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

Optimal. Leaf size=429 \[ -\frac {c^6 d^3 (f x)^{m+7} \left (a+b \cosh ^{-1}(c x)\right )}{f^7 (m+7)}+\frac {3 c^4 d^3 (f x)^{m+5} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (m+5)}-\frac {3 c^2 d^3 (f x)^{m+3} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (m+3)}+\frac {d^3 (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{f (m+1)}-\frac {3 b c d^3 \left (35 m^3+455 m^2+1813 m+2161\right ) \sqrt {1-c^2 x^2} (f x)^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{f^2 (m+1) (m+2) (m+3)^2 (m+5)^2 (m+7)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^3 \left (m^4+27 m^3+284 m^2+1329 m+2271\right ) \left (1-c^2 x^2\right ) (f x)^{m+2}}{f^2 (m+3)^2 (m+5)^2 (m+7)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d^3 \left (1-c^2 x^2\right ) (f x)^{m+6}}{f^6 (m+7)^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d^3 (m+9) (2 m+13) \left (1-c^2 x^2\right ) (f x)^{m+4}}{f^4 (m+5)^2 (m+7)^2 \sqrt {c x-1} \sqrt {c x+1}} \]

[Out]

d^3*(f*x)^(1+m)*(a+b*arccosh(c*x))/f/(1+m)-3*c^2*d^3*(f*x)^(3+m)*(a+b*arccosh(c*x))/f^3/(3+m)+3*c^4*d^3*(f*x)^
(5+m)*(a+b*arccosh(c*x))/f^5/(5+m)-c^6*d^3*(f*x)^(7+m)*(a+b*arccosh(c*x))/f^7/(7+m)-b*c*d^3*(m^4+27*m^3+284*m^
2+1329*m+2271)*(f*x)^(2+m)*(-c^2*x^2+1)/f^2/(7+m)^2/(m^2+8*m+15)^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)+b*c^3*d^3*(9+m)
*(13+2*m)*(f*x)^(4+m)*(-c^2*x^2+1)/f^4/(5+m)^2/(7+m)^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-b*c^5*d^3*(f*x)^(6+m)*(-c^2
*x^2+1)/f^6/(7+m)^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3*b*c*d^3*(35*m^3+455*m^2+1813*m+2161)*(f*x)^(2+m)*hypergeom([
1/2, 1+1/2*m],[2+1/2*m],c^2*x^2)*(-c^2*x^2+1)^(1/2)/f^2/(7+m)^2/(m^2+3*m+2)/(m^2+8*m+15)^2/(c*x-1)^(1/2)/(c*x+
1)^(1/2)

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Rubi [A]  time = 2.76, antiderivative size = 429, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {270, 5731, 12, 1610, 1809, 1267, 459, 365, 364} \[ -\frac {3 c^2 d^3 (f x)^{m+3} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (m+3)}+\frac {3 c^4 d^3 (f x)^{m+5} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (m+5)}-\frac {c^6 d^3 (f x)^{m+7} \left (a+b \cosh ^{-1}(c x)\right )}{f^7 (m+7)}+\frac {d^3 (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{f (m+1)}-\frac {3 b c d^3 \left (35 m^3+455 m^2+1813 m+2161\right ) \sqrt {1-c^2 x^2} (f x)^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{f^2 (m+1) (m+2) (m+3)^2 (m+5)^2 (m+7)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^3 \left (m^4+27 m^3+284 m^2+1329 m+2271\right ) \left (1-c^2 x^2\right ) (f x)^{m+2}}{f^2 (m+3)^2 (m+5)^2 (m+7)^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d^3 (m+9) (2 m+13) \left (1-c^2 x^2\right ) (f x)^{m+4}}{f^4 (m+5)^2 (m+7)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d^3 \left (1-c^2 x^2\right ) (f x)^{m+6}}{f^6 (m+7)^2 \sqrt {c x-1} \sqrt {c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*c*d^3*(2271 + 1329*m + 284*m^2 + 27*m^3 + m^4)*(f*x)^(2 + m)*(1 - c^2*x^2))/(f^2*(3 + m)^2*(5 + m)^2*(7 +
m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (b*c^3*d^3*(9 + m)*(13 + 2*m)*(f*x)^(4 + m)*(1 - c^2*x^2))/(f^4*(5 + m)
^2*(7 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^3*(f*x)^(6 + m)*(1 - c^2*x^2))/(f^6*(7 + m)^2*Sqrt[-1 +
c*x]*Sqrt[1 + c*x]) + (d^3*(f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(f*(1 + m)) - (3*c^2*d^3*(f*x)^(3 + m)*(a + b*A
rcCosh[c*x]))/(f^3*(3 + m)) + (3*c^4*d^3*(f*x)^(5 + m)*(a + b*ArcCosh[c*x]))/(f^5*(5 + m)) - (c^6*d^3*(f*x)^(7
+ m)*(a + b*ArcCosh[c*x]))/(f^7*(7 + m)) - (3*b*c*d^3*(2161 + 1813*m + 455*m^2 + 35*m^3)*(f*x)^(2 + m)*Sqrt[1
- c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(f^2*(1 + m)*(2 + m)*(3 + m)^2*(5 + m)^2*(7
+ m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 12

