3.157 \(\int (f x)^m (\text{d1}+c \text{d1} x)^{5/2} (\text{d2}-c \text{d2} x)^{5/2} (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=817 \[ \frac{(c x \text{d1}+\text{d1})^{5/2} (\text{d2}-c \text{d2} x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) (f x)^{m+1}}{f (m+6)}+\frac{5 \text{d1} \text{d2} (c x \text{d1}+\text{d1})^{3/2} (\text{d2}-c \text{d2} x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) (f x)^{m+1}}{f (m+4) (m+6)}+\frac{15 \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} \left (a+b \cosh ^{-1}(c x)\right ) (f x)^{m+1}}{f (m+6) \left (m^2+6 m+8\right )}+\frac{15 \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} \left (a+b \cosh ^{-1}(c x)\right ) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},c^2 x^2\right ) (f x)^{m+1}}{f (m+4) (m+6) \left (m^2+3 m+2\right ) \sqrt{1-c x} \sqrt{c x+1}}-\frac{15 b c \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+1,\frac{m}{2}+1\right \},\left \{\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2\right \},c^2 x^2\right ) (f x)^{m+2}}{f^2 (m+1) (m+2)^2 (m+4) (m+6) \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+2}}{f^2 (m+2) (m+6) \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 b c \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+2}}{f^2 (m+2) (m+4) (m+6) \sqrt{c x-1} \sqrt{c x+1}}-\frac{15 b c \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+2}}{f^2 (m+2)^2 (m+4) (m+6) \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b c^3 \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+4}}{f^4 (m+4) (m+6) \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b c^3 \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+4}}{f^4 (m+4)^2 (m+6) \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+6}}{f^6 (m+6)^2 \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

-((b*c*d1^2*d2^2*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^2*(2 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1
+ c*x])) - (15*b*c*d1^2*d2^2*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^2*(2 + m)^2*(4 + m)*(6 + m
)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*b*c*d1^2*d2^2*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^2*(2
+ m)*(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*b*c^3*d1^2*d2^2*(f*x)^(4 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d
2 - c*d2*x])/(f^4*(4 + m)^2*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c^3*d1^2*d2^2*(f*x)^(4 + m)*Sqrt[d1 +
c*d1*x]*Sqrt[d2 - c*d2*x])/(f^4*(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d1^2*d2^2*(f*x)^(6 + m
)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^6*(6 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (15*d1^2*d2^2*(f*x)^(1 +
m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*(a + b*ArcCosh[c*x]))/(f*(6 + m)*(8 + 6*m + m^2)) + (5*d1*d2*(f*x)^(1
+ m)*(d1 + c*d1*x)^(3/2)*(d2 - c*d2*x)^(3/2)*(a + b*ArcCosh[c*x]))/(f*(4 + m)*(6 + m)) + ((f*x)^(1 + m)*(d1 +
c*d1*x)^(5/2)*(d2 - c*d2*x)^(5/2)*(a + b*ArcCosh[c*x]))/(f*(6 + m)) + (15*d1^2*d2^2*(f*x)^(1 + m)*Sqrt[d1 + c*
d1*x]*Sqrt[d2 - c*d2*x]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(f*(4 + m)
*(6 + m)*(2 + 3*m + m^2)*Sqrt[1 - c*x]*Sqrt[1 + c*x]) - (15*b*c*d1^2*d2^2*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt
[d2 - c*d2*x]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(f^2*(1 + m)*(2 + m)^2*
(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

________________________________________________________________________________________

Rubi [A]  time = 1.57982, antiderivative size = 827, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {5745, 5743, 5763, 32, 14, 270} \[ \frac{(c x \text{d1}+\text{d1})^{5/2} (\text{d2}-c \text{d2} x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) (f x)^{m+1}}{f (m+6)}+\frac{5 \text{d1} \text{d2} (c x \text{d1}+\text{d1})^{3/2} (\text{d2}-c \text{d2} x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) (f x)^{m+1}}{f (m+4) (m+6)}+\frac{15 \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} \left (a+b \cosh ^{-1}(c x)\right ) (f x)^{m+1}}{f (m+6) \left (m^2+6 m+8\right )}+\frac{15 \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right ) \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right ) (f x)^{m+1}}{f (m+4) (m+6) \left (m^2+3 m+2\right ) (1-c x) (c x+1)}-\frac{15 b c \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} \, _3F_2\left (1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right ) (f x)^{m+2}}{f^2 (m+1) (m+2)^2 (m+4) (m+6) \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+2}}{f^2 (m+2) (m+6) \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 b c \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+2}}{f^2 (m+2) (m+4) (m+6) \sqrt{c x-1} \sqrt{c x+1}}-\frac{15 b c \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+2}}{f^2 (m+2)^2 (m+4) (m+6) \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b c^3 \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+4}}{f^4 (m+4) (m+6) \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b c^3 \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+4}}{f^4 (m+4)^2 (m+6) \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+6}}{f^6 (m+6)^2 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

