Integrand size = 28, antiderivative size = 455 \[ \int (f+g x) \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {b d^2 g x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}-\frac {5 b c d^2 f x^2 \sqrt {d+c^2 d x^2}}{32 \sqrt {1+c^2 x^2}}-\frac {b c d^2 g x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d^2 g x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 g x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {5 b d^2 f \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2}}{96 c}-\frac {b d^2 f \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{24} d f x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{6} f x \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {g \left (d+c^2 d x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^2 d}+\frac {5 d^2 f \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{32 b c \sqrt {1+c^2 x^2}} \] Output:
-1/7*b*d^2*g*x*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-5/32*b*c*d^2*f*x^2* (c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/7*b*c*d^2*g*x^3*(c^2*d*x^2+d)^(1/2 )/(c^2*x^2+1)^(1/2)-3/35*b*c^3*d^2*g*x^5*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^( 1/2)-1/49*b*c^5*d^2*g*x^7*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-5/96*b*d^2 *f*(c^2*x^2+1)^(3/2)*(c^2*d*x^2+d)^(1/2)/c-1/36*b*d^2*f*(c^2*x^2+1)^(5/2)* (c^2*d*x^2+d)^(1/2)/c+5/16*d^2*f*x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))+ 5/24*d*f*x*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))+1/6*f*x*(c^2*d*x^2+d)^(5 /2)*(a+b*arcsinh(c*x))+1/7*g*(c^2*d*x^2+d)^(7/2)*(a+b*arcsinh(c*x))/c^2/d+ 5/32*d^2*f*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/b/c/(c^2*x^2+1)^(1/2)
Time = 0.80 (sec) , antiderivative size = 390, normalized size of antiderivative = 0.86 \[ \int (f+g x) \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {-d^3 \left (1+c^2 x^2\right ) \left (-1680 a \sqrt {1+c^2 x^2} \left (48 g \left (1+c^2 x^2\right )^3+7 c^2 f x \left (33+26 c^2 x^2+8 c^4 x^4\right )\right )+b c \left (2304 g x \left (35+35 c^2 x^2+21 c^4 x^4+5 c^6 x^6\right )+245 f \left (299+792 c^2 x^2+312 c^4 x^4+64 c^6 x^6\right )\right )\right )+88200 b c d^3 f \left (1+c^2 x^2\right ) \text {arcsinh}(c x)^2+176400 a c d^{5/2} f \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+420 b d^3 \left (1+c^2 x^2\right ) \text {arcsinh}(c x) \left (192 g \sqrt {1+c^2 x^2}+576 c^2 g x^2 \sqrt {1+c^2 x^2}+576 c^4 g x^4 \sqrt {1+c^2 x^2}+192 c^6 g x^6 \sqrt {1+c^2 x^2}+315 c f \sinh (2 \text {arcsinh}(c x))+63 c f \sinh (4 \text {arcsinh}(c x))+7 c f \sinh (6 \text {arcsinh}(c x))\right )}{564480 c^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}} \] Input:
Integrate[(f + g*x)*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]
Output:
(-(d^3*(1 + c^2*x^2)*(-1680*a*Sqrt[1 + c^2*x^2]*(48*g*(1 + c^2*x^2)^3 + 7* c^2*f*x*(33 + 26*c^2*x^2 + 8*c^4*x^4)) + b*c*(2304*g*x*(35 + 35*c^2*x^2 + 21*c^4*x^4 + 5*c^6*x^6) + 245*f*(299 + 792*c^2*x^2 + 312*c^4*x^4 + 64*c^6* x^6)))) + 88200*b*c*d^3*f*(1 + c^2*x^2)*ArcSinh[c*x]^2 + 176400*a*c*d^(5/2 )*f*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2 *d*x^2]] + 420*b*d^3*(1 + c^2*x^2)*ArcSinh[c*x]*(192*g*Sqrt[1 + c^2*x^2] + 576*c^2*g*x^2*Sqrt[1 + c^2*x^2] + 576*c^4*g*x^4*Sqrt[1 + c^2*x^2] + 192*c ^6*g*x^6*Sqrt[1 + c^2*x^2] + 315*c*f*Sinh[2*ArcSinh[c*x]] + 63*c*f*Sinh[4* ArcSinh[c*x]] + 7*c*f*Sinh[6*ArcSinh[c*x]]))/(564480*c^2*Sqrt[1 + c^2*x^2] *Sqrt[d + c^2*d*x^2])
Time = 1.13 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.55, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {6260, 6253, 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.
