Integrand size = 24, antiderivative size = 226 \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=-\frac {16 b d^3 \sqrt {1+c^2 x^2}}{1155 c^5}-\frac {8 b d^3 \left (1+c^2 x^2\right )^{3/2}}{3465 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{5/2}}{1925 c^5}-\frac {b d^3 \left (1+c^2 x^2\right )^{7/2}}{1617 c^5}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{9/2}}{297 c^5}-\frac {b d^3 \left (1+c^2 x^2\right )^{11/2}}{121 c^5}+\frac {1}{5} d^3 x^5 (a+b \text {arcsinh}(c x))+\frac {3}{7} c^2 d^3 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^4 d^3 x^9 (a+b \text {arcsinh}(c x))+\frac {1}{11} c^6 d^3 x^{11} (a+b \text {arcsinh}(c x)) \] Output:
-16/1155*b*d^3*(c^2*x^2+1)^(1/2)/c^5-8/3465*b*d^3*(c^2*x^2+1)^(3/2)/c^5-2/ 1925*b*d^3*(c^2*x^2+1)^(5/2)/c^5-1/1617*b*d^3*(c^2*x^2+1)^(7/2)/c^5+4/297* b*d^3*(c^2*x^2+1)^(9/2)/c^5-1/121*b*d^3*(c^2*x^2+1)^(11/2)/c^5+1/5*d^3*x^5 *(a+b*arcsinh(c*x))+3/7*c^2*d^3*x^7*(a+b*arcsinh(c*x))+1/3*c^4*d^3*x^9*(a+ b*arcsinh(c*x))+1/11*c^6*d^3*x^11*(a+b*arcsinh(c*x))
Time = 0.09 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.63 \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^3 \left (3465 a c^5 x^5 \left (231+495 c^2 x^2+385 c^4 x^4+105 c^6 x^6\right )-b \sqrt {1+c^2 x^2} \left (50488-25244 c^2 x^2+18933 c^4 x^4+117625 c^6 x^6+111475 c^8 x^8+33075 c^{10} x^{10}\right )+3465 b c^5 x^5 \left (231+495 c^2 x^2+385 c^4 x^4+105 c^6 x^6\right ) \text {arcsinh}(c x)\right )}{4002075 c^5} \] Input:
Integrate[x^4*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]),x]
Output:
(d^3*(3465*a*c^5*x^5*(231 + 495*c^2*x^2 + 385*c^4*x^4 + 105*c^6*x^6) - b*S qrt[1 + c^2*x^2]*(50488 - 25244*c^2*x^2 + 18933*c^4*x^4 + 117625*c^6*x^6 + 111475*c^8*x^8 + 33075*c^10*x^10) + 3465*b*c^5*x^5*(231 + 495*c^2*x^2 + 3 85*c^4*x^4 + 105*c^6*x^6)*ArcSinh[c*x]))/(4002075*c^5)
Time = 0.69 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6218, 27, 2331, 2123, 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.
