Integrand size = 28, antiderivative size = 106 \[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x) \, dx=-\frac {5 (a+b x)^2}{16 b}-\frac {(a+b x)^4}{16 b}+\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{8 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2} \text {arcsinh}(a+b x)}{4 b}+\frac {3 \text {arcsinh}(a+b x)^2}{16 b} \]
-5/16*(b*x+a)^2/b-1/16*(b*x+a)^4/b+1/4*(b*x+a)*(1+(b*x+a)^2)^(3/2)*arcsinh (b*x+a)/b+3/16*arcsinh(b*x+a)^2/b+3/8*(b*x+a)*arcsinh(b*x+a)*(1+(b*x+a)^2) ^(1/2)/b
Time = 0.07 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.17 \[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x) \, dx=\frac {-b x \left (10 a+4 a^3+5 b x+6 a^2 b x+4 a b^2 x^2+b^3 x^3\right )+2 \sqrt {1+a^2+2 a b x+b^2 x^2} \left (5 a+2 a^3+5 b x+6 a^2 b x+6 a b^2 x^2+2 b^3 x^3\right ) \text {arcsinh}(a+b x)+3 \text {arcsinh}(a+b x)^2}{16 b} \]
(-(b*x*(10*a + 4*a^3 + 5*b*x + 6*a^2*b*x + 4*a*b^2*x^2 + b^3*x^3)) + 2*Sqr t[1 + a^2 + 2*a*b*x + b^2*x^2]*(5*a + 2*a^3 + 5*b*x + 6*a^2*b*x + 6*a*b^2* x^2 + 2*b^3*x^3)*ArcSinh[a + b*x] + 3*ArcSinh[a + b*x]^2)/(16*b)
Time = 0.47 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6275, 6201, 244, 2009, 6200, 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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2+1\right )^{3/2} \text {arcsinh}(a+b x) \, dx\) |
| \(\Big \downarrow \) 6275 |
| \(\displaystyle \frac {\int \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)d(a+b x)}{b}\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle \frac {\frac {3}{4} \int \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)d(a+b x)-\frac {1}{4} \int (a+b x) \left ((a+b x)^2+1\right )d(a+b x)+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)}{b}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {\frac {3}{4} \int \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)d(a+b x)-\frac {1}{4} \int \left ((a+b x)^3+a+b x\right )d(a+b x)+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3}{4} \int \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)d(a+b x)+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)+\frac {1}{4} \left (-\frac {1}{4} (a+b x)^4-\frac {1}{2} (a+b x)^2\right )}{b}\) |
| \(\Big \downarrow \) 6200 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \int \frac {\text {arcsinh}(a+b x)}{\sqrt {(a+b x)^2+1}}d(a+b x)-\frac {1}{2} \int (a+b x)d(a+b x)+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)\right )+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)+\frac {1}{4} \left (-\frac {1}{4} (a+b x)^4-\frac {1}{2} (a+b x)^2\right )}{b}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \int \frac {\text {arcsinh}(a+b x)}{\sqrt {(a+b x)^2+1}}d(a+b x)+\frac {1}{2} \sqrt {(a+b x)^2+1} (a+b x) \text {arcsinh}(a+b x)-\frac {1}{4} (a+b x)^2\right )+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)+\frac {1}{4} \left (-\frac {1}{4} (a+b x)^4-\frac {1}{2} (a+b x)^2\right )}{b}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)+\frac {3}{4} \left (\frac {1}{2} \sqrt {(a+b x)^2+1} (a+b x) \text {arcsinh}(a+b x)+\frac {1}{4} \text {arcsinh}(a+b x)^2-\frac {1}{4} (a+b x)^2\right )+\frac {1}{4} \left (-\frac {1}{4} (a+b x)^4-\frac {1}{2} (a+b x)^2\right )}{b}\) |
((-1/2*(a + b*x)^2 - (a + b*x)^4/4)/4 + ((a + b*x)*(1 + (a + b*x)^2)^(3/2) *ArcSinh[a + b*x])/4 + (3*(-1/4*(a + b*x)^2 + ((a + b*x)*Sqrt[1 + (a + b*x )^2]*ArcSinh[a + b*x])/2 + ArcSinh[a + b*x]^2/4))/4)/b
3.3.68.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
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]
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]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSinh[c*x])^n/(2*p + 1)), x] + (Simp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x ], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[x* (1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(261\) vs. \(2(92)=184\).
