3.6.57 \(\int (d+e x^2) (a+b \cosh ^{-1}(c x))^{3/2} \, dx\) [557]

Optimal. Leaf size=442 \[ -\frac {3 b d \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}}{2 c}-\frac {b e \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}}{3 c^3}-\frac {b e x^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}}{6 c}+d x \left (a+b \cosh ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}-\frac {3 b^{3/2} d e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {3 b^{3/2} e e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 c^3}-\frac {b^{3/2} e e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{96 c^3}+\frac {3 b^{3/2} d e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}+\frac {3 b^{3/2} e e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {b^{3/2} e e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{96 c^3} \]

[Out]

d*x*(a+b*arccosh(c*x))^(3/2)+1/3*e*x^3*(a+b*arccosh(c*x))^(3/2)-1/288*b^(3/2)*e*exp(3*a/b)*erf(3^(1/2)*(a+b*ar
ccosh(c*x))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/c^3+1/288*b^(3/2)*e*erfi(3^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))
*3^(1/2)*Pi^(1/2)/c^3/exp(3*a/b)-3/8*b^(3/2)*d*exp(a/b)*erf((a+b*arccosh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/c-3/32*
b^(3/2)*e*exp(a/b)*erf((a+b*arccosh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/c^3+3/8*b^(3/2)*d*erfi((a+b*arccosh(c*x))^(1
/2)/b^(1/2))*Pi^(1/2)/c/exp(a/b)+3/32*b^(3/2)*e*erfi((a+b*arccosh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/c^3/exp(a/b)-3
/2*b*d*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))^(1/2)/c-1/3*b*e*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh
(c*x))^(1/2)/c^3-1/6*b*e*x^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))^(1/2)/c

________________________________________________________________________________________

Rubi [A]
time = 1.12, antiderivative size = 442, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 12, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5909, 5879, 5915, 5881, 3389, 2211, 2236, 2235, 5884, 5939, 5887, 5556} \begin {gather*} -\frac {3 \sqrt {\pi } b^{3/2} e e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 c^3}-\frac {\sqrt {\frac {\pi }{3}} b^{3/2} e e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{96 c^3}+\frac {3 \sqrt {\pi } b^{3/2} e e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {\sqrt {\frac {\pi }{3}} b^{3/2} e e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{96 c^3}-\frac {3 \sqrt {\pi } b^{3/2} d e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}+\frac {3 \sqrt {\pi } b^{3/2} d e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {b e \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \cosh ^{-1}(c x)}}{3 c^3}+d x \left (a+b \cosh ^{-1}(c x)\right )^{3/2}-\frac {3 b d \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \cosh ^{-1}(c x)}}{2 c}+\frac {1}{3} e x^3 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}-\frac {b e x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \cosh ^{-1}(c x)}}{6 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[a + b*ArcCosh[c*x]])/(2*c) - (b*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[
a + b*ArcCosh[c*x]])/(3*c^3) - (b*e*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[a + b*ArcCosh[c*x]])/(6*c) + d*x*(a
+ b*ArcCosh[c*x])^(3/2) + (e*x^3*(a + b*ArcCosh[c*x])^(3/2))/3 - (3*b^(3/2)*d*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*
ArcCosh[c*x]]/Sqrt[b]])/(8*c) - (3*b^(3/2)*e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(32*c^3)
- (b^(3/2)*e*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(96*c^3) + (3*b^(3/2)*d*S
qrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*c*E^(a/b)) + (3*b^(3/2)*e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[
c*x]]/Sqrt[b]])/(32*c^3*E^(a/b)) + (b^(3/2)*e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(96
*c^3*E^((3*a)/b))

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - c*(
f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 3389

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/E^(I*(e + f*x))
, x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]

Rule 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5879

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 5881

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Sinh[-a/b + x/b], x], x
, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x]

Rule 5884

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

Rule 5887

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/(b*c^(m + 1)), Subst[Int[x^n*Cosh
[-a/b + x/b]^m*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 5909

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcCosh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p
] && (p > 0 || IGtQ[n, 0])

Rule 5915

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sy
mbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Dist[b*
(n/(2*c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Int[(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, p}, x] && EqQ[e1, c
*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]

Rule 5939

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1
*e2*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m + 2*p + 1))), Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)
^p*(a + b*ArcCosh[c*x])^n, x], x] - Dist[b*f*(n/(c*(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, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IG
tQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rubi steps

