\(\int (f+g x)^2 (d+c^2 d x^2)^{5/2} (a+b \text {arcsinh}(c x)) \, dx\) [44]

Optimal result

Integrand size = 30, antiderivative size = 837 \[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {2 b d^2 f g x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}-\frac {5 b c d^2 f^2 x^2 \sqrt {d+c^2 d x^2}}{32 \sqrt {1+c^2 x^2}}-\frac {5 b d^2 g^2 x^2 \sqrt {d+c^2 d x^2}}{256 c \sqrt {1+c^2 x^2}}-\frac {2 b c d^2 f g x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {59 b c d^2 g^2 x^4 \sqrt {d+c^2 d x^2}}{768 \sqrt {1+c^2 x^2}}-\frac {6 b c^3 d^2 f g x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {17 b c^3 d^2 g^2 x^6 \sqrt {d+c^2 d x^2}}{288 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 f g x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 g^2 x^8 \sqrt {d+c^2 d x^2}}{64 \sqrt {1+c^2 x^2}}-\frac {5 b d^2 f^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2}}{96 c}-\frac {b d^2 f^2 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5 d^2 g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{128 c^2}+\frac {5}{64} d^2 g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{24} d f^2 x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {5}{48} d g^2 x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{6} f^2 x \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {1}{8} g^2 x^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {2 f g \left (d+c^2 d x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^2 d}+\frac {5 d^2 f^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{32 b c \sqrt {1+c^2 x^2}}-\frac {5 d^2 g^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{256 b c^3 \sqrt {1+c^2 x^2}} \] Output:

-2/7*b*d^2*f*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^2* 
x^2*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-5/256*b*d^2*g^2*x^2*(c^2*d*x^2+d 
)^(1/2)/c/(c^2*x^2+1)^(1/2)-2/7*b*c*d^2*f*g*x^3*(c^2*d*x^2+d)^(1/2)/(c^2*x 
^2+1)^(1/2)-59/768*b*c*d^2*g^2*x^4*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-6 
/35*b*c^3*d^2*f*g*x^5*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-17/288*b*c^3*d 
^2*g^2*x^6*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-2/49*b*c^5*d^2*f*g*x^7*(c 
^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/64*b*c^5*d^2*g^2*x^8*(c^2*d*x^2+d)^( 
1/2)/(c^2*x^2+1)^(1/2)-5/96*b*d^2*f^2*(c^2*x^2+1)^(3/2)*(c^2*d*x^2+d)^(1/2 
)/c-1/36*b*d^2*f^2*(c^2*x^2+1)^(5/2)*(c^2*d*x^2+d)^(1/2)/c+5/16*d^2*f^2*x* 
(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))+5/128*d^2*g^2*x*(c^2*d*x^2+d)^(1/2) 
*(a+b*arcsinh(c*x))/c^2+5/64*d^2*g^2*x^3*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh( 
c*x))+5/24*d*f^2*x*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))+5/48*d*g^2*x^3*( 
c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))+1/6*f^2*x*(c^2*d*x^2+d)^(5/2)*(a+b*a 
rcsinh(c*x))+1/8*g^2*x^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))+2/7*f*g*(c 
^2*d*x^2+d)^(7/2)*(a+b*arcsinh(c*x))/c^2/d+5/32*d^2*f^2*(c^2*d*x^2+d)^(1/2 
)*(a+b*arcsinh(c*x))^2/b/c/(c^2*x^2+1)^(1/2)-5/256*d^2*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 = 1.57 (sec) , antiderivative size = 555, normalized size of antiderivative = 0.66 \[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {-d^3 \left (1+c^2 x^2\right ) \left (b \left (-87955 g^2+1120 c^2 \left (2093 f^2+4608 f g x+315 g^2 x^2\right )+3360 c^4 x^2 \left (1848 f^2+1536 f g x+413 g^2 x^2\right )+640 c^8 x^6 \left (784 f^2+1152 f g x+441 g^2 x^2\right )+1792 c^6 x^4 \left (1365 f^2+1728 f g x+595 g^2 x^2\right )\right )-6720 a c \sqrt {1+c^2 x^2} \left (768 f g \left (1+c^2 x^2\right )^3+56 c^2 f^2 x \left (33+26 c^2 x^2+8 c^4 x^4\right )+7 g^2 x \left (15+118 c^2 x^2+136 c^4 x^4+48 c^6 x^6\right )\right )\right )+352800 b d^3 \left (8 c^2 f^2-g^2\right ) \left (1+c^2 x^2\right ) \text {arcsinh}(c x)^2+705600 a d^{5/2} \left (8 c^2 f^2-g^2\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+840 b d^3 \left (1+c^2 x^2\right ) \text {arcsinh}(c x) \left (6144 c f g \sqrt {1+c^2 x^2}+18432 c^3 f g x^2 \sqrt {1+c^2 x^2}+18432 c^5 f g x^4 \sqrt {1+c^2 x^2}+6144 c^7 f g x^6 \sqrt {1+c^2 x^2}+336 \left (15 c^2 f^2-g^2\right ) \sinh (2 \text {arcsinh}(c x))+168 \left (6 c^2 f^2+g^2\right ) \sinh (4 \text {arcsinh}(c x))+112 c^2 f^2 \sinh (6 \text {arcsinh}(c x))+112 g^2 \sinh (6 \text {arcsinh}(c x))+21 g^2 \sinh (8 \text {arcsinh}(c x))\right )}{18063360 c^3 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}} \] Input:

Integrate[(f + g*x)^2*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]
 

Output:

(-(d^3*(1 + c^2*x^2)*(b*(-87955*g^2 + 1120*c^2*(2093*f^2 + 4608*f*g*x + 31 
5*g^2*x^2) + 3360*c^4*x^2*(1848*f^2 + 1536*f*g*x + 413*g^2*x^2) + 640*c^8* 
x^6*(784*f^2 + 1152*f*g*x + 441*g^2*x^2) + 1792*c^6*x^4*(1365*f^2 + 1728*f 
*g*x + 595*g^2*x^2)) - 6720*a*c*Sqrt[1 + c^2*x^2]*(768*f*g*(1 + c^2*x^2)^3 
 + 56*c^2*f^2*x*(33 + 26*c^2*x^2 + 8*c^4*x^4) + 7*g^2*x*(15 + 118*c^2*x^2 
+ 136*c^4*x^4 + 48*c^6*x^6)))) + 352800*b*d^3*(8*c^2*f^2 - g^2)*(1 + c^2*x 
^2)*ArcSinh[c*x]^2 + 705600*a*d^(5/2)*(8*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]] + 840*b*d^3*( 
1 + c^2*x^2)*ArcSinh[c*x]*(6144*c*f*g*Sqrt[1 + c^2*x^2] + 18432*c^3*f*g*x^ 
2*Sqrt[1 + c^2*x^2] + 18432*c^5*f*g*x^4*Sqrt[1 + c^2*x^2] + 6144*c^7*f*g*x 
^6*Sqrt[1 + c^2*x^2] + 336*(15*c^2*f^2 - g^2)*Sinh[2*ArcSinh[c*x]] + 168*( 
6*c^2*f^2 + g^2)*Sinh[4*ArcSinh[c*x]] + 112*c^2*f^2*Sinh[6*ArcSinh[c*x]] + 
 112*g^2*Sinh[6*ArcSinh[c*x]] + 21*g^2*Sinh[8*ArcSinh[c*x]]))/(18063360*c^ 
3*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2])
 

Rubi [A] (verified)

Time = 1.56 (sec) , antiderivative size = 470, 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.

\(\displaystyle \int \left (c^2 d x^2+d\right )^{5/2} (f+g x)^2 (a+b \text {arcsinh}(c x)) \, dx\)

\(\Big \downarrow \) 6260

\(\displaystyle \frac {d^2 \sqrt {c^2 d x^2+d} \int (f+g x)^2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))dx}{\sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6253

\(\displaystyle \frac {d^2 \sqrt {c^2 d x^2+d} \int \left (f^2 (a+b \text {arcsinh}(c x)) \left (c^2 x^2+1\right )^{5/2}+g^2 x^2 (a+b \text {arcsinh}(c x)) \left (c^2 x^2+1\right )^{5/2}+2 f g x (a+b \text {arcsinh}(c x)) \left (c^2 x^2+1\right )^{5/2}\right )dx}{\sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^2 \sqrt {c^2 d x^2+d} \left (-\frac {5 g^2 (a+b \text {arcsinh}(c x))^2}{256 b c^3}+\frac {1}{6} f^2 x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{24} f^2 x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {5}{16} f^2 x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {2 f g \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^2}+\frac {5 g^2 x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{128 c^2}+\frac {1}{8} g^2 x^3 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{48} g^2 x^3 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {5}{64} g^2 x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {5 f^2 (a+b \text {arcsinh}(c x))^2}{32 b c}-\frac {2}{49} b c^5 f g x^7-\frac {1}{64} b c^5 g^2 x^8-\frac {5}{96} b c^3 f^2 x^4-\frac {6}{35} b c^3 f g x^5-\frac {17}{288} b c^3 g^2 x^6-\frac {b f^2 \left (c^2 x^2+1\right )^3}{36 c}-\frac {25}{96} b c f^2 x^2-\frac {2}{7} b c f g x^3-\frac {2 b f g x}{7 c}-\frac {59}{768} b c g^2 x^4-\frac {5 b g^2 x^2}{256 c}\right )}{\sqrt {c^2 x^2+1}}\)

Input:

