3.1.97 \(\int x^5 (d-c^2 d x^2)^{5/2} (a+b \cosh ^{-1}(c x)) \, dx\) [97]

Optimal. Leaf size=378 \[ \frac {8 b d^2 x \sqrt {d-c^2 d x^2}}{693 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b d^2 x^3 \sqrt {d-c^2 d x^2}}{2079 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 x^5 \sqrt {d-c^2 d x^2}}{1155 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {113 b c d^2 x^7 \sqrt {d-c^2 d x^2}}{4851 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {23 b c^3 d^2 x^9 \sqrt {d-c^2 d x^2}}{891 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^{11} \sqrt {d-c^2 d x^2}}{121 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{7/2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^6 d}+\frac {2 \left (d-c^2 d x^2\right )^{9/2} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{11/2} \left (a+b \cosh ^{-1}(c x)\right )}{11 c^6 d^3} \]

[Out]

-1/7*(-c^2*d*x^2+d)^(7/2)*(a+b*arccosh(c*x))/c^6/d+2/9*(-c^2*d*x^2+d)^(9/2)*(a+b*arccosh(c*x))/c^6/d^2-1/11*(-
c^2*d*x^2+d)^(11/2)*(a+b*arccosh(c*x))/c^6/d^3+8/693*b*d^2*x*(-c^2*d*x^2+d)^(1/2)/c^5/(c*x-1)^(1/2)/(c*x+1)^(1
/2)+4/2079*b*d^2*x^3*(-c^2*d*x^2+d)^(1/2)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/1155*b*d^2*x^5*(-c^2*d*x^2+d)^(1/2
)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-113/4851*b*c*d^2*x^7*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+23/891*b
*c^3*d^2*x^9*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/121*b*c^5*d^2*x^11*(-c^2*d*x^2+d)^(1/2)/(c*x-1
)^(1/2)/(c*x+1)^(1/2)

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Rubi [A]
time = 0.16, antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {272, 45, 5922, 12, 1167} \begin {gather*} -\frac {\left (d-c^2 d x^2\right )^{11/2} \left (a+b \cosh ^{-1}(c x)\right )}{11 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{9/2} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^6 d}-\frac {113 b c d^2 x^7 \sqrt {d-c^2 d x^2}}{4851 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 x^5 \sqrt {d-c^2 d x^2}}{1155 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 b d^2 x \sqrt {d-c^2 d x^2}}{693 c^5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d^2 x^{11} \sqrt {d-c^2 d x^2}}{121 \sqrt {c x-1} \sqrt {c x+1}}+\frac {23 b c^3 d^2 x^9 \sqrt {d-c^2 d x^2}}{891 \sqrt {c x-1} \sqrt {c x+1}}+\frac {4 b d^2 x^3 \sqrt {d-c^2 d x^2}}{2079 c^3 \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(8*b*d^2*x*Sqrt[d - c^2*d*x^2])/(693*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*b*d^2*x^3*Sqrt[d - c^2*d*x^2])/(20
79*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*x^5*Sqrt[d - c^2*d*x^2])/(1155*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) -
(113*b*c*d^2*x^7*Sqrt[d - c^2*d*x^2])/(4851*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (23*b*c^3*d^2*x^9*Sqrt[d - c^2*d*
x^2])/(891*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*x^11*Sqrt[d - c^2*d*x^2])/(121*Sqrt[-1 + c*x]*Sqrt[1 + c
*x]) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(7*c^6*d) + (2*(d - c^2*d*x^2)^(9/2)*(a + b*ArcCosh[c*x]))
/(9*c^6*d^2) - ((d - c^2*d*x^2)^(11/2)*(a + b*ArcCosh[c*x]))/(11*c^6*d^3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 1167

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rule 5922

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x
^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 +
c*x])], Int[SimplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0
] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])

