\(\int \frac {x^4 (a+b \text {arccosh}(c x))}{(d-c^2 d x^2)^{5/2}} \, dx\) [125]

Optimal result

Integrand size = 27, antiderivative size = 224 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{6 c^5 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {x (a+b \text {arccosh}(c x))}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{2 b c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 b \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5934, 5892, 74, 266, 272, 45} \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{2 b c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {x (a+b \text {arccosh}(c x))}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{6 c^5 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \log \left (1-c^2 x^2\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}} \]

[In]

Int[(x^4*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(5/2),x]

[Out]

(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*c^5*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (x^3*(a + b*ArcCosh[c*x]))/(3
*c^2*d*(d - c^2*d*x^2)^(3/2)) - (x*(a + b*ArcCosh[c*x]))/(c^4*d^2*Sqrt[d - c^2*d*x^2]) + (Sqrt[-1 + c*x]*Sqrt[
1 + c*x]*(a + b*ArcCosh[c*x])^2)/(2*b*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 -
c^2*x^2])/(3*c^5*d^2*Sqrt[d - c^2*d*x^2])

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 74

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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] && Integer
Q[m] && (NeQ[m, -1] || (EqQ[e, 0] && (EqQ[p, 1] ||  !IntegerQ[p])))

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 5892

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])]*(a + b*ArcCosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e,
n}, x] && EqQ[c^2*d + e, 0] && NeQ[n, -1]

Rule 5934

Int[((a_.) + ArcCosh[(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*ArcCosh[c*x])^n/(2*e*(p + 1))), x] + (-Dist[f^2*((m - 1)/(2*e*(p +
1))), Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[b*f*(n/(2*c*(p + 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*A
rcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1]
&& IGtQ[m, 1]

Rubi steps \begin{align*} \text {integral}& = \frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx}{c^2 d}+\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3}{(-1+c x)^2 (1+c x)^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {x (a+b \text {arccosh}(c x))}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {d-c^2 d x^2}} \, dx}{c^4 d^2}+\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{(-1+c x) (1+c x)} \, dx}{c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3}{\left (-1+c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {x (a+b \text {arccosh}(c x))}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{2 b c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{-1+c^2 x^2} \, dx}{c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {x}{\left (-1+c^2 x\right )^2} \, dx,x,x^2\right )}{6 c d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {x (a+b \text {arccosh}(c x))}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{2 b c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{2 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \left (\frac {1}{c^2 \left (-1+c^2 x\right )^2}+\frac {1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{6 c d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b \sqrt {-1+c x} \sqrt {1+c x}}{6 c^5 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {x (a+b \text {arccosh}(c x))}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{2 b c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 b \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 0.72 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {\frac {2 a c x \left (-3+4 c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{\left (-1+c^2 x^2\right )^2}-6 a \sqrt {d} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\frac {b d \left (-8 c x \text {arccosh}(c x)-\frac {\sqrt {\frac {-1+c x}{1+c x}} (1+c x)+2 c x \text {arccosh}(c x)}{-1+c^2 x^2}+\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (3 \text {arccosh}(c x)^2+8 \log \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)\right )\right )\right )}{\sqrt {d-c^2 d x^2}}}{6 c^5 d^3} \]

[In]

Integrate[(x^4*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(5/2),x]

[Out]

((2*a*c*x*(-3 + 4*c^2*x^2)*Sqrt[d - c^2*d*x^2])/(-1 + c^2*x^2)^2 - 6*a*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2]
)/(Sqrt[d]*(-1 + c^2*x^2))] + (b*d*(-8*c*x*ArcCosh[c*x] - (Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) + 2*c*x*ArcCos
h[c*x])/(-1 + c^2*x^2) + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(3*ArcCosh[c*x]^2 + 8*Log[Sqrt[(-1 + c*x)/(1 + c
*x)]*(1 + c*x)])))/Sqrt[d - c^2*d*x^2])/(6*c^5*d^3)

Maple [A] (verified)

Time = 1.20 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.63

method	result	size
default	\(\frac {a \,x^{3}}{3 c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {a x}{c^{4} d^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {a \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{c^{4} d^{2} \sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (3 \operatorname {arccosh}\left (c x \right )^{2} x^{4} c^{4}-8 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{3} x^{3}-8 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )+8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{4} c^{4}-6 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}+6 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c x +16 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-16 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{2} c^{2}-c^{2} x^{2}+3 \operatorname {arccosh}\left (c x \right )^{2}-8 \,\operatorname {arccosh}\left (c x \right )+8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )+1\right )}{6 \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) d^{3} c^{5}}\)	\(366\)
parts	\(\frac {a \,x^{3}}{3 c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {a x}{c^{4} d^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {a \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{c^{4} d^{2} \sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (3 \operatorname {arccosh}\left (c x \right )^{2} x^{4} c^{4}-8 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{3} x^{3}-8 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )+8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{4} c^{4}-6 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}+6 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c x +16 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-16 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{2} c^{2}-c^{2} x^{2}+3 \operatorname {arccosh}\left (c x \right )^{2}-8 \,\operatorname {arccosh}\left (c x \right )+8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )+1\right )}{6 \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) d^{3} c^{5}}\)	\(366\)

[In]

int(x^4*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/3*a*x^3/c^2/d/(-c^2*d*x^2+d)^(3/2)-a/c^4/d^2*x/(-c^2*d*x^2+d)^(1/2)+a/c^4/d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(
1/2)*x/(-c^2*d*x^2+d)^(1/2))-1/6*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^6*x^6-3*c^4*x^4+3*c^2
*x^2-1)/d^3/c^5*(3*arccosh(c*x)^2*x^4*c^4-8*(c*x+1)^(1/2)*arccosh(c*x)*(c*x-1)^(1/2)*c^3*x^3-8*c^4*x^4*arccosh
(c*x)+8*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^4*c^4-6*arccosh(c*x)^2*x^2*c^2+6*(c*x+1)^(1/2)*arccosh(c*x
)*(c*x-1)^(1/2)*c*x+16*c^2*x^2*arccosh(c*x)-16*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^2*c^2-c^2*x^2+3*arc
cosh(c*x)^2-8*arccosh(c*x)+8*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)+1)

Fricas [F]

\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x^4*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

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

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

[In]

integrate(x**4*(a+b*acosh(c*x))/(-c**2*d*x**2+d)**(5/2),x)

[Out]

Integral(x**4*(a + b*acosh(c*x))/(-d*(c*x - 1)*(c*x + 1))**(5/2), x)

Maxima [F]

\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x^4*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

1/3*(x*(3*x^2/((-c^2*d*x^2 + d)^(3/2)*c^2*d) - 2/((-c^2*d*x^2 + d)^(3/2)*c^4*d)) - x/(sqrt(-c^2*d*x^2 + d)*c^4
*d^2) + 3*arcsin(c*x)/(c^5*d^(5/2)))*a + b*integrate(x^4*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(-c^2*d*x^2 +
d)^(5/2), x)

Giac [F]

\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x^4*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)*x^4/(-c^2*d*x^2 + d)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]

[In]

int((x^4*(a + b*acosh(c*x)))/(d - c^2*d*x^2)^(5/2),x)

[Out]

int((x^4*(a + b*acosh(c*x)))/(d - c^2*d*x^2)^(5/2), x)