Integrand size = 30, antiderivative size = 406 \[ \int \frac {(f+g x)^3 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=-\frac {3 b f^2 g x \sqrt {1+c^2 x^2}}{c \sqrt {d+c^2 d x^2}}+\frac {2 b g^3 x \sqrt {1+c^2 x^2}}{3 c^3 \sqrt {d+c^2 d x^2}}-\frac {3 b f g^2 x^2 \sqrt {1+c^2 x^2}}{4 c \sqrt {d+c^2 d x^2}}-\frac {b g^3 x^3 \sqrt {1+c^2 x^2}}{9 c \sqrt {d+c^2 d x^2}}+\frac {3 f^2 g \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{c^2 d}-\frac {2 g^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 c^4 d}+\frac {3 f g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 c^2 d}+\frac {g^3 x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 c^2 d}+\frac {f^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{2 b c \sqrt {d+c^2 d x^2}}-\frac {3 f g^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{4 b c^3 \sqrt {d+c^2 d x^2}} \] Output:
-3*b*f^2*g*x*(c^2*x^2+1)^(1/2)/c/(c^2*d*x^2+d)^(1/2)+2/3*b*g^3*x*(c^2*x^2+ 1)^(1/2)/c^3/(c^2*d*x^2+d)^(1/2)-3/4*b*f*g^2*x^2*(c^2*x^2+1)^(1/2)/c/(c^2* d*x^2+d)^(1/2)-1/9*b*g^3*x^3*(c^2*x^2+1)^(1/2)/c/(c^2*d*x^2+d)^(1/2)+3*f^2 *g*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/c^2/d-2/3*g^3*(c^2*d*x^2+d)^(1/2 )*(a+b*arcsinh(c*x))/c^4/d+3/2*f*g^2*x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c* x))/c^2/d+1/3*g^3*x^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/c^2/d+1/2*f^3 *(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^2/b/c/(c^2*d*x^2+d)^(1/2)-3/4*f*g^2* (c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^2/b/c^3/(c^2*d*x^2+d)^(1/2)
Time = 1.10 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.75 \[ \int \frac {(f+g x)^3 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\frac {4 \sqrt {d} g \left (-2 b c x \sqrt {1+c^2 x^2} \left (-6 g^2+c^2 \left (27 f^2+g^2 x^2\right )\right )+3 a \left (1+c^2 x^2\right ) \left (-4 g^2+c^2 \left (18 f^2+9 f g x+2 g^2 x^2\right )\right )\right )+12 b \sqrt {d} g \left (1+c^2 x^2\right ) \left (-4 g^2+c^2 \left (18 f^2+9 f g x+2 g^2 x^2\right )\right ) \text {arcsinh}(c x)+18 b c \sqrt {d} f \left (2 c^2 f^2-3 g^2\right ) \sqrt {1+c^2 x^2} \text {arcsinh}(c x)^2-27 b c \sqrt {d} f g^2 \sqrt {1+c^2 x^2} \cosh (2 \text {arcsinh}(c x))+36 a c f \left (2 c^2 f^2-3 g^2\right ) \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )}{72 c^4 \sqrt {d} \sqrt {d+c^2 d x^2}} \] Input:
Integrate[((f + g*x)^3*(a + b*ArcSinh[c*x]))/Sqrt[d + c^2*d*x^2],x]
Output:
(4*Sqrt[d]*g*(-2*b*c*x*Sqrt[1 + c^2*x^2]*(-6*g^2 + c^2*(27*f^2 + g^2*x^2)) + 3*a*(1 + c^2*x^2)*(-4*g^2 + c^2*(18*f^2 + 9*f*g*x + 2*g^2*x^2))) + 12*b *Sqrt[d]*g*(1 + c^2*x^2)*(-4*g^2 + c^2*(18*f^2 + 9*f*g*x + 2*g^2*x^2))*Arc Sinh[c*x] + 18*b*c*Sqrt[d]*f*(2*c^2*f^2 - 3*g^2)*Sqrt[1 + c^2*x^2]*ArcSinh [c*x]^2 - 27*b*c*Sqrt[d]*f*g^2*Sqrt[1 + c^2*x^2]*Cosh[2*ArcSinh[c*x]] + 36 *a*c*f*(2*c^2*f^2 - 3*g^2)*Sqrt[d + c^2*d*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]])/(72*c^4*Sqrt[d]*Sqrt[d + c^2*d*x^2])
Time = 0.86 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.63, 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.