Int[(f*x)^m*(d1 + c*d1*x)^(5/2)*(d2 - c*d2*x)^(5/2)*(a + b*ArcCosh[c*x]),x]

[Out]

-((b*c*d1^2*d2^2*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^2*(2 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1
+ c*x])) - (15*b*c*d1^2*d2^2*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^2*(2 + m)^2*(4 + m)*(6 + m
)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*b*c*d1^2*d2^2*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^2*(2
+ m)*(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*b*c^3*d1^2*d2^2*(f*x)^(4 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d
2 - c*d2*x])/(f^4*(4 + m)^2*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c^3*d1^2*d2^2*(f*x)^(4 + m)*Sqrt[d1 +
c*d1*x]*Sqrt[d2 - c*d2*x])/(f^4*(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d1^2*d2^2*(f*x)^(6 + m
)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^6*(6 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (15*d1^2*d2^2*(f*x)^(1 +
m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*(a + b*ArcCosh[c*x]))/(f*(6 + m)*(8 + 6*m + m^2)) + (5*d1*d2*(f*x)^(1
+ m)*(d1 + c*d1*x)^(3/2)*(d2 - c*d2*x)^(3/2)*(a + b*ArcCosh[c*x]))/(f*(4 + m)*(6 + m)) + ((f*x)^(1 + m)*(d1 +
c*d1*x)^(5/2)*(d2 - c*d2*x)^(5/2)*(a + b*ArcCosh[c*x]))/(f*(6 + m)) + (15*d1^2*d2^2*(f*x)^(1 + m)*Sqrt[d1 + c*
d1*x]*Sqrt[d2 - c*d2*x]*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^
2*x^2])/(f*(4 + m)*(6 + m)*(2 + 3*m + m^2)*(1 - c*x)*(1 + c*x)) - (15*b*c*d1^2*d2^2*(f*x)^(2 + m)*Sqrt[d1 + c*
d1*x]*Sqrt[d2 - c*d2*x]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(f^2*(1 + m)*
(2 + m)^2*(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

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 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 5763

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_
)]), x_Symbol] :> Simp[((f*x)^(m + 1)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2,
(3 + m)/2, c^2*x^2])/(f*(m + 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), x] + Simp[(b*c*(f*x)^(m + 2)*Hypergeometric
PFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(Sqrt[-(d1*d2)]*f^2*(m + 1)*(m + 2)), x] /; FreeQ[{
a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[d1, 0] && LtQ[d2, 0] &&  !
IntegerQ[m]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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 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]

