Integrand size = 35, antiderivative size = 817 \[ \int (f x)^m (\text {d1}+c \text {d1} x)^{5/2} (\text {d2}-c \text {d2} x)^{5/2} (a+b \text {arccosh}(c x)) \, dx=-\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} (a+b \text {arccosh}(c x))}{f (6+m) \left (8+6 m+m^2\right )}+\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} (a+b \text {arccosh}(c x))}{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} (a+b \text {arccosh}(c x))}{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} (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) (6+m) \left (2+3 m+m^2\right ) \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} \, _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}} \]
5*d1*d2*(f*x)^(1+m)*(c*d1*x+d1)^(3/2)*(-c*d2*x+d2)^(3/2)*(a+b*arccosh(c*x) )/f/(4+m)/(6+m)+(f*x)^(1+m)*(c*d1*x+d1)^(5/2)*(-c*d2*x+d2)^(5/2)*(a+b*arcc osh(c*x))/f/(6+m)+15*d1^2*d2^2*(f*x)^(1+m)*(a+b*arccosh(c*x))*(c*d1*x+d1)^ (1/2)*(-c*d2*x+d2)^(1/2)/f/(6+m)/(m^2+6*m+8)+15*d1^2*d2^2*(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/(6+m)/(m^3+7*m^2+14*m+8)/(-c*x+1)^(1/2)/(c*x+1 )^(1/2)-b*c*d1^2*d2^2*(f*x)^(2+m)*(c*d1*x+d1)^(1/2)*(-c*d2*x+d2)^(1/2)/f^2 /(2+m)/(6+m)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-15*b*c*d1^2*d2^2*(f*x)^(2+m)*(c*d 1*x+d1)^(1/2)*(-c*d2*x+d2)^(1/2)/f^2/(2+m)^2/(4+m)/(6+m)/(c*x-1)^(1/2)/(c* x+1)^(1/2)-5*b*c*d1^2*d2^2*(f*x)^(2+m)*(c*d1*x+d1)^(1/2)*(-c*d2*x+d2)^(1/2 )/f^2/(6+m)/(m^2+6*m+8)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+5*b*c^3*d1^2*d2^2*(f*x )^(4+m)*(c*d1*x+d1)^(1/2)*(-c*d2*x+d2)^(1/2)/f^4/(4+m)^2/(6+m)/(c*x-1)^(1/ 2)/(c*x+1)^(1/2)+2*b*c^3*d1^2*d2^2*(f*x)^(4+m)*(c*d1*x+d1)^(1/2)*(-c*d2*x+ d2)^(1/2)/f^4/(4+m)/(6+m)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-b*c^5*d1^2*d2^2*(f*x )^(6+m)*(c*d1*x+d1)^(1/2)*(-c*d2*x+d2)^(1/2)/f^6/(6+m)^2/(c*x-1)^(1/2)/(c* x+1)^(1/2)-15*b*c*d1^2*d2^2*(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/(6+m)/(m^2+5*m+4)/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 1.65 (sec) , antiderivative size = 387, normalized size of antiderivative = 0.47 \[ \int (f x)^m (\text {d1}+c \text {d1} x)^{5/2} (\text {d2}-c \text {d2} x)^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {\text {d1}^2 \text {d2}^2 x (f x)^m \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x} \left (-\frac {b c x \left (\frac {1}{2+m}-\frac {2 c^2 x^2}{4+m}+\frac {c^4 x^4}{6+m}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\left (-1+c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {5 \left (\frac {b c x \left (-\frac {1}{2+m}+\frac {c^2 x^2}{4+m}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-(-1+c x) (1+c x) (a+b \text {arccosh}(c x))+\frac {3 \left ((1+m) \left (-b c x \sqrt {-1+c x} \sqrt {1+c x}+a (2+m) \left (-1+c^2 x^2\right )+b (2+m) \left (-1+c^2 x^2\right ) \text {arccosh}(c x)\right )-(2+m) \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )-b c x \sqrt {-1+c x} \sqrt {1+c x} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )\right )}{(1+m) (2+m)^2 (-1+c x) (1+c x)}\right )}{4+m}\right )}{6+m} \]
(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]*Hypergeom etricPFQ[{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)
Time = 2.60 (sec) , antiderivative size = 596, normalized size of antiderivative = 0.73, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.343, Rules used = {6346, 82, 244, 2009, 6346, 25, 82, 244, 2009, 6342, 17, 6364}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (c \text {d1} x+\text {d1})^{5/2} (\text {d2}-c \text {d2} x)^{5/2} (f x)^m (a+b \text {arccosh}(c x)) \, dx\) |
\(\Big \downarrow \) 6346 |
\(\displaystyle \frac {5 \text {d1} \text {d2} \int (f x)^m (c x \text {d1}+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x))dx}{m+6}-\frac {b c \text {d1}^2 \text {d2}^2 \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \int (f x)^{m+1} (1-c x)^2 (c x+1)^2dx}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}+\frac {(c \text {d1} x+\text {d1})^{5/2} (\text {d2}-c \text {d2} x)^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+6)}\) |
\(\Big \downarrow \) 82 |