\begin {align*} \int \left (d+e x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )^{3/2} \, dx &=\int \left (d \left (a+b \cosh ^{-1}(c x)\right )^{3/2}+e x^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}\right ) \, dx\\ &=d \int \left (a+b \cosh ^{-1}(c x)\right )^{3/2} \, dx+e \int x^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2} \, dx\\ &=d x \left (a+b \cosh ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}-\frac {1}{2} (3 b c d) \int \frac {x \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {1}{2} (b c e) \int \frac {x^3 \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {3 b d \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}}{2 c}-\frac {b e x^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}}{6 c}+d x \left (a+b \cosh ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}+\frac {1}{4} \left (3 b^2 d\right ) \int \frac {1}{\sqrt {a+b \cosh ^{-1}(c x)}} \, dx+\frac {1}{12} \left (b^2 e\right ) \int \frac {x^2}{\sqrt {a+b \cosh ^{-1}(c x)}} \, dx-\frac {(b e) \int \frac {x \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 c}\\ &=-\frac {3 b d \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}}{2 c}-\frac {b e \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}}{3 c^3}-\frac {b e x^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}}{6 c}+d x \left (a+b \cosh ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}-\frac {(3 b d) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{4 c}+\frac {\left (b^2 e\right ) \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{12 c^3}+\frac {\left (b^2 e\right ) \int \frac {1}{\sqrt {a+b \cosh ^{-1}(c x)}} \, dx}{6 c^2}\\ &=-\frac {3 b d \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}}{2 c}-\frac {b e \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}}{3 c^3}-\frac {b e x^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}}{6 c}+d x \left (a+b \cosh ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}-\frac {(3 b d) \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{8 c}+\frac {(3 b d) \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{8 c}-\frac {(b e) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{6 c^3}+\frac {\left (b^2 e\right ) \text {Subst}\left (\int \left (\frac {\sinh (x)}{4 \sqrt {a+b x}}+\frac {\sinh (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{12 c^3}\\ &=-\frac {3 b d \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}}{2 c}-\frac {b e \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}}{3 c^3}-\frac {b e x^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}}{6 c}+d x \left (a+b \cosh ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}-\frac {(3 b d) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{4 c}+\frac {(3 b d) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{4 c}-\frac {(b e) \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{12 c^3}+\frac {(b e) \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{12 c^3}+\frac {\left (b^2 e\right ) \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{48 c^3}+\frac {\left (b^2 e\right ) \text {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{48 c^3}\\ &=-\frac {3 b d \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}}{2 c}-\frac {b e \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}}{3 c^3}-\frac {b e x^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}}{6 c}+d x \left (a+b \cosh ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}-\frac {3 b^{3/2} d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}+\frac {3 b^{3/2} d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {(b e) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{6 c^3}+\frac {(b e) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{6 c^3}-\frac {\left (b^2 e\right ) \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{96 c^3}-\frac {\left (b^2 e\right ) \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{96 c^3}+\frac {\left (b^2 e\right ) \text {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{96 c^3}+\frac {\left (b^2 e\right ) \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{96 c^3}\\ &=-\frac {3 b d \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}}{2 c}-\frac {b e \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}}{3 c^3}-\frac {b e x^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}}{6 c}+d x \left (a+b \cosh ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}-\frac {3 b^{3/2} d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {b^{3/2} e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{12 c^3}+\frac {3 b^{3/2} d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}+\frac {b^{3/2} e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{12 c^3}-\frac {(b e) \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{48 c^3}-\frac {(b e) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{48 c^3}+\frac {(b e) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{48 c^3}+\frac {(b e) \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{48 c^3}\\ &=-\frac {3 b d \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}}{2 c}-\frac {b e \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}}{3 c^3}-\frac {b e x^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}}{6 c}+d x \left (a+b \cosh ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}-\frac {3 b^{3/2} d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {3 b^{3/2} e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 c^3}-\frac {b^{3/2} e e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{96 c^3}+\frac {3 b^{3/2} d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}+\frac {3 b^{3/2} e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {b^{3/2} e e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{96 c^3}\\ \end {align*}

