Integrand size = 22, antiderivative size = 124 \[ \int x^4 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=-\frac {2 b d \sqrt {1+c^2 x^2}}{35 c^5}-\frac {b d \left (1+c^2 x^2\right )^{3/2}}{105 c^5}+\frac {8 b d \left (1+c^2 x^2\right )^{5/2}}{175 c^5}-\frac {b d \left (1+c^2 x^2\right )^{7/2}}{49 c^5}+\frac {1}{5} d x^5 (a+b \text {arcsinh}(c x))+\frac {1}{7} c^2 d x^7 (a+b \text {arcsinh}(c x)) \] Output:
-2/35*b*d*(c^2*x^2+1)^(1/2)/c^5-1/105*b*d*(c^2*x^2+1)^(3/2)/c^5+8/175*b*d* (c^2*x^2+1)^(5/2)/c^5-1/49*b*d*(c^2*x^2+1)^(7/2)/c^5+1/5*d*x^5*(a+b*arcsin h(c*x))+1/7*c^2*d*x^7*(a+b*arcsinh(c*x))
Time = 0.07 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.70 \[ \int x^4 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {d \left (105 a x^5 \left (7+5 c^2 x^2\right )-\frac {b \sqrt {1+c^2 x^2} \left (152-76 c^2 x^2+57 c^4 x^4+75 c^6 x^6\right )}{c^5}+105 b x^5 \left (7+5 c^2 x^2\right ) \text {arcsinh}(c x)\right )}{3675} \] Input:
Integrate[x^4*(d + c^2*d*x^2)*(a + b*ArcSinh[c*x]),x]
Output:
(d*(105*a*x^5*(7 + 5*c^2*x^2) - (b*Sqrt[1 + c^2*x^2]*(152 - 76*c^2*x^2 + 5 7*c^4*x^4 + 75*c^6*x^6))/c^5 + 105*b*x^5*(7 + 5*c^2*x^2)*ArcSinh[c*x]))/36 75
Time = 0.32 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6218, 27, 354, 86, 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.
\(\displaystyle \int x^4 \left (c^2 d x^2+d\right ) (a+b \text {arcsinh}(c x)) \, dx\) |
\(\Big \downarrow \) 6218 |
\(\displaystyle -b c \int \frac {d x^5 \left (5 c^2 x^2+7\right )}{35 \sqrt {c^2 x^2+1}}dx+\frac {1}{7} c^2 d x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} d x^5 (a+b \text {arcsinh}(c x))\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{35} b c d \int \frac {x^5 \left (5 c^2 x^2+7\right )}{\sqrt {c^2 x^2+1}}dx+\frac {1}{7} c^2 d x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} d x^5 (a+b \text {arcsinh}(c x))\) |
\(\Big \downarrow \) 354 |
\(\displaystyle -\frac {1}{70} b c d \int \frac {x^4 \left (5 c^2 x^2+7\right )}{\sqrt {c^2 x^2+1}}dx^2+\frac {1}{7} c^2 d x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} d x^5 (a+b \text {arcsinh}(c x))\) |
\(\Big \downarrow \) 86 |
\(\displaystyle -\frac {1}{70} b c d \int \left (\frac {5 \left (c^2 x^2+1\right )^{5/2}}{c^4}-\frac {8 \left (c^2 x^2+1\right )^{3/2}}{c^4}+\frac {\sqrt {c^2 x^2+1}}{c^4}+\frac {2}{c^4 \sqrt {c^2 x^2+1}}\right )dx^2+\frac {1}{7} c^2 d x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} d x^5 (a+b \text {arcsinh}(c x))\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{7} c^2 d x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} d x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c d \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\) |
Input:
Int[x^4*(d + c^2*d*x^2)*(a + b*ArcSinh[c*x]),x]
Output:
-1/70*(b*c*d*((4*Sqrt[1 + c^2*x^2])/c^6 + (2*(1 + c^2*x^2)^(3/2))/(3*c^6) - (16*(1 + c^2*x^2)^(5/2))/(5*c^6) + (10*(1 + c^2*x^2)^(7/2))/(7*c^6))) + (d*x^5*(a + b*ArcSinh[c*x]))/5 + (c^2*d*x^7*(a + b*ArcSinh[c*x]))/7
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ )^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Simp [(a + b*ArcSinh[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