Int[(f + g*x)^2*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]
 

Output:

(d^2*Sqrt[d + c^2*d*x^2]*((-2*b*f*g*x)/(7*c) - (25*b*c*f^2*x^2)/96 - (5*b* 
g^2*x^2)/(256*c) - (2*b*c*f*g*x^3)/7 - (5*b*c^3*f^2*x^4)/96 - (59*b*c*g^2* 
x^4)/768 - (6*b*c^3*f*g*x^5)/35 - (17*b*c^3*g^2*x^6)/288 - (2*b*c^5*f*g*x^ 
7)/49 - (b*c^5*g^2*x^8)/64 - (b*f^2*(1 + c^2*x^2)^3)/(36*c) + (5*f^2*x*Sqr 
t[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/16 + (5*g^2*x*Sqrt[1 + c^2*x^2]*(a + 
b*ArcSinh[c*x]))/(128*c^2) + (5*g^2*x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c 
*x]))/64 + (5*f^2*x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/24 + (5*g^2* 
x^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/48 + (f^2*x*(1 + c^2*x^2)^(5 
/2)*(a + b*ArcSinh[c*x]))/6 + (g^2*x^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[ 
c*x]))/8 + (2*f*g*(1 + c^2*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(7*c^2) + (5*f 
^2*(a + b*ArcSinh[c*x])^2)/(32*b*c) - (5*g^2*(a + b*ArcSinh[c*x])^2)/(256* 
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. \(2308\) vs. \(2(727)=1454\).

Time = 1.45 (sec) , antiderivative size = 2309, normalized size of antiderivative = 2.76

method	result	size
default	\(\text {Expression too large to display}\)	\(2309\)
parts	\(\text {Expression too large to display}\)	\(2309\)

Input:

int((g*x+f)^2*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(x*c)),x,method=_RETURNVERBO 
SE)
 

Output:

a*(f^2*(1/6*x*(c^2*d*x^2+d)^(5/2)+5/6*d*(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))))+g^2*(1/8*x*(c^2*d*x^2+d)^(7/2)/c^2/d-1/8/c^2*(1/6*x*( 
c^2*d*x^2+d)^(5/2)+5/6*d*(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)))))+2/7*f*g*(c^2*d*x^2+d)^(7/2)/c^2/d)+b*(5/256*(d*(c^2*x^2+1))^(1/2)*a 
rcsinh(x*c)^2*(8*c^2*f^2-g^2)*d^2/(c^2*x^2+1)^(1/2)/c^3+1/16384*(d*(c^2*x^ 
2+1))^(1/2)*(128*c^9*x^9+128*(c^2*x^2+1)^(1/2)*x^8*c^8+320*x^7*c^7+256*x^6 
*c^6*(c^2*x^2+1)^(1/2)+272*x^5*c^5+160*x^4*c^4*(c^2*x^2+1)^(1/2)+88*x^3*c^ 
3+32*x^2*c^2*(c^2*x^2+1)^(1/2)+8*x*c+(c^2*x^2+1)^(1/2))*g^2*(-1+8*arcsinh( 
x*c))*d^2/c^3/(c^2*x^2+1)+1/3136*(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)*f*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))*(6*arcsi 
nh(x*c)*c^2*f^2-c^2*f^2+6*arcsinh(x*c)*g^2-g^2)*d^2/c^3/(c^2*x^2+1)+1/320* 
(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)*f*g*(-1 
+5*arcsinh(x*c))*d^2/c^2/(c^2*x^2+1)+1/1024*(d*(c^2*x^2+1))^(1/2)*(8*x^...
 

Fricas [F]

\[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{2} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} \,d x } \] Input:

integrate((g*x+f)^2*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="f 
ricas")
 

Output:

integral((a*c^4*d^2*g^2*x^6 + 2*a*c^4*d^2*f*g*x^5 + 4*a*c^2*d^2*f*g*x^3 + 
2*a*d^2*f*g*x + a*d^2*f^2 + (a*c^4*d^2*f^2 + 2*a*c^2*d^2*g^2)*x^4 + (2*a*c 
^2*d^2*f^2 + a*d^2*g^2)*x^2 + (b*c^4*d^2*g^2*x^6 + 2*b*c^4*d^2*f*g*x^5 + 4 
*b*c^2*d^2*f*g*x^3 + 2*b*d^2*f*g*x + b*d^2*f^2 + (b*c^4*d^2*f^2 + 2*b*c^2* 
d^2*g^2)*x^4 + (2*b*c^2*d^2*f^2 + b*d^2*g^2)*x^2)*arcsinh(c*x))*sqrt(c^2*d 
*x^2 + d), x)
 

Sympy [F]

\[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \left (f + g x\right )^{2}\, dx \] Input:

integrate((g*x+f)**2*(c**2*d*x**2+d)**(5/2)*(a+b*asinh(c*x)),x)
 

