\(\int (d x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx\) [83]

Optimal result

Integrand size = 18, antiderivative size = 111 \[ \int (d x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {2 (d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{5 d}-\frac {8 b c (d x)^{7/2} (a+b \text {arcsinh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{4},\frac {11}{4},-c^2 x^2\right )}{35 d^2}+\frac {16 b^2 c^2 (d x)^{9/2} \, _3F_2\left (1,\frac {9}{4},\frac {9}{4};\frac {11}{4},\frac {13}{4};-c^2 x^2\right )}{315 d^3} \] Output:

2/5*(d*x)^(5/2)*(a+b*arcsinh(c*x))^2/d-8/35*b*c*(d*x)^(7/2)*(a+b*arcsinh(c 
*x))*hypergeom([1/2, 7/4],[11/4],-c^2*x^2)/d^2+16/315*b^2*c^2*(d*x)^(9/2)* 
hypergeom([1, 9/4, 9/4],[11/4, 13/4],-c^2*x^2)/d^3
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.83 \[ \int (d x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {2}{315} x (d x)^{3/2} \left (9 (a+b \text {arcsinh}(c x)) \left (7 (a+b \text {arcsinh}(c x))-4 b c x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{4},\frac {11}{4},-c^2 x^2\right )\right )+8 b^2 c^2 x^2 \, _3F_2\left (1,\frac {9}{4},\frac {9}{4};\frac {11}{4},\frac {13}{4};-c^2 x^2\right )\right ) \] Input:

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

Output:

(2*x*(d*x)^(3/2)*(9*(a + b*ArcSinh[c*x])*(7*(a + b*ArcSinh[c*x]) - 4*b*c*x 
*Hypergeometric2F1[1/2, 7/4, 11/4, -(c^2*x^2)]) + 8*b^2*c^2*x^2*Hypergeome 
tricPFQ[{1, 9/4, 9/4}, {11/4, 13/4}, -(c^2*x^2)]))/315
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.04, 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.

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

\(\Big \downarrow \) 6191

\(\displaystyle \frac {2 (d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{5 d}-\frac {4 b c \int \frac {(d x)^{5/2} (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{5 d}\)

\(\Big \downarrow \) 6232

\(\displaystyle \frac {2 (d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{5 d}-\frac {4 b c \left (\frac {2 (d x)^{7/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{4},\frac {11}{4},-c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{7 d}-\frac {4 b c (d x)^{9/2} \, _3F_2\left (1,\frac {9}{4},\frac {9}{4};\frac {11}{4},\frac {13}{4};-c^2 x^2\right )}{63 d^2}\right )}{5 d}\)

Input:

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

Output:

(2*(d*x)^(5/2)*(a + b*ArcSinh[c*x])^2)/(5*d) - (4*b*c*((2*(d*x)^(7/2)*(a + 
 b*ArcSinh[c*x])*Hypergeometric2F1[1/2, 7/4, 11/4, -(c^2*x^2)])/(7*d) - (4 
*b*c*(d*x)^(9/2)*HypergeometricPFQ[{1, 9/4, 9/4}, {11/4, 13/4}, -(c^2*x^2) 
])/(63*d^2)))/(5*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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

2/5*b^2*d^(3/2)*x^(5/2)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 2/5*(d*x)^(5/2)*a 
^2/d + integrate(2/5*(((5*a*b*d^(3/2) - 2*b^2*d^(3/2))*c^2*x^2 + 5*a*b*d^( 
3/2))*sqrt(c^2*x^2 + 1)*x^(3/2) + ((5*a*b*d^(3/2) - 2*b^2*d^(3/2))*c^3*x^3 
 + (5*a*b*d^(3/2) - 2*b^2*d^(3/2))*c*x)*x^(3/2))*log(c*x + sqrt(c^2*x^2 + 
1))/(c^3*x^3 + c*x + (c^2*x^2 + 1)^(3/2)), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

\[ \int (d x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {\sqrt {d}\, d \left (2 \sqrt {x}\, a^{2} x^{2}+10 \left (\int \sqrt {x}\, \mathit {asinh} \left (c x \right ) x d x \right ) a b +5 \left (\int \sqrt {x}\, \mathit {asinh} \left (c x \right )^{2} x d x \right ) b^{2}\right )}{5} \] Input:

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

Output:

(sqrt(d)*d*(2*sqrt(x)*a**2*x**2 + 10*int(sqrt(x)*asinh(c*x)*x,x)*a*b + 5*i 
nt(sqrt(x)*asinh(c*x)**2*x,x)*b**2))/5