Integrand size = 33, antiderivative size = 155 \[ \int \frac {\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}}{\arcsin (a+b x)^4} \, dx=-\frac {\left (1-(a+b x)^2\right )^2}{3 b \arcsin (a+b x)^3}+\frac {2 (a+b x) \left (1-(a+b x)^2\right )^{3/2}}{3 b \arcsin (a+b x)^2}+\frac {2 \left (1-(a+b x)^2\right )}{3 b \arcsin (a+b x)}-\frac {8 (a+b x)^2 \left (1-(a+b x)^2\right )}{3 b \arcsin (a+b x)}+\frac {2 \text {Si}(2 \arcsin (a+b x))}{3 b}+\frac {4 \text {Si}(4 \arcsin (a+b x))}{3 b} \]
-1/3*(1-(b*x+a)^2)^2/b/arcsin(b*x+a)^3+2/3*(b*x+a)*(1-(b*x+a)^2)^(3/2)/b/a rcsin(b*x+a)^2+2/3*(1-(b*x+a)^2)/b/arcsin(b*x+a)-8/3*(b*x+a)^2*(1-(b*x+a)^ 2)/b/arcsin(b*x+a)+2/3*Si(2*arcsin(b*x+a))/b+4/3*Si(4*arcsin(b*x+a))/b
Time = 0.24 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.92 \[ \int \frac {\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}}{\arcsin (a+b x)^4} \, dx=\frac {\frac {\left (-1+a^2+2 a b x+b^2 x^2\right ) \left (1-a^2-2 a b x-b^2 x^2-2 (a+b x) \sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x)+2 \left (-1+4 a^2+8 a b x+4 b^2 x^2\right ) \arcsin (a+b x)^2\right )}{\arcsin (a+b x)^3}+2 \text {Si}(2 \arcsin (a+b x))+4 \text {Si}(4 \arcsin (a+b x))}{3 b} \]
(((-1 + a^2 + 2*a*b*x + b^2*x^2)*(1 - a^2 - 2*a*b*x - b^2*x^2 - 2*(a + b*x )*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x] + 2*(-1 + 4*a^2 + 8*a* b*x + 4*b^2*x^2)*ArcSin[a + b*x]^2))/ArcSin[a + b*x]^3 + 2*SinIntegral[2*A rcSin[a + b*x]] + 4*SinIntegral[4*ArcSin[a + b*x]])/(3*b)
Time = 1.36 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.12, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.485, Rules used = {5306, 5166, 5214, 5166, 5146, 4906, 27, 3042, 3780, 5214, 5146, 4906, 27, 2009, 3042, 3780}
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.
| \(\displaystyle \int \frac {\left (-a^2-2 a b x-b^2 x^2+1\right )^{3/2}}{\arcsin (a+b x)^4} \, dx\) |
| \(\Big \downarrow \) 5306 |
| \(\displaystyle \frac {\int \frac {\left (1-(a+b x)^2\right )^{3/2}}{\arcsin (a+b x)^4}d(a+b x)}{b}\) |
| \(\Big \downarrow \) 5166 |
| \(\displaystyle \frac {-\frac {4}{3} \int \frac {(a+b x) \left (1-(a+b x)^2\right )}{\arcsin (a+b x)^3}d(a+b x)-\frac {\left (1-(a+b x)^2\right )^2}{3 \arcsin (a+b x)^3}}{b}\) |
| \(\Big \downarrow \) 5214 |
| \(\displaystyle \frac {-\frac {4}{3} \left (\frac {1}{2} \int \frac {\sqrt {1-(a+b x)^2}}{\arcsin (a+b x)^2}d(a+b x)-2 \int \frac {(a+b x)^2 \sqrt {1-(a+b x)^2}}{\arcsin (a+b x)^2}d(a+b x)-\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2}}{2 \arcsin (a+b x)^2}\right )-\frac {\left (1-(a+b x)^2\right )^2}{3 \arcsin (a+b x)^3}}{b}\) |
| \(\Big \downarrow \) 5166 |
| \(\displaystyle \frac {-\frac {4}{3} \left (-2 \int \frac {(a+b x)^2 \sqrt {1-(a+b x)^2}}{\arcsin (a+b x)^2}d(a+b x)+\frac {1}{2} \left (-2 \int \frac {a+b x}{\arcsin (a+b x)}d(a+b x)-\frac {1-(a+b x)^2}{\arcsin (a+b x)}\right )-\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2}}{2 \arcsin (a+b x)^2}\right )-\frac {\left (1-(a+b x)^2\right )^2}{3 \arcsin (a+b x)^3}}{b}\) |
| \(\Big \downarrow \) 5146 |