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

Rule 270

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

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 459

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

Rule 1267

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Si
mp[(c^p*(f*x)^(m + 4*p - 1)*(d + e*x^2)^(q + 1))/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1)), x] + Dist[1/(e*(m + 4*p
+ 2*q + 1)), Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + b*x^2 + c*x^4)^p - c^p*x^(4*p))
- d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] &&
IGtQ[p, 0] &&  !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]

Rule 1610

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

Rule 1809

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[(f*(c*x)^(m + q - 1)*(a + b*x^2)^(p + 1))/(b*c^(q - 1)*(m + q + 2*p + 1)), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 5731

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

Rubi steps

\begin {align*} \int (f x)^m \left (d-c^2 d x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {d^3 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac {3 c^2 d^3 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 c^4 d^3 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {c^6 d^3 (f x)^{7+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^7 (7+m)}-(b c) \int \frac {d^3 (f x)^{1+m} \left (\frac {1}{1+m}-\frac {3 c^2 x^2}{3+m}+\frac {3 c^4 x^4}{5+m}-\frac {c^6 x^6}{7+m}\right )}{f \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {d^3 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac {3 c^2 d^3 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 c^4 d^3 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {c^6 d^3 (f x)^{7+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^7 (7+m)}-\frac {\left (b c d^3\right ) \int \frac {(f x)^{1+m} \left (\frac {1}{1+m}-\frac {3 c^2 x^2}{3+m}+\frac {3 c^4 x^4}{5+m}-\frac {c^6 x^6}{7+m}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{f}\\ &=\frac {d^3 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac {3 c^2 d^3 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 c^4 d^3 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {c^6 d^3 (f x)^{7+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^7 (7+m)}-\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \int \frac {(f x)^{1+m} \left (\frac {1}{1+m}-\frac {3 c^2 x^2}{3+m}+\frac {3 c^4 x^4}{5+m}-\frac {c^6 x^6}{7+m}\right )}{\sqrt {-1+c^2 x^2}} \, dx}{f \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c^5 d^3 (f x)^{6+m} \left (1-c^2 x^2\right )}{f^6 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^3 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac {3 c^2 d^3 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 c^4 d^3 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {c^6 d^3 (f x)^{7+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^7 (7+m)}-\frac {\left (b d^3 \sqrt {-1+c^2 x^2}\right ) \int \frac {(f x)^{1+m} \left (\frac {c^2 (7+m)}{1+m}-\frac {3 c^4 (7+m) x^2}{3+m}+\frac {c^6 (9+m) (13+2 m) x^4}{(5+m) (7+m)}\right )}{\sqrt {-1+c^2 x^2}} \, dx}{c f (7+m) \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b c^3 d^3 (9+m) (13+2 m) (f x)^{4+m} \left (1-c^2 x^2\right )}{f^4 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^3 (f x)^{6+m} \left (1-c^2 x^2\right )}{f^6 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^3 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac {3 c^2 d^3 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 c^4 d^3 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {c^6 d^3 (f x)^{7+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^7 (7+m)}-\frac {\left (b d^3 \sqrt {-1+c^2 x^2}\right ) \int \frac {(f x)^{1+m} \left (\frac {c^4 (5+m) (7+m)}{1+m}-\frac {c^6 \left (2271+1329 m+284 m^2+27 m^3+m^4\right ) x^2}{(3+m) (5+m) (7+m)}\right )}{\sqrt {-1+c^2 x^2}} \, dx}{c^3 f (5+m) (7+m) \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^3 \left (2271+1329 m+284 m^2+27 m^3+m^4\right ) (f x)^{2+m} \left (1-c^2 x^2\right )}{f^2 (3+m)^2 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^3 (9+m) (13+2 m) (f x)^{4+m} \left (1-c^2 x^2\right )}{f^4 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^3 (f x)^{6+m} \left (1-c^2 x^2\right )}{f^6 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^3 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac {3 c^2 d^3 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 c^4 d^3 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {c^6 d^3 (f x)^{7+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^7 (7+m)}-\frac {\left (3 b c d^3 \left (2161+1813 m+455 m^2+35 m^3\right ) \sqrt {-1+c^2 x^2}\right ) \int \frac {(f x)^{1+m}}{\sqrt {-1+c^2 x^2}} \, dx}{f (1+m) (3+m)^2 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^3 \left (2271+1329 m+284 m^2+27 m^3+m^4\right ) (f x)^{2+m} \left (1-c^2 x^2\right )}{f^2 (3+m)^2 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^3 (9+m) (13+2 m) (f x)^{4+m} \left (1-c^2 x^2\right )}{f^4 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^3 (f x)^{6+m} \left (1-c^2 x^2\right )}{f^6 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^3 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac {3 c^2 d^3 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 c^4 d^3 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {c^6 d^3 (f x)^{7+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^7 (7+m)}-\frac {\left (3 b c d^3 \left (2161+1813 m+455 m^2+35 m^3\right ) \sqrt {1-c^2 x^2}\right ) \int \frac {(f x)^{1+m}}{\sqrt {1-c^2 x^2}} \, dx}{f (1+m) (3+m)^2 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^3 \left (2271+1329 m+284 m^2+27 m^3+m^4\right ) (f x)^{2+m} \left (1-c^2 x^2\right )}{f^2 (3+m)^2 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^3 (9+m) (13+2 m) (f x)^{4+m} \left (1-c^2 x^2\right )}{f^4 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^3 (f x)^{6+m} \left (1-c^2 x^2\right )}{f^6 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^3 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac {3 c^2 d^3 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 c^4 d^3 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {c^6 d^3 (f x)^{7+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^7 (7+m)}-\frac {3 b c d^3 \left (2161+1813 m+455 m^2+35 m^3\right ) (f x)^{2+m} \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{f^2 (1+m) (2+m) (3+m)^2 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]  time = 1.18, size = 387, normalized size = 0.90 \[ d^3 x (f x)^m \left (-\frac {c^6 x^6 \left (a+b \cosh ^{-1}(c x)\right )}{m+7}+\frac {3 c^4 x^4 \left (a+b \cosh ^{-1}(c x)\right )}{m+5}-\frac {3 c^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )}{m+3}+\frac {a+b \cosh ^{-1}(c x)}{m+1}-\frac {b c x \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{\left (m^2+3 m+2\right ) \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^7 x^7 \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {m}{2}+4;\frac {m}{2}+5;c^2 x^2\right )}{(m+7) (m+8) \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 b c^5 x^5 \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {m+6}{2};\frac {m+8}{2};c^2 x^2\right )}{(m+5) (m+6) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b c^3 x^3 \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {m+4}{2};\frac {m+6}{2};c^2 x^2\right )}{\left (m^2+7 m+12\right ) \sqrt {c x-1} \sqrt {c x+1}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