| \(\displaystyle \int \frac {(f+g x)^3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}} \, dx\) |
| \(\Big \downarrow \) 6260 |
| \(\displaystyle \frac {\sqrt {c^2 x^2+1} \int \frac {(f+g x)^3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{\sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 6253 |
| \(\displaystyle \frac {\sqrt {c^2 x^2+1} \int \left (\frac {(a+b \text {arcsinh}(c x)) f^3}{\sqrt {c^2 x^2+1}}+\frac {3 g x (a+b \text {arcsinh}(c x)) f^2}{\sqrt {c^2 x^2+1}}+\frac {3 g^2 x^2 (a+b \text {arcsinh}(c x)) f}{\sqrt {c^2 x^2+1}}+\frac {g^3 x^3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}\right )dx}{\sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {c^2 x^2+1} \left (-\frac {3 f g^2 (a+b \text {arcsinh}(c x))^2}{4 b c^3}+\frac {3 f^2 g \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}+\frac {3 f g^2 x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}+\frac {g^3 x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {2 g^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^4}+\frac {f^3 (a+b \text {arcsinh}(c x))^2}{2 b c}+\frac {2 b g^3 x}{3 c^3}-\frac {3 b f^2 g x}{c}-\frac {3 b f g^2 x^2}{4 c}-\frac {b g^3 x^3}{9 c}\right )}{\sqrt {c^2 d x^2+d}}\) |
Input:
Int[((f + g*x)^3*(a + b*ArcSinh[c*x]))/Sqrt[d + c^2*d*x^2],x]
Output:
(Sqrt[1 + c^2*x^2]*((-3*b*f^2*g*x)/c + (2*b*g^3*x)/(3*c^3) - (3*b*f*g^2*x^ 2)/(4*c) - (b*g^3*x^3)/(9*c) + (3*f^2*g*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c *x]))/c^2 - (2*g^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3*c^4) + (3*f* g^2*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2*c^2) + (g^3*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3*c^2) + (f^3*(a + b*ArcSinh[c*x])^2)/(2*b *c) - (3*f*g^2*(a + b*ArcSinh[c*x])^2)/(4*b*c^3)))/Sqrt[d + c^2*d*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. \(784\) vs. \(2(358)=716\).
Time = 1.47 (sec) , antiderivative size = 785, normalized size of antiderivative = 1.93
| method | result | size |
| default | \(a \left (\frac {f^{3} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{\sqrt {c^{2} d}}+g^{3} \left (\frac {x^{2} \sqrt {c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )+3 f \,g^{2} \left (\frac {x \sqrt {c^{2} d \,x^{2}+d}}{2 c^{2} d}-\frac {\ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 c^{2} \sqrt {c^{2} d}}\right )+\frac {3 f^{2} g \sqrt {c^{2} d \,x^{2}+d}}{c^{2} d}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, f \operatorname {arcsinh}\left (x c \right )^{2} \left (2 c^{2} f^{2}-3 g^{2}\right )}{4 \sqrt {c^{2} x^{2}+1}\, d \,c^{3}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) g^{3} \left (-1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{72 c^{4} d \left (c^{2} x^{2}+1\right )}+\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) f \,g^{2} \left (-1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}+1\right )}+\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) g \left (4 \,\operatorname {arcsinh}\left (x c \right ) c^{2} f^{2}-4 c^{2} f^{2}-\operatorname {arcsinh}\left (x c \right ) g^{2}+g^{2}\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}+\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) g \left (4 \,\operatorname {arcsinh}\left (x c \right ) c^{2} f^{2}+4 c^{2} f^{2}-\operatorname {arcsinh}\left (x c \right ) g^{2}-g^{2}\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}+\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) f \,g^{2} \left (1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) g^{3} \left (1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{72 c^{4} d \left (c^{2} x^{2}+1\right )}\right )\) | \(785\) |
| parts | \(a \left (\frac {f^{3} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{\sqrt {c^{2} d}}+g^{3} \left (\frac {x^{2} \sqrt {c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )+3 f \,g^{2} \left (\frac {x \sqrt {c^{2} d \,x^{2}+d}}{2 c^{2} d}-\frac {\ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 c^{2} \sqrt {c^{2} d}}\right )+\frac {3 f^{2} g \sqrt {c^{2} d \,x^{2}+d}}{c^{2} d}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, f \operatorname {arcsinh}\left (x c \right )^{2} \left (2 c^{2} f^{2}-3 g^{2}\right )}{4 \sqrt {c^{2} x^{2}+1}\, d \,c^{3}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) g^{3} \left (-1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{72 c^{4} d \left (c^{2} x^{2}+1\right )}+\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) f \,g^{2} \left (-1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}+1\right )}+\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) g \left (4 \,\operatorname {arcsinh}\left (x c \right ) c^{2} f^{2}-4 c^{2} f^{2}-\operatorname {arcsinh}\left (x c \right ) g^{2}+g^{2}\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}+\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) g \left (4 \,\operatorname {arcsinh}\left (x c \right ) c^{2} f^{2}+4 c^{2} f^{2}-\operatorname {arcsinh}\left (x c \right ) g^{2}-g^{2}\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}+\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) f \,g^{2} \left (1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) g^{3} \left (1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{72 c^{4} d \left (c^{2} x^{2}+1\right )}\right )\) | \(785\) |
