3.3.0 ∫ (fx)m(d − c2dx2)5∕2(a + barccosh(cx))2 dx [233]

Integrand size = 31, antiderivative size = 31 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {10 b^2 c^2 d^2 (f x)^{3+m} \sqrt {d-c^2 d x^2}}{f^3 (4+m)^3 (6+m)}-\frac {2 b^2 c^2 d^2 \left (52+15 m+m^2\right ) (f x)^{3+m} \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{f^3 (4+m)^2 (6+m)^3 (1-c x) (1+c x)}+\frac {2 b^2 c^4 d^2 (f x)^{5+m} \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{f^5 (6+m)^3 (1-c x) (1+c x)}-\frac {2 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {30 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m)^2 (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {10 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m) (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {10 b c^3 d^2 (f x)^{4+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^4 (4+m)^2 (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b c^3 d^2 (f x)^{4+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^4 (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 (f x)^{6+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^6 (6+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {15 d^2 (f x)^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{f (6+m) \left (8+6 m+m^2\right )}+\frac {5 d (f x)^{1+m} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{f (4+m) (6+m)}+\frac {(f x)^{1+m} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{f (6+m)}-\frac {30 b^2 c^2 d^2 (f x)^{3+m} \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{f^3 (2+m)^2 (3+m) (4+m) (6+m) (1-c x) (1+c x)}-\frac {10 b^2 c^2 d^2 (10+3 m) (f x)^{3+m} \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{f^3 (2+m) (3+m) (4+m)^3 (6+m) (1-c x) (1+c x)}-\frac {2 b^2 c^2 d^2 \left (264+130 m+15 m^2\right ) (f x)^{3+m} \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{f^3 (2+m) (3+m) (4+m)^2 (6+m)^3 (1-c x) (1+c x)}+\frac {15 d^3 \text {Int}\left (\frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}},x\right )}{(6+m) \left (8+6 m+m^2\right )} \]

5*d*(f*x)^(1+m)*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2/f/(4+m)/(6+m)+(f*x)^(1+m)*(-c^2*d*x^2+d)^(5/2)*(a+b*

arccosh(c*x))^2/f/(6+m)-10*b^2*c^2*d^2*(f*x)^(3+m)*(-c^2*d*x^2+d)^(1/2)/f^3/(4+m)^3/(6+m)-2*b^2*c^2*d^2*(m^2+1

5*m+52)*(f*x)^(3+m)*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/f^3/(4+m)^2/(6+m)^3/(-c*x+1)/(c*x+1)+2*b^2*c^4*d^2*(f*x)

^(5+m)*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/f^5/(6+m)^3/(-c*x+1)/(c*x+1)+15*d^2*(f*x)^(1+m)*(a+b*arccosh(c*x))^2*

(-c^2*d*x^2+d)^(1/2)/f/(6+m)/(m^2+6*m+8)-2*b*c*d^2*(f*x)^(2+m)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/f^2/(2+

m)/(6+m)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-30*b*c*d^2*(f*x)^(2+m)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/f^2/(2+m)^

2/(4+m)/(6+m)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-10*b*c*d^2*(f*x)^(2+m)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/f^2/(

6+m)/(m^2+6*m+8)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+10*b*c^3*d^2*(f*x)^(4+m)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/

f^4/(4+m)^2/(6+m)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+4*b*c^3*d^2*(f*x)^(4+m)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/

f^4/(4+m)/(6+m)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2*b*c^5*d^2*(f*x)^(6+m)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/f^

6/(6+m)^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-30*b^2*c^2*d^2*(f*x)^(3+m)*hypergeom([1/2, 3/2+1/2*m],[5/2+1/2*m],c^2*x^

2)*(-c^2*x^2+1)^(1/2)*(-c^2*d*x^2+d)^(1/2)/f^3/(2+m)^2/(3+m)/(4+m)/(6+m)/(-c*x+1)/(c*x+1)-10*b^2*c^2*d^2*(10+3

