3.3.60 \(\int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x)^3 \, dx\) [260]

3.3.60.1 Optimal result

Integrand size = 30, antiderivative size = 131 \[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x)^3 \, dx=-\frac {3 (a+b x)^2}{8 b}+\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{4 b}-\frac {3 \text {arcsinh}(a+b x)^2}{8 b}-\frac {3 (a+b x)^2 \text {arcsinh}(a+b x)^2}{4 b}+\frac {(a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^3}{2 b}+\frac {\text {arcsinh}(a+b x)^4}{8 b} \]

output

-3/8*(b*x+a)^2/b-3/8*arcsinh(b*x+a)^2/b-3/4*(b*x+a)^2*arcsinh(b*x+a)^2/b+1 
/8*arcsinh(b*x+a)^4/b+3/4*(b*x+a)*arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2)/b+1/2 
*(b*x+a)*arcsinh(b*x+a)^3*(1+(b*x+a)^2)^(1/2)/b
 

3.3.60.2 Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.97 \[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x)^3 \, dx=\frac {-3 b x (2 a+b x)+6 (a+b x) \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x)-3 \left (1+2 a^2+4 a b x+2 b^2 x^2\right ) \text {arcsinh}(a+b x)^2+4 (a+b x) \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x)^3+\text {arcsinh}(a+b x)^4}{8 b} \]

input

Integrate[Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x]^3,x]
 

output

(-3*b*x*(2*a + b*x) + 6*(a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSin 
h[a + b*x] - 3*(1 + 2*a^2 + 4*a*b*x + 2*b^2*x^2)*ArcSinh[a + b*x]^2 + 4*(a 
 + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x]^3 + ArcSinh[a + 
 b*x]^4)/(8*b)
 

3.3.60.3 Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.93, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {6275, 6200, 6191, 6198, 6227, 15, 6198}

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.

\(\displaystyle \int \sqrt {a^2+2 a b x+b^2 x^2+1} \text {arcsinh}(a+b x)^3 \, dx\)

\(\Big \downarrow \) 6275

\(\displaystyle \frac {\int \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^3d(a+b x)}{b}\)

\(\Big \downarrow \) 6200

\(\displaystyle \frac {-\frac {3}{2} \int (a+b x) \text {arcsinh}(a+b x)^2d(a+b x)+\frac {1}{2} \int \frac {\text {arcsinh}(a+b x)^3}{\sqrt {(a+b x)^2+1}}d(a+b x)+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^3}{b}\)

\(\Big \downarrow \) 6191

\(\displaystyle \frac {-\frac {3}{2} \left (\frac {1}{2} (a+b x)^2 \text {arcsinh}(a+b x)^2-\int \frac {(a+b x)^2 \text {arcsinh}(a+b x)}{\sqrt {(a+b x)^2+1}}d(a+b x)\right )+\frac {1}{2} \int \frac {\text {arcsinh}(a+b x)^3}{\sqrt {(a+b x)^2+1}}d(a+b x)+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^3}{b}\)

\(\Big \downarrow \) 6198

\(\displaystyle \frac {-\frac {3}{2} \left (\frac {1}{2} (a+b x)^2 \text {arcsinh}(a+b x)^2-\int \frac {(a+b x)^2 \text {arcsinh}(a+b x)}{\sqrt {(a+b x)^2+1}}d(a+b x)\right )+\frac {1}{8} \text {arcsinh}(a+b x)^4+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^3}{b}\)

\(\Big \downarrow \) 6227

\(\displaystyle \frac {-\frac {3}{2} \left (\frac {1}{2} \int \frac {\text {arcsinh}(a+b x)}{\sqrt {(a+b x)^2+1}}d(a+b x)+\frac {1}{2} \int (a+b x)d(a+b x)+\frac {1}{2} (a+b x)^2 \text {arcsinh}(a+b x)^2-\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)\right )+\frac {1}{8} \text {arcsinh}(a+b x)^4+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^3}{b}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {-\frac {3}{2} \left (\frac {1}{2} \int \frac {\text {arcsinh}(a+b x)}{\sqrt {(a+b x)^2+1}}d(a+b x)+\frac {1}{2} (a+b x)^2 \text {arcsinh}(a+b x)^2-\frac {1}{2} \sqrt {(a+b x)^2+1} (a+b x) \text {arcsinh}(a+b x)+\frac {1}{4} (a+b x)^2\right )+\frac {1}{8} \text {arcsinh}(a+b x)^4+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^3}{b}\)

\(\Big \downarrow \) 6198

\(\displaystyle \frac {\frac {1}{8} \text {arcsinh}(a+b x)^4+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^3-\frac {3}{2} \left (\frac {1}{2} (a+b x)^2 \text {arcsinh}(a+b x)^2-\frac {1}{2} \sqrt {(a+b x)^2+1} (a+b x) \text {arcsinh}(a+b x)+\frac {1}{4} \text {arcsinh}(a+b x)^2+\frac {1}{4} (a+b x)^2\right )}{b}\)

input

Int[Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x]^3,x]
 

output

(((a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^3)/2 + ArcSinh[a + b*x] 
^4/8 - (3*((a + b*x)^2/4 - ((a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b* 
x])/2 + ArcSinh[a + b*x]^2/4 + ((a + b*x)^2*ArcSinh[a + b*x]^2)/2))/2)/b
 