Input:
int((g*x+f)^3*(a+b*arcsinh(x*c))/(c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBO SE)
Output:
a*(f^3*ln(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/ 3*x^2/c^2/d*(c^2*d*x^2+d)^(1/2)-2/3/d/c^4*(c^2*d*x^2+d)^(1/2))+3*f*g^2*(1/ 2*x/c^2/d*(c^2*d*x^2+d)^(1/2)-1/2/c^2*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+ d)^(1/2))/(c^2*d)^(1/2))+3*f^2*g/c^2/d*(c^2*d*x^2+d)^(1/2))+b*(1/4*(d*(c^2 *x^2+1))^(1/2)*f*arcsinh(x*c)^2*(2*c^2*f^2-3*g^2)/(c^2*x^2+1)^(1/2)/d/c^3+ 1/72*(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^3*(-1+3*arcsinh(x*c))/c^4/d/(c^2*x^2+1)+3/1 6*(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*g^2*(-1+2*arcsinh(x*c))/d/c^3/(c^2*x^2+1)+3/8*(d*(c^2*x^2+ 1))^(1/2)*(c^2*x^2+(c^2*x^2+1)^(1/2)*x*c+1)*g*(4*arcsinh(x*c)*c^2*f^2-4*c^ 2*f^2-arcsinh(x*c)*g^2+g^2)/c^4/d/(c^2*x^2+1)+3/8*(d*(c^2*x^2+1))^(1/2)*(c ^2*x^2-(c^2*x^2+1)^(1/2)*x*c+1)*g*(4*arcsinh(x*c)*c^2*f^2+4*c^2*f^2-arcsin h(x*c)*g^2-g^2)/c^4/d/(c^2*x^2+1)+3/16*(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*g^2*(1+2*arcsinh(x*c) )/d/c^3/(c^2*x^2+1)+1/72*(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^3*(1+3*arcsinh(x*c))/c^ 4/d/(c^2*x^2+1))
\[ \int \frac {(f+g x)^3 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{\sqrt {c^{2} d x^{2} + d}} \,d x } \] Input:
integrate((g*x+f)^3*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(1/2),x, algorithm="f ricas")
Output:
integral((a*g^3*x^3 + 3*a*f*g^2*x^2 + 3*a*f^2*g*x + a*f^3 + (b*g^3*x^3 + 3 *b*f*g^2*x^2 + 3*b*f^2*g*x + b*f^3)*arcsinh(c*x))/sqrt(c^2*d*x^2 + d), x)
\[ \int \frac {(f+g x)^3 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \left (f + g x\right )^{3}}{\sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \] Input:
integrate((g*x+f)**3*(a+b*asinh(c*x))/(c**2*d*x**2+d)**(1/2),x)
Output:
Integral((a + b*asinh(c*x))*(f + g*x)**3/sqrt(d*(c**2*x**2 + 1)), x)
Exception generated. \[ \int \frac {(f+g x)^3 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((g*x+f)^3*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(1/2),x, algorithm="m axima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
\[ \int \frac {(f+g x)^3 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{\sqrt {c^{2} d x^{2} + d}} \,d x } \] Input:
integrate((g*x+f)^3*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(1/2),x, algorithm="g iac")
Output:
integrate((g*x + f)^3*(b*arcsinh(c*x) + a)/sqrt(c^2*d*x^2 + d), x)
Timed out. \[ \int \frac {(f+g x)^3 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\int \frac {{\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{\sqrt {d\,c^2\,x^2+d}} \,d x \] Input:
int(((f + g*x)^3*(a + b*asinh(c*x)))/(d + c^2*d*x^2)^(1/2),x)
Output:
int(((f + g*x)^3*(a + b*asinh(c*x)))/(d + c^2*d*x^2)^(1/2), x)
\[ \int \frac {(f+g x)^3 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\frac {3 \mathit {asinh} \left (c x \right )^{2} b \,c^{3} f^{3}+18 \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) b \,c^{2} f^{2} g +18 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} f^{2} g +9 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} f \,g^{2} x +2 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} g^{3} x^{2}-4 \sqrt {c^{2} x^{2}+1}\, a \,g^{3}+6 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{3}}{\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{4} g^{3}+18 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{2}}{\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{4} f \,g^{2}+6 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a \,c^{3} f^{3}-9 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a c f \,g^{2}-18 b \,c^{3} f^{2} g x}{6 \sqrt {d}\, c^{4}} \] Input:
int((g*x+f)^3*(a+b*asinh(c*x))/(c^2*d*x^2+d)^(1/2),x)
Output:
(3*asinh(c*x)**2*b*c**3*f**3 + 18*sqrt(c**2*x**2 + 1)*asinh(c*x)*b*c**2*f* *2*g + 18*sqrt(c**2*x**2 + 1)*a*c**2*f**2*g + 9*sqrt(c**2*x**2 + 1)*a*c**2 *f*g**2*x + 2*sqrt(c**2*x**2 + 1)*a*c**2*g**3*x**2 - 4*sqrt(c**2*x**2 + 1) *a*g**3 + 6*int((asinh(c*x)*x**3)/sqrt(c**2*x**2 + 1),x)*b*c**4*g**3 + 18* int((asinh(c*x)*x**2)/sqrt(c**2*x**2 + 1),x)*b*c**4*f*g**2 + 6*log(sqrt(c* *2*x**2 + 1) + c*x)*a*c**3*f**3 - 9*log(sqrt(c**2*x**2 + 1) + c*x)*a*c*f*g **2 - 18*b*c**3*f**2*g*x)/(6*sqrt(d)*c**4)