Integrand size = 28, antiderivative size = 331 \[ \int (f+g x) \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {b d g x \sqrt {d+c^2 d x^2}}{5 c \sqrt {1+c^2 x^2}}-\frac {3 b c d f x^2 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {2 b c d g x^3 \sqrt {d+c^2 d x^2}}{15 \sqrt {1+c^2 x^2}}-\frac {b c^3 d g x^5 \sqrt {d+c^2 d x^2}}{25 \sqrt {1+c^2 x^2}}-\frac {b d f \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2}}{16 c}+\frac {3}{8} d f x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{4} f x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {g \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2 d}+\frac {3 d f \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 b c \sqrt {1+c^2 x^2}} \] Output:
-1/5*b*d*g*x*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-3/16*b*c*d*f*x^2*(c^2 *d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-2/15*b*c*d*g*x^3*(c^2*d*x^2+d)^(1/2)/(c^ 2*x^2+1)^(1/2)-1/25*b*c^3*d*g*x^5*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/ 16*b*d*f*(c^2*x^2+1)^(3/2)*(c^2*d*x^2+d)^(1/2)/c+3/8*d*f*x*(c^2*d*x^2+d)^( 1/2)*(a+b*arcsinh(c*x))+1/4*f*x*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))+1/5 *g*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))/c^2/d+3/16*d*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 = 0.60 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.87 \[ \int (f+g x) \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {-d^2 \left (1+c^2 x^2\right ) \left (-240 a \sqrt {1+c^2 x^2} \left (8 g \left (1+c^2 x^2\right )^2+5 c^2 f x \left (5+2 c^2 x^2\right )\right )+b c \left (128 g x \left (15+10 c^2 x^2+3 c^4 x^4\right )+75 f \left (17+40 c^2 x^2+8 c^4 x^4\right )\right )\right )+1800 b c d^2 f \left (1+c^2 x^2\right ) \text {arcsinh}(c x)^2+3600 a c d^{3/2} f \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+60 b d^2 \left (1+c^2 x^2\right ) \text {arcsinh}(c x) \left (32 g \left (1+c^2 x^2\right )^{5/2}+40 c f \sinh (2 \text {arcsinh}(c x))+5 c f \sinh (4 \text {arcsinh}(c x))\right )}{9600 c^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}} \] Input:
Integrate[(f + g*x)*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]),x]
Output:
(-(d^2*(1 + c^2*x^2)*(-240*a*Sqrt[1 + c^2*x^2]*(8*g*(1 + c^2*x^2)^2 + 5*c^ 2*f*x*(5 + 2*c^2*x^2)) + b*c*(128*g*x*(15 + 10*c^2*x^2 + 3*c^4*x^4) + 75*f *(17 + 40*c^2*x^2 + 8*c^4*x^4)))) + 1800*b*c*d^2*f*(1 + c^2*x^2)*ArcSinh[c *x]^2 + 3600*a*c*d^(3/2)*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]] + 60*b*d^2*(1 + c^2*x^2)*ArcSinh[c*x]*(32* g*(1 + c^2*x^2)^(5/2) + 40*c*f*Sinh[2*ArcSinh[c*x]] + 5*c*f*Sinh[4*ArcSinh [c*x]]))/(9600*c^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2])
Time = 0.66 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.57, 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.
| \(\displaystyle \int \left (c^2 d x^2+d\right )^{3/2} (f+g x) (a+b \text {arcsinh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6260 |
| \(\displaystyle \frac {d \sqrt {c^2 d x^2+d} \int (f+g x) \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx}{\sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6253 |
| \(\displaystyle \frac {d \sqrt {c^2 d x^2+d} \int \left (f (a+b \text {arcsinh}(c x)) \left (c^2 x^2+1\right )^{3/2}+g x (a+b \text {arcsinh}(c x)) \left (c^2 x^2+1\right )^{3/2}\right )dx}{\sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \sqrt {c^2 d x^2+d} \left (\frac {1}{4} f x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{8} f x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {g \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}+\frac {3 f (a+b \text {arcsinh}(c x))^2}{16 b c}-\frac {1}{16} b c^3 f x^4-\frac {1}{25} b c^3 g x^5-\frac {5}{16} b c f x^2-\frac {2}{15} b c g x^3-\frac {b g x}{5 c}\right )}{\sqrt {c^2 x^2+1}}\) |
Input:
Int[(f + g*x)*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]),x]
Output:
(d*Sqrt[d + c^2*d*x^2]*(-1/5*(b*g*x)/c - (5*b*c*f*x^2)/16 - (2*b*c*g*x^3)/ 15 - (b*c^3*f*x^4)/16 - (b*c^3*g*x^5)/25 + (3*f*x*Sqrt[1 + c^2*x^2]*(a + b *ArcSinh[c*x]))/8 + (f*x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/4 + (g* (1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(5*c^2) + (3*f*(a + b*ArcSinh[c* x])^2)/(16*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. \(1064\) vs. \(2(283)=566\).
