\(\int \frac {(f x)^m (a+b \text {arccosh}(c x))}{(d-c^2 d x^2)^3} \, dx\) [150]

Optimal result

Integrand size = 27, antiderivative size = 27 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {(f x)^{1+m} (a+b \text {arccosh}(c x))}{4 d^3 f \left (1-c^2 x^2\right )^2}+\frac {(3-m) (f x)^{1+m} (a+b \text {arccosh}(c x))}{8 d^3 f \left (1-c^2 x^2\right )}-\frac {b c (3-m) (f x)^{2+m} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{8 d^3 f^2 (2+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c (f x)^{2+m} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{4 d^3 f^2 (2+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(1-m) (3-m) \text {Int}\left (\frac {(f x)^m (a+b \text {arccosh}(c x))}{d-c^2 d x^2},x\right )}{8 d^2} \]

[Out]

1/4*(f*x)^(1+m)*(a+b*arccosh(c*x))/d^3/f/(-c^2*x^2+1)^2+1/8*(3-m)*(f*x)^(1+m)*(a+b*arccosh(c*x))/d^3/f/(-c^2*x
^2+1)-1/8*b*c*(3-m)*(f*x)^(2+m)*hypergeom([3/2, 1+1/2*m],[2+1/2*m],c^2*x^2)*(-c^2*x^2+1)^(1/2)/d^3/f^2/(2+m)/(
c*x-1)^(1/2)/(c*x+1)^(1/2)-1/4*b*c*(f*x)^(2+m)*hypergeom([5/2, 1+1/2*m],[2+1/2*m],c^2*x^2)*(-c^2*x^2+1)^(1/2)/
d^3/f^2/(2+m)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/8*(1-m)*(3-m)*Unintegrable((f*x)^m*(a+b*arccosh(c*x))/(-c^2*d*x^2+
d),x)/d^2

Rubi [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx \]

[In]

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

[Out]

((f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(4*d^3*f*(1 - c^2*x^2)^2) + ((3 - m)*(f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/
(8*d^3*f*(1 - c^2*x^2)) - (b*c*(3 - m)*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[3/2, (2 + m)/2, (4 +
m)/2, c^2*x^2])/(8*d^3*f^2*(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2]*Hyperg
eometric2F1[5/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(4*d^3*f^2*(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((1 - m)*(
3 - m)*Defer[Int][((f*x)^m*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2), x])/(8*d^2)

Rubi steps \begin{align*} \text {integral}& = \frac {(f x)^{1+m} (a+b \text {arccosh}(c x))}{4 d^3 f \left (1-c^2 x^2\right )^2}-\frac {(b c) \int \frac {(f x)^{1+m}}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{4 d^3 f}+\frac {(3-m) \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx}{4 d} \\ & = \frac {(f x)^{1+m} (a+b \text {arccosh}(c x))}{4 d^3 f \left (1-c^2 x^2\right )^2}+\frac {(3-m) (f x)^{1+m} (a+b \text {arccosh}(c x))}{8 d^3 f \left (1-c^2 x^2\right )}+\frac {(b c (3-m)) \int \frac {(f x)^{1+m}}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{8 d^3 f}+\frac {((1-m) (3-m)) \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx}{8 d^2}-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {(f x)^{1+m}}{\left (-1+c^2 x^2\right )^{5/2}} \, dx}{4 d^3 f \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {(f x)^{1+m} (a+b \text {arccosh}(c x))}{4 d^3 f \left (1-c^2 x^2\right )^2}+\frac {(3-m) (f x)^{1+m} (a+b \text {arccosh}(c x))}{8 d^3 f \left (1-c^2 x^2\right )}+\frac {((1-m) (3-m)) \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx}{8 d^2}-\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \int \frac {(f x)^{1+m}}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{4 d^3 f \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c (3-m) \sqrt {-1+c^2 x^2}\right ) \int \frac {(f x)^{1+m}}{\left (-1+c^2 x^2\right )^{3/2}} \, dx}{8 d^3 f \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {(f x)^{1+m} (a+b \text {arccosh}(c x))}{4 d^3 f \left (1-c^2 x^2\right )^2}+\frac {(3-m) (f x)^{1+m} (a+b \text {arccosh}(c x))}{8 d^3 f \left (1-c^2 x^2\right )}-\frac {b c (f x)^{2+m} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{4 d^3 f^2 (2+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {((1-m) (3-m)) \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx}{8 d^2}-\frac {\left (b c (3-m) \sqrt {1-c^2 x^2}\right ) \int \frac {(f x)^{1+m}}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{8 d^3 f \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {(f x)^{1+m} (a+b \text {arccosh}(c x))}{4 d^3 f \left (1-c^2 x^2\right )^2}+\frac {(3-m) (f x)^{1+m} (a+b \text {arccosh}(c x))}{8 d^3 f \left (1-c^2 x^2\right )}-\frac {b c (3-m) (f x)^{2+m} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{8 d^3 f^2 (2+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c (f x)^{2+m} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{4 d^3 f^2 (2+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {((1-m) (3-m)) \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx}{8 d^2} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 12.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.11 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]

[In]

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

[Out]

integral(-(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. \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.72 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 3.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (f\,x\right )}^m}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]

[In]

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

[Out]

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