\(\int (f x)^m (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x)) \, dx\) [158]

Optimal result

Integrand size = 35, antiderivative size = 503 \[ \int (f x)^m (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {3 b c \text {d1} \text {d2} (f x)^{2+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}{f^2 (2+m)^2 (4+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c \text {d1} \text {d2} (f x)^{2+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}{f^2 (2+m) (4+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 \text {d1} \text {d2} (f x)^{4+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}{f^4 (4+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 \text {d1} \text {d2} (f x)^{1+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x} (a+b \text {arccosh}(c x))}{f \left (8+6 m+m^2\right )}+\frac {(f x)^{1+m} (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x))}{f (4+m)}+\frac {3 \text {d1} \text {d2} (f x)^{1+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{f (4+m) \left (2+3 m+m^2\right ) \sqrt {1-c x} \sqrt {1+c x}}-\frac {3 b c \text {d1} \text {d2} (f x)^{2+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )}{f^2 (1+m) (2+m)^2 (4+m) \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

(f*x)^(1+m)*(c*d1*x+d1)^(3/2)*(-c*d2*x+d2)^(3/2)*(a+b*arccosh(c*x))/f/(4+m)+3*d1*d2*(f*x)^(1+m)*(a+b*arccosh(c
*x))*(c*d1*x+d1)^(1/2)*(-c*d2*x+d2)^(1/2)/f/(m^2+6*m+8)+3*d1*d2*(f*x)^(1+m)*(a+b*arccosh(c*x))*hypergeom([1/2,
1/2+1/2*m],[3/2+1/2*m],c^2*x^2)*(c*d1*x+d1)^(1/2)*(-c*d2*x+d2)^(1/2)/f/(m^3+7*m^2+14*m+8)/(-c*x+1)^(1/2)/(c*x
+1)^(1/2)-3*b*c*d1*d2*(f*x)^(2+m)*(c*d1*x+d1)^(1/2)*(-c*d2*x+d2)^(1/2)/f^2/(2+m)^2/(4+m)/(c*x-1)^(1/2)/(c*x+1)
^(1/2)-b*c*d1*d2*(f*x)^(2+m)*(c*d1*x+d1)^(1/2)*(-c*d2*x+d2)^(1/2)/f^2/(2+m)/(4+m)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+
b*c^3*d1*d2*(f*x)^(4+m)*(c*d1*x+d1)^(1/2)*(-c*d2*x+d2)^(1/2)/f^4/(4+m)^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3*b*c*d1*
d2*(f*x)^(2+m)*hypergeom([1, 1+1/2*m, 1+1/2*m],[2+1/2*m, 3/2+1/2*m],c^2*x^2)*(c*d1*x+d1)^(1/2)*(-c*d2*x+d2)^(1
/2)/f^2/(2+m)^2/(m^2+5*m+4)/(c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 503, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {5931, 5927, 5949, 32, 74, 14} \[ \int (f x)^m (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {3 b c \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+2} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;c^2 x^2\right )}{f^2 (m+1) (m+2)^2 (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},c^2 x^2\right ) (a+b \text {arccosh}(c x))}{f (m+4) \left (m^2+3 m+2\right ) \sqrt {1-c x} \sqrt {c x+1}}+\frac {3 \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f \left (m^2+6 m+8\right )}+\frac {(c \text {d1} x+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+4)}+\frac {b c^3 \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+4}}{f^4 (m+4)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+2}}{f^2 (m+2) (m+4) \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 b c \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+2}}{f^2 (m+2)^2 (m+4) \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

Int[(f*x)^m*(d1 + c*d1*x)^(3/2)*(d2 - c*d2*x)^(3/2)*(a + b*ArcCosh[c*x]),x]

[Out]

(-3*b*c*d1*d2*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^2*(2 + m)^2*(4 + m)*Sqrt[-1 + c*x]*Sqrt[1
+ c*x]) - (b*c*d1*d2*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^2*(2 + m)*(4 + m)*Sqrt[-1 + c*x]*Sq
rt[1 + c*x]) + (b*c^3*d1*d2*(f*x)^(4 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^4*(4 + m)^2*Sqrt[-1 + c*x]*S
qrt[1 + c*x]) + (3*d1*d2*(f*x)^(1 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*(a + b*ArcCosh[c*x]))/(f*(8 + 6*m +
m^2)) + ((f*x)^(1 + m)*(d1 + c*d1*x)^(3/2)*(d2 - c*d2*x)^(3/2)*(a + b*ArcCosh[c*x]))/(f*(4 + m)) + (3*d1*d2*(
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)*(2 + 3*m + m^2)*Sqrt[1 - c*x]*Sqrt[1 + c*x]) - (3*b*c*d1*d2*(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)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

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 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 74

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

Rule 5927

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[(1/(m + 2))*Simp[Sqrt[d1 + e1*x]/Sqrt[1 + c*x]]*Simp[Sqrt[d2 + e2*x]/Sqrt[-1 + c*x]], Int[(f*x)^m*(
(a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] - Dist[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d1 + e1*x]
/Sqrt[1 + c*x]]*Simp[Sqrt[d2 + e2*x]/Sqrt[-1 + c*x]], Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x])
/; FreeQ[{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[n, 0] && (IGtQ[m, -2
] || EqQ[n, 1])

Rule 5931

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/(f*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p
], Int[(f*x)^(m + 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(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] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m,
-1]

