\(\int (c e+d e x)^{3/2} (a+b \text {arccosh}(c+d x)) \, dx\) [200]

Optimal result

Integrand size = 23, antiderivative size = 145 \[ \int (c e+d e x)^{3/2} (a+b \text {arccosh}(c+d x)) \, dx=-\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{3/2} \sqrt {1+c+d x}}{25 d}+\frac {2 (e (c+d x))^{5/2} (a+b \text {arccosh}(c+d x))}{5 d e}-\frac {12 b e \sqrt {1-c-d x} \sqrt {e (c+d x)} E\left (\left .\arcsin \left (\frac {\sqrt {1+c+d x}}{\sqrt {2}}\right )\right |2\right )}{25 d \sqrt {-c-d x} \sqrt {-1+c+d x}} \]

[Out]

2/5*(e*(d*x+c))^(5/2)*(a+b*arccosh(d*x+c))/d/e-12/25*b*e*EllipticE(1/2*(d*x+c+1)^(1/2)*2^(1/2),2^(1/2))*(-d*x-
c+1)^(1/2)*(e*(d*x+c))^(1/2)/d/(-d*x-c)^(1/2)/(d*x+c-1)^(1/2)-4/25*b*(e*(d*x+c))^(3/2)*(d*x+c-1)^(1/2)*(d*x+c+
1)^(1/2)/d

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5996, 5883, 104, 12, 115, 114} \[ \int (c e+d e x)^{3/2} (a+b \text {arccosh}(c+d x)) \, dx=\frac {2 (e (c+d x))^{5/2} (a+b \text {arccosh}(c+d x))}{5 d e}-\frac {12 b e \sqrt {-c-d x+1} \sqrt {e (c+d x)} E\left (\left .\arcsin \left (\frac {\sqrt {c+d x+1}}{\sqrt {2}}\right )\right |2\right )}{25 d \sqrt {-c-d x} \sqrt {c+d x-1}}-\frac {4 b \sqrt {c+d x-1} \sqrt {c+d x+1} (e (c+d x))^{3/2}}{25 d} \]

[In]

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

[Out]

(-4*b*Sqrt[-1 + c + d*x]*(e*(c + d*x))^(3/2)*Sqrt[1 + c + d*x])/(25*d) + (2*(e*(c + d*x))^(5/2)*(a + b*ArcCosh
[c + d*x]))/(5*d*e) - (12*b*e*Sqrt[1 - c - d*x]*Sqrt[e*(c + d*x)]*EllipticE[ArcSin[Sqrt[1 + c + d*x]/Sqrt[2]],
2])/(25*d*Sqrt[-c - d*x]*Sqrt[-1 + c + d*x])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 104

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 1))), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2/b)*Rt[-(b
*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /;
FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !LtQ[-(b*c - a*d)/d, 0] &&
!(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)/b, 0])

Rule 115

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[Sqrt[e + f*x
]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d*x]*Sqrt[b*((e + f*x)/(b*e - a*f))])), Int[Sqrt[b*(e/(b*e - a*f)
) + b*f*(x/(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]), x], x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !(GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0]) &&  !LtQ[-(b*c - a*d)/d, 0]

Rule 5883

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcC
osh[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5996

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (e x)^{3/2} (a+b \text {arccosh}(x)) \, dx,x,c+d x\right )}{d} \\ & = \frac {2 (e (c+d x))^{5/2} (a+b \text {arccosh}(c+d x))}{5 d e}-\frac {(2 b) \text {Subst}\left (\int \frac {(e x)^{5/2}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{5 d e} \\ & = -\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{3/2} \sqrt {1+c+d x}}{25 d}+\frac {2 (e (c+d x))^{5/2} (a+b \text {arccosh}(c+d x))}{5 d e}-\frac {(4 b) \text {Subst}\left (\int \frac {3 e^2 \sqrt {e x}}{2 \sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{25 d e} \\ & = -\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{3/2} \sqrt {1+c+d x}}{25 d}+\frac {2 (e (c+d x))^{5/2} (a+b \text {arccosh}(c+d x))}{5 d e}-\frac {(6 b e) \text {Subst}\left (\int \frac {\sqrt {e x}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{25 d} \\ & = -\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{3/2} \sqrt {1+c+d x}}{25 d}+\frac {2 (e (c+d x))^{5/2} (a+b \text {arccosh}(c+d x))}{5 d e}-\frac {\left (3 \sqrt {2} b e \sqrt {1-c-d x} \sqrt {e (c+d x)}\right ) \text {Subst}\left (\int \frac {\sqrt {-x}}{\sqrt {\frac {1}{2}-\frac {x}{2}} \sqrt {1+x}} \, dx,x,c+d x\right )}{25 d \sqrt {-c-d x} \sqrt {-1+c+d x}} \\ & = -\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{3/2} \sqrt {1+c+d x}}{25 d}+\frac {2 (e (c+d x))^{5/2} (a+b \text {arccosh}(c+d x))}{5 d e}-\frac {12 b e \sqrt {1-c-d x} \sqrt {e (c+d x)} E\left (\left .\arcsin \left (\frac {\sqrt {1+c+d x}}{\sqrt {2}}\right )\right |2\right )}{25 d \sqrt {-c-d x} \sqrt {-1+c+d x}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.33 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.75 \[ \int (c e+d e x)^{3/2} (a+b \text {arccosh}(c+d x)) \, dx=\frac {2 (e (c+d x))^{3/2} \left (5 (c+d x) (a+b \text {arccosh}(c+d x))-\frac {2 b \left (-1+c^2+2 c d x+d^2 x^2+\sqrt {1-(c+d x)^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},(c+d x)^2\right )\right )}{\sqrt {-1+c+d x} \sqrt {1+c+d x}}\right )}{25 d} \]

