\(\int (f+g x)^3 (d+c^2 d x^2)^{3/2} (a+b \text {arcsinh}(c x)) \, dx\) [39]

Optimal result

Integrand size = 30, antiderivative size = 868 \[ \int (f+g x)^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {3 b d f^2 g x \sqrt {d+c^2 d x^2}}{5 c \sqrt {1+c^2 x^2}}+\frac {2 b d g^3 x \sqrt {d+c^2 d x^2}}{35 c^3 \sqrt {1+c^2 x^2}}-\frac {3 b c d f^3 x^2 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {3 b d f g^2 x^2 \sqrt {d+c^2 d x^2}}{32 c \sqrt {1+c^2 x^2}}-\frac {2 b c d f^2 g x^3 \sqrt {d+c^2 d x^2}}{5 \sqrt {1+c^2 x^2}}-\frac {b d g^3 x^3 \sqrt {d+c^2 d x^2}}{105 c \sqrt {1+c^2 x^2}}-\frac {7 b c d f g^2 x^4 \sqrt {d+c^2 d x^2}}{32 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d f^2 g x^5 \sqrt {d+c^2 d x^2}}{25 \sqrt {1+c^2 x^2}}-\frac {8 b c d g^3 x^5 \sqrt {d+c^2 d x^2}}{175 \sqrt {1+c^2 x^2}}-\frac {b c^3 d f g^2 x^6 \sqrt {d+c^2 d x^2}}{12 \sqrt {1+c^2 x^2}}-\frac {b c^3 d g^3 x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {b d f^3 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2}}{16 c}+\frac {3}{8} d f^3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3 d f g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{16 c^2}+\frac {3}{8} d f g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{4} f^3 x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{2} f g^2 x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3 f^2 g \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2 d}-\frac {g^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4 d}+\frac {g^3 \left (d+c^2 d x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4 d^2}+\frac {3 d f^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 b c \sqrt {1+c^2 x^2}}-\frac {3 d f g^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{32 b c^3 \sqrt {1+c^2 x^2}} \] Output:

-3/5*b*d*f^2*g*x*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)+2/35*b*d*g^3*x*(c 
^2*d*x^2+d)^(1/2)/c^3/(c^2*x^2+1)^(1/2)-3/16*b*c*d*f^3*x^2*(c^2*d*x^2+d)^( 
1/2)/(c^2*x^2+1)^(1/2)-3/32*b*d*f*g^2*x^2*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1 
)^(1/2)-2/5*b*c*d*f^2*g*x^3*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/105*b* 
d*g^3*x^3*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-7/32*b*c*d*f*g^2*x^4*(c^ 
2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-3/25*b*c^3*d*f^2*g*x^5*(c^2*d*x^2+d)^(1 
/2)/(c^2*x^2+1)^(1/2)-8/175*b*c*d*g^3*x^5*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^ 
(1/2)-1/12*b*c^3*d*f*g^2*x^6*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/49*b* 
c^3*d*g^3*x^7*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/16*b*d*f^3*(c^2*x^2+ 
1)^(3/2)*(c^2*d*x^2+d)^(1/2)/c+3/8*d*f^3*x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsin 
h(c*x))+3/16*d*f*g^2*x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/c^2+3/8*d*f* 
g^2*x^3*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))+1/4*f^3*x*(c^2*d*x^2+d)^(3/ 
2)*(a+b*arcsinh(c*x))+1/2*f*g^2*x^3*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x)) 
+3/5*f^2*g*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))/c^2/d-1/5*g^3*(c^2*d*x^2 
+d)^(5/2)*(a+b*arcsinh(c*x))/c^4/d+1/7*g^3*(c^2*d*x^2+d)^(7/2)*(a+b*arcsin 
h(c*x))/c^4/d^2+3/16*d*f^3*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/b/c/(c 
^2*x^2+1)^(1/2)-3/32*d*f*g^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/b/c^ 
3/(c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 1.50 (sec) , antiderivative size = 536, normalized size of antiderivative = 0.62 \[ \int (f+g x)^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {-d^2 \left (1+c^2 x^2\right ) \left (-1680 a \sqrt {1+c^2 x^2} \left (-32 g^3+c^2 g \left (336 f^2+105 f g x+16 g^2 x^2\right )+4 c^6 x^3 \left (35 f^3+84 f^2 g x+70 f g^2 x^2+20 g^3 x^3\right )+2 c^4 x \left (175 f^3+336 f^2 g x+245 f g^2 x^2+64 g^3 x^3\right )\right )+b \left (-35 c g^2 (245 f+1536 g x)+70 c^3 \left (1785 f^3+8064 f^2 g x+1260 f g^2 x^2+128 g^3 x^3\right )+168 c^5 x^2 \left (1750 f^3+2240 f^2 g x+1225 f g^2 x^2+256 g^3 x^3\right )+16 c^7 x^4 \left (3675 f^3+7056 f^2 g x+4900 f g^2 x^2+1200 g^3 x^3\right )\right )\right )+88200 b c d^2 f \left (2 c^2 f^2-g^2\right ) \left (1+c^2 x^2\right ) \text {arcsinh}(c x)^2+176400 a c d^{3/2} f \left (2 c^2 f^2-g^2\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+420 b d^2 \left (1+c^2 x^2\right ) \text {arcsinh}(c x) \left (35 c f \left (16 c^2 f^2-3 g^2\right ) \sinh (2 \text {arcsinh}(c x))+35 c f \left (2 c^2 f^2+3 g^2\right ) \sinh (4 \text {arcsinh}(c x))+g \left (64 \left (1+c^2 x^2\right )^{5/2} \left (-2 g^2+c^2 \left (21 f^2+5 g^2 x^2\right )\right )+35 c f g \sinh (6 \text {arcsinh}(c x))\right )\right )}{940800 c^4 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}} \] Input:

