3.82 \(\int x^3 (d-c^2 d x^2)^{3/2} (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=243 \[ \frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^4 d^2}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^4 d}+\frac{b c^3 d x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{c x-1} \sqrt{c x+1}}-\frac{8 b c d x^5 \sqrt{d-c^2 d x^2}}{175 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d x^3 \sqrt{d-c^2 d x^2}}{105 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b d x \sqrt{d-c^2 d x^2}}{35 c^3 \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

(2*b*d*x*Sqrt[d - c^2*d*x^2])/(35*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d*x^3*Sqrt[d - c^2*d*x^2])/(105*c*Sqr
t[-1 + c*x]*Sqrt[1 + c*x]) - (8*b*c*d*x^5*Sqrt[d - c^2*d*x^2])/(175*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*x
^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(5*c^
4*d) + ((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(7*c^4*d^2)

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Rubi [A]  time = 0.411973, antiderivative size = 272, normalized size of antiderivative = 1.12, number of steps used = 5, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5798, 100, 12, 74, 5733, 373} \[ -\frac{d x^2 (1-c x)^2 (c x+1)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}-\frac{2 d (1-c x)^2 (c x+1)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 c^4}+\frac{b c^3 d x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{c x-1} \sqrt{c x+1}}-\frac{8 b c d x^5 \sqrt{d-c^2 d x^2}}{175 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d x^3 \sqrt{d-c^2 d x^2}}{105 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b d x \sqrt{d-c^2 d x^2}}{35 c^3 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*d*x*Sqrt[d - c^2*d*x^2])/(35*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d*x^3*Sqrt[d - c^2*d*x^2])/(105*c*Sqr
t[-1 + c*x]*Sqrt[1 + c*x]) - (8*b*c*d*x^5*Sqrt[d - c^2*d*x^2])/(175*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*x
^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*d*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(
a + b*ArcCosh[c*x]))/(35*c^4) - (d*x^2*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(7*c^
2)

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]

Rule 100

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

Rule 12

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

Rule 74

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

Rule 5733

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sym
bol] :> With[{u = IntHide[x^m*(1 + c*x)^p*(-1 + c*x)^p, x]}, Dist[(-(d1*d2))^p*(a + b*ArcCosh[c*x]), u, x] - D
ist[b*c*(-(d1*d2))^p, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d
1, e1, d2, e2}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p - 1/2] && (IGtQ[(m + 1)/2, 0] || IL
tQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -2^(-1)] && GtQ[d1, 0] && LtQ[d2, 0]

Rule 373

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

Rubi steps

\begin{align*} \int x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int x^3 (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{2 d (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 c^4}-\frac{d x^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}+\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (1-c^2 x^2\right )^2 \left (2+5 c^2 x^2\right )}{35 c^4} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{2 d (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 c^4}-\frac{d x^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}+\frac{\left (b d \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^2 \left (2+5 c^2 x^2\right ) \, dx}{35 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{2 d (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 c^4}-\frac{d x^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}+\frac{\left (b d \sqrt{d-c^2 d x^2}\right ) \int \left (2+c^2 x^2-8 c^4 x^4+5 c^6 x^6\right ) \, dx}{35 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{2 b d x \sqrt{d-c^2 d x^2}}{35 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d x^3 \sqrt{d-c^2 d x^2}}{105 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{8 b c d x^5 \sqrt{d-c^2 d x^2}}{175 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 d (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 c^4}-\frac{d x^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}\\ \end{align*}

Mathematica [A]  time = 0.221202, size = 150, normalized size = 0.62 \[ -\frac{d \sqrt{d-c^2 d x^2} \left (5 c^2 x^2 (c x-1)^{5/2} (c x+1)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+2 (c x-1)^{5/2} (c x+1)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{5}{7} b c x \left (c^2 x^2-1\right )^3-\frac{19}{7} b c \left (\frac{c^4 x^5}{5}-\frac{2 c^2 x^3}{3}+x\right )\right )}{35 c^4 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*Sqrt[d - c^2*d*x^2]*((-5*b*c*x*(-1 + c^2*x^2)^3)/7 - (19*b*c*(x - (2*c^2*x^3)/3 + (c^4*x^5)/5))/7 + 2*(-1
+ c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x]) + 5*c^2*x^2*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[
c*x])))/(35*c^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

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Maple [B]  time = 0.31, size = 966, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*(-1/7*x^2*(-c^2*d*x^2+d)^(5/2)/c^2/d-2/35/d/c^4*(-c^2*d*x^2+d)^(5/2))+b*(-1/6272*(-d*(c^2*x^2-1))^(1/2)*(64*
c^8*x^8-144*c^6*x^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*arccosh(c*x))*d/
(c*x+1)/c^4/(c*x-1)+1/3200*(-d*(c^2*x^2-1))^(1/2)*(16*c^6*x^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
/(c*x+1)/c^4/(c*x-1)+1/384*(-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/(c*x+1)/c^4/(c*x-1)-3/128*(-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/(c*x+1)/c^4/(c*x-1)-3/128*(-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/(c*x+1)/c^4/(c*x-1)+1/384*(-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*arcco
sh(c*x))*d/(c*x+1)/c^4/(c*x-1)+1/3200*(-d*(c^2*x^2-1))^(1/2)*(-16*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5+16*c^6*x
^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*arcc
osh(c*x))*d/(c*x+1)/c^4/(c*x-1)-1/6272*(-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*c^6*x^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/(c*x+1)/c^4/(c*x-1))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.21809, size = 482, normalized size = 1.98 \begin{align*} -\frac{105 \,{\left (5 \, b c^{8} d x^{8} - 13 \, b c^{6} d x^{6} + 9 \, b c^{4} d x^{4} + b c^{2} d x^{2} - 2 \, b d\right )} \sqrt{-c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (75 \, b c^{7} d x^{7} - 168 \, b c^{5} d x^{5} + 35 \, b c^{3} d x^{3} + 210 \, b c d x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} + 105 \,{\left (5 \, a c^{8} d x^{8} - 13 \, a c^{6} d x^{6} + 9 \, a c^{4} d x^{4} + a c^{2} d x^{2} - 2 \, a d\right )} \sqrt{-c^{2} d x^{2} + d}}{3675 \,{\left (c^{6} x^{2} - c^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3675*(105*(5*b*c^8*d*x^8 - 13*b*c^6*d*x^6 + 9*b*c^4*d*x^4 + b*c^2*d*x^2 - 2*b*d)*sqrt(-c^2*d*x^2 + d)*log(c
*x + sqrt(c^2*x^2 - 1)) - (75*b*c^7*d*x^7 - 168*b*c^5*d*x^5 + 35*b*c^3*d*x^3 + 210*b*c*d*x)*sqrt(-c^2*d*x^2 +
d)*sqrt(c^2*x^2 - 1) + 105*(5*a*c^8*d*x^8 - 13*a*c^6*d*x^6 + 9*a*c^4*d*x^4 + a*c^2*d*x^2 - 2*a*d)*sqrt(-c^2*d*
x^2 + d))/(c^6*x^2 - c^4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError