3.88 \(\int x^2 (d-c^2 d x^2)^{5/2} (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=371 \[ -\frac {5 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{256 b c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2}}{256 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2}}{768 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {c x-1} \sqrt {c x+1}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2}}{288 \sqrt {c x-1} \sqrt {c x+1}} \]

[Out]

5/48*d*x^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))+1/8*x^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))-5/128*d^2*x
*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+5/64*d^2*x^3*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)+5/256*b*d^2*
x^2*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-59/768*b*c*d^2*x^4*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(
c*x+1)^(1/2)+17/288*b*c^3*d^2*x^6*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/64*b*c^5*d^2*x^8*(-c^2*d*
x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-5/256*d^2*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c^3/(c*x-1)^(1/
2)/(c*x+1)^(1/2)

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Rubi [A]  time = 1.17, antiderivative size = 402, normalized size of antiderivative = 1.08, number of steps used = 13, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5798, 5745, 5743, 5759, 5676, 30, 14, 266, 43} \[ \frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} d^2 x^3 (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{48} d^2 x^3 (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {5 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^2}-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{256 b c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {c x-1} \sqrt {c x+1}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2}}{288 \sqrt {c x-1} \sqrt {c x+1}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2}}{768 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2}}{256 c \sqrt {c x-1} \sqrt {c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*b*d^2*x^2*Sqrt[d - c^2*d*x^2])/(256*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (59*b*c*d^2*x^4*Sqrt[d - c^2*d*x^2])/
(768*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (17*b*c^3*d^2*x^6*Sqrt[d - c^2*d*x^2])/(288*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
- (b*c^5*d^2*x^8*Sqrt[d - c^2*d*x^2])/(64*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b
*ArcCosh[c*x]))/(128*c^2) + (5*d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/64 + (5*d^2*x^3*(1 - c*x)*(1
+ c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/48 + (d^2*x^3*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a
+ b*ArcCosh[c*x]))/8 - (5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(256*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c
*x])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 43

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 266

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 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -1]

Rule 5743

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*
(x_)], x_Symbol] :> Simp[((f*x)^(m + 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(f*(m + 2)), x
] + (-Dist[(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/((m + 2)*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[((f*x)^m*(a + b*ArcCo
sh[c*x])^n)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dist[(b*c*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f*(m + 2)*S
qrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e
1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] &&  !LtQ[m, -1] && (RationalQ[m] |
| EqQ[n, 1])

Rule 5745

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> Simp[((f*x)^(m + 1)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n)/(f*(m + 2*p + 1)
), x] + (Dist[(2*d1*d2*p)/(m + 2*p + 1), Int[(f*x)^m*(d1 + e1*x)^(p - 1)*(d2 + e2*x)^(p - 1)*(a + b*ArcCosh[c*
x])^n, x], x] - Dist[(b*c*n*(-(d1*d2))^(p - 1/2)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f*(m + 2*p + 1)*Sqrt[1 + c*
x]*Sqrt[-1 + c*x]), Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[
{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !L
tQ[m, -1] && IntegerQ[p - 1/2] && (RationalQ[m] || EqQ[n, 1])

Rule 5759

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(e1*e2*m
), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*
x]), x], x] + Dist[(b*f*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(c*d1*d2*m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)
^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0]
&& EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 5798

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

Rubi steps

\begin {align*} \int x^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (-1+c^2 x^2\right )^2 \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {5}{48} d^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int x \left (-1+c^2 x\right )^2 \, dx,x,x^2\right )}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (-1+c^2 x^2\right ) \, dx}{48 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{48} d^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{64 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (x-2 c^2 x^2+c^4 x^3\right ) \, dx,x,x^2\right )}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{64 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-x^3+c^2 x^5\right ) \, dx}{48 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2}}{768 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2}}{288 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{48} d^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{128 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b d^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{128 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2}}{256 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2}}{768 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2}}{288 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{48} d^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{256 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]  time = 4.55, size = 415, normalized size = 1.12 \[ \frac {-2880 a d^{5/2} \sqrt {\frac {c x-1}{c x+1}} (c x+1) \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )+192 a c d^2 x \sqrt {\frac {c x-1}{c x+1}} (c x+1) \left (48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right ) \sqrt {d-c^2 d x^2}-576 b d^2 \sqrt {d-c^2 d x^2} \left (8 \cosh ^{-1}(c x)^2+\cosh \left (4 \cosh ^{-1}(c x)\right )-4 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )-64 b d^2 \sqrt {d-c^2 d x^2} \left (-72 \cosh ^{-1}(c x)^2+18 \cosh \left (2 \cosh ^{-1}(c x)\right )-9 \cosh \left (4 \cosh ^{-1}(c x)\right )-2 \cosh \left (6 \cosh ^{-1}(c x)\right )+12 \cosh ^{-1}(c x) \left (-3 \sinh \left (2 \cosh ^{-1}(c x)\right )+3 \sinh \left (4 \cosh ^{-1}(c x)\right )+\sinh \left (6 \cosh ^{-1}(c x)\right )\right )\right )+b d^2 \sqrt {d-c^2 d x^2} \left (-1440 \cosh ^{-1}(c x)^2+576 \cosh \left (2 \cosh ^{-1}(c x)\right )-144 \cosh \left (4 \cosh ^{-1}(c x)\right )-64 \cosh \left (6 \cosh ^{-1}(c x)\right )-9 \cosh \left (8 \cosh ^{-1}(c x)\right )+24 \cosh ^{-1}(c x) \left (-48 \sinh \left (2 \cosh ^{-1}(c x)\right )+24 \sinh \left (4 \cosh ^{-1}(c x)\right )+16 \sinh \left (6 \cosh ^{-1}(c x)\right )+3 \sinh \left (8 \cosh ^{-1}(c x)\right )\right )\right )}{73728 c^3 \sqrt {\frac {c x-1}{c x+1}} (c x+1)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(192*a*c*d^2*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(-15 + 118*c^2*x^2 - 136*c^4*x^4 + 48*
c^6*x^6) - 2880*a*d^(5/2)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 +
c^2*x^2))] - 576*b*d^2*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*A
rcCosh[c*x]]) - 64*b*d^2*Sqrt[d - c^2*d*x^2]*(-72*ArcCosh[c*x]^2 + 18*Cosh[2*ArcCosh[c*x]] - 9*Cosh[4*ArcCosh[
c*x]] - 2*Cosh[6*ArcCosh[c*x]] + 12*ArcCosh[c*x]*(-3*Sinh[2*ArcCosh[c*x]] + 3*Sinh[4*ArcCosh[c*x]] + Sinh[6*Ar
cCosh[c*x]])) + b*d^2*Sqrt[d - c^2*d*x^2]*(-1440*ArcCosh[c*x]^2 + 576*Cosh[2*ArcCosh[c*x]] - 144*Cosh[4*ArcCos
h[c*x]] - 64*Cosh[6*ArcCosh[c*x]] - 9*Cosh[8*ArcCosh[c*x]] + 24*ArcCosh[c*x]*(-48*Sinh[2*ArcCosh[c*x]] + 24*Si
nh[4*ArcCosh[c*x]] + 16*Sinh[6*ArcCosh[c*x]] + 3*Sinh[8*ArcCosh[c*x]])))/(73728*c^3*Sqrt[(-1 + c*x)/(1 + c*x)]
*(1 + c*x))

