\(\int \frac {(f x)^m (a+b \text {arccosh}(c x))}{(d-c^2 d x^2)^{5/2}} \, dx\) [156]

Optimal result

Integrand size = 29, antiderivative size = 450 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {(f x)^{1+m} (a+b \text {arccosh}(c x))}{3 d f \left (d-c^2 d x^2\right )^{3/2}}+\frac {(2-m) (f x)^{1+m} (a+b \text {arccosh}(c x))}{3 d^2 f \sqrt {d-c^2 d x^2}}-\frac {(2-m) m (f x)^{1+m} \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{3 d^2 f (1+m) \sqrt {d-c^2 d x^2}}+\frac {b c (2-m) (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{3 d^2 f^2 (2+m) \sqrt {d-c^2 d x^2}}+\frac {b c (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} \operatorname {Hypergeometric2F1}\left (2,\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{3 d^2 f^2 (2+m) \sqrt {d-c^2 d x^2}}-\frac {b c (2-m) m (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )}{3 d^2 f^2 (1+m) (2+m) \sqrt {d-c^2 d x^2}} \]

[Out]

1/3*(f*x)^(1+m)*(a+b*arccosh(c*x))/d/f/(-c^2*d*x^2+d)^(3/2)+1/3*(2-m)*(f*x)^(1+m)*(a+b*arccosh(c*x))/d^2/f/(-c
^2*d*x^2+d)^(1/2)+1/3*b*c*(2-m)*(f*x)^(2+m)*hypergeom([1, 1+1/2*m],[2+1/2*m],c^2*x^2)*(c*x-1)^(1/2)*(c*x+1)^(1
/2)/d^2/f^2/(2+m)/(-c^2*d*x^2+d)^(1/2)+1/3*b*c*(f*x)^(2+m)*hypergeom([2, 1+1/2*m],[2+1/2*m],c^2*x^2)*(c*x-1)^(
1/2)*(c*x+1)^(1/2)/d^2/f^2/(2+m)/(-c^2*d*x^2+d)^(1/2)-1/3*b*c*(2-m)*m*(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*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/f^2/(1+m)/(2+m)/(-c^2*d*x^2+d)^(1/2)-1/3*(2-
m)*m*(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^2*x^2+1)^(1/2)/d^2/f/(
1+m)/(-c^2*d*x^2+d)^(1/2)

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {5936, 5948, 74, 371} \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b c (2-m) m \sqrt {c x-1} \sqrt {c x+1} (f x)^{m+2} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;c^2 x^2\right )}{3 d^2 f^2 (m+1) (m+2) \sqrt {d-c^2 d x^2}}-\frac {(2-m) m \sqrt {1-c^2 x^2} (f x)^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},c^2 x^2\right ) (a+b \text {arccosh}(c x))}{3 d^2 f (m+1) \sqrt {d-c^2 d x^2}}+\frac {(2-m) (f x)^{m+1} (a+b \text {arccosh}(c x))}{3 d^2 f \sqrt {d-c^2 d x^2}}+\frac {(f x)^{m+1} (a+b \text {arccosh}(c x))}{3 d f \left (d-c^2 d x^2\right )^{3/2}}+\frac {b c (2-m) \sqrt {c x-1} \sqrt {c x+1} (f x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{3 d^2 f^2 (m+2) \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} (f x)^{m+2} \operatorname {Hypergeometric2F1}\left (2,\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{3 d^2 f^2 (m+2) \sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

((f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(3*d*f*(d - c^2*d*x^2)^(3/2)) + ((2 - m)*(f*x)^(1 + m)*(a + b*ArcCosh[c*x
]))/(3*d^2*f*Sqrt[d - c^2*d*x^2]) - ((2 - m)*m*(f*x)^(1 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeome
tric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(3*d^2*f*(1 + m)*Sqrt[d - c^2*d*x^2]) + (b*c*(2 - m)*(f*x)^(2 + m
)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, c^2*x^2])/(3*d^2*f^2*(2 + m)*Sqrt[d
- c^2*d*x^2]) + (b*c*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Hypergeometric2F1[2, (2 + m)/2, (4 + m)/2, c^2
*x^2])/(3*d^2*f^2*(2 + m)*Sqrt[d - c^2*d*x^2]) - (b*c*(2 - m)*m*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Hyp
ergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(3*d^2*f^2*(1 + m)*(2 + m)*Sqrt[d - c^2*
d*x^2])

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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 5936

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
(-(f*x)^(m + 1))*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*d*f*(p + 1))), x] + (Dist[(m + 2*p + 3)/(2*d*(
p + 1)), Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[b*c*(n/(2*f*(p + 1)))*Simp[(d +
e*x^2)^p/((1 + c*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*ArcCo
sh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] &
&  !GtQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])

Rule 5948

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((f*x
)^(m + 1)/(f*(m + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(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[-1 +
c*x]/Sqrt[d + e*x^2])]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2], x] /; FreeQ[{
a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] &&  !IntegerQ[m]

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

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.71 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {x (f x)^m \sqrt {-1+c x} \sqrt {1+c x} \left (-\frac {a+b \text {arccosh}(c x)}{(-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b c x \operatorname {Hypergeometric2F1}\left (2,1+\frac {m}{2},2+\frac {m}{2},c^2 x^2\right )}{2+m}+\frac {(-2+m) \left (m (2+m) \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )-(1+m) \left ((2+m) (a+b \text {arccosh}(c x))+b c x \sqrt {-1+c x} \sqrt {1+c x} \operatorname {Hypergeometric2F1}\left (1,1+\frac {m}{2},2+\frac {m}{2},c^2 x^2\right )\right )+b c m x \sqrt {-1+c x} \sqrt {1+c x} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )\right )}{(1+m) (2+m) \sqrt {-1+c x} \sqrt {1+c x}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

(x*(f*x)^m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/((-1 + c*x)^(3/2)*(1 + c*x)^(3/2))) + (b*c*x*H
ypergeometric2F1[2, 1 + m/2, 2 + m/2, c^2*x^2])/(2 + m) + ((-2 + m)*(m*(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcCos
h[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2] - (1 + m)*((2 + m)*(a + b*ArcCosh[c*x]) + b*c*x*
Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Hypergeometric2F1[1, 1 + m/2, 2 + m/2, c^2*x^2]) + b*c*m*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)*Sqrt[-1 + c*
x]*Sqrt[1 + c*x])))/(3*d^2*Sqrt[d - c^2*d*x^2])

Maple [F]

\[\int \frac {\left (f x \right )^{m} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (f\,x\right )}^m}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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