Rubi steps

\begin {align*} \int x^5 \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^5 (-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 {8 d^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{693 c^6}-\frac {4 d^2 x^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{99 c^4}-\frac {d^2 x^4 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{11 c^2}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^3 \left (-8-28 c^2 x^2-63 c^4 x^4\right )}{693 c^6} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {8 d^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{693 c^6}-\frac {4 d^2 x^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{99 c^4}-\frac {d^2 x^4 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{11 c^2}-\frac {\left (b d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^3 \left (-8-28 c^2 x^2-63 c^4 x^4\right ) \, dx}{693 c^5 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {8 d^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{693 c^6}-\frac {4 d^2 x^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{99 c^4}-\frac {d^2 x^4 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{11 c^2}-\frac {\left (b d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-8-4 c^2 x^2-3 c^4 x^4+113 c^6 x^6-161 c^8 x^8+63 c^{10} x^{10}\right ) \, dx}{693 c^5 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {8 b d^2 x \sqrt {d-c^2 d x^2}}{693 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b d^2 x^3 \sqrt {d-c^2 d x^2}}{2079 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 x^5 \sqrt {d-c^2 d x^2}}{1155 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {113 b c d^2 x^7 \sqrt {d-c^2 d x^2}}{4851 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {23 b c^3 d^2 x^9 \sqrt {d-c^2 d x^2}}{891 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^{11} \sqrt {d-c^2 d x^2}}{121 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {8 d^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{693 c^6}-\frac {4 d^2 x^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{99 c^4}-\frac {d^2 x^4 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{11 c^2}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 175, normalized size = 0.46 \begin {gather*} \frac {d^2 \sqrt {d-c^2 d x^2} \left (b \left (8 x+\frac {4 c^2 x^3}{3}+\frac {3 c^4 x^5}{5}-\frac {113 c^6 x^7}{7}+\frac {161 c^8 x^9}{9}-\frac {63 c^{10} x^{11}}{11}\right )+63 c^3 x^4 (-1+c x)^{7/2} (1+c x)^{7/2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {4 (-1+c x)^{7/2} (1+c x)^{7/2} \left (2+7 c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c}\right )}{693 c^5 \sqrt {-1+c x} \sqrt {1+c x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*Sqrt[d - c^2*d*x^2]*(b*(8*x + (4*c^2*x^3)/3 + (3*c^4*x^5)/5 - (113*c^6*x^7)/7 + (161*c^8*x^9)/9 - (63*c^1
0*x^11)/11) + 63*c^3*x^4*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x]) + (4*(-1 + c*x)^(7/2)*(1 + c*x)
^(7/2)*(2 + 7*c^2*x^2)*(a + b*ArcCosh[c*x]))/c))/(693*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1839\) vs. \(2(318)=636\).
time = 3.38, size = 1840, normalized size = 4.87