| \(\displaystyle \frac {-\frac {4}{3} \left (-2 \int \frac {(a+b x)^2 \sqrt {1-(a+b x)^2}}{\arcsin (a+b x)^2}d(a+b x)+\frac {1}{2} \left (-2 \int \frac {(a+b x) \sqrt {1-(a+b x)^2}}{\arcsin (a+b x)}d\arcsin (a+b x)-\frac {1-(a+b x)^2}{\arcsin (a+b x)}\right )-\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2}}{2 \arcsin (a+b x)^2}\right )-\frac {\left (1-(a+b x)^2\right )^2}{3 \arcsin (a+b x)^3}}{b}\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \frac {-\frac {4}{3} \left (-2 \int \frac {(a+b x)^2 \sqrt {1-(a+b x)^2}}{\arcsin (a+b x)^2}d(a+b x)+\frac {1}{2} \left (-2 \int \frac {\sin (2 \arcsin (a+b x))}{2 \arcsin (a+b x)}d\arcsin (a+b x)-\frac {1-(a+b x)^2}{\arcsin (a+b x)}\right )-\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2}}{2 \arcsin (a+b x)^2}\right )-\frac {\left (1-(a+b x)^2\right )^2}{3 \arcsin (a+b x)^3}}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {4}{3} \left (-2 \int \frac {(a+b x)^2 \sqrt {1-(a+b x)^2}}{\arcsin (a+b x)^2}d(a+b x)+\frac {1}{2} \left (-\int \frac {\sin (2 \arcsin (a+b x))}{\arcsin (a+b x)}d\arcsin (a+b x)-\frac {1-(a+b x)^2}{\arcsin (a+b x)}\right )-\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2}}{2 \arcsin (a+b x)^2}\right )-\frac {\left (1-(a+b x)^2\right )^2}{3 \arcsin (a+b x)^3}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {4}{3} \left (-2 \int \frac {(a+b x)^2 \sqrt {1-(a+b x)^2}}{\arcsin (a+b x)^2}d(a+b x)+\frac {1}{2} \left (-\int \frac {\sin (2 \arcsin (a+b x))}{\arcsin (a+b x)}d\arcsin (a+b x)-\frac {1-(a+b x)^2}{\arcsin (a+b x)}\right )-\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2}}{2 \arcsin (a+b x)^2}\right )-\frac {\left (1-(a+b x)^2\right )^2}{3 \arcsin (a+b x)^3}}{b}\) |
| \(\Big \downarrow \) 3780 |
| \(\displaystyle \frac {-\frac {4}{3} \left (-2 \int \frac {(a+b x)^2 \sqrt {1-(a+b x)^2}}{\arcsin (a+b x)^2}d(a+b x)+\frac {1}{2} \left (-\text {Si}(2 \arcsin (a+b x))-\frac {1-(a+b x)^2}{\arcsin (a+b x)}\right )-\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2}}{2 \arcsin (a+b x)^2}\right )-\frac {\left (1-(a+b x)^2\right )^2}{3 \arcsin (a+b x)^3}}{b}\) |
| \(\Big \downarrow \) 5214 |
| \(\displaystyle \frac {-\frac {4}{3} \left (-2 \left (2 \int \frac {a+b x}{\arcsin (a+b x)}d(a+b x)-4 \int \frac {(a+b x)^3}{\arcsin (a+b x)}d(a+b x)-\frac {\left (1-(a+b x)^2\right ) (a+b x)^2}{\arcsin (a+b x)}\right )+\frac {1}{2} \left (-\text {Si}(2 \arcsin (a+b x))-\frac {1-(a+b x)^2}{\arcsin (a+b x)}\right )-\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2}}{2 \arcsin (a+b x)^2}\right )-\frac {\left (1-(a+b x)^2\right )^2}{3 \arcsin (a+b x)^3}}{b}\) |
| \(\Big \downarrow \) 5146 |
| \(\displaystyle \frac {-\frac {4}{3} \left (-2 \left (2 \int \frac {(a+b x) \sqrt {1-(a+b x)^2}}{\arcsin (a+b x)}d\arcsin (a+b x)-4 \int \frac {(a+b x)^3 \sqrt {1-(a+b x)^2}}{\arcsin (a+b x)}d\arcsin (a+b x)-\frac {\left (1-(a+b x)^2\right ) (a+b x)^2}{\arcsin (a+b x)}\right )+\frac {1}{2} \left (-\text {Si}(2 \arcsin (a+b x))-\frac {1-(a+b x)^2}{\arcsin (a+b x)}\right )-\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2}}{2 \arcsin (a+b x)^2}\right )-\frac {\left (1-(a+b x)^2\right )^2}{3 \arcsin (a+b x)^3}}{b}\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \frac {-\frac {4}{3} \left (-2 \left (2 \int \frac {\sin (2 \arcsin (a+b x))}{2 \arcsin (a+b x)}d\arcsin (a+b x)-4 \int \left (\frac {\sin (2 \arcsin (a+b x))}{4 \arcsin (a+b x)}-\frac {\sin (4 \arcsin (a+b x))}{8 \arcsin (a+b x)}\right )d\arcsin (a+b x)-\frac {\left (1-(a+b x)^2\right ) (a+b x)^2}{\arcsin (a+b x)}\right )+\frac {1}{2} \left (-\text {Si}(2 \arcsin (a+b x))-\frac {1-(a+b x)^2}{\arcsin (a+b x)}\right )-\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2}}{2 \arcsin (a+b x)^2}\right )-\frac {\left (1-(a+b x)^2\right )^2}{3 \arcsin (a+b x)^3}}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {4}{3} \left (-2 \left (\int \frac {\sin (2 \arcsin (a+b x))}{\arcsin (a+b x)}d\arcsin (a+b x)-4 \int \left (\frac {\sin (2 \arcsin (a+b x))}{4 \arcsin (a+b x)}-\frac {\sin (4 \arcsin (a+b x))}{8 \arcsin (a+b x)}\right )d\arcsin (a+b x)-\frac {\left (1-(a+b x)^2\right ) (a+b x)^2}{\arcsin (a+b x)}\right )+\frac {1}{2} \left (-\text {Si}(2 \arcsin (a+b x))-\frac {1-(a+b x)^2}{\arcsin (a+b x)}\right )-\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2}}{2 \arcsin (a+b x)^2}\right )-\frac {\left (1-(a+b x)^2\right )^2}{3 \arcsin (a+b x)^3}}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {4}{3} \left (-2 \left (\int \frac {\sin (2 \arcsin (a+b x))}{\arcsin (a+b x)}d\arcsin (a+b x)-4 \left (\frac {1}{4} \text {Si}(2 \arcsin (a+b x))-\frac {1}{8} \text {Si}(4 \arcsin (a+b x))\right )-\frac {\left (1-(a+b x)^2\right ) (a+b x)^2}{\arcsin (a+b x)}\right )+\frac {1}{2} \left (-\text {Si}(2 \arcsin (a+b x))-\frac {1-(a+b x)^2}{\arcsin (a+b x)}\right )-\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2}}{2 \arcsin (a+b x)^2}\right )-\frac {\left (1-(a+b x)^2\right )^2}{3 \arcsin (a+b x)^3}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {4}{3} \left (-2 \left (\int \frac {\sin (2 \arcsin (a+b x))}{\arcsin (a+b x)}d\arcsin (a+b x)-4 \left (\frac {1}{4} \text {Si}(2 \arcsin (a+b x))-\frac {1}{8} \text {Si}(4 \arcsin (a+b x))\right )-\frac {\left (1-(a+b x)^2\right ) (a+b x)^2}{\arcsin (a+b x)}\right )+\frac {1}{2} \left (-\text {Si}(2 \arcsin (a+b x))-\frac {1-(a+b x)^2}{\arcsin (a+b x)}\right )-\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2}}{2 \arcsin (a+b x)^2}\right )-\frac {\left (1-(a+b x)^2\right )^2}{3 \arcsin (a+b x)^3}}{b}\) |
| \(\Big \downarrow \) 3780 |
| \(\displaystyle \frac {-\frac {4}{3} \left (\frac {1}{2} \left (-\text {Si}(2 \arcsin (a+b x))-\frac {1-(a+b x)^2}{\arcsin (a+b x)}\right )-2 \left (\text {Si}(2 \arcsin (a+b x))-4 \left (\frac {1}{4} \text {Si}(2 \arcsin (a+b x))-\frac {1}{8} \text {Si}(4 \arcsin (a+b x))\right )-\frac {\left (1-(a+b x)^2\right ) (a+b x)^2}{\arcsin (a+b x)}\right )-\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2}}{2 \arcsin (a+b x)^2}\right )-\frac {\left (1-(a+b x)^2\right )^2}{3 \arcsin (a+b x)^3}}{b}\) |
(-1/3*(1 - (a + b*x)^2)^2/ArcSin[a + b*x]^3 - (4*(-1/2*((a + b*x)*(1 - (a + b*x)^2)^(3/2))/ArcSin[a + b*x]^2 + (-((1 - (a + b*x)^2)/ArcSin[a + b*x]) - SinIntegral[2*ArcSin[a + b*x]])/2 - 2*(-(((a + b*x)^2*(1 - (a + b*x)^2) )/ArcSin[a + b*x]) + SinIntegral[2*ArcSin[a + b*x]] - 4*(SinIntegral[2*Arc Sin[a + b*x]]/4 - SinIntegral[4*ArcSin[a + b*x]]/8))))/3)/b
3.4.26.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x ]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG tQ[p, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[1 /(b*c^(m + 1)) Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[Sqrt[1 - c^2*x^2]*(d + e*x^2)^p*((a + b*ArcSin[c*x])^(n + 1 )/(b*c*(n + 1))), x] + Simp[c*((2*p + 1)/(b*(n + 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*ArcSin[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_. )*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^m*Sqrt[1 - c^2*x^2]*(d + e*x^2)^p* ((a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1))), x] + (-Simp[f*(m/(b*c*(n + 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*ArcSin[c*x])^(n + 1), x], x] + Simp[c*((m + 2*p + 1)/(b*f*(n + 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*ArcSin[c*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d, e, f }, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 2*p + 1 , 0] && IGtQ[m, -3]
