\(\int \frac {x^2 (a+b \text {arcsinh}(c x))}{(d+c^2 d x^2)^{3/2}} \, dx\) [166]

Optimal result

Integrand size = 26, antiderivative size = 130 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {x (a+b \text {arcsinh}(c x))}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{2 b c^3 d \sqrt {d+c^2 d x^2}}+\frac {b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{2 c^3 d \sqrt {d+c^2 d x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.12 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {a x \sqrt {d \left (1+c^2 x^2\right )}}{c^2 d^2 \left (1+c^2 x^2\right )}+\frac {b \left (-2 c x \text {arcsinh}(c x)+\sqrt {1+c^2 x^2} \left (\text {arcsinh}(c x)^2+2 \log \left (\sqrt {1+c^2 x^2}\right )\right )\right )}{2 c^3 d \sqrt {d \left (1+c^2 x^2\right )}}+\frac {a \log \left (c d x+\sqrt {d} \sqrt {d \left (1+c^2 x^2\right )}\right )}{c^3 d^{3/2}} \] Input:

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

Output:

-((a*x*Sqrt[d*(1 + c^2*x^2)])/(c^2*d^2*(1 + c^2*x^2))) + (b*(-2*c*x*ArcSin 
h[c*x] + Sqrt[1 + c^2*x^2]*(ArcSinh[c*x]^2 + 2*Log[Sqrt[1 + c^2*x^2]])))/( 
2*c^3*d*Sqrt[d*(1 + c^2*x^2)]) + (a*Log[c*d*x + Sqrt[d]*Sqrt[d*(1 + c^2*x^ 
2)]])/(c^3*d^(3/2))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {6225, 240, 6198}

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.

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

\(\Big \downarrow \) 6225

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

\(\Big \downarrow \) 240

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

\(\Big \downarrow \) 6198

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
 

rule 6198

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( 
a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c 
^2*d] && NeQ[n, -1]
 

rule 6225

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a 
+ b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) 
   Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] - S 
imp[b*f*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int[(f*x)^( 
m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; Fre 
eQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && IG 
tQ[m, 1]
 

Maple [A] (verified)

Time = 0.90 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.78

method	result	size
default	\(-\frac {a x}{c^{2} d \sqrt {c^{2} d \,x^{2}+d}}+\frac {a \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{c^{2} d \sqrt {c^{2} d}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d^{2} c^{3}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )}{\sqrt {c^{2} x^{2}+1}\, d^{2} c^{3}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) x}{d^{2} c^{2} \left (c^{2} x^{2}+1\right )}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, d^{2} c^{3}}\)	\(232\)
parts	\(-\frac {a x}{c^{2} d \sqrt {c^{2} d \,x^{2}+d}}+\frac {a \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{c^{2} d \sqrt {c^{2} d}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d^{2} c^{3}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )}{\sqrt {c^{2} x^{2}+1}\, d^{2} c^{3}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) x}{d^{2} c^{2} \left (c^{2} x^{2}+1\right )}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, d^{2} c^{3}}\)	\(232\)

Input:

int(x^2*(a+b*arcsinh(x*c))/(c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-a*x/c^2/d/(c^2*d*x^2+d)^(1/2)+a/c^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)+1/2*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^2 
/c^3*arcsinh(x*c)^2-b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^2/c^3*arcs 
inh(x*c)-b*(d*(c^2*x^2+1))^(1/2)*arcsinh(x*c)/d^2/c^2/(c^2*x^2+1)*x+b*(d*( 
c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^2/c^3*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2 
)
 

Fricas [F]

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

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

Output:

integral(sqrt(c^2*d*x^2 + d)*(b*x^2*arcsinh(c*x) + a*x^2)/(c^4*d^2*x^4 + 2 
*c^2*d^2*x^2 + d^2), x)
 

Sympy [F]

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

integrate(x**2*(a+b*asinh(c*x))/(c**2*d*x**2+d)**(3/2),x)
 

Output:

Integral(x**2*(a + b*asinh(c*x))/(d*(c**2*x**2 + 1))**(3/2), x)
 

Maxima [F]

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

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

Output:

-a*(x/(sqrt(c^2*d*x^2 + d)*c^2*d) - arcsinh(c*x)/(c^3*d^(3/2))) + b*integr 
ate(x^2*log(c*x + sqrt(c^2*x^2 + 1))/(c^2*d*x^2 + d)^(3/2), x)
 

Giac [F]

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

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

Output:

integrate((b*arcsinh(c*x) + a)*x^2/(c^2*d*x^2 + d)^(3/2), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {-\sqrt {c^{2} x^{2}+1}\, a c x +\left (\int \frac {\mathit {asinh} \left (c x \right ) x^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{5} x^{2}+\left (\int \frac {\mathit {asinh} \left (c x \right ) x^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{3}+\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a \,c^{2} x^{2}+\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a -a \,c^{2} x^{2}-a}{\sqrt {d}\, c^{3} d \left (c^{2} x^{2}+1\right )} \] Input:

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

Output:

( - sqrt(c**2*x**2 + 1)*a*c*x + int((asinh(c*x)*x**2)/(sqrt(c**2*x**2 + 1) 
*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b*c**5*x**2 + int((asinh(c*x)*x**2)/( 
sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b*c**3 + log(sqrt( 
c**2*x**2 + 1) + c*x)*a*c**2*x**2 + log(sqrt(c**2*x**2 + 1) + c*x)*a - a*c 
**2*x**2 - a)/(sqrt(d)*c**3*d*(c**2*x**2 + 1))