Integrand size = 29, antiderivative size = 454 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {3 b c d (f x)^{2+m} \sqrt {d-c^2 d x^2}}{f^2 (2+m)^2 (4+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d (f x)^{2+m} \sqrt {d-c^2 d x^2}}{f^2 \left (8+6 m+m^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d (f x)^{4+m} \sqrt {d-c^2 d x^2}}{f^4 (4+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 d (f x)^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f \left (8+6 m+m^2\right )}+\frac {(f x)^{1+m} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{f (4+m)}+\frac {3 d^2 (f x)^{1+m} \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{f \left (8+14 m+7 m^2+m^3\right ) \sqrt {d-c^2 d x^2}}+\frac {3 b c d^2 (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )}{f^2 (1+m) (2+m)^2 (4+m) \sqrt {d-c^2 d x^2}} \] Output:
-3*b*c*d*(f*x)^(2+m)*(-c^2*d*x^2+d)^(1/2)/f^2/(2+m)^2/(4+m)/(c*x-1)^(1/2)/ (c*x+1)^(1/2)-b*c*d*(f*x)^(2+m)*(-c^2*d*x^2+d)^(1/2)/f^2/(m^2+6*m+8)/(c*x- 1)^(1/2)/(c*x+1)^(1/2)+b*c^3*d*(f*x)^(4+m)*(-c^2*d*x^2+d)^(1/2)/f^4/(4+m)^ 2/(c*x-1)^(1/2)/(c*x+1)^(1/2)+3*d*(f*x)^(1+m)*(-c^2*d*x^2+d)^(1/2)*(a+b*ar ccosh(c*x))/f/(m^2+6*m+8)+(f*x)^(1+m)*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c* x))/f/(4+m)+3*d^2*(f*x)^(1+m)*(-c^2*x^2+1)^(1/2)*(a+b*arccosh(c*x))*hyperg eom([1/2, 1/2+1/2*m],[3/2+1/2*m],c^2*x^2)/f/(m^3+7*m^2+14*m+8)/(-c^2*d*x^2 +d)^(1/2)+3*b*c*d^2*(f*x)^(2+m)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*hypergeom([1, 1+1/2*m, 1+1/2*m],[2+1/2*m, 3/2+1/2*m],c^2*x^2)/f^2/(1+m)/(2+m)^2/(4+m)/(- c^2*d*x^2+d)^(1/2)
Time = 0.50 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.60 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {d x (f x)^m \sqrt {d-c^2 d x^2} \left (\frac {3 b c x}{(2+m)^2}+b c x \left (\frac {1}{2+m}-\frac {c^2 x^2}{4+m}\right )-\frac {3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{2+m}+(-1+c x)^{3/2} (1+c x)^{3/2} (a+b \text {arccosh}(c x))+\frac {3 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{(1+m) (2+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c x \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )}{(1+m) (2+m)^2}\right )}{(4+m) \sqrt {-1+c x} \sqrt {1+c x}} \] Input:
Integrate[(f*x)^m*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]),x]
Output:
-((d*x*(f*x)^m*Sqrt[d - c^2*d*x^2]*((3*b*c*x)/(2 + m)^2 + b*c*x*((2 + m)^( -1) - (c^2*x^2)/(4 + m)) - (3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[ c*x]))/(2 + m) + (-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]) + ( 3*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/((1 + m)*(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3* b*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)^2)))/((4 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))
Time = 1.17 (sec) , antiderivative size = 396, normalized size of antiderivative = 0.87, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {6345, 25, 82, 244, 2009, 6341, 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.
