\(\int \frac {(a+b \text {arcsinh}(c x))^2}{(d x)^{3/2}} \, dx\) [86]

Optimal result

Integrand size = 18, antiderivative size = 107 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d x)^{3/2}} \, dx=-\frac {2 (a+b \text {arcsinh}(c x))^2}{d \sqrt {d x}}+\frac {8 b c \sqrt {d x} (a+b \text {arcsinh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-c^2 x^2\right )}{d^2}-\frac {16 b^2 c^2 (d x)^{3/2} \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};-c^2 x^2\right )}{3 d^3} \] Output:

-2*(a+b*arcsinh(c*x))^2/d/(d*x)^(1/2)+8*b*c*(d*x)^(1/2)*(a+b*arcsinh(c*x)) 
*hypergeom([1/4, 1/2],[5/4],-c^2*x^2)/d^2-16/3*b^2*c^2*(d*x)^(3/2)*hyperge 
om([3/4, 3/4, 1],[5/4, 7/4],-c^2*x^2)/d^3
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d x)^{3/2}} \, dx=-\frac {2 x \left (3 (a+b \text {arcsinh}(c x)) \left (a+b \text {arcsinh}(c x)-4 b c x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-c^2 x^2\right )\right )+8 b^2 c^2 x^2 \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};-c^2 x^2\right )\right )}{3 (d x)^{3/2}} \] Input:

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

Output:

(-2*x*(3*(a + b*ArcSinh[c*x])*(a + b*ArcSinh[c*x] - 4*b*c*x*Hypergeometric 
2F1[1/4, 1/2, 5/4, -(c^2*x^2)]) + 8*b^2*c^2*x^2*HypergeometricPFQ[{3/4, 3/ 
4, 1}, {5/4, 7/4}, -(c^2*x^2)]))/(3*(d*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.02, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6191, 6232}

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.

\(\displaystyle \int \frac {(a+b \text {arcsinh}(c x))^2}{(d x)^{3/2}} \, dx\)

\(\Big \downarrow \) 6191

\(\displaystyle \frac {4 b c \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {d x} \sqrt {c^2 x^2+1}}dx}{d}-\frac {2 (a+b \text {arcsinh}(c x))^2}{d \sqrt {d x}}\)

\(\Big \downarrow \) 6232

\(\displaystyle \frac {4 b c \left (\frac {2 \sqrt {d x} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{d}-\frac {4 b c (d x)^{3/2} \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};-c^2 x^2\right )}{3 d^2}\right )}{d}-\frac {2 (a+b \text {arcsinh}(c x))^2}{d \sqrt {d x}}\)

Input:

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

Output:

(-2*(a + b*ArcSinh[c*x])^2)/(d*Sqrt[d*x]) + (4*b*c*((2*Sqrt[d*x]*(a + b*Ar 
cSinh[c*x])*Hypergeometric2F1[1/4, 1/2, 5/4, -(c^2*x^2)])/d - (4*b*c*(d*x) 
^(3/2)*HypergeometricPFQ[{3/4, 3/4, 1}, {5/4, 7/4}, -(c^2*x^2)])/(3*d^2))) 
/d
 

Defintions of rubi rules used

rule 6191

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
 :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* 
(n/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + 
c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
 

rule 6232

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_ 
.)*(x_)^2], x_Symbol] :> Simp[((f*x)^(m + 1)/(f*(m + 1)))*Simp[Sqrt[1 + c^2 
*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[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^2*x^2]/Sqrt[d + e*x^2]]*HypergeometricPFQ[{1, 1 + m/2, 
1 + m/2}, {3/2 + m/2, 2 + m/2}, (-c^2)*x^2], x] /; FreeQ[{a, b, c, d, e, f, 
 m}, x] && EqQ[e, c^2*d] &&  !IntegerQ[m]
 

Maple [F]

\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{\left (d x \right )^{\frac {3}{2}}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d x)^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{\left (d x\right )^{\frac {3}{2}}} \,d x } \] Input:

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

Output:

integral((b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)*sqrt(d*x)/(d^2*x^ 
2), x)
 

Sympy [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d x)^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (d x\right )^{\frac {3}{2}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d x)^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{\left (d x\right )^{\frac {3}{2}}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d x)^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{\left (d x\right )^{\frac {3}{2}}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d x)^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,x\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d x)^{3/2}} \, dx=\frac {2 \sqrt {x}\, \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {x}\, x}d x \right ) a b +\sqrt {x}\, \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{\sqrt {x}\, x}d x \right ) b^{2}-2 a^{2}}{\sqrt {x}\, \sqrt {d}\, d} \] Input:

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

Output:

(2*sqrt(x)*int(asinh(c*x)/(sqrt(x)*x),x)*a*b + sqrt(x)*int(asinh(c*x)**2/( 
sqrt(x)*x),x)*b**2 - 2*a**2)/(sqrt(x)*sqrt(d)*d)