method	result	size
default	\(\text {Expression too large to display}\)	\(1840\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*(-1/11*x^4*(-c^2*d*x^2+d)^(7/2)/c^2/d+4/11/c^2*(-1/9*x^2*(-c^2*d*x^2+d)^(7/2)/c^2/d-2/63/d/c^4*(-c^2*d*x^2+d
)^(7/2)))+b*(1/247808*(-d*(c^2*x^2-1))^(1/2)*(1-11*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+2816*(c*x+1)^(1/2)*(c*x-1)^
(1/2)*x^7*c^7-1232*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5+1024*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^11*c^11-2816*(c*x+1)
^(1/2)*(c*x-1)^(1/2)*x^9*c^9-61*c^2*x^2+220*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+1024*c^12*x^12-3328*c^10*x^10+
620*c^4*x^4-2352*x^6*c^6+4096*c^8*x^8)*(-1+11*arccosh(c*x))*d^2/(c*x+1)/c^6/(c*x-1)-1/165888*(-d*(c^2*x^2-1))^
(1/2)*(256*c^10*x^10-704*c^8*x^8+256*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^9*c^9+688*x^6*c^6-576*(c*x+1)^(1/2)*(c*x-1)
^(1/2)*x^7*c^7-280*c^4*x^4+432*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5+41*c^2*x^2-120*(c*x+1)^(1/2)*(c*x-1)^(1/2)*
x^3*c^3+9*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-1)*(-1+9*arccosh(c*x))*d^2/(c*x+1)/c^6/(c*x-1)-5/100352*(-d*(c^2*x^2
-1))^(1/2)*(64*c^8*x^8-144*x^6*c^6+64*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7+104*c^4*x^4-112*(c*x+1)^(1/2)*(c*x-1
)^(1/2)*x^5*c^5-25*c^2*x^2+56*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-7*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+1)*(-1+7*a
rccosh(c*x))*d^2/(c*x+1)/c^6/(c*x-1)+1/10240*(-d*(c^2*x^2-1))^(1/2)*(16*x^6*c^6-28*c^4*x^4+16*(c*x+1)^(1/2)*(c
*x-1)^(1/2)*x^5*c^5+13*c^2*x^2-20*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+5*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-1)*(-1
+5*arccosh(c*x))*d^2/(c*x+1)/c^6/(c*x-1)+5/9216*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2+4*(c*x+1)^(1/2)*(c
*x-1)^(1/2)*x^3*c^3-3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+1)*(-1+3*arccosh(c*x))*d^2/(c*x+1)/c^6/(c*x-1)-5/1024*(-
d*(c^2*x^2-1))^(1/2)*((c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*(-1+arccosh(c*x))*d^2/(c*x+1)/c^6/(c*x-1)-5/1
024*(-d*(c^2*x^2-1))^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*(1+arccosh(c*x))*d^2/(c*x+1)/c^6/(c*x-
1)+5/9216*(-d*(c^2*x^2-1))^(1/2)*(-4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+4*c^4*x^4+3*(c*x+1)^(1/2)*(c*x-1)^(1/
2)*x*c-5*c^2*x^2+1)*(1+3*arccosh(c*x))*d^2/(c*x+1)/c^6/(c*x-1)+1/10240*(-d*(c^2*x^2-1))^(1/2)*(-16*(c*x+1)^(1/
2)*(c*x-1)^(1/2)*x^5*c^5+16*x^6*c^6+20*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-28*c^4*x^4-5*(c*x+1)^(1/2)*(c*x-1)^
(1/2)*x*c+13*c^2*x^2-1)*(1+5*arccosh(c*x))*d^2/(c*x+1)/c^6/(c*x-1)-5/100352*(-d*(c^2*x^2-1))^(1/2)*(-64*(c*x+1
)^(1/2)*(c*x-1)^(1/2)*x^7*c^7+64*c^8*x^8+112*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5-144*x^6*c^6-56*(c*x+1)^(1/2)*
(c*x-1)^(1/2)*x^3*c^3+104*c^4*x^4+7*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-25*c^2*x^2+1)*(1+7*arccosh(c*x))*d^2/(c*x+
1)/c^6/(c*x-1)-1/165888*(-d*(c^2*x^2-1))^(1/2)*(-256*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^9*c^9+256*c^10*x^10+576*(c*
x+1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7-704*c^8*x^8-432*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5+688*x^6*c^6+120*(c*x+1)^(
1/2)*(c*x-1)^(1/2)*x^3*c^3-280*c^4*x^4-9*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+41*c^2*x^2-1)*(1+9*arccosh(c*x))*d^2/
(c*x+1)/c^6/(c*x-1)+1/247808*(-d*(c^2*x^2-1))^(1/2)*(-1024*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^11*c^11+1024*c^12*x^1
2+2816*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^9*c^9-3328*c^10*x^10-2816*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7+4096*c^8*x^
8+1232*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5-2352*x^6*c^6-220*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+620*c^4*x^4+11
*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-61*c^2*x^2+1)*(1+11*arccosh(c*x))*d^2/(c*x+1)/c^6/(c*x-1))