Integrate[(f + g*x)^3*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]),x]
 

Output:

(-(d^2*(1 + c^2*x^2)*(-1680*a*Sqrt[1 + c^2*x^2]*(-32*g^3 + c^2*g*(336*f^2 
+ 105*f*g*x + 16*g^2*x^2) + 4*c^6*x^3*(35*f^3 + 84*f^2*g*x + 70*f*g^2*x^2 
+ 20*g^3*x^3) + 2*c^4*x*(175*f^3 + 336*f^2*g*x + 245*f*g^2*x^2 + 64*g^3*x^ 
3)) + b*(-35*c*g^2*(245*f + 1536*g*x) + 70*c^3*(1785*f^3 + 8064*f^2*g*x + 
1260*f*g^2*x^2 + 128*g^3*x^3) + 168*c^5*x^2*(1750*f^3 + 2240*f^2*g*x + 122 
5*f*g^2*x^2 + 256*g^3*x^3) + 16*c^7*x^4*(3675*f^3 + 7056*f^2*g*x + 4900*f* 
g^2*x^2 + 1200*g^3*x^3)))) + 88200*b*c*d^2*f*(2*c^2*f^2 - g^2)*(1 + c^2*x^ 
2)*ArcSinh[c*x]^2 + 176400*a*c*d^(3/2)*f*(2*c^2*f^2 - g^2)*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^ 
2*(1 + c^2*x^2)*ArcSinh[c*x]*(35*c*f*(16*c^2*f^2 - 3*g^2)*Sinh[2*ArcSinh[c 
*x]] + 35*c*f*(2*c^2*f^2 + 3*g^2)*Sinh[4*ArcSinh[c*x]] + g*(64*(1 + c^2*x^ 
2)^(5/2)*(-2*g^2 + c^2*(21*f^2 + 5*g^2*x^2)) + 35*c*f*g*Sinh[6*ArcSinh[c*x 
]])))/(940800*c^4*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2])
 

Rubi [A] (verified)

Time = 2.01 (sec) , antiderivative size = 488, normalized size of antiderivative = 0.56, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.

\(\displaystyle \int \left (c^2 d x^2+d\right )^{3/2} (f+g x)^3 (a+b \text {arcsinh}(c x)) \, dx\)

\(\Big \downarrow \) 6260

\(\displaystyle \frac {d \sqrt {c^2 d x^2+d} \int (f+g x)^3 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx}{\sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6253

\(\displaystyle \frac {d \sqrt {c^2 d x^2+d} \int \left (\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x)) f^3+3 g x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x)) f^2+3 g^2 x^2 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x)) f+g^3 x^3 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))\right )dx}{\sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d \sqrt {c^2 d x^2+d} \left (-\frac {3 f g^2 (a+b \text {arcsinh}(c x))^2}{32 b c^3}+\frac {1}{4} f^3 x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{8} f^3 x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {3 f^2 g \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}+\frac {3 f g^2 x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{16 c^2}+\frac {1}{2} f g^2 x^3 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{8} f g^2 x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {g^3 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {g^3 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {3 f^3 (a+b \text {arcsinh}(c x))^2}{16 b c}-\frac {1}{16} b c^3 f^3 x^4-\frac {3}{25} b c^3 f^2 g x^5-\frac {1}{12} b c^3 f g^2 x^6-\frac {1}{49} b c^3 g^3 x^7+\frac {2 b g^3 x}{35 c^3}-\frac {5}{16} b c f^3 x^2-\frac {2}{5} b c f^2 g x^3-\frac {3 b f^2 g x}{5 c}-\frac {7}{32} b c f g^2 x^4-\frac {3 b f g^2 x^2}{32 c}-\frac {8}{175} b c g^3 x^5-\frac {b g^3 x^3}{105 c}\right )}{\sqrt {c^2 x^2+1}}\)