Output:

Integral((d*(c**2*x**2 + 1))**(5/2)*(a + b*asinh(c*x))*(f + g*x)**2, x)
 

Maxima [F(-2)]

Exception generated. \[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((g*x+f)^2*(c^2*d*x^2+d)^(5/2)*(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. \[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((g*x+f)^2*(c^2*d*x^2+d)^(5/2)*(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. \[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int {\left (f+g\,x\right )}^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{5/2} \,d x \] Input:

int((f + g*x)^2*(a + b*asinh(c*x))*(d + c^2*d*x^2)^(5/2),x)
 

Output:

int((f + g*x)^2*(a + b*asinh(c*x))*(d + c^2*d*x^2)^(5/2), x)
 

Reduce [F]

\[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {d}\, d^{2} \left (448 \sqrt {c^{2} x^{2}+1}\, a \,c^{7} f^{2} x^{5}+768 \sqrt {c^{2} x^{2}+1}\, a \,c^{7} f g \,x^{6}+336 \sqrt {c^{2} x^{2}+1}\, a \,c^{7} g^{2} x^{7}+1456 \sqrt {c^{2} x^{2}+1}\, a \,c^{5} f^{2} x^{3}+2304 \sqrt {c^{2} x^{2}+1}\, a \,c^{5} f g \,x^{4}+952 \sqrt {c^{2} x^{2}+1}\, a \,c^{5} g^{2} x^{5}+1848 \sqrt {c^{2} x^{2}+1}\, a \,c^{3} f^{2} x +2304 \sqrt {c^{2} x^{2}+1}\, a \,c^{3} f g \,x^{2}+826 \sqrt {c^{2} x^{2}+1}\, a \,c^{3} g^{2} x^{3}+768 \sqrt {c^{2} x^{2}+1}\, a c f g +105 \sqrt {c^{2} x^{2}+1}\, a c \,g^{2} x +2688 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{6}d x \right ) b \,c^{7} g^{2}+5376 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{5}d x \right ) b \,c^{7} f g +2688 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{4}d x \right ) b \,c^{7} f^{2}+5376 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{4}d x \right ) b \,c^{5} g^{2}+10752 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{3}d x \right ) b \,c^{5} f g +5376 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{2}d x \right ) b \,c^{5} f^{2}+2688 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{2}d x \right ) b \,c^{3} g^{2}+5376 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x d x \right ) b \,c^{3} f g +2688 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )d x \right ) b \,c^{3} f^{2}+840 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a \,c^{2} f^{2}-105 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a \,g^{2}\right )}{2688 c^{3}} \] Input:

int((g*x+f)^2*(c^2*d*x^2+d)^(5/2)*(a+b*asinh(c*x)),x)
 

Output:

(sqrt(d)*d**2*(448*sqrt(c**2*x**2 + 1)*a*c**7*f**2*x**5 + 768*sqrt(c**2*x* 
*2 + 1)*a*c**7*f*g*x**6 + 336*sqrt(c**2*x**2 + 1)*a*c**7*g**2*x**7 + 1456* 
sqrt(c**2*x**2 + 1)*a*c**5*f**2*x**3 + 2304*sqrt(c**2*x**2 + 1)*a*c**5*f*g 
*x**4 + 952*sqrt(c**2*x**2 + 1)*a*c**5*g**2*x**5 + 1848*sqrt(c**2*x**2 + 1 
)*a*c**3*f**2*x + 2304*sqrt(c**2*x**2 + 1)*a*c**3*f*g*x**2 + 826*sqrt(c**2 
*x**2 + 1)*a*c**3*g**2*x**3 + 768*sqrt(c**2*x**2 + 1)*a*c*f*g + 105*sqrt(c 
**2*x**2 + 1)*a*c*g**2*x + 2688*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**6,x) 
*b*c**7*g**2 + 5376*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**5,x)*b*c**7*f*g 
+ 2688*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**4,x)*b*c**7*f**2 + 5376*int(s 
qrt(c**2*x**2 + 1)*asinh(c*x)*x**4,x)*b*c**5*g**2 + 10752*int(sqrt(c**2*x* 
*2 + 1)*asinh(c*x)*x**3,x)*b*c**5*f*g + 5376*int(sqrt(c**2*x**2 + 1)*asinh 
(c*x)*x**2,x)*b*c**5*f**2 + 2688*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**2,x 
)*b*c**3*g**2 + 5376*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x,x)*b*c**3*f*g + 
2688*int(sqrt(c**2*x**2 + 1)*asinh(c*x),x)*b*c**3*f**2 + 840*log(sqrt(c**2 
*x**2 + 1) + c*x)*a*c**2*f**2 - 105*log(sqrt(c**2*x**2 + 1) + c*x)*a*g**2) 
)/(2688*c**3)