\(\int \frac {x^2 (a+b \text {arccosh}(c x))}{(d-c^2 d x^2)^{5/2}} \, dx\) [122]

Optimal result

Integrand size = 27, antiderivative size = 133 \[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b}{6 c^3 d^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{6 c^3 d^2 \sqrt {d-c^2 d x^2}} \] Output:

-1/6*b/c^3/d^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+1/3*x^3*(a 
+b*arccosh(c*x))/d/(-c^2*d*x^2+d)^(3/2)+1/6*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)* 
ln(-c^2*x^2+1)/c^3/d^2/(-c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.76 \[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (-\frac {2 x^3 (a+b \text {arccosh}(c x))}{(-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b \left (\frac {1}{1-c^2 x^2}+\log \left (1-c^2 x^2\right )\right )}{c^3}\right )}{6 d^2 \sqrt {d-c^2 d x^2}} \] Input:

Integrate[(x^2*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(5/2),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.81, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6332, 82, 243, 49, 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.

\(\displaystyle \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 6332

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

\(\Big \downarrow \) 82

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

\(\Big \downarrow \) 243

\(\displaystyle \frac {b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^2}{\left (1-c^2 x^2\right )^2}dx^2}{6 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 49

\(\displaystyle \frac {b c \sqrt {c x-1} \sqrt {c x+1} \int \left (\frac {1}{c^2 \left (c^2 x^2-1\right )}+\frac {1}{c^2 \left (c^2 x^2-1\right )^2}\right )dx^2}{6 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x^3 (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{c^4 \left (1-c^2 x^2\right )}+\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{6 d^2 \sqrt {d-c^2 d x^2}}\)

Input:

Int[(x^2*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(5/2),x]
 

Output:

(x^3*(a + b*ArcCosh[c*x]))/(3*d*(d - c^2*d*x^2)^(3/2)) + (b*c*Sqrt[-1 + c* 
x]*Sqrt[1 + c*x]*(1/(c^4*(1 - c^2*x^2)) + Log[1 - c^2*x^2]/c^4))/(6*d^2*Sq 
rt[d - c^2*d*x^2])
 

Defintions of rubi rules used

rule 49

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 82

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_) 
)^(p_.), x_] :> Int[(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, 
 e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && IntegerQ[m]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6332

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + 
 b*ArcCosh[c*x])^n/(d*f*(m + 1))), x] + Simp[b*c*(n/(f*(m + 1)))*Simp[(d + 
e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]   Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2 
)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3 
, 0] && NeQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(457\) vs. \(2(113)=226\).

Time = 0.49 (sec) , antiderivative size = 458, normalized size of antiderivative = 3.44

method	result	size
default	\(a \left (\frac {x}{2 c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {\frac {x}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}}{2 c^{2}}\right )+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{5} c^{5}+2 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{6} c^{6}+6 \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}+2 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{3} c^{3}-6 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{4} c^{4}+c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}-c^{4} x^{4}-6 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+6 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{2} c^{2}+2 c^{2} x^{2}+2 \,\operatorname {arccosh}\left (c x \right )-2 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )-1\right )}{6 \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right ) d^{3} c^{3}}\)	\(458\)
parts	\(a \left (\frac {x}{2 c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {\frac {x}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}}{2 c^{2}}\right )+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{5} c^{5}+2 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{6} c^{6}+6 \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}+2 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{3} c^{3}-6 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{4} c^{4}+c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}-c^{4} x^{4}-6 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+6 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{2} c^{2}+2 c^{2} x^{2}+2 \,\operatorname {arccosh}\left (c x \right )-2 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )-1\right )}{6 \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right ) d^{3} c^{3}}\)	\(458\)