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

+m)^3/(6+m)/(m^2+5*m+6)/(-c*x+1)/(c*x+1)-2*b^2*c^2*d^2*(15*m^2+130*m+264)*(f*x)^(3+m)*hypergeom([1/2, 3/2+1/2*

m],[5/2+1/2*m],c^2*x^2)*(-c^2*x^2+1)^(1/2)*(-c^2*d*x^2+d)^(1/2)/f^3/(4+m)^2/(6+m)^3/(m^2+5*m+6)/(-c*x+1)/(c*x+

1)+15*d^3*Unintegrable((f*x)^m*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(1/2),x)/(6+m)/(m^2+6*m+8)

Rubi [N/A]

Time = 1.48 (sec) , antiderivative size = 31, 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 = {} \[ \int (f x)^m \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\int (f x)^m \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx \]

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

(-10*b^2*c^2*d^2*(f*x)^(3 + m)*Sqrt[d - c^2*d*x^2])/(f^3*(4 + m)^3*(6 + m)) - (2*b^2*c^2*d^2*(52 + 15*m + m^2)

*(f*x)^(3 + m)*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(f^3*(4 + m)^2*(6 + m)^3*(1 - c*x)*(1 + c*x)) + (2*b^2*c^4*d

^2*(f*x)^(5 + m)*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(f^5*(6 + m)^3*(1 - c*x)*(1 + c*x)) - (2*b*c*d^2*(f*x)^(2

+ m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^2*(2 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (30*b*c*d^

2*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^2*(2 + m)^2*(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1

+ c*x]) - (10*b*c*d^2*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^2*(2 + m)*(4 + m)*(6 + m)*Sq

rt[-1 + c*x]*Sqrt[1 + c*x]) + (10*b*c^3*d^2*(f*x)^(4 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^4*(4 +

m)^2*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*b*c^3*d^2*(f*x)^(4 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x

]))/(f^4*(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c^5*d^2*(f*x)^(6 + m)*Sqrt[d - c^2*d*x^2]*(a + b

*ArcCosh[c*x]))/(f^6*(6 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (15*d^2*(f*x)^(1 + m)*Sqrt[d - c^2*d*x^2]*(a +

b*ArcCosh[c*x])^2)/(f*(6 + m)*(8 + 6*m + m^2)) + (5*d*(f*x)^(1 + m)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])

^2)/(f*(4 + m)*(6 + m)) + ((f*x)^(1 + m)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/(f*(6 + m)) - (30*b^2*c

^2*d^2*(f*x)^(3 + m)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^

2])/(f^3*(2 + m)^2*(3 + m)*(4 + m)*(6 + m)*(1 - c*x)*(1 + c*x)) - (10*b^2*c^2*d^2*(10 + 3*m)*(f*x)^(3 + m)*Sqr

t[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/(f^3*(2 + m)*(3 + m)

*(4 + m)^3*(6 + m)*(1 - c*x)*(1 + c*x)) - (2*b^2*c^2*d^2*(264 + 130*m + 15*m^2)*(f*x)^(3 + m)*Sqrt[1 - c^2*x^2

]*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/(f^3*(2 + m)*(3 + m)*(4 + m)^2*(6

+ m)^3*(1 - c*x)*(1 + c*x)) + (15*d^3*Defer[Int][((f*x)^m*(a + b*ArcCosh[c*x])^2)/Sqrt[d - c^2*d*x^2], x])/((