| \(\displaystyle \int x^4 \left (c^2 d x^2+d\right )^3 (a+b \text {arcsinh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6218 |
| \(\displaystyle -b c \int \frac {d^3 x^5 \left (105 c^6 x^6+385 c^4 x^4+495 c^2 x^2+231\right )}{1155 \sqrt {c^2 x^2+1}}dx+\frac {1}{11} c^6 d^3 x^{11} (a+b \text {arcsinh}(c x))+\frac {1}{3} c^4 d^3 x^9 (a+b \text {arcsinh}(c x))+\frac {3}{7} c^2 d^3 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} d^3 x^5 (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c d^3 \int \frac {x^5 \left (105 c^6 x^6+385 c^4 x^4+495 c^2 x^2+231\right )}{\sqrt {c^2 x^2+1}}dx}{1155}+\frac {1}{11} c^6 d^3 x^{11} (a+b \text {arcsinh}(c x))+\frac {1}{3} c^4 d^3 x^9 (a+b \text {arcsinh}(c x))+\frac {3}{7} c^2 d^3 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} d^3 x^5 (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle -\frac {b c d^3 \int \frac {x^4 \left (105 c^6 x^6+385 c^4 x^4+495 c^2 x^2+231\right )}{\sqrt {c^2 x^2+1}}dx^2}{2310}+\frac {1}{11} c^6 d^3 x^{11} (a+b \text {arcsinh}(c x))+\frac {1}{3} c^4 d^3 x^9 (a+b \text {arcsinh}(c x))+\frac {3}{7} c^2 d^3 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} d^3 x^5 (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle -\frac {b c d^3 \int \left (\frac {105 \left (c^2 x^2+1\right )^{9/2}}{c^4}-\frac {140 \left (c^2 x^2+1\right )^{7/2}}{c^4}+\frac {5 \left (c^2 x^2+1\right )^{5/2}}{c^4}+\frac {6 \left (c^2 x^2+1\right )^{3/2}}{c^4}+\frac {8 \sqrt {c^2 x^2+1}}{c^4}+\frac {16}{c^4 \sqrt {c^2 x^2+1}}\right )dx^2}{2310}+\frac {1}{11} c^6 d^3 x^{11} (a+b \text {arcsinh}(c x))+\frac {1}{3} c^4 d^3 x^9 (a+b \text {arcsinh}(c x))+\frac {3}{7} c^2 d^3 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} d^3 x^5 (a+b \text {arcsinh}(c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{11} c^6 d^3 x^{11} (a+b \text {arcsinh}(c x))+\frac {1}{3} c^4 d^3 x^9 (a+b \text {arcsinh}(c x))+\frac {3}{7} c^2 d^3 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} d^3 x^5 (a+b \text {arcsinh}(c x))-\frac {b c d^3 \left (\frac {210 \left (c^2 x^2+1\right )^{11/2}}{11 c^6}-\frac {280 \left (c^2 x^2+1\right )^{9/2}}{9 c^6}+\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}+\frac {12 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {16 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {32 \sqrt {c^2 x^2+1}}{c^6}\right )}{2310}\) |
Input:
Int[x^4*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]),x]
Output:
-1/2310*(b*c*d^3*((32*Sqrt[1 + c^2*x^2])/c^6 + (16*(1 + c^2*x^2)^(3/2))/(3 *c^6) + (12*(1 + c^2*x^2)^(5/2))/(5*c^6) + (10*(1 + c^2*x^2)^(7/2))/(7*c^6 ) - (280*(1 + c^2*x^2)^(9/2))/(9*c^6) + (210*(1 + c^2*x^2)^(11/2))/(11*c^6 ))) + (d^3*x^5*(a + b*ArcSinh[c*x]))/5 + (3*c^2*d^3*x^7*(a + b*ArcSinh[c*x ]))/7 + (c^4*d^3*x^9*(a + b*ArcSinh[c*x]))/3 + (c^6*d^3*x^11*(a + b*ArcSin h[c*x]))/11
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2 S ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && 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 = 0.45 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.89
| method | result | size |