\(\displaystyle \frac {5 \text {d1} \text {d2} \int (f x)^m (c x \text {d1}+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x))dx}{m+6}-\frac {b c \text {d1}^2 \text {d2}^2 \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \int (f x)^{m+1} \left (1-c^2 x^2\right )^2dx}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}+\frac {(c \text {d1} x+\text {d1})^{5/2} (\text {d2}-c \text {d2} x)^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+6)}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {5 \text {d1} \text {d2} \int (f x)^m (c x \text {d1}+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x))dx}{m+6}-\frac {b c \text {d1}^2 \text {d2}^2 \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \int \left ((f x)^{m+1}-\frac {2 c^2 (f x)^{m+3}}{f^2}+\frac {c^4 (f x)^{m+5}}{f^4}\right )dx}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}+\frac {(c \text {d1} x+\text {d1})^{5/2} (\text {d2}-c \text {d2} x)^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+6)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {5 \text {d1} \text {d2} \int (f x)^m (c x \text {d1}+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x))dx}{m+6}+\frac {(c \text {d1} x+\text {d1})^{5/2} (\text {d2}-c \text {d2} x)^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+6)}-\frac {b c \text {d1}^2 \text {d2}^2 \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \left (\frac {c^4 (f x)^{m+6}}{f^5 (m+6)}-\frac {2 c^2 (f x)^{m+4}}{f^3 (m+4)}+\frac {(f x)^{m+2}}{f (m+2)}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 6346 |
\(\displaystyle \frac {5 \text {d1} \text {d2} \left (\frac {3 \text {d1} \text {d2} \int (f x)^m \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (a+b \text {arccosh}(c x))dx}{m+4}+\frac {b c \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \int -(f x)^{m+1} (1-c x) (c x+1)dx}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {(c \text {d1} x+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+4)}\right )}{m+6}+\frac {(c \text {d1} x+\text {d1})^{5/2} (\text {d2}-c \text {d2} x)^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+6)}-\frac {b c \text {d1}^2 \text {d2}^2 \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \left (\frac {c^4 (f x)^{m+6}}{f^5 (m+6)}-\frac {2 c^2 (f x)^{m+4}}{f^3 (m+4)}+\frac {(f x)^{m+2}}{f (m+2)}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {5 \text {d1} \text {d2} \left (\frac {3 \text {d1} \text {d2} \int (f x)^m \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (a+b \text {arccosh}(c x))dx}{m+4}-\frac {b c \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \int (f x)^{m+1} (1-c x) (c x+1)dx}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {(c \text {d1} x+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+4)}\right )}{m+6}+\frac {(c \text {d1} x+\text {d1})^{5/2} (\text {d2}-c \text {d2} x)^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+6)}-\frac {b c \text {d1}^2 \text {d2}^2 \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \left (\frac {c^4 (f x)^{m+6}}{f^5 (m+6)}-\frac {2 c^2 (f x)^{m+4}}{f^3 (m+4)}+\frac {(f x)^{m+2}}{f (m+2)}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 82 |
\(\displaystyle \frac {5 \text {d1} \text {d2} \left (\frac {3 \text {d1} \text {d2} \int (f x)^m \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (a+b \text {arccosh}(c x))dx}{m+4}-\frac {b c \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \int (f x)^{m+1} \left (1-c^2 x^2\right )dx}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {(c \text {d1} x+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+4)}\right )}{m+6}+\frac {(c \text {d1} x+\text {d1})^{5/2} (\text {d2}-c \text {d2} x)^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+6)}-\frac {b c \text {d1}^2 \text {d2}^2 \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \left (\frac {c^4 (f x)^{m+6}}{f^5 (m+6)}-\frac {2 c^2 (f x)^{m+4}}{f^3 (m+4)}+\frac {(f x)^{m+2}}{f (m+2)}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {5 \text {d1} \text {d2} \left (\frac {3 \text {d1} \text {d2} \int (f x)^m \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (a+b \text {arccosh}(c x))dx}{m+4}-\frac {b c \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \int \left ((f x)^{m+1}-\frac {c^2 (f x)^{m+3}}{f^2}\right )dx}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {(c \text {d1} x+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+4)}\right )}{m+6}+\frac {(c \text {d1} x+\text {d1})^{5/2} (\text {d2}-c \text {d2} x)^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+6)}-\frac {b c \text {d1}^2 \text {d2}^2 \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \left (\frac {c^4 (f x)^{m+6}}{f^5 (m+6)}-\frac {2 c^2 (f x)^{m+4}}{f^3 (m+4)}+\frac {(f x)^{m+2}}{f (m+2)}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {5 \text {d1} \text {d2} \left (\frac {3 \text {d1} \text {d2} \int (f x)^m \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (a+b \text {arccosh}(c x))dx}{m+4}+\frac {(c \text {d1} x+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+4)}-\frac {b c \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \left (\frac {(f x)^{m+2}}{f (m+2)}-\frac {c^2 (f x)^{m+4}}{f^3 (m+4)}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}\right )}{m+6}+\frac {(c \text {d1} x+\text {d1})^{5/2} (\text {d2}-c \text {d2} x)^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+6)}-\frac {b c \text {d1}^2 \text {d2}^2 \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \left (\frac {c^4 (f x)^{m+6}}{f^5 (m+6)}-\frac {2 c^2 (f x)^{m+4}}{f^3 (m+4)}+\frac {(f x)^{m+2}}{f (m+2)}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 6342 |
\(\displaystyle \frac {5 \text {d1} \text {d2} \left (\frac {3 \text {d1} \text {d2} \left (-\frac {\sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{(m+2) \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \int (f x)^{m+1}dx}{f (m+2) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+2)}\right )}{m+4}+\frac {(c \text {d1} x+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+4)}-\frac {b c \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \left (\frac {(f x)^{m+2}}{f (m+2)}-\frac {c^2 (f x)^{m+4}}{f^3 (m+4)}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}\right )}{m+6}+\frac {(c \text {d1} x+\text {d1})^{5/2} (\text {d2}-c \text {d2} x)^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+6)}-\frac {b c \text {d1}^2 \text {d2}^2 \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \left (\frac {c^4 (f x)^{m+6}}{f^5 (m+6)}-\frac {2 c^2 (f x)^{m+4}}{f^3 (m+4)}+\frac {(f x)^{m+2}}{f (m+2)}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 17 |
\(\displaystyle \frac {5 \text {d1} \text {d2} \left (\frac {3 \text {d1} \text {d2} \left (-\frac {\sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{(m+2) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+2}}{f^2 (m+2)^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{m+4}+\frac {(c \text {d1} x+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+4)}-\frac {b c \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \left (\frac {(f x)^{m+2}}{f (m+2)}-\frac {c^2 (f x)^{m+4}}{f^3 (m+4)}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}\right )}{m+6}+\frac {(c \text {d1} x+\text {d1})^{5/2} (\text {d2}-c \text {d2} x)^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+6)}-\frac {b c \text {d1}^2 \text {d2}^2 \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \left (\frac {c^4 (f x)^{m+6}}{f^5 (m+6)}-\frac {2 c^2 (f x)^{m+4}}{f^3 (m+4)}+\frac {(f x)^{m+2}}{f (m+2)}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 6364 |
\(\displaystyle \frac {5 \text {d1} \text {d2} \left (\frac {3 \text {d1} \text {d2} \left (-\frac {\sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \left (\frac {b c (f x)^{m+2} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;c^2 x^2\right )}{f^2 (m+1) (m+2)}+\frac {\sqrt {1-c x} (f x)^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},c^2 x^2\right ) (a+b \text {arccosh}(c x))}{f (m+1) \sqrt {c x-1}}\right )}{(m+2) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+2}}{f^2 (m+2)^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{m+4}+\frac {(c \text {d1} x+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+4)}-\frac {b c \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \left (\frac {(f x)^{m+2}}{f (m+2)}-\frac {c^2 (f x)^{m+4}}{f^3 (m+4)}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}\right )}{m+6}+\frac {(c \text {d1} x+\text {d1})^{5/2} (\text {d2}-c \text {d2} x)^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+6)}-\frac {b c \text {d1}^2 \text {d2}^2 \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \left (\frac {c^4 (f x)^{m+6}}{f^5 (m+6)}-\frac {2 c^2 (f x)^{m+4}}{f^3 (m+4)}+\frac {(f x)^{m+2}}{f (m+2)}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}\) |
