1/2)* (35*arcsinh(x*c)*x^8*c^8+140*arcsinh(x*c)*x^6*c^6-5*x^7*c^7*(c^2*x^2+1)^(1 /2)+210*arcsinh(x*c)*c^4*x^4-21*(c^2*x^2+1)^(1/2)*x^5*c^5+140*arcsinh(x*c) *c^2*x^2-35*(c^2*x^2+1)^(1/2)*c^3*x^3+35*arcsinh(x*c)-35*(c^2*x^2+1)^(1/2) *x*c)
Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (73) = 146\).
Time = 0.10 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.42 \[ \int x \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {35 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (\pi ^{2} b c^{8} x^{8} + 4 \, \pi ^{2} b c^{6} x^{6} + 6 \, \pi ^{2} b c^{4} x^{4} + 4 \, \pi ^{2} b c^{2} x^{2} + \pi ^{2} b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (35 \, \pi ^{2} a c^{8} x^{8} + 140 \, \pi ^{2} a c^{6} x^{6} + 210 \, \pi ^{2} a c^{4} x^{4} + 140 \, \pi ^{2} a c^{2} x^{2} + 35 \, \pi ^{2} a - {\left (5 \, \pi ^{2} b c^{7} x^{7} + 21 \, \pi ^{2} b c^{5} x^{5} + 35 \, \pi ^{2} b c^{3} x^{3} + 35 \, \pi ^{2} b c x\right )} \sqrt {c^{2} x^{2} + 1}\right )}}{245 \, {\left (c^{4} x^{2} + c^{2}\right )}} \] Input:
integrate(x*(pi*c^2*x^2+pi)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="fricas" )
Output:
1/245*(35*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)) + sq rt(pi + pi*c^2*x^2)*(35*pi^2*a*c^8*x^8 + 140*pi^2*a*c^6*x^6 + 210*pi^2*a*c ^4*x^4 + 140*pi^2*a*c^2*x^2 + 35*pi^2*a - (5*pi^2*b*c^7*x^7 + 21*pi^2*b*c^ 5*x^5 + 35*pi^2*b*c^3*x^3 + 35*pi^2*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. 299 vs. \(2 (85) = 170\).
Time = 28.47 (sec) , antiderivative size = 299, normalized size of antiderivative = 3.22 \[ \int x \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {\pi ^{\frac {5}{2}} a c^{4} x^{6} \sqrt {c^{2} x^{2} + 1}}{7} + \frac {3 \pi ^{\frac {5}{2}} a c^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{7} + \frac {3 \pi ^{\frac {5}{2}} a x^{2} \sqrt {c^{2} x^{2} + 1}}{7} + \frac {\pi ^{\frac {5}{2}} a \sqrt {c^{2} x^{2} + 1}}{7 c^{2}} - \frac {\pi ^{\frac {5}{2}} b c^{5} x^{7}}{49} + \frac {\pi ^{\frac {5}{2}} b c^{4} x^{6} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {3 \pi ^{\frac {5}{2}} b c^{3} x^{5}}{35} + \frac {3 \pi ^{\frac {5}{2}} b c^{2} x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {\pi ^{\frac {5}{2}} b c x^{3}}{7} + \frac {3 \pi ^{\frac {5}{2}} b x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {\pi ^{\frac {5}{2}} b x}{7 c} + \frac {\pi ^{\frac {5}{2}} b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{7 c^{2}} & \text {for}\: c \neq 0 \\\frac {\pi ^{\frac {5}{2}} a x^{2}}{2} & \text {otherwise} \end {cases} \] Input:
integrate(x*(pi*c**2*x**2+pi)**(5/2)*(a+b*asinh(c*x)),x)
Output:
Piecewise((pi**(5/2)*a*c**4*x**6*sqrt(c**2*x**2 + 1)/7 + 3*pi**(5/2)*a*c** 2*x**4*sqrt(c**2*x**2 + 1)/7 + 3*pi**(5/2)*a*x**2*sqrt(c**2*x**2 + 1)/7 + pi**(5/2)*a*sqrt(c**2*x**2 + 1)/(7*c**2) - pi**(5/2)*b*c**5*x**7/49 + pi** (5/2)*b*c**4*x**6*sqrt(c**2*x**2 + 1)*asinh(c*x)/7 - 3*pi**(5/2)*b*c**3*x* *5/35 + 3*pi**(5/2)*b*c**2*x**4*sqrt(c**2*x**2 + 1)*asinh(c*x)/7 - pi**(5/ 2)*b*c*x**3/7 + 3*pi**(5/2)*b*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/7 - pi** (5/2)*b*x/(7*c) + pi**(5/2)*b*sqrt(c**2*x**2 + 1)*asinh(c*x)/(7*c**2), Ne( c, 0)), (pi**(5/2)*a*x**2/2, True))
Time = 0.04 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.03 \[ \int x \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}} b \operatorname {arsinh}\left (c x\right )}{7 \, \pi c^{2}} + \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}} a}{7 \, \pi c^{2}} - \frac {{\left (5 \, \pi ^{\frac {7}{2}} c^{6} x^{7} + 21 \, \pi ^{\frac {7}{2}} c^{4} x^{5} + 35 \, \pi ^{\frac {7}{2}} c^{2} x^{3} + 35 \, \pi ^{\frac {7}{2}} x\right )} b}{245 \, \pi c} \] Input:
integrate(x*(pi*c^2*x^2+pi)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="maxima" )
Output:
1/7*(pi + pi*c^2*x^2)^(7/2)*b*arcsinh(c*x)/(pi*c^2) + 1/7*(pi + pi*c^2*x^2 )^(7/2)*a/(pi*c^2) - 1/245*(5*pi^(7/2)*c^6*x^7 + 21*pi^(7/2)*c^4*x^5 + 35* pi^(7/2)*c^2*x^3 + 35*pi^(7/2)*x)*b/(pi*c)
Exception generated. \[ \int x \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x*(pi*c^2*x^2+pi)^(5/2)*(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. \[ \int x \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2} \,d x \] Input:
int(x*(a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(5/2),x)
Output:
int(x*(a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(5/2), x)
\[ \int x \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {\pi }\, \pi ^{2} \left (\sqrt {c^{2} x^{2}+1}\, a \,c^{6} x^{6}+3 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} x^{4}+3 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, a +7 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{5}d x \right ) b \,c^{6}+14 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{3}d x \right ) b \,c^{4}+7 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x d x \right ) b \,c^{2}\right )}{7 c^{2}} \] Input:
int(x*(Pi*c^2*x^2+Pi)^(5/2)*(a+b*asinh(c*x)),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**5,x)*b*c**6 + 14*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**3,x)*b*c**4 + 7*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x,x )*b*c**2))/(7*c**2)