Integrand size = 30, antiderivative size = 901 \[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {2 b d^2 f g x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}-\frac {25 b c d^2 f^2 x^2 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {5 b d^2 g^2 x^2 \sqrt {d+c^2 d x^2}}{256 c \sqrt {1+c^2 x^2}}-\frac {2 b c d^2 f g x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {5 b c^3 d^2 f^2 x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {59 b c d^2 g^2 x^4 \sqrt {d+c^2 d x^2}}{768 \sqrt {1+c^2 x^2}}-\frac {6 b c^3 d^2 f g x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {17 b c^3 d^2 g^2 x^6 \sqrt {d+c^2 d x^2}}{288 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 f g x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 g^2 x^8 \sqrt {d+c^2 d x^2}}{64 \sqrt {1+c^2 x^2}}-\frac {b d^2 f^2 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5 d^2 g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{128 c^2}+\frac {5}{64} d^2 g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{24} d^2 f^2 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{48} d^2 g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} d^2 f^2 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{8} d^2 g^2 x^3 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {2 d^2 f g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^2}+\frac {5 d^2 f^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{32 b c \sqrt {1+c^2 x^2}}-\frac {5 d^2 g^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{256 b c^3 \sqrt {1+c^2 x^2}} \]
-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*(a+ b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+5/128*d^2*g^2*x*(a+b*arcsinh(c*x))*(c^ 2*d*x^2+d)^(1/2)/c^2+5/64*d^2*g^2*x^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/ 2)+5/24*d^2*f^2*x*(c^2*x^2+1)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+5/48* d^2*g^2*x^3*(c^2*x^2+1)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+1/6*d^2*f^2 *x*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+1/8*d^2*g^2*x^3*(c ^2*x^2+1)^2*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+2/7*d^2*f*g*(c^2*x^2+1) ^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^2-2/7*b*d^2*f*g*x*(c^2*d*x^2+d )^(1/2)/c/(c^2*x^2+1)^(1/2)-25/96*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)-5/96*b*c^3*d^2*f ^2*x^4*(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)+5/32*d^2*f^2*(a+b* arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/b/c/(c^2*x^2+1)^(1/2)-5/256*d^2*g^2*(a +b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/b/c^3/(c^2*x^2+1)^(1/2)
Time = 1.73 (sec) , antiderivative size = 555, normalized size of antiderivative = 0.62 \[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {-d^3 \left (1+c^2 x^2\right ) \left (b \left (-87955 g^2+1120 c^2 \left (2093 f^2+4608 f g x+315 g^2 x^2\right )+3360 c^4 x^2 \left (1848 f^2+1536 f g x+413 g^2 x^2\right )+640 c^8 x^6 \left (784 f^2+1152 f g x+441 g^2 x^2\right )+1792 c^6 x^4 \left (1365 f^2+1728 f g x+595 g^2 x^2\right )\right )-6720 a c \sqrt {1+c^2 x^2} \left (768 f g \left (1+c^2 x^2\right )^3+56 c^2 f^2 x \left (33+26 c^2 x^2+8 c^4 x^4\right )+7 g^2 x \left (15+118 c^2 x^2+136 c^4 x^4+48 c^6 x^6\right )\right )\right )+352800 b d^3 \left (8 c^2 f^2-g^2\right ) \left (1+c^2 x^2\right ) \text {arcsinh}(c x)^2+705600 a d^{5/2} \left (8 c^2 f^2-g^2\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+840 b d^3 \left (1+c^2 x^2\right ) \text {arcsinh}(c x) \left (6144 c f g \sqrt {1+c^2 x^2}+18432 c^3 f g x^2 \sqrt {1+c^2 x^2}+18432 c^5 f g x^4 \sqrt {1+c^2 x^2}+6144 c^7 f g x^6 \sqrt {1+c^2 x^2}+336 \left (15 c^2 f^2-g^2\right ) \sinh (2 \text {arcsinh}(c x))+168 \left (6 c^2 f^2+g^2\right ) \sinh (4 \text {arcsinh}(c x))+112 c^2 f^2 \sinh (6 \text {arcsinh}(c x))+112 g^2 \sinh (6 \text {arcsinh}(c x))+21 g^2 \sinh (8 \text {arcsinh}(c x))\right )}{18063360 c^3 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}} \]