Time = 1.87 (sec) , antiderivative size = 1065, normalized size of antiderivative = 3.22
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1065\) |
| parts | \(\text {Expression too large to display}\) | \(1065\) |
Input:
int((g*x+f)*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(x*c)),x,method=_RETURNVERBOSE )
Output:
1/4*a*f*x*(c^2*d*x^2+d)^(3/2)+3/8*a*f*d*x*(c^2*d*x^2+d)^(1/2)+3/8*a*f*d^2* ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+1/5*a*g*(c^2*d *x^2+d)^(5/2)/c^2/d+b*(3/16*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c*f*ar csinh(x*c)^2*d+1/800*(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/c^2/(c^2*x^2+1)+1/256*(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/(c^2*x^2+1)/c +1/96*(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/c^2/(c^2*x^2+1)+1/16 *(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*arcsinh(x*c))*d/(c^2*x^2+1)/c+1/16*(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/c^2/(c^2*x^2+1) +1/16*(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/c^2/(c^2*x^2+1)+1/16*(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*arcsinh(x*c))*d/(c^2*x^2+ 1)/c+1/96*(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/c^2/(c^2*x^2+1)+1 /256*(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(...
\[ \int (f+g x) \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} \,d x } \] Input:
integrate((g*x+f)*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x)),x, algorithm="fri cas")
Output:
integral((a*c^2*d*g*x^3 + a*c^2*d*f*x^2 + a*d*g*x + a*d*f + (b*c^2*d*g*x^3 + b*c^2*d*f*x^2 + b*d*g*x + b*d*f)*arcsinh(c*x))*sqrt(c^2*d*x^2 + d), x)
\[ \int (f+g x) \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \left (f + g x\right )\, dx \] Input:
integrate((g*x+f)*(c**2*d*x**2+d)**(3/2)*(a+b*asinh(c*x)),x)
Output:
Integral((d*(c**2*x**2 + 1))**(3/2)*(a + b*asinh(c*x))*(f + g*x), x)
Exception generated. \[ \int (f+g x) \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((g*x+f)*(c^2*d*x^2+d)^(3/2)*(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. \[ \int (f+g x) \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((g*x+f)*(c^2*d*x^2+d)^(3/2)*(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. \[ \int (f+g x) \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int \left (f+g\,x\right )\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \] Input:
int((f + g*x)*(a + b*asinh(c*x))*(d + c^2*d*x^2)^(3/2),x)
Output:
int((f + g*x)*(a + b*asinh(c*x))*(d + c^2*d*x^2)^(3/2), x)
\[ \int (f+g x) \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {d}\, d \left (10 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} f \,x^{3}+8 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} g \,x^{4}+25 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} f x +16 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} g \,x^{2}+8 \sqrt {c^{2} x^{2}+1}\, a g +40 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{3}d x \right ) b \,c^{4} g +40 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{2}d x \right ) b \,c^{4} f +40 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x d x \right ) b \,c^{2} g +40 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )d x \right ) b \,c^{2} f +15 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a c f \right )}{40 c^{2}} \] Input:
int((g*x+f)*(c^2*d*x^2+d)^(3/2)*(a+b*asinh(c*x)),x)
Output:
(sqrt(d)*d*(10*sqrt(c**2*x**2 + 1)*a*c**4*f*x**3 + 8*sqrt(c**2*x**2 + 1)*a *c**4*g*x**4 + 25*sqrt(c**2*x**2 + 1)*a*c**2*f*x + 16*sqrt(c**2*x**2 + 1)* a*c**2*g*x**2 + 8*sqrt(c**2*x**2 + 1)*a*g + 40*int(sqrt(c**2*x**2 + 1)*asi nh(c*x)*x**3,x)*b*c**4*g + 40*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**2,x)*b *c**4*f + 40*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x,x)*b*c**2*g + 40*int(sqr t(c**2*x**2 + 1)*asinh(c*x),x)*b*c**2*f + 15*log(sqrt(c**2*x**2 + 1) + c*x )*a*c*f))/(40*c**2)