Integrand size = 35, antiderivative size = 336 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{(\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}} \, dx=\frac {(f x)^{1+m} (a+b \text {arccosh}(c x))}{\text {d1} \text {d2} f \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}-\frac {m (f x)^{1+m} \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{\text {d1} \text {d2} f (1+m) \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}+\frac {b c (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{\text {d1} \text {d2} f^2 (2+m) \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}-\frac {b c m (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )}{\text {d1} \text {d2} f^2 (1+m) (2+m) \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}} \]
(f*x)^(1+m)*(a+b*arccosh(c*x))/d1/d2/f/(c*d1*x+d1)^(1/2)/(-c*d2*x+d2)^(1/2 )+b*c*(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)/d1/d2/f^2/(2+m)/(c*d1*x+d1)^(1/2)/(-c*d2*x+d2)^(1/2)-b*c*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)/d1/d2/f^2/(1+m)/(2+m)/(c*d1*x+d1)^(1/2)/(-c*d2* x+d2)^(1/2)-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)/d1/d2/f/(1+m)/(c*d1*x+d1)^(1/2)/(-c* d2*x+d2)^(1/2)
Time = 0.32 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.67 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{(\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}} \, dx=\frac {x (f x)^m \left (-m (2+m) \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )+(1+m) \left ((2+m) (a+b \text {arccosh}(c x))+b c x \sqrt {-1+c x} \sqrt {1+c x} \operatorname {Hypergeometric2F1}\left (1,1+\frac {m}{2},2+\frac {m}{2},c^2 x^2\right )\right )-b c m x \sqrt {-1+c x} \sqrt {1+c x} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )\right )}{\text {d1} \text {d2} (1+m) (2+m) \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}} \]
(x*(f*x)^m*(-(m*(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeomet ric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2]) + (1 + m)*((2 + m)*(a + b*ArcC osh[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]*Hypergeometri cPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2]))/(d1*d2*(1 + m )*(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])
Time = 1.19 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6352, 25, 82, 278, 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.
| \(\displaystyle \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{(c \text {d1} x+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6352 |
| \(\displaystyle -\frac {m \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}dx}{\text {d1} \text {d2}}-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \int -\frac {(f x)^{m+1}}{(1-c x) (c x+1)}dx}{\text {d1} \text {d2} f \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}+\frac {(f x)^{m+1} (a+b \text {arccosh}(c x))}{\text {d1} \text {d2} f \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {m \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}dx}{\text {d1} \text {d2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {(f x)^{m+1}}{(1-c x) (c x+1)}dx}{\text {d1} \text {d2} f \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}+\frac {(f x)^{m+1} (a+b \text {arccosh}(c x))}{\text {d1} \text {d2} f \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}\) |
| \(\Big \downarrow \) 82 |
| \(\displaystyle -\frac {m \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}dx}{\text {d1} \text {d2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {(f x)^{m+1}}{1-c^2 x^2}dx}{\text {d1} \text {d2} f \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}+\frac {(f x)^{m+1} (a+b \text {arccosh}(c x))}{\text {d1} \text {d2} f \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle -\frac {m \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}dx}{\text {d1} \text {d2}}+\frac {(f x)^{m+1} (a+b \text {arccosh}(c x))}{\text {d1} \text {d2} f \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} (f x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{\text {d1} \text {d2} f^2 (m+2) \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}\) |
| \(\Big \downarrow \) 6364 |
| \(\displaystyle -\frac {m \left (\frac {b c \sqrt {c x-1} \sqrt {c x+1} (f x)^{m+2} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;c^2 x^2\right )}{f^2 (m+1) (m+2) \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}+\frac {\sqrt {1-c^2 x^2} (f x)^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},c^2 x^2\right ) (a+b \text {arccosh}(c x))}{f (m+1) \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}\right )}{\text {d1} \text {d2}}+\frac {(f x)^{m+1} (a+b \text {arccosh}(c x))}{\text {d1} \text {d2} f \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} (f x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{\text {d1} \text {d2} f^2 (m+2) \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}\) |
((f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(d1*d2*f*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]) + (b*c*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Hypergeometric2 F1[1, (2 + m)/2, (4 + m)/2, c^2*x^2])/(d1*d2*f^2*(2 + m)*Sqrt[d1 + c*d1*x] *Sqrt[d2 - c*d2*x]) - (m*(((f*x)^(1 + 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])/(f*(1 + m)*Sq rt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]) + (b*c*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqr t[1 + c*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)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])))/(d1* d2)
3.2.61.3.1 Defintions of rubi rules used
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] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
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 + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*d1*d 2*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d1*d2*(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/(2*f*(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] && LtQ[p, -1] && ! GtQ[m, 1] && (IntegerQ[m] || EqQ[n, 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]
\[\int \frac {\left (f x \right )^{m} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{\left (c \operatorname {d1} x +\operatorname {d1} \right )^{\frac {3}{2}} \left (-c \operatorname {d2} x +\operatorname {d2} \right )^{\frac {3}{2}}}d x\]
\[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{(\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{{\left (c d_{1} x + d_{1}\right )}^{\frac {3}{2}} {\left (-c d_{2} x + d_{2}\right )}^{\frac {3}{2}}} \,d x } \]
integrate((f*x)^m*(a+b*arccosh(c*x))/(c*d1*x+d1)^(3/2)/(-c*d2*x+d2)^(3/2), x, algorithm="fricas")
integral(sqrt(c*d1*x + d1)*sqrt(-c*d2*x + d2)*(b*arccosh(c*x) + a)*(f*x)^m /(c^4*d1^2*d2^2*x^4 - 2*c^2*d1^2*d2^2*x^2 + d1^2*d2^2), x)
Timed out. \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{(\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{(\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{{\left (c d_{1} x + d_{1}\right )}^{\frac {3}{2}} {\left (-c d_{2} x + d_{2}\right )}^{\frac {3}{2}}} \,d x } \]
integrate((f*x)^m*(a+b*arccosh(c*x))/(c*d1*x+d1)^(3/2)/(-c*d2*x+d2)^(3/2), x, algorithm="maxima")
\[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{(\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{{\left (c d_{1} x + d_{1}\right )}^{\frac {3}{2}} {\left (-c d_{2} x + d_{2}\right )}^{\frac {3}{2}}} \,d x } \]
integrate((f*x)^m*(a+b*arccosh(c*x))/(c*d1*x+d1)^(3/2)/(-c*d2*x+d2)^(3/2), x, algorithm="giac")
Timed out. \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{(\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (f\,x\right )}^m}{{\left (d_{1}+c\,d_{1}\,x\right )}^{3/2}\,{\left (d_{2}-c\,d_{2}\,x\right )}^{3/2}} \,d x \]