Rubi steps

\begin{align*} \int (f x)^m (\text{d1}+c \text{d1} x)^{5/2} (\text{d2}-c \text{d2} x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac{(f x)^{1+m} (\text{d1}+c \text{d1} x)^{5/2} (\text{d2}-c \text{d2} x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{f (6+m)}+\frac{(5 \text{d1} \text{d2}) \int (f x)^m (\text{d1}+c \text{d1} x)^{3/2} (\text{d2}-c \text{d2} x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{6+m}-\frac{\left (b c \text{d1}^2 \text{d2}^2 \sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x}\right ) \int (f x)^{1+m} \left (-1+c^2 x^2\right )^2 \, dx}{f (6+m) \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{5 \text{d1} \text{d2} (f x)^{1+m} (\text{d1}+c \text{d1} x)^{3/2} (\text{d2}-c \text{d2} x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{f (4+m) (6+m)}+\frac{(f x)^{1+m} (\text{d1}+c \text{d1} x)^{5/2} (\text{d2}-c \text{d2} x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{f (6+m)}+\frac{\left (15 \text{d1}^2 \text{d2}^2\right ) \int (f x)^m \sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{(4+m) (6+m)}-\frac{\left (b c \text{d1}^2 \text{d2}^2 \sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x}\right ) \int \left ((f x)^{1+m}-\frac{2 c^2 (f x)^{3+m}}{f^2}+\frac{c^4 (f x)^{5+m}}{f^4}\right ) \, dx}{f (6+m) \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 b c \text{d1}^2 \text{d2}^2 \sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x}\right ) \int (f x)^{1+m} \left (-1+c^2 x^2\right ) \, dx}{f (4+m) (6+m) \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c \text{d1}^2 \text{d2}^2 (f x)^{2+m} \sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x}}{f^2 (2+m) (6+m) \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c^3 \text{d1}^2 \text{d2}^2 (f x)^{4+m} \sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x}}{f^4 (4+m) (6+m) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 \text{d1}^2 \text{d2}^2 (f x)^{6+m} \sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x}}{f^6 (6+m)^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{15 \text{d1}^2 \text{d2}^2 (f x)^{1+m} \sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x} \left (a+b \cosh ^{-1}(c x)\right )}{f (2+m) (4+m) (6+m)}+\frac{5 \text{d1} \text{d2} (f x)^{1+m} (\text{d1}+c \text{d1} x)^{3/2} (\text{d2}-c \text{d2} x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{f (4+m) (6+m)}+\frac{(f x)^{1+m} (\text{d1}+c \text{d1} x)^{5/2} (\text{d2}-c \text{d2} x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{f (6+m)}+\frac{\left (5 b c \text{d1}^2 \text{d2}^2 \sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x}\right ) \int \left (-(f x)^{1+m}+\frac{c^2 (f x)^{3+m}}{f^2}\right ) \, dx}{f (4+m) (6+m) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (15 \text{d1}^2 \text{d2}^2 \sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x}\right ) \int \frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{(2+m) (4+m) (6+m) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (15 b c \text{d1}^2 \text{d2}^2 \sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x}\right ) \int (f x)^{1+m} \, dx}{f (2+m) (4+m) (6+m) \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c \text{d1}^2 \text{d2}^2 (f x)^{2+m} \sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x}}{f^2 (2+m) (6+m) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{15 b c \text{d1}^2 \text{d2}^2 (f x)^{2+m} \sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x}}{f^2 (2+m)^2 (4+m) (6+m) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{5 b c \text{d1}^2 \text{d2}^2 (f x)^{2+m} \sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x}}{f^2 (2+m) (4+m) (6+m) \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 b c^3 \text{d1}^2 \text{d2}^2 (f x)^{4+m} \sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x}}{f^4 (4+m)^2 (6+m) \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c^3 \text{d1}^2 \text{d2}^2 (f x)^{4+m} \sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x}}{f^4 (4+m) (6+m) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 \text{d1}^2 \text{d2}^2 (f x)^{6+m} \sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x}}{f^6 (6+m)^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{15 \text{d1}^2 \text{d2}^2 (f x)^{1+m} \sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x} \left (a+b \cosh ^{-1}(c x)\right )}{f (2+m) (4+m) (6+m)}+\frac{5 \text{d1} \text{d2} (f x)^{1+m} (\text{d1}+c \text{d1} x)^{3/2} (\text{d2}-c \text{d2} x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{f (4+m) (6+m)}+\frac{(f x)^{1+m} (\text{d1}+c \text{d1} x)^{5/2} (\text{d2}-c \text{d2} x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{f (6+m)}+\frac{15 \text{d1}^2 \text{d2}^2 (f x)^{1+m} \sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x} \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right ) \, _2F_1\left (\frac{1}{2},\frac{1+m}{2};\frac{3+m}{2};c^2 x^2\right )}{f (1+m) (2+m) (4+m) (6+m) (1-c x) (1+c x)}-\frac{15 b c \text{d1}^2 \text{d2}^2 (f x)^{2+m} \sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x} \, _3F_2\left (1,1+\frac{m}{2},1+\frac{m}{2};\frac{3}{2}+\frac{m}{2},2+\frac{m}{2};c^2 x^2\right )}{f^2 (1+m) (2+m)^2 (4+m) (6+m) \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 2.57504, size = 387, normalized size = 0.47 \[ \frac{\text{d1}^2 \text{d2}^2 x \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^m \left (\frac{5 \left (\frac{3 \left (-b c x \sqrt{c x-1} \sqrt{c x+1} \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+1,\frac{m}{2}+1\right \},\left \{\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2\right \},c^2 x^2\right )-(m+2) \sqrt{1-c^2 x^2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+(m+1) \left (a (m+2) \left (c^2 x^2-1\right )+b (m+2) \left (c^2 x^2-1\right ) \cosh ^{-1}(c x)-b c x \sqrt{c x-1} \sqrt{c x+1}\right )\right )}{(m+1) (m+2)^2 (c x-1) (c x+1)}-(c x-1) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )+\frac{b c x \left (\frac{c^2 x^2}{m+4}-\frac{1}{m+2}\right )}{\sqrt{c x-1} \sqrt{c x+1}}\right )}{m+4}+\left (c^2 x^2-1\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac{b c x \left (\frac{c^4 x^4}{m+6}-\frac{2 c^2 x^2}{m+4}+\frac{1}{m+2}\right )}{\sqrt{c x-1} \sqrt{c x+1}}\right )}{m+6} \]