d^3*x*(f*x)^m*((a + b*ArcCosh[c*x])/(1 + m) - (3*c^2*x^2*(a + b*ArcCosh[c*x]))/(3 + m) + (3*c^4*x^4*(a + b*Arc
Cosh[c*x]))/(5 + m) - (c^6*x^6*(a + b*ArcCosh[c*x]))/(7 + m) + (b*c^7*x^7*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[
1/2, 4 + m/2, 5 + m/2, c^2*x^2])/((7 + m)*(8 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*x*Sqrt[1 - c^2*x^2]*Hyp
ergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((2 + 3*m + m^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b*c^3*
x^3*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, c^2*x^2])/((12 + 7*m + m^2)*Sqrt[-1 + c*x]*
Sqrt[1 + c*x]) - (3*b*c^5*x^5*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (6 + m)/2, (8 + m)/2, c^2*x^2])/((5 + m
)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))

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fricas [F]  time = 1.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a c^{6} d^{3} x^{6} - 3 \, a c^{4} d^{3} x^{4} + 3 \, a c^{2} d^{3} x^{2} - a d^{3} + {\left (b c^{6} d^{3} x^{6} - 3 \, b c^{4} d^{3} x^{4} + 3 \, b c^{2} d^{3} x^{2} - b d^{3}\right )} \operatorname {arcosh}\left (c x\right )\right )} \left (f x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(a*c^6*d^3*x^6 - 3*a*c^4*d^3*x^4 + 3*a*c^2*d^3*x^2 - a*d^3 + (b*c^6*d^3*x^6 - 3*b*c^4*d^3*x^4 + 3*b*
c^2*d^3*x^2 - b*d^3)*arccosh(c*x))*(f*x)^m, x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{m} \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {a c^{6} d^{3} f^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, a c^{4} d^{3} f^{m} x^{5} x^{m}}{m + 5} - \frac {3 \, a c^{2} d^{3} f^{m} x^{3} x^{m}}{m + 3} + \frac {\left (f x\right )^{m + 1} a d^{3}}{f {\left (m + 1\right )}} - \frac {{\left ({\left (m^{3} + 9 \, m^{2} + 23 \, m + 15\right )} b c^{6} d^{3} f^{m} x^{7} - 3 \, {\left (m^{3} + 11 \, m^{2} + 31 \, m + 21\right )} b c^{4} d^{3} f^{m} x^{5} + 3 \, {\left (m^{3} + 13 \, m^{2} + 47 \, m + 35\right )} b c^{2} d^{3} f^{m} x^{3} - {\left (m^{3} + 15 \, m^{2} + 71 \, m + 105\right )} b d^{3} f^{m} x\right )} x^{m} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} - \int \frac {{\left ({\left (m^{3} + 9 \, m^{2} + 23 \, m + 15\right )} b c^{7} d^{3} f^{m} x^{7} - 3 \, {\left (m^{3} + 11 \, m^{2} + 31 \, m + 21\right )} b c^{5} d^{3} f^{m} x^{5} + 3 \, {\left (m^{3} + 13 \, m^{2} + 47 \, m + 35\right )} b c^{3} d^{3} f^{m} x^{3} - {\left (m^{3} + 15 \, m^{2} + 71 \, m + 105\right )} b c d^{3} f^{m} x\right )} x^{m}}{{\left (m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105\right )} c^{3} x^{3} - {\left (m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105\right )} c x + {\left ({\left (m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105\right )} c^{2} x^{2} - m^{4} - 16 \, m^{3} - 86 \, m^{2} - 176 \, m - 105\right )} \sqrt {c x + 1} \sqrt {c x - 1}}\,{d x} + \int \frac {{\left ({\left (m^{3} + 9 \, m^{2} + 23 \, m + 15\right )} b c^{8} d^{3} f^{m} x^{8} - 3 \, {\left (m^{3} + 11 \, m^{2} + 31 \, m + 21\right )} b c^{6} d^{3} f^{m} x^{6} + 3 \, {\left (m^{3} + 13 \, m^{2} + 47 \, m + 35\right )} b c^{4} d^{3} f^{m} x^{4} - {\left (m^{3} + 15 \, m^{2} + 71 \, m + 105\right )} b c^{2} d^{3} f^{m} x^{2}\right )} x^{m}}{{\left (m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105\right )} c^{2} x^{2} - m^{4} - 16 \, m^{3} - 86 \, m^{2} - 176 \, m - 105}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*c^6*d^3*f^m*x^7*x^m/(m + 7) + 3*a*c^4*d^3*f^m*x^5*x^m/(m + 5) - 3*a*c^2*d^3*f^m*x^3*x^m/(m + 3) + (f*x)^(m
+ 1)*a*d^3/(f*(m + 1)) - ((m^3 + 9*m^2 + 23*m + 15)*b*c^6*d^3*f^m*x^7 - 3*(m^3 + 11*m^2 + 31*m + 21)*b*c^4*d^3
*f^m*x^5 + 3*(m^3 + 13*m^2 + 47*m + 35)*b*c^2*d^3*f^m*x^3 - (m^3 + 15*m^2 + 71*m + 105)*b*d^3*f^m*x)*x^m*log(c
*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(m^4 + 16*m^3 + 86*m^2 + 176*m + 105) - integrate(((m^3 + 9*m^2 + 23*m + 15)
*b*c^7*d^3*f^m*x^7 - 3*(m^3 + 11*m^2 + 31*m + 21)*b*c^5*d^3*f^m*x^5 + 3*(m^3 + 13*m^2 + 47*m + 35)*b*c^3*d^3*f
^m*x^3 - (m^3 + 15*m^2 + 71*m + 105)*b*c*d^3*f^m*x)*x^m/((m^4 + 16*m^3 + 86*m^2 + 176*m + 105)*c^3*x^3 - (m^4
+ 16*m^3 + 86*m^2 + 176*m + 105)*c*x + ((m^4 + 16*m^3 + 86*m^2 + 176*m + 105)*c^2*x^2 - m^4 - 16*m^3 - 86*m^2
- 176*m - 105)*sqrt(c*x + 1)*sqrt(c*x - 1)), x) + integrate(((m^3 + 9*m^2 + 23*m + 15)*b*c^8*d^3*f^m*x^8 - 3*(
m^3 + 11*m^2 + 31*m + 21)*b*c^6*d^3*f^m*x^6 + 3*(m^3 + 13*m^2 + 47*m + 35)*b*c^4*d^3*f^m*x^4 - (m^3 + 15*m^2 +
71*m + 105)*b*c^2*d^3*f^m*x^2)*x^m/((m^4 + 16*m^3 + 86*m^2 + 176*m + 105)*c^2*x^2 - m^4 - 16*m^3 - 86*m^2 - 1
76*m - 105), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3\,{\left (f\,x\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(c*x))*(d - c^2*d*x^2)^3*(f*x)^m,x)