Rubi steps \begin{align*} \text {integral}& = \frac {(f x)^{1+m} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{f (6+m)}+\frac {(5 d) \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx}{6+m}-\frac {\left (2 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int (f x)^{1+m} (-1+c x)^2 (1+c x)^2 (a+b \text {arccosh}(c x)) \, dx}{f (6+m) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {5 d (f x)^{1+m} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{f (4+m) (6+m)}+\frac {(f x)^{1+m} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{f (6+m)}+\frac {\left (15 d^2\right ) \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx}{(4+m) (6+m)}-\frac {\left (2 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int (f x)^{1+m} \left (-1+c^2 x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx}{f (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (10 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int (f x)^{1+m} (-1+c x) (1+c x) (a+b \text {arccosh}(c x)) \, dx}{f (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {2 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b c^3 d^2 (f x)^{4+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^4 (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 (f x)^{6+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^6 (6+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {15 d^2 (f x)^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{f (2+m) (4+m) (6+m)}+\frac {5 d (f x)^{1+m} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{f (4+m) (6+m)}+\frac {(f x)^{1+m} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{f (6+m)}+\frac {\left (15 d^3\right ) \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx}{(2+m) (4+m) (6+m)}+\frac {\left (2 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(f x)^{2+m} \left (\frac {1}{2+m}-\frac {2 c^2 x^2}{4+m}+\frac {c^4 x^4}{6+m}\right )}{f \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{f (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (10 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int (f x)^{1+m} \left (-1+c^2 x^2\right ) (a+b \text {arccosh}(c x)) \, dx}{f (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (30 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int (f x)^{1+m} (a+b \text {arccosh}(c x)) \, dx}{f (2+m) (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {2 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {30 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m)^2 (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {10 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m) (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {10 b c^3 d^2 (f x)^{4+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^4 (4+m)^2 (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b c^3 d^2 (f x)^{4+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^4 (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 (f x)^{6+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^6 (6+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {15 d^2 (f x)^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{f (2+m) (4+m) (6+m)}+\frac {5 d (f x)^{1+m} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{f (4+m) (6+m)}+\frac {(f x)^{1+m} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{f (6+m)}+\frac {\left (15 d^3\right ) \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx}{(2+m) (4+m) (6+m)}+\frac {\left (2 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(f x)^{2+m} \left (\frac {1}{2+m}-\frac {2 c^2 x^2}{4+m}+\frac {c^4 x^4}{6+m}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{f^2 (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (10 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(f x)^{2+m} \left (-\frac {1}{2+m}+\frac {c^2 x^2}{4+m}\right )}{f \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{f (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (30 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(f x)^{2+m}}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{f^2 (2+m)^2 (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {2 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {30 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m)^2 (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {10 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m) (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {10 b c^3 d^2 (f x)^{4+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^4 (4+m)^2 (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b c^3 d^2 (f x)^{4+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^4 (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 (f x)^{6+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^6 (6+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {15 d^2 (f x)^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{f (2+m) (4+m) (6+m)}+\frac {5 d (f x)^{1+m} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{f (4+m) (6+m)}+\frac {(f x)^{1+m} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{f (6+m)}+\frac {\left (15 d^3\right ) \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx}{(2+m) (4+m) (6+m)}-\frac {\left (10 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(f x)^{2+m} \left (-\frac {1}{2+m}+\frac {c^2 x^2}{4+m}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{f^2 (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b^2 c^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {(f x)^{2+m} \left (\frac {1}{2+m}-\frac {2 c^2 x^2}{4+m}+\frac {c^4 x^4}{6+m}\right )}{\sqrt {-1+c^2 x^2}} \, dx}{f^2 (6+m) (-1+c x) (1+c x)}+\frac {\left (30 b^2 c^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {(f x)^{2+m}}{\sqrt {-1+c^2 x^2}} \, dx}{f^2 (2+m)^2 (4+m) (6+m) (-1+c x) (1+c x)} \\ & = -\frac {10 b^2 c^2 d^2 (f x)^{3+m} \sqrt {d-c^2 d x^2}}{f^3 (4+m)^3 (6+m)}+\frac {2 b^2 c^4 d^2 (f x)^{5+m} \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{f^5 (6+m)^3 (1-c x) (1+c x)}-\frac {2 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {30 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m)^2 (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {10 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m) (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {10 b c^3 d^2 (f x)^{4+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^4 (4+m)^2 (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b c^3 d^2 (f x)^{4+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^4 (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 (f x)^{6+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^6 (6+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {15 d^2 (f x)^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{f (2+m) (4+m) (6+m)}+\frac {5 d (f x)^{1+m} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{f (4+m) (6+m)}+\frac {(f x)^{1+m} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{f (6+m)}+\frac {\left (15 d^3\right ) \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx}{(2+m) (4+m) (6+m)}+\frac {\left (10 b^2 c^2 d^2 (10+3 m) \sqrt {d-c^2 d x^2}\right ) \int \frac {(f x)^{2+m}}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{f^2 (2+m) (4+m)^3 (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (30 b^2 c^2 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {(f x)^{2+m}}{\sqrt {1-c^2 x^2}} \, dx}{f^2 (2+m)^2 (4+m) (6+m) (-1+c x) (1+c x)}+\frac {\left (2 b^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {(f x)^{2+m} \left (\frac {c^2 (6+m)}{2+m}-\frac {c^4 \left (52+15 m+m^2\right ) x^2}{(4+m) (6+m)}\right )}{\sqrt {-1+c^2 x^2}} \, dx}{f^2 (6+m)^2 (-1+c x) (1+c x)} \\ & = -\frac {10 b^2 c^2 d^2 (f x)^{3+m} \sqrt {d-c^2 d x^2}}{f^3 (4+m)^3 (6+m)}-\frac {2 b^2 c^2 d^2 \left (52+15 m+m^2\right ) (f x)^{3+m} \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{f^3 (4+m)^2 (6+m)^3 (1-c x) (1+c x)}+\frac {2 b^2 c^4 d^2 (f x)^{5+m} \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{f^5 (6+m)^3 (1-c x) (1+c x)}-\frac {2 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {30 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m)^2 (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {10 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m) (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {10 b c^3 d^2 (f x)^{4+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^4 (4+m)^2 (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b c^3 d^2 (f x)^{4+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^4 (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 (f x)^{6+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^6 (6+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {15 d^2 (f x)^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{f (2+m) (4+m) (6+m)}+\frac {5 d (f x)^{1+m} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{f (4+m) (6+m)}+\frac {(f x)^{1+m} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{f (6+m)}-\frac {30 b^2 c^2 d^2 (f x)^{3+m} \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{f^3 (2+m)^2 (3+m) (4+m) (6+m) (1-c x) (1+c x)}+\frac {\left (15 d^3\right ) \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx}{(2+m) (4+m) (6+m)}+\frac {\left (10 b^2 c^2 d^2 (10+3 m) \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {(f x)^{2+m}}{\sqrt {-1+c^2 x^2}} \, dx}{f^2 (2+m) (4+m)^3 (6+m) (-1+c x) (1+c x)}+\frac {\left (2 b^2 c^2 d^2 \left (264+130 m+15 m^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {(f x)^{2+m}}{\sqrt {-1+c^2 x^2}} \, dx}{f^2 (2+m) (4+m)^2 (6+m)^3 (-1+c x) (1+c x)} \\ & = -\frac {10 b^2 c^2 d^2 (f x)^{3+m} \sqrt {d-c^2 d x^2}}{f^3 (4+m)^3 (6+m)}-\frac {2 b^2 c^2 d^2 \left (52+15 m+m^2\right ) (f x)^{3+m} \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{f^3 (4+m)^2 (6+m)^3 (1-c x) (1+c x)}+\frac {2 b^2 c^4 d^2 (f x)^{5+m} \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{f^5 (6+m)^3 (1-c x) (1+c x)}-\frac {2 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {30 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m)^2 (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {10 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m) (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {10 