| parts | \(d^{3} a \left (\frac {1}{11} c^{6} x^{11}+\frac {1}{3} c^{4} x^{9}+\frac {3}{7} c^{2} x^{7}+\frac {1}{5} x^{5}\right )+\frac {d^{3} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{11} c^{11}}{11}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{9} c^{9}}{3}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) x^{7} c^{7}}{7}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}-\frac {6311 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{1334025}+\frac {25244 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {50488 \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {4705 x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{160083}-\frac {91 x^{8} c^{8} \sqrt {c^{2} x^{2}+1}}{3267}-\frac {x^{10} c^{10} \sqrt {c^{2} x^{2}+1}}{121}\right )}{c^{5}}\) | \(202\) |
| derivativedivides | \(\frac {d^{3} a \left (\frac {1}{11} x^{11} c^{11}+\frac {1}{3} c^{9} x^{9}+\frac {3}{7} x^{7} c^{7}+\frac {1}{5} x^{5} c^{5}\right )+d^{3} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{11} c^{11}}{11}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{9} c^{9}}{3}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) x^{7} c^{7}}{7}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}-\frac {6311 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{1334025}+\frac {25244 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {50488 \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {4705 x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{160083}-\frac {91 x^{8} c^{8} \sqrt {c^{2} x^{2}+1}}{3267}-\frac {x^{10} c^{10} \sqrt {c^{2} x^{2}+1}}{121}\right )}{c^{5}}\) | \(206\) |
| default | \(\frac {d^{3} a \left (\frac {1}{11} x^{11} c^{11}+\frac {1}{3} c^{9} x^{9}+\frac {3}{7} x^{7} c^{7}+\frac {1}{5} x^{5} c^{5}\right )+d^{3} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{11} c^{11}}{11}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{9} c^{9}}{3}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) x^{7} c^{7}}{7}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}-\frac {6311 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{1334025}+\frac {25244 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {50488 \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {4705 x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{160083}-\frac {91 x^{8} c^{8} \sqrt {c^{2} x^{2}+1}}{3267}-\frac {x^{10} c^{10} \sqrt {c^{2} x^{2}+1}}{121}\right )}{c^{5}}\) | \(206\) |
| orering | \(\frac {\left (694575 c^{12} x^{12}+2581075 c^{10} x^{10}+3337325 c^{8} x^{8}+1460245 c^{6} x^{6}-176708 c^{4} x^{4}+403904 c^{2} x^{2}+201952\right ) \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{4002075 c^{6} \left (c^{2} x^{2}+1\right )^{3} x}-\frac {\left (33075 c^{10} x^{10}+111475 c^{8} x^{8}+117625 c^{6} x^{6}+18933 c^{4} x^{4}-25244 c^{2} x^{2}+50488\right ) \left (4 x^{3} \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )+6 x^{5} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{2} d +\frac {x^{4} \left (c^{2} d \,x^{2}+d \right )^{3} b c}{\sqrt {c^{2} x^{2}+1}}\right )}{4002075 c^{6} \left (c^{2} x^{2}+1\right )^{2} x^{4}}\) | \(236\) |
Input:
int(x^4*(c^2*d*x^2+d)^3*(a+b*arcsinh(x*c)),x,method=_RETURNVERBOSE)
Output:
d^3*a*(1/11*c^6*x^11+1/3*c^4*x^9+3/7*c^2*x^7+1/5*x^5)+d^3*b/c^5*(1/11*arcs inh(x*c)*x^11*c^11+1/3*arcsinh(x*c)*x^9*c^9+3/7*arcsinh(x*c)*x^7*c^7+1/5*a rcsinh(x*c)*x^5*c^5-6311/1334025*x^4*c^4*(c^2*x^2+1)^(1/2)+25244/4002075*x ^2*c^2*(c^2*x^2+1)^(1/2)-50488/4002075*(c^2*x^2+1)^(1/2)-4705/160083*x^6*c ^6*(c^2*x^2+1)^(1/2)-91/3267*x^8*c^8*(c^2*x^2+1)^(1/2)-1/121*x^10*c^10*(c^ 2*x^2+1)^(1/2))
Time = 0.10 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.89 \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {363825 \, a c^{11} d^{3} x^{11} + 1334025 \, a c^{9} d^{3} x^{9} + 1715175 \, a c^{7} d^{3} x^{7} + 800415 \, a c^{5} d^{3} x^{5} + 3465 \, {\left (105 \, b c^{11} d^{3} x^{11} + 385 \, b c^{9} d^{3} x^{9} + 495 \, b c^{7} d^{3} x^{7} + 231 \, b c^{5} d^{3} x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (33075 \, b c^{10} d^{3} x^{10} + 111475 \, b c^{8} d^{3} x^{8} + 117625 \, b c^{6} d^{3} x^{6} + 18933 \, b c^{4} d^{3} x^{4} - 25244 \, b c^{2} d^{3} x^{2} + 50488 \, b d^{3}\right )} \sqrt {c^{2} x^{2} + 1}}{4002075 \, c^{5}} \] Input:
integrate(x^4*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x)),x, algorithm="fricas")
Output:
1/4002075*(363825*a*c^11*d^3*x^11 + 1334025*a*c^9*d^3*x^9 + 1715175*a*c^7* d^3*x^7 + 800415*a*c^5*d^3*x^5 + 3465*(105*b*c^11*d^3*x^11 + 385*b*c^9*d^3 *x^9 + 495*b*c^7*d^3*x^7 + 231*b*c^5*d^3*x^5)*log(c*x + sqrt(c^2*x^2 + 1)) - (33075*b*c^10*d^3*x^10 + 111475*b*c^8*d^3*x^8 + 117625*b*c^6*d^3*x^6 + 18933*b*c^4*d^3*x^4 - 25244*b*c^2*d^3*x^2 + 50488*b*d^3)*sqrt(c^2*x^2 + 1) )/c^5
Time = 2.25 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.28 \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {a c^{6} d^{3} x^{11}}{11} + \frac {a c^{4} d^{3} x^{9}}{3} + \frac {3 a c^{2} d^{3} x^{7}}{7} + \frac {a d^{3} x^{5}}{5} + \frac {b c^{6} d^{3} x^{11} \operatorname {asinh}{\left (c x \right )}}{11} - \frac {b c^{5} d^{3} x^{10} \sqrt {c^{2} x^{2} + 1}}{121} + \frac {b c^{4} d^{3} x^{9} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {91 b c^{3} d^{3} x^{8} \sqrt {c^{2} x^{2} + 1}}{3267} + \frac {3 b c^{2} d^{3} x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {4705 b c d^{3} x^{6} \sqrt {c^{2} x^{2} + 1}}{160083} + \frac {b d^{3} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {6311 b d^{3} x^{4} \sqrt {c^{2} x^{2} + 1}}{1334025 c} + \frac {25244 b d^{3} x^{2} \sqrt {c^{2} x^{2} + 1}}{4002075 c^{3}} - \frac {50488 b d^{3} \sqrt {c^{2} x^{2} + 1}}{4002075 c^{5}} & \text {for}\: c \neq 0 \\\frac {a d^{3} x^{5}}{5} & \text {otherwise} \end {cases} \] Input:
integrate(x**4*(c**2*d*x**2+d)**3*(a+b*asinh(c*x)),x)
Output:
Piecewise((a*c**6*d**3*x**11/11 + a*c**4*d**3*x**9/3 + 3*a*c**2*d**3*x**7/ 7 + a*d**3*x**5/5 + b*c**6*d**3*x**11*asinh(c*x)/11 - b*c**5*d**3*x**10*sq rt(c**2*x**2 + 1)/121 + b*c**4*d**3*x**9*asinh(c*x)/3 - 91*b*c**3*d**3*x** 8*sqrt(c**2*x**2 + 1)/3267 + 3*b*c**2*d**3*x**7*asinh(c*x)/7 - 4705*b*c*d* *3*x**6*sqrt(c**2*x**2 + 1)/160083 + b*d**3*x**5*asinh(c*x)/5 - 6311*b*d** 3*x**4*sqrt(c**2*x**2 + 1)/(1334025*c) + 25244*b*d**3*x**2*sqrt(c**2*x**2 + 1)/(4002075*c**3) - 50488*b*d**3*sqrt(c**2*x**2 + 1)/(4002075*c**5), Ne( c, 0)), (a*d**3*x**5/5, True))
Leaf count of result is larger than twice the leaf count of optimal. 465 vs. \(2 (194) = 388\).