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Mathematica [A]
time = 2.33, size = 812, normalized size = 1.84 \begin {gather*} \frac {a d e^{-\frac {a}{b}} \sqrt {a+b \cosh ^{-1}(c x)} \left (\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\cosh ^{-1}(c x)\right )}{\sqrt {\frac {a}{b}+\cosh ^{-1}(c x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{\sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}}}\right )}{2 c}+\frac {a e e^{-\frac {3 a}{b}} \sqrt {a+b \cosh ^{-1}(c x)} \left (9 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\cosh ^{-1}(c x)\right )+\sqrt {3} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \Gamma \left (\frac {3}{2},-\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+9 e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \Gamma \left (\frac {3}{2},-\frac {a+b \cosh ^{-1}(c x)}{b}\right )+\sqrt {3} e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \Gamma \left (\frac {3}{2},\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )}{72 c^3 \sqrt {-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}}}+\frac {b d \left (-12 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {a+b \cosh ^{-1}(c x)}+8 c x \cosh ^{-1}(c x) \sqrt {a+b \cosh ^{-1}(c x)}+\frac {(2 a+3 b) \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )}{\sqrt {b}}+\frac {(2 a-3 b) \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )}{\sqrt {b}}\right )}{8 c}+\frac {\sqrt {b} e \left (9 \left (-12 \sqrt {b} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {a+b \cosh ^{-1}(c x)}+8 \sqrt {b} c x \cosh ^{-1}(c x) \sqrt {a+b \cosh ^{-1}(c x)}+(2 a+3 b) \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )+(2 a-3 b) \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )\right )+(2 a+b) \sqrt {3 \pi } \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {3 a}{b}\right )-\sinh \left (\frac {3 a}{b}\right )\right )+(2 a-b) \sqrt {3 \pi } \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {3 a}{b}\right )+\sinh \left (\frac {3 a}{b}\right )\right )+12 \sqrt {b} \sqrt {a+b \cosh ^{-1}(c x)} \left (2 \cosh ^{-1}(c x) \cosh \left (3 \cosh ^{-1}(c x)\right )-\sinh \left (3 \cosh ^{-1}(c x)\right )\right )\right )}{288 c^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*d*Sqrt[a + b*ArcCosh[c*x]]*((E^((2*a)/b)*Gamma[3/2, a/b + ArcCosh[c*x]])/Sqrt[a/b + ArcCosh[c*x]] + Gamma[3
/2, -((a + b*ArcCosh[c*x])/b)]/Sqrt[-((a + b*ArcCosh[c*x])/b)]))/(2*c*E^(a/b)) + (a*e*Sqrt[a + b*ArcCosh[c*x]]
*(9*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[3/2, a/b + ArcCosh[c*x]] + Sqrt[3]*Sqrt[a/b + ArcCosh[c*
x]]*Gamma[3/2, (-3*(a + b*ArcCosh[c*x]))/b] + 9*E^((2*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[3/2, -((a + b*ArcCo
sh[c*x])/b)] + Sqrt[3]*E^((6*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[3/2, (3*(a + b*ArcCosh[c*x]))/b]))/(7
2*c^3*E^((3*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])^2/b^2)]) + (b*d*(-12*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[a
+ b*ArcCosh[c*x]] + 8*c*x*ArcCosh[c*x]*Sqrt[a + b*ArcCosh[c*x]] + ((2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCo
sh[c*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]))/Sqrt[b] + ((2*a - 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b
]]*(Cosh[a/b] + Sinh[a/b]))/Sqrt[b]))/(8*c) + (Sqrt[b]*e*(9*(-12*Sqrt[b]*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*
Sqrt[a + b*ArcCosh[c*x]] + 8*Sqrt[b]*c*x*ArcCosh[c*x]*Sqrt[a + b*ArcCosh[c*x]] + (2*a + 3*b)*Sqrt[Pi]*Erfi[Sqr
t[a + b*ArcCosh[c*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (2*a - 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqr
t[b]]*(Cosh[a/b] + Sinh[a/b])) + (2*a + b)*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]]*(Cosh[(
3*a)/b] - Sinh[(3*a)/b]) + (2*a - b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]]*(Cosh[(3*a)/b]
+ Sinh[(3*a)/b]) + 12*Sqrt[b]*Sqrt[a + b*ArcCosh[c*x]]*(2*ArcCosh[c*x]*Cosh[3*ArcCosh[c*x]] - Sinh[3*ArcCosh[
c*x]])))/(288*c^3)

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (e \,x^{2}+d \right ) \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{\frac {3}{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{\frac {3}{2}} \left (d + e x^{2}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{3/2}\,\left (e\,x^2+d\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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