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fricas [F]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a c^{4} d^{2} x^{6} - 2 \, a c^{2} d^{2} x^{4} + a d^{2} x^{2} + {\left (b c^{4} d^{2} x^{6} - 2 \, b c^{2} d^{2} x^{4} + b d^{2} x^{2}\right )} \operatorname {arcosh}\left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.49, size = 581, normalized size = 1.57 \[ -\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{8 c^{2} d}+\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{48 c^{2}}+\frac {5 a d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{192 c^{2}}+\frac {5 a \,d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{128 c^{2}}+\frac {5 a \,d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{128 c^{2} \sqrt {c^{2} d}}-\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2} d^{2}}{256 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} c^{5} x^{8}}{64 \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {17 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} c^{3} x^{6}}{288 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {59 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} c \,x^{4}}{768 \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} x^{2}}{256 \sqrt {c x +1}\, c \sqrt {c x -1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} c^{6} \mathrm {arccosh}\left (c x \right ) x^{9}}{8 \left (c x +1\right ) \left (c x -1\right )}-\frac {23 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} c^{4} \mathrm {arccosh}\left (c x \right ) x^{7}}{48 \left (c x +1\right ) \left (c x -1\right )}+\frac {127 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} c^{2} \mathrm {arccosh}\left (c x \right ) x^{5}}{192 \left (c x +1\right ) \left (c x -1\right )}-\frac {133 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \mathrm {arccosh}\left (c x \right ) x^{3}}{384 \left (c x +1\right ) \left (c x -1\right )}+\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \mathrm {arccosh}\left (c x \right ) x}{128 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {359 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2}}{73728 \sqrt {c x +1}\, c^{3} \sqrt {c x -1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*a*x*(-c^2*d*x^2+d)^(7/2)/c^2/d+1/48*a/c^2*x*(-c^2*d*x^2+d)^(5/2)+5/192*a/c^2*d*x*(-c^2*d*x^2+d)^(3/2)+5/1
28*a/c^2*d^2*x*(-c^2*d*x^2+d)^(1/2)+5/128*a/c^2*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))
-5/256*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^3*arccosh(c*x)^2*d^2-1/64*b*(-d*(c^2*x^2-1))^(1/
2)*d^2/(c*x+1)^(1/2)*c^5/(c*x-1)^(1/2)*x^8+17/288*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)*c^3/(c*x-1)^(1/2)
*x^6-59/768*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)*c/(c*x-1)^(1/2)*x^4+5/256*b*(-d*(c^2*x^2-1))^(1/2)*d^2/
(c*x+1)^(1/2)/c/(c*x-1)^(1/2)*x^2+1/8*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)*c^6/(c*x-1)*arccosh(c*x)*x^9-23/48*
b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)*c^4/(c*x-1)*arccosh(c*x)*x^7+127/192*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)
*c^2/(c*x-1)*arccosh(c*x)*x^5-133/384*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/(c*x-1)*arccosh(c*x)*x^3+5/128*b*(-
d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/c^2/(c*x-1)*arccosh(c*x)*x+359/73728*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/
2)/c^3/(c*x-1)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{384} \, {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x}{c^{2}} - \frac {48 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x}{c^{2} d} + \frac {10 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x}{c^{2}} + \frac {15 \, \sqrt {-c^{2} d x^{2} + d} d^{2} x}{c^{2}} + \frac {15 \, d^{\frac {5}{2}} \arcsin \left (c x\right )}{c^{3}}\right )} a + b \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(8*(-c^2*d*x^2 + d)^(5/2)*x/c^2 - 48*(-c^2*d*x^2 + d)^(7/2)*x/(c^2*d) + 10*(-c^2*d*x^2 + d)^(3/2)*d*x/c^
2 + 15*sqrt(-c^2*d*x^2 + d)*d^2*x/c^2 + 15*d^(5/2)*arcsin(c*x)/c^3)*a + b*integrate((-c^2*d*x^2 + d)^(5/2)*x^2
*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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