| \(\displaystyle \int \left (c^2 d x^2+d\right )^{5/2} (f+g x) (a+b \text {arcsinh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6260 |
| \(\displaystyle \frac {d^2 \sqrt {c^2 d x^2+d} \int (f+g x) \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))dx}{\sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6253 |
| \(\displaystyle \frac {d^2 \sqrt {c^2 d x^2+d} \int \left (f (a+b \text {arcsinh}(c x)) \left (c^2 x^2+1\right )^{5/2}+g x (a+b \text {arcsinh}(c x)) \left (c^2 x^2+1\right )^{5/2}\right )dx}{\sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{6} f x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{24} f x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {5}{16} f x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {g \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^2}+\frac {5 f (a+b \text {arcsinh}(c x))^2}{32 b c}-\frac {1}{49} b c^5 g x^7-\frac {5}{96} b c^3 f x^4-\frac {3}{35} b c^3 g x^5-\frac {b f \left (c^2 x^2+1\right )^3}{36 c}-\frac {25}{96} b c f x^2-\frac {1}{7} b c g x^3-\frac {b g x}{7 c}\right )}{\sqrt {c^2 x^2+1}}\) |
Input:
Int[(f + g*x)*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]
Output:
(d^2*Sqrt[d + c^2*d*x^2]*(-1/7*(b*g*x)/c - (25*b*c*f*x^2)/96 - (b*c*g*x^3) /7 - (5*b*c^3*f*x^4)/96 - (3*b*c^3*g*x^5)/35 - (b*c^5*g*x^7)/49 - (b*f*(1 + c^2*x^2)^3)/(36*c) + (5*f*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/16 + (5*f*x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/24 + (f*x*(1 + c^2*x^2)^ (5/2)*(a + b*ArcSinh[c*x]))/6 + (g*(1 + c^2*x^2)^(7/2)*(a + b*ArcSinh[c*x] ))/(7*c^2) + (5*f*(a + b*ArcSinh[c*x])^2)/(32*b*c)))/Sqrt[1 + c^2*x^2]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d _) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n , 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] || EqQ[m, 1] || (EqQ[m, 2] && LtQ[p, -2]))
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d _) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[Simp[(d + e*x^2)^p/(1 + c^2*x^2) ^p] Int[(f + g*x)^m*(1 + c^2*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] /; Fre eQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e, c^2*d] && IntegerQ[m] && IntegerQ [p - 1/2] && !GtQ[d, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1678\) vs. \(2(391)=782\).
Time = 2.01 (sec) , antiderivative size = 1679, normalized size of antiderivative = 3.69
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1679\) |
| parts | \(\text {Expression too large to display}\) | \(1679\) |
Input:
int((g*x+f)*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(x*c)),x,method=_RETURNVERBOSE )
Output:
1/6*a*f*x*(c^2*d*x^2+d)^(5/2)+5/24*a*f*d*x*(c^2*d*x^2+d)^(3/2)+5/16*a*f*d^ 2*x*(c^2*d*x^2+d)^(1/2)+5/16*a*f*d^3*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d )^(1/2))/(c^2*d)^(1/2)+1/7*a*g*(c^2*d*x^2+d)^(7/2)/c^2/d+b*(5/32*(d*(c^2*x ^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c*f*arcsinh(x*c)^2*d^2+1/6272*(d*(c^2*x^2+1 ))^(1/2)*(64*c^8*x^8+64*x^7*c^7*(c^2*x^2+1)^(1/2)+144*c^6*x^6+112*(c^2*x^2 +1)^(1/2)*x^5*c^5+104*c^4*x^4+56*(c^2*x^2+1)^(1/2)*c^3*x^3+25*c^2*x^2+7*(c ^2*x^2+1)^(1/2)*x*c+1)*g*(-1+7*arcsinh(x*c))*d^2/c^2/(c^2*x^2+1)+1/2304*(d *(c^2*x^2+1))^(1/2)*(32*x^7*c^7+32*x^6*c^6*(c^2*x^2+1)^(1/2)+64*x^5*c^5+48 *x^4*c^4*(c^2*x^2+1)^(1/2)+38*x^3*c^3+18*x^2*c^2*(c^2*x^2+1)^(1/2)+6*x*c+( c^2*x^2+1)^(1/2))*f*(-1+6*arcsinh(x*c))*d^2/(c^2*x^2+1)/c+1/640*(d*(c^2*x^ 2+1))^(1/2)*(16*c^6*x^6+16*(c^2*x^2+1)^(1/2)*x^5*c^5+28*c^4*x^4+20*(c^2*x^ 2+1)^(1/2)*c^3*x^3+13*c^2*x^2+5*(c^2*x^2+1)^(1/2)*x*c+1)*g*(-1+5*arcsinh(x *c))*d^2/c^2/(c^2*x^2+1)+3/512*(d*(c^2*x^2+1))^(1/2)*(8*x^5*c^5+8*x^4*c^4* (c^2*x^2+1)^(1/2)+12*x^3*c^3+8*x^2*c^2*(c^2*x^2+1)^(1/2)+4*x*c+(c^2*x^2+1) ^(1/2))*f*(-1+4*arcsinh(x*c))*d^2/(c^2*x^2+1)/c+1/128*(d*(c^2*x^2+1))^(1/2 )*(4*c^4*x^4+4*(c^2*x^2+1)^(1/2)*c^3*x^3+5*c^2*x^2+3*(c^2*x^2+1)^(1/2)*x*c +1)*g*(-1+3*arcsinh(x*c))*d^2/c^2/(c^2*x^2+1)+15/256*(d*(c^2*x^2+1))^(1/2) *(2*x^3*c^3+2*x^2*c^2*(c^2*x^2+1)^(1/2)+2*x*c+(c^2*x^2+1)^(1/2))*f*(-1+2*a rcsinh(x*c))*d^2/(c^2*x^2+1)/c+5/128*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+(c^2*x ^2+1)^(1/2)*x*c+1)*g*(arcsinh(x*c)-1)*d^2/c^2/(c^2*x^2+1)+5/128*(d*(c^2...