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Maxima [A]
time = 0.48, size = 249, normalized size = 0.66 \begin {gather*} -\frac {1}{693} \, {\left (\frac {63 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{4}}{c^{2} d} + \frac {28 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{4} d} + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{6} d}\right )} b \operatorname {arcosh}\left (c x\right ) - \frac {1}{693} \, {\left (\frac {63 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{4}}{c^{2} d} + \frac {28 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{4} d} + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{6} d}\right )} a - \frac {{\left (19845 \, c^{10} \sqrt {-d} d^{2} x^{11} - 61985 \, c^{8} \sqrt {-d} d^{2} x^{9} + 55935 \, c^{6} \sqrt {-d} d^{2} x^{7} - 2079 \, c^{4} \sqrt {-d} d^{2} x^{5} - 4620 \, c^{2} \sqrt {-d} d^{2} x^{3} - 27720 \, \sqrt {-d} d^{2} x\right )} b}{2401245 \, c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/693*(63*(-c^2*d*x^2 + d)^(7/2)*x^4/(c^2*d) + 28*(-c^2*d*x^2 + d)^(7/2)*x^2/(c^4*d) + 8*(-c^2*d*x^2 + d)^(7/
2)/(c^6*d))*b*arccosh(c*x) - 1/693*(63*(-c^2*d*x^2 + d)^(7/2)*x^4/(c^2*d) + 28*(-c^2*d*x^2 + d)^(7/2)*x^2/(c^4
*d) + 8*(-c^2*d*x^2 + d)^(7/2)/(c^6*d))*a - 1/2401245*(19845*c^10*sqrt(-d)*d^2*x^11 - 61985*c^8*sqrt(-d)*d^2*x
^9 + 55935*c^6*sqrt(-d)*d^2*x^7 - 2079*c^4*sqrt(-d)*d^2*x^5 - 4620*c^2*sqrt(-d)*d^2*x^3 - 27720*sqrt(-d)*d^2*x
)*b/c^5

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Fricas [A]
time = 0.37, size = 317, normalized size = 0.84 \begin {gather*} \frac {3465 \, {\left (63 \, b c^{12} d^{2} x^{12} - 224 \, b c^{10} d^{2} x^{10} + 274 \, b c^{8} d^{2} x^{8} - 116 \, b c^{6} d^{2} x^{6} - b c^{4} d^{2} x^{4} - 4 \, b c^{2} d^{2} x^{2} + 8 \, b d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (19845 \, b c^{11} d^{2} x^{11} - 61985 \, b c^{9} d^{2} x^{9} + 55935 \, b c^{7} d^{2} x^{7} - 2079 \, b c^{5} d^{2} x^{5} - 4620 \, b c^{3} d^{2} x^{3} - 27720 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 3465 \, {\left (63 \, a c^{12} d^{2} x^{12} - 224 \, a c^{10} d^{2} x^{10} + 274 \, a c^{8} d^{2} x^{8} - 116 \, a c^{6} d^{2} x^{6} - a c^{4} d^{2} x^{4} - 4 \, a c^{2} d^{2} x^{2} + 8 \, a d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{2401245 \, {\left (c^{8} x^{2} - c^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2401245*(3465*(63*b*c^12*d^2*x^12 - 224*b*c^10*d^2*x^10 + 274*b*c^8*d^2*x^8 - 116*b*c^6*d^2*x^6 - b*c^4*d^2*
x^4 - 4*b*c^2*d^2*x^2 + 8*b*d^2)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - (19845*b*c^11*d^2*x^11 -
61985*b*c^9*d^2*x^9 + 55935*b*c^7*d^2*x^7 - 2079*b*c^5*d^2*x^5 - 4620*b*c^3*d^2*x^3 - 27720*b*c*d^2*x)*sqrt(-c
^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 3465*(63*a*c^12*d^2*x^12 - 224*a*c^10*d^2*x^10 + 274*a*c^8*d^2*x^8 - 116*a*c
^6*d^2*x^6 - a*c^4*d^2*x^4 - 4*a*c^2*d^2*x^2 + 8*a*d^2)*sqrt(-c^2*d*x^2 + d))/(c^8*x^2 - c^6)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4845 deep

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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