\(\displaystyle \int \left (d-c^2 d x^2\right )^{3/2} (f x)^m (a+b \text {arccosh}(c x)) \, dx\) |
\(\Big \downarrow \) 6345 |
\(\displaystyle \frac {3 d \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx}{m+4}+\frac {b c d \sqrt {d-c^2 d x^2} \int -(f x)^{m+1} (1-c x) (c x+1)dx}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+4)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {3 d \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx}{m+4}-\frac {b c d \sqrt {d-c^2 d x^2} \int (f x)^{m+1} (1-c x) (c x+1)dx}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+4)}\) |
\(\Big \downarrow \) 82 |
\(\displaystyle \frac {3 d \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx}{m+4}-\frac {b c d \sqrt {d-c^2 d x^2} \int (f x)^{m+1} \left (1-c^2 x^2\right )dx}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+4)}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {3 d \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx}{m+4}-\frac {b c d \sqrt {d-c^2 d x^2} \int \left ((f x)^{m+1}-\frac {c^2 (f x)^{m+3}}{f^2}\right )dx}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+4)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 d \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx}{m+4}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+4)}-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {(f x)^{m+2}}{f (m+2)}-\frac {c^2 (f x)^{m+4}}{f^3 (m+4)}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 6341 |
\(\displaystyle \frac {3 d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{(m+2) \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \int (f x)^{m+1}dx}{f (m+2) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\sqrt {d-c^2 d x^2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+2)}\right )}{m+4}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+4)}-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {(f x)^{m+2}}{f (m+2)}-\frac {c^2 (f x)^{m+4}}{f^3 (m+4)}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 17 |
\(\displaystyle \frac {3 d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{(m+2) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\sqrt {d-c^2 d x^2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \sqrt {d-c^2 d x^2} (f x)^{m+2}}{f^2 (m+2)^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{m+4}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+4)}-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {(f x)^{m+2}}{f (m+2)}-\frac {c^2 (f x)^{m+4}}{f^3 (m+4)}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 6364 |
\(\displaystyle \frac {3 d \left (-\frac {\sqrt {d-c^2 d x^2} \left (\frac {b c (f x)^{m+2} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;c^2 x^2\right )}{f^2 (m+1) (m+2)}+\frac {\sqrt {1-c x} (f x)^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},c^2 x^2\right ) (a+b \text {arccosh}(c x))}{f (m+1) \sqrt {c x-1}}\right )}{(m+2) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\sqrt {d-c^2 d x^2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \sqrt {d-c^2 d x^2} (f x)^{m+2}}{f^2 (m+2)^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{m+4}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+4)}-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {(f x)^{m+2}}{f (m+2)}-\frac {c^2 (f x)^{m+4}}{f^3 (m+4)}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}\) |
Input:
Int[(f*x)^m*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]),x]
Output:
-((b*c*d*Sqrt[d - c^2*d*x^2]*((f*x)^(2 + m)/(f*(2 + m)) - (c^2*(f*x)^(4 + m))/(f^3*(4 + m))))/(f*(4 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + ((f*x)^(1 + m)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(f*(4 + m)) + (3*d*(-((b* c*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2])/(f^2*(2 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + ((f*x)^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f*(2 + m)) - (Sqrt[d - c^2*d*x^2]*(((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)*Sq rt[-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)
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[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc Cosh[c*x])^n/(f*(m + 2))), x] + (-Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/(Sq rt[1 + c*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[d + e* x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])] Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x ])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
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*((a + b*Arc Cosh[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1)) Int[(f* x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*c*(n/(f*(m + 2*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*ArcCosh[c*x])^(n - 1) , x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && 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]
\[\int \left (f x \right )^{m} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )d x\]
Input:
int((f*x)^m*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),x)
Output:
int((f*x)^m*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),x)
\[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \] Input:
integrate((f*x)^m*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),x, algorithm="fr icas")
Output:
integral(-(a*c^2*d*x^2 - a*d + (b*c^2*d*x^2 - b*d)*arccosh(c*x))*sqrt(-c^2 *d*x^2 + d)*(f*x)^m, x)
Timed out. \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\text {Timed out} \] Input:
integrate((f*x)**m*(-c**2*d*x**2+d)**(3/2)*(a+b*acosh(c*x)),x)
Output:
Timed out
\[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \] Input:
integrate((f*x)^m*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),x, algorithm="ma xima")
Output:
integrate((-c^2*d*x^2 + d)^(3/2)*(b*arccosh(c*x) + a)*(f*x)^m, x)
Exception generated. \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \] Input:
integrate((f*x)^m*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),x, algorithm="gi ac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}\,{\left (f\,x\right )}^m \,d x \] Input:
int((a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2)*(f*x)^m,x)
Output:
int((a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2)*(f*x)^m, x)
\[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=f^{m} \sqrt {d}\, d \left (-\left (\int x^{m} \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{2}d x \right ) b \,c^{2}+\left (\int x^{m} \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )d x \right ) b -\left (\int x^{m} \sqrt {-c^{2} x^{2}+1}\, x^{2}d x \right ) a \,c^{2}+\left (\int x^{m} \sqrt {-c^{2} x^{2}+1}d x \right ) a \right ) \] Input:
int((f*x)^m*(-c^2*d*x^2+d)^(3/2)*(a+b*acosh(c*x)),x)
Output:
f**m*sqrt(d)*d*( - int(x**m*sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**2,x)*b*c* *2 + int(x**m*sqrt( - c**2*x**2 + 1)*acosh(c*x),x)*b - int(x**m*sqrt( - c* *2*x**2 + 1)*x**2,x)*a*c**2 + int(x**m*sqrt( - c**2*x**2 + 1),x)*a)