Input:

int(x^2*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

a*(1/2*x/c^2/d/(-c^2*d*x^2+d)^(3/2)-1/2/c^2*(1/3*x/d/(-c^2*d*x^2+d)^(3/2)+ 
2/3/d^2*x/(-c^2*d*x^2+d)^(1/2)))+1/6*b*(-d*(c^2*x^2-1))^(1/2)*(c^3*x^3+(c* 
x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1)^(1/2))*(-2*(c*x-1)^ 
(1/2)*(c*x+1)^(1/2)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^5*c^5+2*ln 
((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^6*c^6+6*arccosh(c*x)*c^4*x^4+2*( 
c*x-1)^(1/2)*(c*x+1)^(1/2)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^3*c 
^3-6*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^4*c^4+c^3*x^3*(c*x-1)^(1/ 
2)*(c*x+1)^(1/2)-c^4*x^4-6*c^2*x^2*arccosh(c*x)+6*ln((c*x+(c*x-1)^(1/2)*(c 
*x+1)^(1/2))^2-1)*x^2*c^2+2*c^2*x^2+2*arccosh(c*x)-2*ln((c*x+(c*x-1)^(1/2) 
*(c*x+1)^(1/2))^2-1)-1)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/d^3/c 
^3
 

Fricas [F]

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

integrate(x^2*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas 
")
 

Output:

integral(-sqrt(-c^2*d*x^2 + d)*(b*x^2*arccosh(c*x) + a*x^2)/(c^6*d^3*x^6 - 
 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
                                                                                    
                                                                                    
 

Sympy [F]

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

integrate(x**2*(a+b*acosh(c*x))/(-c**2*d*x**2+d)**(5/2),x)
 

Output:

Integral(x**2*(a + b*acosh(c*x))/(-d*(c*x - 1)*(c*x + 1))**(5/2), x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.27 \[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {1}{6} \, b c {\left (\frac {\sqrt {-d}}{c^{6} d^{3} x^{2} - c^{4} d^{3}} - \frac {\sqrt {-d} \log \left (c x + 1\right )}{c^{4} d^{3}} - \frac {\sqrt {-d} \log \left (c x - 1\right )}{c^{4} d^{3}}\right )} - \frac {1}{3} \, b {\left (\frac {x}{\sqrt {-c^{2} d x^{2} + d} c^{2} d^{2}} - \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {1}{3} \, a {\left (\frac {x}{\sqrt {-c^{2} d x^{2} + d} c^{2} d^{2}} - \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d}\right )} \] Input:

integrate(x^2*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima 
")
 

Output:

1/6*b*c*(sqrt(-d)/(c^6*d^3*x^2 - c^4*d^3) - sqrt(-d)*log(c*x + 1)/(c^4*d^3 
) - sqrt(-d)*log(c*x - 1)/(c^4*d^3)) - 1/3*b*(x/(sqrt(-c^2*d*x^2 + d)*c^2* 
d^2) - x/((-c^2*d*x^2 + d)^(3/2)*c^2*d))*arccosh(c*x) - 1/3*a*(x/(sqrt(-c^ 
2*d*x^2 + d)*c^2*d^2) - x/((-c^2*d*x^2 + d)^(3/2)*c^2*d))
 

Giac [F]

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

integrate(x^2*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")
 

Output:

integrate((b*arccosh(c*x) + a)*x^2/(-c^2*d*x^2 + d)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \] Input:

int((x^2*(a + b*acosh(c*x)))/(d - c^2*d*x^2)^(5/2),x)
 

Output:

int((x^2*(a + b*acosh(c*x)))/(d - c^2*d*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{2}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{2} x^{2}-3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{2}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b -a \,x^{3}}{3 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} \left (c^{2} x^{2}-1\right )} \] Input:

int(x^2*(a+b*acosh(c*x))/(-c^2*d*x^2+d)^(5/2),x)
 

Output:

(3*sqrt( - c**2*x**2 + 1)*int((acosh(c*x)*x**2)/(sqrt( - c**2*x**2 + 1)*c* 
*4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x**2 + 1)),x)* 
b*c**2*x**2 - 3*sqrt( - c**2*x**2 + 1)*int((acosh(c*x)*x**2)/(sqrt( - c**2 
*x**2 + 1)*c**4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x 
**2 + 1)),x)*b - a*x**3)/(3*sqrt(d)*sqrt( - c**2*x**2 + 1)*d**2*(c**2*x**2 
 - 1))