3.3.60.3.1 Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

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 6198

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

rule 6200

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/2), x] + (Simp[(1 
/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   Int[(a + b*ArcSinh[c*x])^n/Sq 
rt[1 + c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2* 
x^2]]   Int[x*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[e, c^2*d] && GtQ[n, 0]
 

rule 6227

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

rule 6275

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((A_.) + (B_.)*(x_) + 
(C_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/d   Subst[Int[(C/d^2 + (C/d^2)*x^2 
)^p*(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, A, B, C 
, n, p}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
 

3.3.60.4 Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.56

method	result	size
default	\(\frac {4 \operatorname {arcsinh}\left (b x +a \right )^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b x -6 \operatorname {arcsinh}\left (b x +a \right )^{2} b^{2} x^{2}+4 \operatorname {arcsinh}\left (b x +a \right )^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a -12 \operatorname {arcsinh}\left (b x +a \right )^{2} a b x +\operatorname {arcsinh}\left (b x +a \right )^{4}-6 a^{2} \operatorname {arcsinh}\left (b x +a \right )^{2}+6 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b x -3 b^{2} x^{2}+6 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a -6 a b x -3 \operatorname {arcsinh}\left (b x +a \right )^{2}-3 a^{2}-3}{8 b}\)	\(204\)

input

int(arcsinh(b*x+a)^3*(b^2*x^2+2*a*b*x+a^2+1)^(1/2),x,method=_RETURNVERBOSE 
)
 

output

1/8*(4*arcsinh(b*x+a)^3*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*b*x-6*arcsinh(b*x+a) 
^2*b^2*x^2+4*arcsinh(b*x+a)^3*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a-12*arcsinh(b 
*x+a)^2*a*b*x+arcsinh(b*x+a)^4-6*a^2*arcsinh(b*x+a)^2+6*arcsinh(b*x+a)*(b^ 
2*x^2+2*a*b*x+a^2+1)^(1/2)*b*x-3*b^2*x^2+6*arcsinh(b*x+a)*(b^2*x^2+2*a*b*x 
+a^2+1)^(1/2)*a-6*a*b*x-3*arcsinh(b*x+a)^2-3*a^2-3)/b
 

3.3.60.5 Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.52 \[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x)^3 \, dx=\frac {4 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b x + a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{3} - 3 \, b^{2} x^{2} + \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{4} - 6 \, a b x - 3 \, {\left (2 \, b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} + 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2} + 6 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b x + a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}{8 \, b} \]

input

integrate(arcsinh(b*x+a)^3*(b^2*x^2+2*a*b*x+a^2+1)^(1/2),x, algorithm="fri 
cas")
 

output

1/8*(4*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(b*x + a)*log(b*x + a + sqrt(b^2* 
x^2 + 2*a*b*x + a^2 + 1))^3 - 3*b^2*x^2 + log(b*x + a + sqrt(b^2*x^2 + 2*a 
*b*x + a^2 + 1))^4 - 6*a*b*x - 3*(2*b^2*x^2 + 4*a*b*x + 2*a^2 + 1)*log(b*x 
 + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2 + 6*sqrt(b^2*x^2 + 2*a*b*x + a 
^2 + 1)*(b*x + a)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)))/b
 

3.3.60.6 Sympy [F]

\[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x)^3 \, dx=\int \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a + b x \right )}\, dx \]

input

integrate(asinh(b*x+a)**3*(b**2*x**2+2*a*b*x+a**2+1)**(1/2),x)
 

output

Integral(sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)**3, x)
 

3.3.60.7 Maxima [F]

\[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x)^3 \, dx=\int { \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \operatorname {arsinh}\left (b x + a\right )^{3} \,d x } \]

input

integrate(arcsinh(b*x+a)^3*(b^2*x^2+2*a*b*x+a^2+1)^(1/2),x, algorithm="max 
ima")
 

output

integrate(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*arcsinh(b*x + a)^3, x)
 

3.3.60.8 Giac [F]

\[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x)^3 \, dx=\int { \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \operatorname {arsinh}\left (b x + a\right )^{3} \,d x } \]

input

integrate(arcsinh(b*x+a)^3*(b^2*x^2+2*a*b*x+a^2+1)^(1/2),x, algorithm="gia 
c")
 

output

integrate(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*arcsinh(b*x + a)^3, x)
 

3.3.60.9 Mupad [F(-1)]

Timed out. \[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x)^3 \, dx=\int {\mathrm {asinh}\left (a+b\,x\right )}^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1} \,d x \]

input

int(asinh(a + b*x)^3*(a^2 + b^2*x^2 + 2*a*b*x + 1)^(1/2),x)
 

output

int(asinh(a + b*x)^3*(a^2 + b^2*x^2 + 2*a*b*x + 1)^(1/2), x)