Int[((a_.) + ArcSin[(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*ArcSin[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]
Time = 1.84 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.95
| method | result | size |
| default | \(\frac {16 \,\operatorname {Si}\left (2 \arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )^{3}+32 \,\operatorname {Si}\left (4 \arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )^{3}+8 \cos \left (2 \arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )^{2}+8 \cos \left (4 \arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )^{2}+4 \sin \left (2 \arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )+2 \sin \left (4 \arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )-4 \cos \left (2 \arcsin \left (b x +a \right )\right )-\cos \left (4 \arcsin \left (b x +a \right )\right )-3}{24 b \arcsin \left (b x +a \right )^{3}}\) | \(148\) |
1/24/b*(16*Si(2*arcsin(b*x+a))*arcsin(b*x+a)^3+32*Si(4*arcsin(b*x+a))*arcs in(b*x+a)^3+8*cos(2*arcsin(b*x+a))*arcsin(b*x+a)^2+8*cos(4*arcsin(b*x+a))* arcsin(b*x+a)^2+4*sin(2*arcsin(b*x+a))*arcsin(b*x+a)+2*sin(4*arcsin(b*x+a) )*arcsin(b*x+a)-4*cos(2*arcsin(b*x+a))-cos(4*arcsin(b*x+a))-3)/arcsin(b*x+ a)^3
\[ \int \frac {\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}}{\arcsin (a+b x)^4} \, dx=\int { \frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}}}{\arcsin \left (b x + a\right )^{4}} \,d x } \]
\[ \int \frac {\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}}{\arcsin (a+b x)^4} \, dx=\int \frac {\left (- \left (a + b x - 1\right ) \left (a + b x + 1\right )\right )^{\frac {3}{2}}}{\operatorname {asin}^{4}{\left (a + b x \right )}}\, dx \]
Timed out. \[ \int \frac {\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}}{\arcsin (a+b x)^4} \, dx=\text {Timed out} \]
Time = 0.40 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.05 \[ \int \frac {\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}}{\arcsin (a+b x)^4} \, dx=\frac {8 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{2}}{3 \, b \arcsin \left (b x + a\right )} + \frac {4 \, \operatorname {Si}\left (4 \, \arcsin \left (b x + a\right )\right )}{3 \, b} + \frac {2 \, \operatorname {Si}\left (2 \, \arcsin \left (b x + a\right )\right )}{3 \, b} + \frac {2 \, {\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} {\left (b x + a\right )}}{3 \, b \arcsin \left (b x + a\right )^{2}} + \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}}{b \arcsin \left (b x + a\right )} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{2}}{3 \, b \arcsin \left (b x + a\right )^{3}} \]
8/3*(b^2*x^2 + 2*a*b*x + a^2 - 1)^2/(b*arcsin(b*x + a)) + 4/3*sin_integral (4*arcsin(b*x + a))/b + 2/3*sin_integral(2*arcsin(b*x + a))/b + 2/3*(-b^2* x^2 - 2*a*b*x - a^2 + 1)^(3/2)*(b*x + a)/(b*arcsin(b*x + a)^2) + 2*(b^2*x^ 2 + 2*a*b*x + a^2 - 1)/(b*arcsin(b*x + a)) - 1/3*(b^2*x^2 + 2*a*b*x + a^2 - 1)^2/(b*arcsin(b*x + a)^3)
Timed out. \[ \int \frac {\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}}{\arcsin (a+b x)^4} \, dx=\int \frac {{\left (-a^2-2\,a\,b\,x-b^2\,x^2+1\right )}^{3/2}}{{\mathrm {asin}\left (a+b\,x\right )}^4} \,d x \]