Time = 0.04 (sec) , antiderivative size = 465, normalized size of antiderivative = 2.06 \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{11} \, a c^{6} d^{3} x^{11} + \frac {1}{3} \, a c^{4} d^{3} x^{9} + \frac {3}{7} \, a c^{2} d^{3} x^{7} + \frac {1}{7623} \, {\left (693 \, x^{11} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {63 \, \sqrt {c^{2} x^{2} + 1} x^{10}}{c^{2}} - \frac {70 \, \sqrt {c^{2} x^{2} + 1} x^{8}}{c^{4}} + \frac {80 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{6}} - \frac {96 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{10}} - \frac {256 \, \sqrt {c^{2} x^{2} + 1}}{c^{12}}\right )} c\right )} b c^{6} d^{3} + \frac {1}{945} \, {\left (315 \, x^{9} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {35 \, \sqrt {c^{2} x^{2} + 1} x^{8}}{c^{2}} - \frac {40 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{6}} - \frac {64 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} b c^{4} d^{3} + \frac {1}{5} \, a d^{3} x^{5} + \frac {3}{245} \, {\left (35 \, x^{7} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b c^{2} d^{3} + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d^{3} \] Input:
integrate(x^4*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x)),x, algorithm="maxima")
Output:
1/11*a*c^6*d^3*x^11 + 1/3*a*c^4*d^3*x^9 + 3/7*a*c^2*d^3*x^7 + 1/7623*(693* x^11*arcsinh(c*x) - (63*sqrt(c^2*x^2 + 1)*x^10/c^2 - 70*sqrt(c^2*x^2 + 1)* x^8/c^4 + 80*sqrt(c^2*x^2 + 1)*x^6/c^6 - 96*sqrt(c^2*x^2 + 1)*x^4/c^8 + 12 8*sqrt(c^2*x^2 + 1)*x^2/c^10 - 256*sqrt(c^2*x^2 + 1)/c^12)*c)*b*c^6*d^3 + 1/945*(315*x^9*arcsinh(c*x) - (35*sqrt(c^2*x^2 + 1)*x^8/c^2 - 40*sqrt(c^2* x^2 + 1)*x^6/c^4 + 48*sqrt(c^2*x^2 + 1)*x^4/c^6 - 64*sqrt(c^2*x^2 + 1)*x^2 /c^8 + 128*sqrt(c^2*x^2 + 1)/c^10)*c)*b*c^4*d^3 + 1/5*a*d^3*x^5 + 3/245*(3 5*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 ^3 + 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^3
Exception generated. \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x^4*(c^2*d*x^2+d)^3*(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. \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\int x^4\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^3 \,d x \] Input:
int(x^4*(a + b*asinh(c*x))*(d + c^2*d*x^2)^3,x)
Output:
int(x^4*(a + b*asinh(c*x))*(d + c^2*d*x^2)^3, x)
Time = 0.19 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.91 \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^{3} \left (363825 \mathit {asinh} \left (c x \right ) b \,c^{11} x^{11}+1334025 \mathit {asinh} \left (c x \right ) b \,c^{9} x^{9}+1715175 \mathit {asinh} \left (c x \right ) b \,c^{7} x^{7}+800415 \mathit {asinh} \left (c x \right ) b \,c^{5} x^{5}-33075 \sqrt {c^{2} x^{2}+1}\, b \,c^{10} x^{10}-111475 \sqrt {c^{2} x^{2}+1}\, b \,c^{8} x^{8}-117625 \sqrt {c^{2} x^{2}+1}\, b \,c^{6} x^{6}-18933 \sqrt {c^{2} x^{2}+1}\, b \,c^{4} x^{4}+25244 \sqrt {c^{2} x^{2}+1}\, b \,c^{2} x^{2}-50488 \sqrt {c^{2} x^{2}+1}\, b +363825 a \,c^{11} x^{11}+1334025 a \,c^{9} x^{9}+1715175 a \,c^{7} x^{7}+800415 a \,c^{5} x^{5}\right )}{4002075 c^{5}} \] Input:
int(x^4*(c^2*d*x^2+d)^3*(a+b*asinh(c*x)),x)
Output:
(d**3*(363825*asinh(c*x)*b*c**11*x**11 + 1334025*asinh(c*x)*b*c**9*x**9 + 1715175*asinh(c*x)*b*c**7*x**7 + 800415*asinh(c*x)*b*c**5*x**5 - 33075*sqr t(c**2*x**2 + 1)*b*c**10*x**10 - 111475*sqrt(c**2*x**2 + 1)*b*c**8*x**8 - 117625*sqrt(c**2*x**2 + 1)*b*c**6*x**6 - 18933*sqrt(c**2*x**2 + 1)*b*c**4* x**4 + 25244*sqrt(c**2*x**2 + 1)*b*c**2*x**2 - 50488*sqrt(c**2*x**2 + 1)*b + 363825*a*c**11*x**11 + 1334025*a*c**9*x**9 + 1715175*a*c**7*x**7 + 8004 15*a*c**5*x**5))/(4002075*c**5)