b c^3 d^2 (f x)^{4+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^4 (4+m)^2 (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b c^3 d^2 (f x)^{4+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^4 (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 (f x)^{6+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^6 (6+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {15 d^2 (f x)^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{f (2+m) (4+m) (6+m)}+\frac {5 d (f x)^{1+m} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{f (4+m) (6+m)}+\frac {(f x)^{1+m} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{f (6+m)}-\frac {30 b^2 c^2 d^2 (f x)^{3+m} \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{f^3 (2+m)^2 (3+m) (4+m) (6+m) (1-c x) (1+c x)}+\frac {\left (15 d^3\right ) \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx}{(2+m) (4+m) (6+m)}+\frac {\left (10 b^2 c^2 d^2 (10+3 m) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {(f x)^{2+m}}{\sqrt {1-c^2 x^2}} \, dx}{f^2 (2+m) (4+m)^3 (6+m) (-1+c x) (1+c x)}+\frac {\left (2 b^2 c^2 d^2 \left (264+130 m+15 m^2\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {(f x)^{2+m}}{\sqrt {1-c^2 x^2}} \, dx}{f^2 (2+m) (4+m)^2 (6+m)^3 (-1+c x) (1+c x)} \\ & = -\frac {10 b^2 c^2 d^2 (f x)^{3+m} \sqrt {d-c^2 d x^2}}{f^3 (4+m)^3 (6+m)}-\frac {2 b^2 c^2 d^2 \left (52+15 m+m^2\right ) (f x)^{3+m} \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{f^3 (4+m)^2 (6+m)^3 (1-c x) (1+c x)}+\frac {2 b^2 c^4 d^2 (f x)^{5+m} \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{f^5 (6+m)^3 (1-c x) (1+c x)}-\frac {2 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {30 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m)^2 (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {10 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m) (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {10 b c^3 d^2 (f x)^{4+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^4 (4+m)^2 (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b c^3 d^2 (f x)^{4+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^4 (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 (f x)^{6+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^6 (6+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {15 d^2 (f x)^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{f (2+m) (4+m) (6+m)}+\frac {5 d (f x)^{1+m} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{f (4+m) (6+m)}+\frac {(f x)^{1+m} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{f (6+m)}-\frac {30 b^2 c^2 d^2 (f x)^{3+m} \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{f^3 (2+m)^2 (3+m) (4+m) (6+m) (1-c x) (1+c x)}-\frac {10 b^2 c^2 d^2 (10+3 m) (f x)^{3+m} \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{f^3 (2+m) (3+m) (4+m)^3 (6+m) (1-c x) (1+c x)}-\frac {2 b^2 c^2 d^2 \left (264+130 m+15 m^2\right ) (f x)^{3+m} \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{f^3 (2+m) (3+m) (4+m)^2 (6+m)^3 (1-c x) (1+c x)}+\frac {\left (15 d^3\right ) \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx}{(2+m) (4+m) (6+m)} \\ \end{align*}

Mathematica [N/A]

Time = 2.23 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\int (f x)^m \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx \]

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

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

Maple [N/A] (veriﬁed)

Time = 2.79 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94

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

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

Fricas [N/A]

Time = 0.26 (sec) , antiderivative size = 136, normalized size of antiderivative = 4.39 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \left (f x\right )^{m} \,d x } \]

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

integral((a^2*c^4*d^2*x^4 - 2*a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^4*d^2*x^4 - 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arcc

osh(c*x)^2 + 2*(a*b*c^4*d^2*x^4 - 2*a*b*c^2*d^2*x^2 + a*b*d^2)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d)*(f*x)^m, x)

Sympy [F(-1)]

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

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

Maxima [N/A]

Time = 0.42 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \left (f x\right )^{m} \,d x } \]

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

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

Giac [F(-2)]

Exception generated. \[ \int (f x)^m \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]

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

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP

UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [N/A]

Time = 3.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}\,{\left (f\,x\right )}^m \,d x \]

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

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

\(\int (f x)^m (d-c^2 d x^2)^{5/2} (a+b \text {arccosh}(c x))^2 \, dx\) [233]

Optimal result