Antiderivative was successfully verified.

[In]

Integrate[(f*x)^m*(d1 + c*d1*x)^(5/2)*(d2 - c*d2*x)^(5/2)*(a + b*ArcCosh[c*x]),x]

[Out]

(d1^2*d2^2*x*(f*x)^m*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*(-((b*c*x*((2 + m)^(-1) - (2*c^2*x^2)/(4 + m) + (c^4*
x^4)/(6 + m)))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (-1 + c^2*x^2)^2*(a + b*ArcCosh[c*x]) + (5*((b*c*x*(-(2 + m)^
(-1) + (c^2*x^2)/(4 + m)))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (-1 + c*x)*(1 + c*x)*(a + b*ArcCosh[c*x]) + (3*((1
+ m)*(-(b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + a*(2 + m)*(-1 + c^2*x^2) + b*(2 + m)*(-1 + c^2*x^2)*ArcCosh[c*x
]) - (2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2] - b*
c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2]))/((1
+ m)*(2 + m)^2*(-1 + c*x)*(1 + c*x))))/(4 + m)))/(6 + m)

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Maple [F]  time = 2.239, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{m} \left ( c{\it d1}\,x+{\it d1} \right ) ^{{\frac{5}{2}}} \left ( -c{\it d2}\,x+{\it d2} \right ) ^{{\frac{5}{2}}} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^m*(c*d1*x+d1)^(5/2)*(-c*d2*x+d2)^(5/2)*(a+b*arccosh(c*x)),x)

[Out]

int((f*x)^m*(c*d1*x+d1)^(5/2)*(-c*d2*x+d2)^(5/2)*(a+b*arccosh(c*x)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c d_{1} x + d_{1}\right )}^{\frac{5}{2}}{\left (-c d_{2} x + d_{2}\right )}^{\frac{5}{2}}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*(c*d1*x+d1)^(5/2)*(-c*d2*x+d2)^(5/2)*(a+b*arccosh(c*x)),x, algorithm="maxima")

[Out]

integrate((c*d1*x + d1)^(5/2)*(-c*d2*x + d2)^(5/2)*(b*arccosh(c*x) + a)*(f*x)^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a c^{4} d_{1}^{2} d_{2}^{2} x^{4} - 2 \, a c^{2} d_{1}^{2} d_{2}^{2} x^{2} + a d_{1}^{2} d_{2}^{2} +{\left (b c^{4} d_{1}^{2} d_{2}^{2} x^{4} - 2 \, b c^{2} d_{1}^{2} d_{2}^{2} x^{2} + b d_{1}^{2} d_{2}^{2}\right )} \operatorname{arcosh}\left (c x\right )\right )} \sqrt{c d_{1} x + d_{1}} \sqrt{-c d_{2} x + d_{2}} \left (f x\right )^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*(c*d1*x+d1)^(5/2)*(-c*d2*x+d2)^(5/2)*(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

integral((a*c^4*d1^2*d2^2*x^4 - 2*a*c^2*d1^2*d2^2*x^2 + a*d1^2*d2^2 + (b*c^4*d1^2*d2^2*x^4 - 2*b*c^2*d1^2*d2^2
*x^2 + b*d1^2*d2^2)*arccosh(c*x))*sqrt(c*d1*x + d1)*sqrt(-c*d2*x + d2)*(f*x)^m, x)

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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((f*x)**m*(c*d1*x+d1)**(5/2)*(-c*d2*x+d2)**(5/2)*(a+b*acosh(c*x)),x)

[Out]

Timed out

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Giac [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((f*x)^m*(c*d1*x+d1)^(5/2)*(-c*d2*x+d2)^(5/2)*(a+b*arccosh(c*x)),x, algorithm="giac")

[Out]

Timed out