[Out]

int((a + b*acosh(c*x))*(d - c^2*d*x^2)^3*(f*x)^m, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - d^{3} \left (\int \left (- a \left (f x\right )^{m}\right )\, dx + \int \left (- b \left (f x\right )^{m} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int 3 a c^{2} x^{2} \left (f x\right )^{m}\, dx + \int \left (- 3 a c^{4} x^{4} \left (f x\right )^{m}\right )\, dx + \int a c^{6} x^{6} \left (f x\right )^{m}\, dx + \int 3 b c^{2} x^{2} \left (f x\right )^{m} \operatorname {acosh}{\left (c x \right )}\, dx + \int \left (- 3 b c^{4} x^{4} \left (f x\right )^{m} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{6} x^{6} \left (f x\right )^{m} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d**3*(Integral(-a*(f*x)**m, x) + Integral(-b*(f*x)**m*acosh(c*x), x) + Integral(3*a*c**2*x**2*(f*x)**m, x) +
Integral(-3*a*c**4*x**4*(f*x)**m, x) + Integral(a*c**6*x**6*(f*x)**m, x) + Integral(3*b*c**2*x**2*(f*x)**m*aco
sh(c*x), x) + Integral(-3*b*c**4*x**4*(f*x)**m*acosh(c*x), x) + Integral(b*c**6*x**6*(f*x)**m*acosh(c*x), x))

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