Input:

Int[(f + g*x)^3*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]),x]
 

Output:

(d*Sqrt[d + c^2*d*x^2]*((-3*b*f^2*g*x)/(5*c) + (2*b*g^3*x)/(35*c^3) - (5*b 
*c*f^3*x^2)/16 - (3*b*f*g^2*x^2)/(32*c) - (2*b*c*f^2*g*x^3)/5 - (b*g^3*x^3 
)/(105*c) - (b*c^3*f^3*x^4)/16 - (7*b*c*f*g^2*x^4)/32 - (3*b*c^3*f^2*g*x^5 
)/25 - (8*b*c*g^3*x^5)/175 - (b*c^3*f*g^2*x^6)/12 - (b*c^3*g^3*x^7)/49 + ( 
3*f^3*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/8 + (3*f*g^2*x*Sqrt[1 + c^ 
2*x^2]*(a + b*ArcSinh[c*x]))/(16*c^2) + (3*f*g^2*x^3*Sqrt[1 + c^2*x^2]*(a 
+ b*ArcSinh[c*x]))/8 + (f^3*x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/4 
+ (f*g^2*x^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/2 + (3*f^2*g*(1 + c 
^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(5*c^2) - (g^3*(1 + c^2*x^2)^(5/2)*(a 
+ b*ArcSinh[c*x]))/(5*c^4) + (g^3*(1 + c^2*x^2)^(7/2)*(a + b*ArcSinh[c*x]) 
)/(7*c^4) + (3*f^3*(a + b*ArcSinh[c*x])^2)/(16*b*c) - (3*f*g^2*(a + b*ArcS 
inh[c*x])^2)/(32*b*c^3)))/Sqrt[1 + c^2*x^2]
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6253

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]))
 

rule 6260

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2078\) vs. \(2(752)=1504\).

Time = 1.43 (sec) , antiderivative size = 2079, normalized size of antiderivative = 2.40

method	result	size
default	\(\text {Expression too large to display}\)	\(2079\)
parts	\(\text {Expression too large to display}\)	\(2079\)

Input:

int((g*x+f)^3*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(x*c)),x,method=_RETURNVERBO 
SE)
 

Output:

a*(f^3*(1/4*x*(c^2*d*x^2+d)^(3/2)+3/4*d*(1/2*x*(c^2*d*x^2+d)^(1/2)+1/2*d*l 
n(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)))+g^3*(1/7*x^2* 
(c^2*d*x^2+d)^(5/2)/c^2/d-2/35/d/c^4*(c^2*d*x^2+d)^(5/2))+3*f*g^2*(1/6*x*( 
c^2*d*x^2+d)^(5/2)/c^2/d-1/6/c^2*(1/4*x*(c^2*d*x^2+d)^(3/2)+3/4*d*(1/2*x*( 
c^2*d*x^2+d)^(1/2)+1/2*d*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^ 
2*d)^(1/2))))+3/5*f^2*g*(c^2*d*x^2+d)^(5/2)/c^2/d)+b*(3/32*(d*(c^2*x^2+1)) 
^(1/2)*f*arcsinh(x*c)^2*(2*c^2*f^2-g^2)*d/(c^2*x^2+1)^(1/2)/c^3+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+11 
2*(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^3*(-1+7*arcsinh(x*c))*d/c^4/(c^2*x^2+1) 
+1/768*(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*g^2*(-1+6*arcsinh(x*c))*d/c^3/(c^2*x^2+1)+1/3 
200*(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*( 
60*arcsinh(x*c)*c^2*f^2-12*c^2*f^2+5*arcsinh(x*c)*g^2-g^2)*d/c^4/(c^2*x^2+ 
1)+1/512*(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*(8*arcsinh(x 
*c)*c^2*f^2-2*c^2*f^2+12*arcsinh(x*c)*g^2-3*g^2)*d/c^3/(c^2*x^2+1)+1/384*( 
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...
 

Fricas [F]

\[ \int (f+g x)^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{3} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} \,d x } \] Input:

integrate((g*x+f)^3*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x)),x, algorithm="f 
ricas")
 

Output:

integral((a*c^2*d*g^3*x^5 + 3*a*c^2*d*f*g^2*x^4 + 3*a*d*f^2*g*x + a*d*f^3 
+ (3*a*c^2*d*f^2*g + a*d*g^3)*x^3 + (a*c^2*d*f^3 + 3*a*d*f*g^2)*x^2 + (b*c 
^2*d*g^3*x^5 + 3*b*c^2*d*f*g^2*x^4 + 3*b*d*f^2*g*x + b*d*f^3 + (3*b*c^2*d* 
f^2*g + b*d*g^3)*x^3 + (b*c^2*d*f^3 + 3*b*d*f*g^2)*x^2)*arcsinh(c*x))*sqrt 
(c^2*d*x^2 + d), x)
 