Rule 5949

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

Rubi steps \begin{align*} \text {integral}& = \frac {(f x)^{1+m} (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x))}{f (4+m)}+\frac {(3 \text {d1} \text {d2}) \int (f x)^m \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x} (a+b \text {arccosh}(c x)) \, dx}{4+m}+\frac {\left (b c \text {d1} \text {d2} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}\right ) \int (f x)^{1+m} (-1+c x) (1+c x) \, dx}{f (4+m) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {3 \text {d1} \text {d2} (f x)^{1+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x} (a+b \text {arccosh}(c x))}{f \left (8+6 m+m^2\right )}+\frac {(f x)^{1+m} (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x))}{f (4+m)}+\frac {\left (b c \text {d1} \text {d2} \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) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 \text {d1} \text {d2} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}\right ) \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{(2+m) (4+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 b c \text {d1} \text {d2} \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) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {3 b c \text {d1} \text {d2} (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) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 \text {d1} \text {d2} (f x)^{1+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x} (a+b \text {arccosh}(c x))}{f \left (8+6 m+m^2\right )}+\frac {(f x)^{1+m} (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x))}{f (4+m)}+\frac {3 \text {d1} \text {d2} (f x)^{1+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{f (1+m) (2+m) (4+m) \sqrt {1-c x} \sqrt {1+c x}}-\frac {3 b c \text {d1} \text {d2} (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) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c \text {d1} \text {d2} \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) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {3 b c \text {d1} \text {d2} (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) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c \text {d1} \text {d2} (f x)^{2+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}{f^2 (2+m) (4+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 \text {d1} \text {d2} (f x)^{4+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}{f^4 (4+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 \text {d1} \text {d2} (f x)^{1+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x} (a+b \text {arccosh}(c x))}{f \left (8+6 m+m^2\right )}+\frac {(f x)^{1+m} (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x))}{f (4+m)}+\frac {3 \text {d1} \text {d2} (f x)^{1+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{f (1+m) (2+m) (4+m) \sqrt {1-c x} \sqrt {1+c x}}-\frac {3 b c \text {d1} \text {d2} (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) \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.57 \[ \int (f x)^m (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {\text {d1} \text {d2} x (f x)^m \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x} \left (-\frac {3 b c x}{(2+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c x \left (-\frac {1}{2+m}+\frac {c^2 x^2}{4+m}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 (a+b \text {arccosh}(c x))}{2+m}-(-1+c x) (1+c x) (a+b \text {arccosh}(c x))-\frac {3 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{(1+m) (2+m) (-1+c x) (1+c x)}-\frac {3 b c x \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )}{(1+m) (2+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}\right )}{4+m} \]

[In]

Integrate[(f*x)^m*(d1 + c*d1*x)^(3/2)*(d2 - c*d2*x)^(3/2)*(a + b*ArcCosh[c*x]),x]

[Out]

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

Maple [F]

\[\int \left (f x \right )^{m} \left (c \operatorname {d1} x +\operatorname {d1} \right )^{\frac {3}{2}} \left (-c \operatorname {d2} x +\operatorname {d2} \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )d x\]

[In]

int((f*x)^m*(c*d1*x+d1)^(3/2)*(-c*d2*x+d2)^(3/2)*(a+b*arccosh(c*x)),x)

[Out]

int((f*x)^m*(c*d1*x+d1)^(3/2)*(-c*d2*x+d2)^(3/2)*(a+b*arccosh(c*x)),x)

Fricas [F]

\[ \int (f x)^m (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (c d_{1} x + d_{1}\right )}^{\frac {3}{2}} {\left (-c d_{2} x + d_{2}\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]

[In]

integrate((f*x)^m*(c*d1*x+d1)^(3/2)*(-c*d2*x+d2)^(3/2)*(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int (f x)^m (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x)) \, dx=\text {Timed out} \]

[In]

integrate((f*x)**m*(c*d1*x+d1)**(3/2)*(-c*d2*x+d2)**(3/2)*(a+b*acosh(c*x)),x)

[Out]

Timed out

Maxima [F]

\[ \int (f x)^m (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (c d_{1} x + d_{1}\right )}^{\frac {3}{2}} {\left (-c d_{2} x + d_{2}\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]

[In]

integrate((f*x)^m*(c*d1*x+d1)^(3/2)*(-c*d2*x+d2)^(3/2)*(a+b*arccosh(c*x)),x, algorithm="maxima")

[Out]

integrate((c*d1*x + d1)^(3/2)*(-c*d2*x + d2)^(3/2)*(b*arccosh(c*x) + a)*(f*x)^m, x)

Giac [F]

\[ \int (f x)^m (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (c d_{1} x + d_{1}\right )}^{\frac {3}{2}} {\left (-c d_{2} x + d_{2}\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]

[In]

integrate((f*x)^m*(c*d1*x+d1)^(3/2)*(-c*d2*x+d2)^(3/2)*(a+b*arccosh(c*x)),x, algorithm="giac")

[Out]

integrate((c*d1*x + d1)^(3/2)*(-c*d2*x + d2)^(3/2)*(b*arccosh(c*x) + a)*(f*x)^m, x)

Mupad [F(-1)]

Timed out. \[ \int (f x)^m (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (f\,x\right )}^m\,{\left (d_{1}+c\,d_{1}\,x\right )}^{3/2}\,{\left (d_{2}-c\,d_{2}\,x\right )}^{3/2} \,d x \]

[In]

int((a + b*acosh(c*x))*(f*x)^m*(d1 + c*d1*x)^(3/2)*(d2 - c*d2*x)^(3/2),x)

[Out]

int((a + b*acosh(c*x))*(f*x)^m*(d1 + c*d1*x)^(3/2)*(d2 - c*d2*x)^(3/2), x)