\(\int \sqrt {f x} (a+b \text {arccosh}(c x))^2 \, dx\) [163]

Optimal result

Integrand size = 18, antiderivative size = 128 \[ \int \sqrt {f x} (a+b \text {arccosh}(c x))^2 \, dx=\frac {2 (f x)^{3/2} (a+b \text {arccosh}(c x))^2}{3 f}-\frac {8 b c (f x)^{5/2} \sqrt {1-c x} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {9}{4},c^2 x^2\right )}{15 f^2 \sqrt {-1+c x}}-\frac {16 b^2 c^2 (f x)^{7/2} \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};c^2 x^2\right )}{105 f^3} \]

[Out]

2/3*(f*x)^(3/2)*(a+b*arccosh(c*x))^2/f-16/105*b^2*c^2*(f*x)^(7/2)*hypergeom([1, 7/4, 7/4],[9/4, 11/4],c^2*x^2)
/f^3-8/15*b*c*(f*x)^(5/2)*(a+b*arccosh(c*x))*hypergeom([1/2, 5/4],[9/4],c^2*x^2)*(-c*x+1)^(1/2)/f^2/(c*x-1)^(1
/2)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5883, 5949} \[ \int \sqrt {f x} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {16 b^2 c^2 (f x)^{7/2} \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};c^2 x^2\right )}{105 f^3}-\frac {8 b c \sqrt {1-c x} (f x)^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {9}{4},c^2 x^2\right ) (a+b \text {arccosh}(c x))}{15 f^2 \sqrt {c x-1}}+\frac {2 (f x)^{3/2} (a+b \text {arccosh}(c x))^2}{3 f} \]

[In]

Int[Sqrt[f*x]*(a + b*ArcCosh[c*x])^2,x]

[Out]

(2*(f*x)^(3/2)*(a + b*ArcCosh[c*x])^2)/(3*f) - (8*b*c*(f*x)^(5/2)*Sqrt[1 - c*x]*(a + b*ArcCosh[c*x])*Hypergeom
etric2F1[1/2, 5/4, 9/4, c^2*x^2])/(15*f^2*Sqrt[-1 + c*x]) - (16*b^2*c^2*(f*x)^(7/2)*HypergeometricPFQ[{1, 7/4,
7/4}, {9/4, 11/4}, c^2*x^2])/(105*f^3)

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 5949

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]

Rubi steps \begin{align*} \text {integral}& = \frac {2 (f x)^{3/2} (a+b \text {arccosh}(c x))^2}{3 f}-\frac {(4 b c) \int \frac {(f x)^{3/2} (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 f} \\ & = \frac {2 (f x)^{3/2} (a+b \text {arccosh}(c x))^2}{3 f}-\frac {8 b c (f x)^{5/2} \sqrt {1-c x} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {9}{4},c^2 x^2\right )}{15 f^2 \sqrt {-1+c x}}-\frac {16 b^2 c^2 (f x)^{7/2} \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};c^2 x^2\right )}{105 f^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.92 \[ \int \sqrt {f x} (a+b \text {arccosh}(c x))^2 \, dx=\frac {2}{105} x \sqrt {f x} \left (35 (a+b \text {arccosh}(c x))^2-4 b c x \left (\frac {7 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {9}{4},c^2 x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+2 b c x \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};c^2 x^2\right )\right )\right ) \]

[In]

Integrate[Sqrt[f*x]*(a + b*ArcCosh[c*x])^2,x]

[Out]

(2*x*Sqrt[f*x]*(35*(a + b*ArcCosh[c*x])^2 - 4*b*c*x*((7*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2
F1[1/2, 5/4, 9/4, c^2*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + 2*b*c*x*HypergeometricPFQ[{1, 7/4, 7/4}, {9/4, 11
/4}, c^2*x^2])))/105

Maple [F]

\[\int \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} \sqrt {f x}d x\]

[In]

int((a+b*arccosh(c*x))^2*(f*x)^(1/2),x)

[Out]

int((a+b*arccosh(c*x))^2*(f*x)^(1/2),x)

Fricas [F]

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

[In]

integrate((a+b*arccosh(c*x))^2*(f*x)^(1/2),x, algorithm="fricas")

[Out]

integral((b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2)*sqrt(f*x), x)

Sympy [F]

\[ \int \sqrt {f x} (a+b \text {arccosh}(c x))^2 \, dx=\int \sqrt {f x} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}\, dx \]

[In]

integrate((a+b*acosh(c*x))**2*(f*x)**(1/2),x)

[Out]

Integral(sqrt(f*x)*(a + b*acosh(c*x))**2, x)

Maxima [F]

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

[In]

integrate((a+b*arccosh(c*x))^2*(f*x)^(1/2),x, algorithm="maxima")

[Out]

2/3*b^2*sqrt(f)*x^(3/2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2 + 2/3*(f*x)^(3/2)*a^2/f + integrate(2/3*(((3*
a*b*c^2*sqrt(f) - 2*b^2*c^2*sqrt(f))*x^2 - 3*a*b*sqrt(f))*sqrt(c*x + 1)*sqrt(c*x - 1)*sqrt(x) + ((3*a*b*c^3*sq
rt(f) - 2*b^2*c^3*sqrt(f))*x^3 - (3*a*b*c*sqrt(f) - 2*b^2*c*sqrt(f))*x)*sqrt(x))*log(c*x + sqrt(c*x + 1)*sqrt(
c*x - 1))/(c^3*x^3 + (c^2*x^2 - 1)*sqrt(c*x + 1)*sqrt(c*x - 1) - c*x), x)

Giac [F(-2)]

Exception generated. \[ \int \sqrt {f x} (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((a+b*arccosh(c*x))^2*(f*x)^(1/2),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

Mupad [F(-1)]

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

[In]

int((a + b*acosh(c*x))^2*(f*x)^(1/2),x)

[Out]

int((a + b*acosh(c*x))^2*(f*x)^(1/2), x)