\(\int \frac {(f+g x)^2 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx\) [48]

Optimal result

Integrand size = 30, antiderivative size = 246 \[ \int \frac {(f+g x)^2 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=-\frac {2 b f g x \sqrt {1+c^2 x^2}}{c \sqrt {d+c^2 d x^2}}-\frac {b g^2 x^2 \sqrt {1+c^2 x^2}}{4 c \sqrt {d+c^2 d x^2}}+\frac {2 f g \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{c^2 d}+\frac {g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 c^2 d}+\frac {f^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{2 b c \sqrt {d+c^2 d x^2}}-\frac {g^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{4 b c^3 \sqrt {d+c^2 d x^2}} \] Output:

-2*b*f*g*x*(c^2*x^2+1)^(1/2)/c/(c^2*d*x^2+d)^(1/2)-1/4*b*g^2*x^2*(c^2*x^2+ 
1)^(1/2)/c/(c^2*d*x^2+d)^(1/2)+2*f*g*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x) 
)/c^2/d+1/2*g^2*x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/c^2/d+1/2*f^2*(c^ 
2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^2/b/c/(c^2*d*x^2+d)^(1/2)-1/4*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)
 

Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.95 \[ \int \frac {(f+g x)^2 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\frac {4 c \sqrt {d} g \left (-4 b c f x \sqrt {1+c^2 x^2}+a (4 f+g x) \left (1+c^2 x^2\right )\right )+4 b c \sqrt {d} g (4 f+g x) \left (1+c^2 x^2\right ) \text {arcsinh}(c x)+2 b \sqrt {d} \left (2 c^2 f^2-g^2\right ) \sqrt {1+c^2 x^2} \text {arcsinh}(c x)^2-b \sqrt {d} g^2 \sqrt {1+c^2 x^2} \cosh (2 \text {arcsinh}(c x))+4 a \left (2 c^2 f^2-g^2\right ) \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )}{8 c^3 \sqrt {d} \sqrt {d+c^2 d x^2}} \] Input:

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

Output:

(4*c*Sqrt[d]*g*(-4*b*c*f*x*Sqrt[1 + c^2*x^2] + a*(4*f + g*x)*(1 + c^2*x^2) 
) + 4*b*c*Sqrt[d]*g*(4*f + g*x)*(1 + c^2*x^2)*ArcSinh[c*x] + 2*b*Sqrt[d]*( 
2*c^2*f^2 - g^2)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2 - b*Sqrt[d]*g^2*Sqrt[1 + 
 c^2*x^2]*Cosh[2*ArcSinh[c*x]] + 4*a*(2*c^2*f^2 - g^2)*Sqrt[d + c^2*d*x^2] 
*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]])/(8*c^3*Sqrt[d]*Sqrt[d + c^2*d*x 
^2])
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.64, 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.

\(\displaystyle \int \frac {(f+g x)^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}} \, dx\)

\(\Big \downarrow \) 6260

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

\(\Big \downarrow \) 6253

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {c^2 x^2+1} \left (-\frac {g^2 (a+b \text {arcsinh}(c x))^2}{4 b c^3}+\frac {2 f g \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}+\frac {g^2 x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}+\frac {f^2 (a+b \text {arcsinh}(c x))^2}{2 b c}-\frac {2 b f g x}{c}-\frac {b g^2 x^2}{4 c}\right )}{\sqrt {c^2 d x^2+d}}\)

Input:

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

Output:

(Sqrt[1 + c^2*x^2]*((-2*b*f*g*x)/c - (b*g^2*x^2)/(4*c) + (2*f*g*Sqrt[1 + c 
^2*x^2]*(a + b*ArcSinh[c*x]))/c^2 + (g^2*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSin 
h[c*x]))/(2*c^2) + (f^2*(a + b*ArcSinh[c*x])^2)/(2*b*c) - (g^2*(a + b*ArcS 
inh[c*x])^2)/(4*b*c^3)))/Sqrt[d + c^2*d*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. \(483\) vs. \(2(218)=436\).