Time = 0.72 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.47
| method | result | size |
| default | \(\frac {4 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b^{3} x^{3}-b^{4} x^{4}+12 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a \,b^{2} x^{2}-4 a \,b^{3} x^{3}+12 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{2} b x -6 a^{2} b^{2} x^{2}+4 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{3}-4 a^{3} b x +10 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b x -a^{4}-5 b^{2} x^{2}+10 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a -10 a b x +3 \operatorname {arcsinh}\left (b x +a \right )^{2}-5 a^{2}-4}{16 b}\) | \(262\) |
1/16*(4*arcsinh(b*x+a)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*b^3*x^3-b^4*x^4+12*ar csinh(b*x+a)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a*b^2*x^2-4*a*b^3*x^3+12*arcsin h(b*x+a)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a^2*b*x-6*a^2*b^2*x^2+4*arcsinh(b*x +a)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a^3-4*a^3*b*x+10*arcsinh(b*x+a)*(b^2*x^2 +2*a*b*x+a^2+1)^(1/2)*b*x-a^4-5*b^2*x^2+10*arcsinh(b*x+a)*(b^2*x^2+2*a*b*x +a^2+1)^(1/2)*a-10*a*b*x+3*arcsinh(b*x+a)^2-5*a^2-4)/b
Time = 0.27 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.51 \[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x) \, dx=-\frac {b^{4} x^{4} + 4 \, a b^{3} x^{3} + {\left (6 \, a^{2} + 5\right )} b^{2} x^{2} + 2 \, {\left (2 \, a^{3} + 5 \, a\right )} b x - 2 \, {\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} + {\left (6 \, a^{2} + 5\right )} b x + 5 \, a\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 3 \, \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2}}{16 \, b} \]
-1/16*(b^4*x^4 + 4*a*b^3*x^3 + (6*a^2 + 5)*b^2*x^2 + 2*(2*a^3 + 5*a)*b*x - 2*(2*b^3*x^3 + 6*a*b^2*x^2 + 2*a^3 + (6*a^2 + 5)*b*x + 5*a)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - 3* log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2)/b
Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (95) = 190\).
Time = 0.49 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.81 \[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x) \, dx=\begin {cases} - \frac {a^{3} x}{4} + \frac {a^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{4 b} - \frac {3 a^{2} b x^{2}}{8} + \frac {3 a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{4} - \frac {a b^{2} x^{3}}{4} + \frac {3 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{4} - \frac {5 a x}{8} + \frac {5 a \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{8 b} - \frac {b^{3} x^{4}}{16} + \frac {b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{4} - \frac {5 b x^{2}}{16} + \frac {5 x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{8} + \frac {3 \operatorname {asinh}^{2}{\left (a + b x \right )}}{16 b} & \text {for}\: b \neq 0 \\x \left (a^{2} + 1\right )^{\frac {3}{2}} \operatorname {asinh}{\left (a \right )} & \text {otherwise} \end {cases} \]
Piecewise((-a**3*x/4 + a**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)/(4*b) - 3*a**2*b*x**2/8 + 3*a**2*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)/4 - a*b**2*x**3/4 + 3*a*b*x**2*sqrt(a**2 + 2*a*b*x + b* *2*x**2 + 1)*asinh(a + b*x)/4 - 5*a*x/8 + 5*a*sqrt(a**2 + 2*a*b*x + b**2*x **2 + 1)*asinh(a + b*x)/(8*b) - b**3*x**4/16 + b**2*x**3*sqrt(a**2 + 2*a*b *x + b**2*x**2 + 1)*asinh(a + b*x)/4 - 5*b*x**2/16 + 5*x*sqrt(a**2 + 2*a*b *x + b**2*x**2 + 1)*asinh(a + b*x)/8 + 3*asinh(a + b*x)**2/(16*b), Ne(b, 0 )), (x*(a**2 + 1)**(3/2)*asinh(a), True))
Leaf count of result is larger than twice the leaf count of optimal. 394 vs. \(2 (92) = 184\).
Time = 0.28 (sec) , antiderivative size = 394, normalized size of antiderivative = 3.72 \[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x) \, dx=-\frac {1}{16} \, {\left (b^{2} x^{4} + 4 \, a b x^{3} + 6 \, a^{2} x^{2} + \frac {4 \, a^{3} x}{b} + 5 \, x^{2} + \frac {10 \, a x}{b} + \frac {6 \, \operatorname {arsinh}\left (b x + a\right ) \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b^{2}} - \frac {3 \, \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )^{2}}{b^{2}}\right )} b + \frac {1}{8} \, {\left (2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} x + \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a}{b} + \frac {3 \, {\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} a^{2} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b^{3}} - \frac {3 \, {\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x}{b^{2}} - \frac {3 \, {\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} {\left (a^{2} + 1\right )} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b^{3}} - \frac {3 \, {\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{b^{3}}\right )} \operatorname {arsinh}\left (b x + a\right ) \]
-1/16*(b^2*x^4 + 4*a*b*x^3 + 6*a^2*x^2 + 4*a^3*x/b + 5*x^2 + 10*a*x/b + 6* arcsinh(b*x + a)*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2 ))/b^2 - 3*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))^2/b ^2)*b + 1/8*(2*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)*x + 2*(b^2*x^2 + 2*a*b* x + a^2 + 1)^(3/2)*a/b + 3*(a^2*b^2 - (a^2 + 1)*b^2)*a^2*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^3 - 3*(a^2*b^2 - (a^2 + 1)*b^ 2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*x/b^2 - 3*(a^2*b^2 - (a^2 + 1)*b^2)*( a^2 + 1)*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^3 - 3*(a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a/b^3)*arcs inh(b*x + a)
\[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x) \, dx=\int { {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (b x + a\right ) \,d x } \]
Timed out. \[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x) \, dx=\int \mathrm {asinh}\left (a+b\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}^{3/2} \,d x \]