-((b*c*d1^2*d2^2*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*((f*x)^(2 + m)/(f*(2 + m)) - (2*c^2*(f*x)^(4 + m))/(f^3*(4 + m)) + (c^4*(f*x)^(6 + m))/(f^5*(6 + m))))/(f*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + ((f*x)^(1 + m)*(d1 + c *d1*x)^(5/2)*(d2 - c*d2*x)^(5/2)*(a + b*ArcCosh[c*x]))/(f*(6 + m)) + (5*d1 *d2*(-((b*c*d1*d2*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*((f*x)^(2 + m)/(f*(2 + m)) - (c^2*(f*x)^(4 + m))/(f^3*(4 + m))))/(f*(4 + m)*Sqrt[-1 + c*x]*Sqr t[1 + c*x])) + ((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*(-((b*c*(f*x)^(2 + m)*Sqrt[d1 + c *d1*x]*Sqrt[d2 - c*d2*x])/(f^2*(2 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + ((f*x)^(1 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*(a + b*ArcCosh[c*x]))/( f*(2 + m)) - (Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*(((f*x)^(1 + m)*Sqrt[1 - c*x]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^ 2*x^2])/(f*(1 + m)*Sqrt[-1 + c*x]) + (b*c*(f*x)^(2 + m)*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 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/(4 + m)))/(6 + m)
3.2.57.3.1 Defintions of rubi rules used
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_) )^(p_.), x_] :> 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] && IntegerQ[m]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
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)*S qrt[d1 + e1*x]*Sqrt[d2 + e2*x]*((a + b*ArcCosh[c*x])^n/(f*(m + 2))), x] + ( -Simp[(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] - Simp[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*ArcC osh[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 ])
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 1_.)*(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] + (Simp[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] - Simp[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]
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]
\[\int \left (f x \right )^{m} \left (c \operatorname {d1} x +\operatorname {d1} \right )^{\frac {5}{2}} \left (-c \operatorname {d2} x +\operatorname {d2} \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )d x\]
\[ \int (f x)^m (\text {d1}+c \text {d1} x)^{5/2} (\text {d2}-c \text {d2} x)^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (c d_{1} x + d_{1}\right )}^{\frac {5}{2}} {\left (-c d_{2} x + d_{2}\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]
integrate((f*x)^m*(c*d1*x+d1)^(5/2)*(-c*d2*x+d2)^(5/2)*(a+b*arccosh(c*x)), x, algorithm="fricas")
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)
Timed out. \[ \int (f x)^m (\text {d1}+c \text {d1} x)^{5/2} (\text {d2}-c \text {d2} x)^{5/2} (a+b \text {arccosh}(c x)) \, dx=\text {Timed out} \]
\[ \int (f x)^m (\text {d1}+c \text {d1} x)^{5/2} (\text {d2}-c \text {d2} x)^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (c d_{1} x + d_{1}\right )}^{\frac {5}{2}} {\left (-c d_{2} x + d_{2}\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]
integrate((f*x)^m*(c*d1*x+d1)^(5/2)*(-c*d2*x+d2)^(5/2)*(a+b*arccosh(c*x)), x, algorithm="maxima")
\[ \int (f x)^m (\text {d1}+c \text {d1} x)^{5/2} (\text {d2}-c \text {d2} x)^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (c d_{1} x + d_{1}\right )}^{\frac {5}{2}} {\left (-c d_{2} x + d_{2}\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]
integrate((f*x)^m*(c*d1*x+d1)^(5/2)*(-c*d2*x+d2)^(5/2)*(a+b*arccosh(c*x)), x, algorithm="giac")
Timed out. \[ \int (f x)^m (\text {d1}+c \text {d1} x)^{5/2} (\text {d2}-c \text {d2} x)^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (f\,x\right )}^m\,{\left (d_{1}+c\,d_{1}\,x\right )}^{5/2}\,{\left (d_{2}-c\,d_{2}\,x\right )}^{5/2} \,d x \]