Time = 1.43 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.97

method	result	size
default	\(a \left (\frac {f^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{\sqrt {c^{2} d}}+g^{2} \left (\frac {x \sqrt {c^{2} d \,x^{2}+d}}{2 c^{2} d}-\frac {\ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 c^{2} \sqrt {c^{2} d}}\right )+\frac {2 f g \sqrt {c^{2} d \,x^{2}+d}}{c^{2} d}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2} \left (2 c^{2} f^{2}-g^{2}\right )}{4 \sqrt {c^{2} x^{2}+1}\, d \,c^{3}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) g^{2} \left (-1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) f g \left (\operatorname {arcsinh}\left (x c \right )-1\right )}{c^{2} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) f g \left (\operatorname {arcsinh}\left (x c \right )+1\right )}{c^{2} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) g^{2} \left (1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}+1\right )}\right )\)	\(484\)
parts	\(a \left (\frac {f^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{\sqrt {c^{2} d}}+g^{2} \left (\frac {x \sqrt {c^{2} d \,x^{2}+d}}{2 c^{2} d}-\frac {\ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 c^{2} \sqrt {c^{2} d}}\right )+\frac {2 f g \sqrt {c^{2} d \,x^{2}+d}}{c^{2} d}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2} \left (2 c^{2} f^{2}-g^{2}\right )}{4 \sqrt {c^{2} x^{2}+1}\, d \,c^{3}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) g^{2} \left (-1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) f g \left (\operatorname {arcsinh}\left (x c \right )-1\right )}{c^{2} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) f g \left (\operatorname {arcsinh}\left (x c \right )+1\right )}{c^{2} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) g^{2} \left (1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}+1\right )}\right )\)	\(484\)

Input:

int((g*x+f)^2*(a+b*arcsinh(x*c))/(c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBO 
SE)
 

Output:

a*(f^2*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/ 
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))+2*f*g/c^2/d*(c^2*d*x^2+d)^(1/2))+b*(1/4*(d*(c^2*x 
^2+1))^(1/2)*arcsinh(x*c)^2*(2*c^2*f^2-g^2)/(c^2*x^2+1)^(1/2)/d/c^3+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))*g^2*(-1+2*arcsinh(x*c))/d/c^3/(c^2*x^2+1)+(d*(c^2*x^2+1))^(1/2) 
*(c^2*x^2+(c^2*x^2+1)^(1/2)*x*c+1)*f*g*(arcsinh(x*c)-1)/c^2/d/(c^2*x^2+1)+ 
(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-(c^2*x^2+1)^(1/2)*x*c+1)*f*g*(arcsinh(x*c)+ 
1)/c^2/d/(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))*g^2*(1+2*arcsinh(x*c))/d/c^3/(c^2*x^ 
2+1))
 

Fricas [F]

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

integrate((g*x+f)^2*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(1/2),x, algorithm="f 
ricas")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((g*x+f)^2*(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.
 

Giac [F]

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

integrate((g*x+f)^2*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(1/2),x, algorithm="g 
iac")
 

Output:

integrate((g*x + f)^2*(b*arcsinh(c*x) + a)/sqrt(c^2*d*x^2 + d), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(f+g x)^2 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\frac {\mathit {asinh} \left (c x \right )^{2} b \,c^{2} f^{2}+4 \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) b c f g +4 \sqrt {c^{2} x^{2}+1}\, a c f g +\sqrt {c^{2} x^{2}+1}\, a c \,g^{2} x +2 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{2}}{\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{3} g^{2}+2 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a \,c^{2} f^{2}-\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a \,g^{2}-4 b \,c^{2} f g x}{2 \sqrt {d}\, c^{3}} \] Input:

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

Output:

(asinh(c*x)**2*b*c**2*f**2 + 4*sqrt(c**2*x**2 + 1)*asinh(c*x)*b*c*f*g + 4* 
sqrt(c**2*x**2 + 1)*a*c*f*g + sqrt(c**2*x**2 + 1)*a*c*g**2*x + 2*int((asin 
h(c*x)*x**2)/sqrt(c**2*x**2 + 1),x)*b*c**3*g**2 + 2*log(sqrt(c**2*x**2 + 1 
) + c*x)*a*c**2*f**2 - log(sqrt(c**2*x**2 + 1) + c*x)*a*g**2 - 4*b*c**2*f* 
g*x)/(2*sqrt(d)*c**3)