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

Optimal. Leaf size=218 \[ -\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2 d}-\frac{b c^5 d^2 x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b c^3 d^2 x^5 \sqrt{d-c^2 d x^2}}{35 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^2 x^3 \sqrt{d-c^2 d x^2}}{7 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d^2 x \sqrt{d-c^2 d x^2}}{7 c \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

(b*d^2*x*Sqrt[d - c^2*d*x^2])/(7*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^2*x^3*Sqrt[d - c^2*d*x^2])/(7*Sqrt[-
1 + c*x]*Sqrt[1 + c*x]) + (3*b*c^3*d^2*x^5*Sqrt[d - c^2*d*x^2])/(35*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2
*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(7*
c^2*d)

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Rubi [A]  time = 0.281578, antiderivative size = 233, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {5798, 5718, 194} \[ -\frac{d^2 (1-c x)^3 (c x+1)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}-\frac{b c^5 d^2 x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b c^3 d^2 x^5 \sqrt{d-c^2 d x^2}}{35 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^2 x^3 \sqrt{d-c^2 d x^2}}{7 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d^2 x \sqrt{d-c^2 d x^2}}{7 c \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*d^2*x*Sqrt[d - c^2*d*x^2])/(7*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^2*x^3*Sqrt[d - c^2*d*x^2])/(7*Sqrt[-
1 + c*x]*Sqrt[1 + c*x]) + (3*b*c^3*d^2*x^5*Sqrt[d - c^2*d*x^2])/(35*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2
*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d^2*(1 - c*x)^3*(1 + c*x)^3*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 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[
(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-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, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

Rule 194

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

Rubi steps

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

Mathematica [A]  time = 0.232665, size = 117, normalized size = 0.54 \[ \frac{d^2 \sqrt{d-c^2 d x^2} \left (35 a \left (c^2 x^2-1\right )^4+b c x \sqrt{c x-1} \sqrt{c x+1} \left (-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35\right )+35 b \left (c^2 x^2-1\right )^4 \cosh ^{-1}(c x)\right )}{245 c^2 \left (c^2 x^2-1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*Sqrt[d - c^2*d*x^2]*(35*a*(-1 + c^2*x^2)^4 + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(35 - 35*c^2*x^2 + 21*c^4
*x^4 - 5*c^6*x^6) + 35*b*(-1 + c^2*x^2)^4*ArcCosh[c*x]))/(245*c^2*(-1 + c^2*x^2))

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Maple [B]  time = 0.267, size = 956, normalized size = 4.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*a/c^2/d*(-c^2*d*x^2+d)^(7/2)+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^2/(c*x+1)/c^2/(c*x-1)-1/640*(-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^2/(c*x+1)/c^2/(c*x-1)+1/128*(-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^2/(c*x-1)-5/128*(-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^2/(c*x+1)/c^2/(c*x-1)-5/128*(-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^2/(c*x+1)/c^2/(c*x-1)+1/128*(-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^2/(c*x-
1)-1/640*(-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*arccosh(c*x))*d^2/(c*x+1)/c^2/(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^2/(c*x+1)/c^2/(c*x-1))

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Maxima [A]  time = 1.22272, size = 159, normalized size = 0.73 \begin{align*} -\frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{7}{2}} b \operatorname{arcosh}\left (c x\right )}{7 \, c^{2} d} - \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{7}{2}} a}{7 \, c^{2} d} - \frac{{\left (5 \, c^{6} \sqrt{-d} d^{3} x^{7} - 21 \, c^{4} \sqrt{-d} d^{3} x^{5} + 35 \, c^{2} \sqrt{-d} d^{3} x^{3} - 35 \, \sqrt{-d} d^{3} x\right )} b}{245 \, c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*(-c^2*d*x^2 + d)^(7/2)*b*arccosh(c*x)/(c^2*d) - 1/7*(-c^2*d*x^2 + d)^(7/2)*a/(c^2*d) - 1/245*(5*c^6*sqrt(
-d)*d^3*x^7 - 21*c^4*sqrt(-d)*d^3*x^5 + 35*c^2*sqrt(-d)*d^3*x^3 - 35*sqrt(-d)*d^3*x)*b/(c*d)

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Fricas [A]  time = 2.12957, size = 502, normalized size = 2.3 \begin{align*} \frac{35 \,{\left (b c^{8} d^{2} x^{8} - 4 \, b c^{6} d^{2} x^{6} + 6 \, b c^{4} d^{2} x^{4} - 4 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \sqrt{-c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (5 \, b c^{7} d^{2} x^{7} - 21 \, b c^{5} d^{2} x^{5} + 35 \, b c^{3} d^{2} x^{3} - 35 \, b c d^{2} x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} + 35 \,{\left (a c^{8} d^{2} x^{8} - 4 \, a c^{6} d^{2} x^{6} + 6 \, a c^{4} d^{2} x^{4} - 4 \, a c^{2} d^{2} x^{2} + a d^{2}\right )} \sqrt{-c^{2} d x^{2} + d}}{245 \,{\left (c^{4} x^{2} - c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/245*(35*(b*c^8*d^2*x^8 - 4*b*c^6*d^2*x^6 + 6*b*c^4*d^2*x^4 - 4*b*c^2*d^2*x^2 + b*d^2)*sqrt(-c^2*d*x^2 + d)*l
og(c*x + sqrt(c^2*x^2 - 1)) - (5*b*c^7*d^2*x^7 - 21*b*c^5*d^2*x^5 + 35*b*c^3*d^2*x^3 - 35*b*c*d^2*x)*sqrt(-c^2
*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 35*(a*c^8*d^2*x^8 - 4*a*c^6*d^2*x^6 + 6*a*c^4*d^2*x^4 - 4*a*c^2*d^2*x^2 + a*d^
2)*sqrt(-c^2*d*x^2 + d))/(c^4*x^2 - c^2)

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

[Out]

Exception raised: NotImplementedError