[In]

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

[Out]

(2*(e*(c + d*x))^(3/2)*(5*(c + d*x)*(a + b*ArcCosh[c + d*x]) - (2*b*(-1 + c^2 + 2*c*d*x + d^2*x^2 + Sqrt[1 - (
c + d*x)^2]*Hypergeometric2F1[1/2, 3/4, 7/4, (c + d*x)^2]))/(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])))/(25*d)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.07 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.74

method	result	size
derivativedivides	\(\frac {\frac {2 \left (d e x +c e \right )^{\frac {5}{2}} a}{5}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {5}{2}} \operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )}{5}-\frac {2 \left (\sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {7}{2}}+3 \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {-d e x -c e +e}{e}}\, e^{3} \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )-3 e^{3} \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {-d e x -c e +e}{e}}\, \operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )-\sqrt {-\frac {1}{e}}\, e^{2} \left (d e x +c e \right )^{\frac {3}{2}}\right )}{25 e \sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\)	\(253\)
default	\(\frac {\frac {2 \left (d e x +c e \right )^{\frac {5}{2}} a}{5}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {5}{2}} \operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )}{5}-\frac {2 \left (\sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {7}{2}}+3 \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {-d e x -c e +e}{e}}\, e^{3} \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )-3 e^{3} \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {-d e x -c e +e}{e}}\, \operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )-\sqrt {-\frac {1}{e}}\, e^{2} \left (d e x +c e \right )^{\frac {3}{2}}\right )}{25 e \sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\)	\(253\)
parts	\(\frac {2 a \left (d e x +c e \right )^{\frac {5}{2}}}{5 d e}+\frac {2 b \left (\frac {\left (d e x +c e \right )^{\frac {5}{2}} \operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )}{5}-\frac {2 \left (\sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {7}{2}}-\sqrt {-\frac {1}{e}}\, e^{2} \left (d e x +c e \right )^{\frac {3}{2}}+3 \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {d e x +c e -e}{e}}\, e^{3} \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )-3 e^{3} \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {d e x +c e -e}{e}}\, \operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )\right )}{25 e \sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {d e x +c e -e}{e}}}\right )}{d e}\)	\(259\)

[In]

int((d*e*x+c*e)^(3/2)*(a+b*arccosh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

2/d/e*(1/5*(d*e*x+c*e)^(5/2)*a+b*(1/5*(d*e*x+c*e)^(5/2)*arccosh(1/e*(d*e*x+c*e))-2/25/e*((-1/e)^(1/2)*(d*e*x+c
*e)^(7/2)+3*((d*e*x+c*e+e)/e)^(1/2)*((-d*e*x-c*e+e)/e)^(1/2)*e^3*EllipticF((d*e*x+c*e)^(1/2)*(-1/e)^(1/2),I)-3
*e^3*((d*e*x+c*e+e)/e)^(1/2)*((-d*e*x-c*e+e)/e)^(1/2)*EllipticE((d*e*x+c*e)^(1/2)*(-1/e)^(1/2),I)-(-1/e)^(1/2)
*e^2*(d*e*x+c*e)^(3/2))/(-1/e)^(1/2)/((d*e*x+c*e+e)/e)^(1/2)/(-(-d*e*x-c*e+e)/e)^(1/2)))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.10 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.28 \[ \int (c e+d e x)^{3/2} (a+b \text {arccosh}(c+d x)) \, dx=\frac {2 \, {\left (6 \, \sqrt {d^{3} e} b e {\rm weierstrassZeta}\left (\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right ) + 5 \, {\left (b d^{3} e x^{2} + 2 \, b c d^{2} e x + b c^{2} d e\right )} \sqrt {d e x + c e} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 2 \, {\left (b d^{2} e x + b c d e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} \sqrt {d e x + c e} + 5 \, {\left (a d^{3} e x^{2} + 2 \, a c d^{2} e x + a c^{2} d e\right )} \sqrt {d e x + c e}\right )}}{25 \, d^{2}} \]

[In]

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

[Out]

2/25*(6*sqrt(d^3*e)*b*e*weierstrassZeta(4/d^2, 0, weierstrassPInverse(4/d^2, 0, (d*x + c)/d)) + 5*(b*d^3*e*x^2
+ 2*b*c*d^2*e*x + b*c^2*d*e)*sqrt(d*e*x + c*e)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) - 2*(b*d^2*e*
x + b*c*d*e)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*sqrt(d*e*x + c*e) + 5*(a*d^3*e*x^2 + 2*a*c*d^2*e*x + a*c^2*d*e)
*sqrt(d*e*x + c*e))/d^2

Sympy [F]

\[ \int (c e+d e x)^{3/2} (a+b \text {arccosh}(c+d x)) \, dx=\int \left (e \left (c + d x\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c + d x \right )}\right )\, dx \]

[In]

integrate((d*e*x+c*e)**(3/2)*(a+b*acosh(d*x+c)),x)

[Out]

Integral((e*(c + d*x))**(3/2)*(a + b*acosh(c + d*x)), x)

Maxima [F(-2)]

Exception generated. \[ \int (c e+d e x)^{3/2} (a+b \text {arccosh}(c+d x)) \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int (c e+d e x)^{3/2} (a+b \text {arccosh}(c+d x)) \, dx=\int { {\left (d e x + c e\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (c e+d e x)^{3/2} (a+b \text {arccosh}(c+d x)) \, dx=\int {\left (c\,e+d\,e\,x\right )}^{3/2}\,\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right ) \,d x \]

[In]

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

[Out]

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