Time = 0.41 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.97
method | result | size |
parts | \(a d \left (\frac {1}{7} c^{2} x^{7}+\frac {1}{5} x^{5}\right )+\frac {b d \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{7} c^{7}}{7}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}-\frac {19 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{1225}+\frac {76 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{3675}-\frac {152 \sqrt {c^{2} x^{2}+1}}{3675}-\frac {x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{49}\right )}{c^{5}}\) | \(120\) |
derivativedivides | \(\frac {a d \left (\frac {1}{7} x^{7} c^{7}+\frac {1}{5} x^{5} c^{5}\right )+b d \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{7} c^{7}}{7}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}-\frac {19 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{1225}+\frac {76 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{3675}-\frac {152 \sqrt {c^{2} x^{2}+1}}{3675}-\frac {x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{49}\right )}{c^{5}}\) | \(124\) |
default | \(\frac {a d \left (\frac {1}{7} x^{7} c^{7}+\frac {1}{5} x^{5} c^{5}\right )+b d \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{7} c^{7}}{7}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}-\frac {19 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{1225}+\frac {76 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{3675}-\frac {152 \sqrt {c^{2} x^{2}+1}}{3675}-\frac {x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{49}\right )}{c^{5}}\) | \(124\) |
orering | \(\frac {\left (975 c^{8} x^{8}+1377 c^{6} x^{6}-228 c^{4} x^{4}+608 c^{2} x^{2}+608\right ) \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{3675 c^{6} x \left (c^{2} x^{2}+1\right )}-\frac {\left (75 c^{6} x^{6}+57 c^{4} x^{4}-76 c^{2} x^{2}+152\right ) \left (4 x^{3} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )+2 c^{2} d \,x^{5} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )+\frac {x^{4} \left (c^{2} d \,x^{2}+d \right ) b c}{\sqrt {c^{2} x^{2}+1}}\right )}{3675 c^{6} x^{4}}\) | \(175\) |
Input:
int(x^4*(c^2*d*x^2+d)*(a+b*arcsinh(x*c)),x,method=_RETURNVERBOSE)
Output:
a*d*(1/7*c^2*x^7+1/5*x^5)+b*d/c^5*(1/7*arcsinh(x*c)*x^7*c^7+1/5*arcsinh(x* c)*x^5*c^5-19/1225*x^4*c^4*(c^2*x^2+1)^(1/2)+76/3675*x^2*c^2*(c^2*x^2+1)^( 1/2)-152/3675*(c^2*x^2+1)^(1/2)-1/49*x^6*c^6*(c^2*x^2+1)^(1/2))
Time = 0.09 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.91 \[ \int x^4 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {525 \, a c^{7} d x^{7} + 735 \, a c^{5} d x^{5} + 105 \, {\left (5 \, b c^{7} d x^{7} + 7 \, b c^{5} d x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (75 \, b c^{6} d x^{6} + 57 \, b c^{4} d x^{4} - 76 \, b c^{2} d x^{2} + 152 \, b d\right )} \sqrt {c^{2} x^{2} + 1}}{3675 \, c^{5}} \] Input:
integrate(x^4*(c^2*d*x^2+d)*(a+b*arcsinh(c*x)),x, algorithm="fricas")
Output:
1/3675*(525*a*c^7*d*x^7 + 735*a*c^5*d*x^5 + 105*(5*b*c^7*d*x^7 + 7*b*c^5*d *x^5)*log(c*x + sqrt(c^2*x^2 + 1)) - (75*b*c^6*d*x^6 + 57*b*c^4*d*x^4 - 76 *b*c^2*d*x^2 + 152*b*d)*sqrt(c^2*x^2 + 1))/c^5
Time = 0.69 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.22 \[ \int x^4 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {a c^{2} d x^{7}}{7} + \frac {a d x^{5}}{5} + \frac {b c^{2} d x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {b c d x^{6} \sqrt {c^{2} x^{2} + 1}}{49} + \frac {b d x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {19 b d x^{4} \sqrt {c^{2} x^{2} + 1}}{1225 c} + \frac {76 b d x^{2} \sqrt {c^{2} x^{2} + 1}}{3675 c^{3}} - \frac {152 b d \sqrt {c^{2} x^{2} + 1}}{3675 c^{5}} & \text {for}\: c \neq 0 \\\frac {a d x^{5}}{5} & \text {otherwise} \end {cases} \] Input:
integrate(x**4*(c**2*d*x**2+d)*(a+b*asinh(c*x)),x)
Output:
Piecewise((a*c**2*d*x**7/7 + a*d*x**5/5 + b*c**2*d*x**7*asinh(c*x)/7 - b*c *d*x**6*sqrt(c**2*x**2 + 1)/49 + b*d*x**5*asinh(c*x)/5 - 19*b*d*x**4*sqrt( c**2*x**2 + 1)/(1225*c) + 76*b*d*x**2*sqrt(c**2*x**2 + 1)/(3675*c**3) - 15 2*b*d*sqrt(c**2*x**2 + 1)/(3675*c**5), Ne(c, 0)), (a*d*x**5/5, True))
Time = 0.04 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.48 \[ \int x^4 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{7} \, a c^{2} d x^{7} + \frac {1}{5} \, a d x^{5} + \frac {1}{245} \, {\left (35 \, x^{7} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b c^{2} d + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d \] Input:
integrate(x^4*(c^2*d*x^2+d)*(a+b*arcsinh(c*x)),x, algorithm="maxima")
Output:
1/7*a*c^2*d*x^7 + 1/5*a*d*x^5 + 1/245*(35*x^7*arcsinh(c*x) - (5*sqrt(c^2*x ^2 + 1)*x^6/c^2 - 6*sqrt(c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(c^2*x^2 + 1)*x^2/c^ 6 - 16*sqrt(c^2*x^2 + 1)/c^8)*c)*b*c^2*d + 1/75*(15*x^5*arcsinh(c*x) - (3* sqrt(c^2*x^2 + 1)*x^4/c^2 - 4*sqrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c)*b*d
Exception generated. \[ \int x^4 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x^4*(c^2*d*x^2+d)*(a+b*arcsinh(c*x)),x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int x^4 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\int x^4\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\left (d\,c^2\,x^2+d\right ) \,d x \] Input:
int(x^4*(a + b*asinh(c*x))*(d + c^2*d*x^2),x)
Output:
int(x^4*(a + b*asinh(c*x))*(d + c^2*d*x^2), x)
Time = 0.19 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.98 \[ \int x^4 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {d \left (525 \mathit {asinh} \left (c x \right ) b \,c^{7} x^{7}+735 \mathit {asinh} \left (c x \right ) b \,c^{5} x^{5}-75 \sqrt {c^{2} x^{2}+1}\, b \,c^{6} x^{6}-57 \sqrt {c^{2} x^{2}+1}\, b \,c^{4} x^{4}+76 \sqrt {c^{2} x^{2}+1}\, b \,c^{2} x^{2}-152 \sqrt {c^{2} x^{2}+1}\, b +525 a \,c^{7} x^{7}+735 a \,c^{5} x^{5}\right )}{3675 c^{5}} \] Input:
int(x^4*(c^2*d*x^2+d)*(a+b*asinh(c*x)),x)
Output:
(d*(525*asinh(c*x)*b*c**7*x**7 + 735*asinh(c*x)*b*c**5*x**5 - 75*sqrt(c**2 *x**2 + 1)*b*c**6*x**6 - 57*sqrt(c**2*x**2 + 1)*b*c**4*x**4 + 76*sqrt(c**2 *x**2 + 1)*b*c**2*x**2 - 152*sqrt(c**2*x**2 + 1)*b + 525*a*c**7*x**7 + 735 *a*c**5*x**5))/(3675*c**5)