Sympy [F]

\[ \int (f+g x)^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \left (f + g x\right )^{3}\, dx \] Input:

integrate((g*x+f)**3*(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)**3, x)
 

Maxima [F(-2)]

Exception generated. \[ \int (f+g x)^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((g*x+f)^3*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x)),x, algorithm="m 
axima")
 

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F(-2)]

Exception generated. \[ \int (f+g x)^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((g*x+f)^3*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x)),x, algorithm="g 
iac")
 

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
 

Mupad [F(-1)]

Timed out. \[ \int (f+g x)^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int {\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \] Input:

int((f + g*x)^3*(a + b*asinh(c*x))*(d + c^2*d*x^2)^(3/2),x)
 

Output:

int((f + g*x)^3*(a + b*asinh(c*x))*(d + c^2*d*x^2)^(3/2), x)
 

Reduce [F]

\[ \int (f+g x)^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {d}\, d \left (140 \sqrt {c^{2} x^{2}+1}\, a \,c^{6} f^{3} x^{3}+336 \sqrt {c^{2} x^{2}+1}\, a \,c^{6} f^{2} g \,x^{4}+280 \sqrt {c^{2} x^{2}+1}\, a \,c^{6} f \,g^{2} x^{5}+80 \sqrt {c^{2} x^{2}+1}\, a \,c^{6} g^{3} x^{6}+350 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} f^{3} x +672 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} f^{2} g \,x^{2}+490 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} f \,g^{2} x^{3}+128 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} g^{3} x^{4}+336 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} f^{2} g +105 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} f \,g^{2} x +16 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} g^{3} x^{2}-32 \sqrt {c^{2} x^{2}+1}\, a \,g^{3}+560 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{5}d x \right ) b \,c^{6} g^{3}+1680 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{4}d x \right ) b \,c^{6} f \,g^{2}+1680 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{3}d x \right ) b \,c^{6} f^{2} g +560 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{3}d x \right ) b \,c^{4} g^{3}+560 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{2}d x \right ) b \,c^{6} f^{3}+1680 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{2}d x \right ) b \,c^{4} f \,g^{2}+1680 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x d x \right ) b \,c^{4} f^{2} g +560 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )d x \right ) b \,c^{4} f^{3}+210 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a \,c^{3} f^{3}-105 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a c f \,g^{2}\right )}{560 c^{4}} \] Input:

int((g*x+f)^3*(c^2*d*x^2+d)^(3/2)*(a+b*asinh(c*x)),x)
 

Output:

(sqrt(d)*d*(140*sqrt(c**2*x**2 + 1)*a*c**6*f**3*x**3 + 336*sqrt(c**2*x**2 
+ 1)*a*c**6*f**2*g*x**4 + 280*sqrt(c**2*x**2 + 1)*a*c**6*f*g**2*x**5 + 80* 
sqrt(c**2*x**2 + 1)*a*c**6*g**3*x**6 + 350*sqrt(c**2*x**2 + 1)*a*c**4*f**3 
*x + 672*sqrt(c**2*x**2 + 1)*a*c**4*f**2*g*x**2 + 490*sqrt(c**2*x**2 + 1)* 
a*c**4*f*g**2*x**3 + 128*sqrt(c**2*x**2 + 1)*a*c**4*g**3*x**4 + 336*sqrt(c 
**2*x**2 + 1)*a*c**2*f**2*g + 105*sqrt(c**2*x**2 + 1)*a*c**2*f*g**2*x + 16 
*sqrt(c**2*x**2 + 1)*a*c**2*g**3*x**2 - 32*sqrt(c**2*x**2 + 1)*a*g**3 + 56 
0*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**5,x)*b*c**6*g**3 + 1680*int(sqrt(c 
**2*x**2 + 1)*asinh(c*x)*x**4,x)*b*c**6*f*g**2 + 1680*int(sqrt(c**2*x**2 + 
 1)*asinh(c*x)*x**3,x)*b*c**6*f**2*g + 560*int(sqrt(c**2*x**2 + 1)*asinh(c 
*x)*x**3,x)*b*c**4*g**3 + 560*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**2,x)*b 
*c**6*f**3 + 1680*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**2,x)*b*c**4*f*g**2 
 + 1680*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x,x)*b*c**4*f**2*g + 560*int(sq 
rt(c**2*x**2 + 1)*asinh(c*x),x)*b*c**4*f**3 + 210*log(sqrt(c**2*x**2 + 1) 
+ c*x)*a*c**3*f**3 - 105*log(sqrt(c**2*x**2 + 1) + c*x)*a*c*f*g**2))/(560* 
c**4)