\[ \int (f+g x) \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} \,d x } \] Input:
integrate((g*x+f)*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="fri cas")
Output:
integral((a*c^4*d^2*g*x^5 + a*c^4*d^2*f*x^4 + 2*a*c^2*d^2*g*x^3 + 2*a*c^2* d^2*f*x^2 + a*d^2*g*x + a*d^2*f + (b*c^4*d^2*g*x^5 + b*c^4*d^2*f*x^4 + 2*b *c^2*d^2*g*x^3 + 2*b*c^2*d^2*f*x^2 + b*d^2*g*x + b*d^2*f)*arcsinh(c*x))*sq rt(c^2*d*x^2 + d), x)
Timed out. \[ \int (f+g x) \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Timed out} \] Input:
integrate((g*x+f)*(c**2*d*x**2+d)**(5/2)*(a+b*asinh(c*x)),x)
Output:
Timed out
Exception generated. \[ \int (f+g x) \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((g*x+f)*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="max ima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Exception generated. \[ \int (f+g x) \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((g*x+f)*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="gia c")
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. \[ \int (f+g x) \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int \left (f+g\,x\right )\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{5/2} \,d x \] Input:
int((f + g*x)*(a + b*asinh(c*x))*(d + c^2*d*x^2)^(5/2),x)
Output:
int((f + g*x)*(a + b*asinh(c*x))*(d + c^2*d*x^2)^(5/2), x)
\[ \int (f+g x) \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {d}\, d^{2} \left (56 \sqrt {c^{2} x^{2}+1}\, a \,c^{6} f \,x^{5}+48 \sqrt {c^{2} x^{2}+1}\, a \,c^{6} g \,x^{6}+182 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} f \,x^{3}+144 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} g \,x^{4}+231 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} f x +144 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} g \,x^{2}+48 \sqrt {c^{2} x^{2}+1}\, a g +336 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{5}d x \right ) b \,c^{6} g +336 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{4}d x \right ) b \,c^{6} f +672 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{3}d x \right ) b \,c^{4} g +672 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{2}d x \right ) b \,c^{4} f +336 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x d x \right ) b \,c^{2} g +336 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )d x \right ) b \,c^{2} f +105 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a c f \right )}{336 c^{2}} \] Input:
int((g*x+f)*(c^2*d*x^2+d)^(5/2)*(a+b*asinh(c*x)),x)
Output:
(sqrt(d)*d**2*(56*sqrt(c**2*x**2 + 1)*a*c**6*f*x**5 + 48*sqrt(c**2*x**2 + 1)*a*c**6*g*x**6 + 182*sqrt(c**2*x**2 + 1)*a*c**4*f*x**3 + 144*sqrt(c**2*x **2 + 1)*a*c**4*g*x**4 + 231*sqrt(c**2*x**2 + 1)*a*c**2*f*x + 144*sqrt(c** 2*x**2 + 1)*a*c**2*g*x**2 + 48*sqrt(c**2*x**2 + 1)*a*g + 336*int(sqrt(c**2 *x**2 + 1)*asinh(c*x)*x**5,x)*b*c**6*g + 336*int(sqrt(c**2*x**2 + 1)*asinh (c*x)*x**4,x)*b*c**6*f + 672*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**3,x)*b* c**4*g + 672*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**2,x)*b*c**4*f + 336*int (sqrt(c**2*x**2 + 1)*asinh(c*x)*x,x)*b*c**2*g + 336*int(sqrt(c**2*x**2 + 1 )*asinh(c*x),x)*b*c**2*f + 105*log(sqrt(c**2*x**2 + 1) + c*x)*a*c*f))/(336 *c**2)