(-(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])
Time = 1.19 (sec) , antiderivative size = 470, normalized size of antiderivative = 0.52, 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 |
| \(\displaystyle \frac {d^2 \sqrt {c^2 d x^2+d} \left (-\frac {5 g^2 (a+b \text {arcsinh}(c x))^2}{256 b c^3}+\frac {1}{6} f^2 x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{24} f^2 x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {5}{16} f^2 x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {2 f g \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^2}+\frac {5 g^2 x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{128 c^2}+\frac {1}{8} g^2 x^3 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{48} g^2 x^3 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {5}{64} g^2 x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {5 f^2 (a+b \text {arcsinh}(c x))^2}{32 b c}-\frac {2}{49} b c^5 f g x^7-\frac {1}{64} b c^5 g^2 x^8-\frac {5}{96} b c^3 f^2 x^4-\frac {6}{35} b c^3 f g x^5-\frac {17}{288} b c^3 g^2 x^6-\frac {b f^2 \left (c^2 x^2+1\right )^3}{36 c}-\frac {25}{96} b c f^2 x^2-\frac {2}{7} b c f g x^3-\frac {2 b f g x}{7 c}-\frac {59}{768} b c g^2 x^4-\frac {5 b g^2 x^2}{256 c}\right )}{\sqrt {c^2 x^2+1}}\) |
(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]
3.1.44.3.1 Defintions of rubi rules used
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. \(2308\) vs. \(2(791)=1582\).
Time = 1.03 (sec) , antiderivative size = 2309, normalized size of antiderivative = 2.56
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2309\) |
| parts | \(\text {Expression too large to display}\) | \(2309\) |
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(c^2*d*x/(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(c^2*d*x/(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(c*x)^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^8*x^8*(c^2*x^2+1)^(1/2)+320*c^7*x^7+256*c^6 *x^6*(c^2*x^2+1)^(1/2)+272*c^5*x^5+160*c^4*x^4*(c^2*x^2+1)^(1/2)+88*c^3*x^ 3+32*c^2*x^2*(c^2*x^2+1)^(1/2)+8*c*x+(c^2*x^2+1)^(1/2))*g^2*(-1+8*arcsinh( c*x))*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*c^7* x^7*(c^2*x^2+1)^(1/2)+144*c^6*x^6+112*c^5*x^5*(c^2*x^2+1)^(1/2)+104*c^4*x^ 4+56*c^3*x^3*(c^2*x^2+1)^(1/2)+25*c^2*x^2+7*c*x*(c^2*x^2+1)^(1/2)+1)*f*g*( -1+7*arcsinh(c*x))*d^2/c^2/(c^2*x^2+1)+1/2304*(d*(c^2*x^2+1))^(1/2)*(32*c^ 7*x^7+32*c^6*x^6*(c^2*x^2+1)^(1/2)+64*c^5*x^5+48*c^4*x^4*(c^2*x^2+1)^(1/2) +38*c^3*x^3+18*c^2*x^2*(c^2*x^2+1)^(1/2)+6*c*x+(c^2*x^2+1)^(1/2))*(6*arcsi nh(c*x)*c^2*f^2-c^2*f^2+6*arcsinh(c*x)*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^5*x^5*(c^2*x^2+1)^(1/2)+28*c^4*x^4+ 20*c^3*x^3*(c^2*x^2+1)^(1/2)+13*c^2*x^2+5*c*x*(c^2*x^2+1)^(1/2)+1)*f*g*(-1 +5*arcsinh(c*x))*d^2/c^2/(c^2*x^2+1)+1/1024*(d*(c^2*x^2+1))^(1/2)*(8*c^...
\[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{2} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} \,d x } \]
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)
\[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \left (f + g x\right )^{2}\, dx \]
Exception generated. \[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
Exception generated. \[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int {\left (f+g\,x\right )}^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{5/2} \,d x \]