Integrand size = 14, antiderivative size = 98 \[ \int (d+e x) (a+b \arcsin (c x)) \, dx=\frac {3 b d \sqrt {1-c^2 x^2}}{4 c}+\frac {b (d+e x) \sqrt {1-c^2 x^2}}{4 c}-\frac {b \left (2 d^2+\frac {e^2}{c^2}\right ) \arcsin (c x)}{4 e}+\frac {(d+e x)^2 (a+b \arcsin (c x))}{2 e} \]
-1/4*b*(2*d^2+e^2/c^2)*arcsin(c*x)/e+1/2*(e*x+d)^2*(a+b*arcsin(c*x))/e+3/4 *b*d*(-c^2*x^2+1)^(1/2)/c+1/4*b*(e*x+d)*(-c^2*x^2+1)^(1/2)/c
Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.94 \[ \int (d+e x) (a+b \arcsin (c x)) \, dx=a d x+\frac {1}{2} a e x^2+\frac {b d \sqrt {1-c^2 x^2}}{c}+\frac {b e x \sqrt {1-c^2 x^2}}{4 c}-\frac {b e \arcsin (c x)}{4 c^2}+b d x \arcsin (c x)+\frac {1}{2} b e x^2 \arcsin (c x) \]
a*d*x + (a*e*x^2)/2 + (b*d*Sqrt[1 - c^2*x^2])/c + (b*e*x*Sqrt[1 - c^2*x^2] )/(4*c) - (b*e*ArcSin[c*x])/(4*c^2) + b*d*x*ArcSin[c*x] + (b*e*x^2*ArcSin[ c*x])/2
Time = 0.23 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5242, 497, 25, 455, 223}
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.
\(\displaystyle \int (d+e x) (a+b \arcsin (c x)) \, dx\) |
\(\Big \downarrow \) 5242 |
\(\displaystyle \frac {(d+e x)^2 (a+b \arcsin (c x))}{2 e}-\frac {b c \int \frac {(d+e x)^2}{\sqrt {1-c^2 x^2}}dx}{2 e}\) |
\(\Big \downarrow \) 497 |
\(\displaystyle \frac {(d+e x)^2 (a+b \arcsin (c x))}{2 e}-\frac {b c \left (-\frac {\int -\frac {2 d^2 c^2+3 d e x c^2+e^2}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {e \sqrt {1-c^2 x^2} (d+e x)}{2 c^2}\right )}{2 e}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {(d+e x)^2 (a+b \arcsin (c x))}{2 e}-\frac {b c \left (\frac {\int \frac {2 d^2 c^2+3 d e x c^2+e^2}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {e \sqrt {1-c^2 x^2} (d+e x)}{2 c^2}\right )}{2 e}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {(d+e x)^2 (a+b \arcsin (c x))}{2 e}-\frac {b c \left (\frac {\left (2 c^2 d^2+e^2\right ) \int \frac {1}{\sqrt {1-c^2 x^2}}dx-3 d e \sqrt {1-c^2 x^2}}{2 c^2}-\frac {e \sqrt {1-c^2 x^2} (d+e x)}{2 c^2}\right )}{2 e}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {(d+e x)^2 (a+b \arcsin (c x))}{2 e}-\frac {b c \left (\frac {\frac {\arcsin (c x) \left (2 c^2 d^2+e^2\right )}{c}-3 d e \sqrt {1-c^2 x^2}}{2 c^2}-\frac {e \sqrt {1-c^2 x^2} (d+e x)}{2 c^2}\right )}{2 e}\) |
((d + e*x)^2*(a + b*ArcSin[c*x]))/(2*e) - (b*c*(-1/2*(e*(d + e*x)*Sqrt[1 - c^2*x^2])/c^2 + (-3*d*e*Sqrt[1 - c^2*x^2] + ((2*c^2*d^2 + e^2)*ArcSin[c*x ])/c)/(2*c^2)))/(2*e)
3.1.3.3.1 Defintions of rubi rules used
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[1/(b *(n + 2*p + 1)) Int[(c + d*x)^(n - 2)*(a + b*x^2)^p*Simp[b*c^2*(n + 2*p + 1) - a*d^2*(n - 1) + 2*b*c*d*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, n , p}, x] && If[RationalQ[n], GtQ[n, 1], SumSimplerQ[n, -2]] && NeQ[n + 2*p + 1, 0] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(m_.), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 1))), x] - Simp[b*c*(n/(e*(m + 1))) Int[(d + e*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Time = 0.02 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.88
method | result | size |
parts | \(a \left (\frac {1}{2} e \,x^{2}+d x \right )+\frac {b \left (\frac {c \arcsin \left (c x \right ) x^{2} e}{2}+\arcsin \left (c x \right ) d c x -\frac {e \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )-2 d c \sqrt {-c^{2} x^{2}+1}}{2 c}\right )}{c}\) | \(86\) |
derivativedivides | \(\frac {\frac {a \left (d \,c^{2} x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b \left (\arcsin \left (c x \right ) c^{2} x d +\frac {\arcsin \left (c x \right ) c^{2} e \,x^{2}}{2}-\frac {e \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{2}+d c \sqrt {-c^{2} x^{2}+1}\right )}{c}}{c}\) | \(97\) |
default | \(\frac {\frac {a \left (d \,c^{2} x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b \left (\arcsin \left (c x \right ) c^{2} x d +\frac {\arcsin \left (c x \right ) c^{2} e \,x^{2}}{2}-\frac {e \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{2}+d c \sqrt {-c^{2} x^{2}+1}\right )}{c}}{c}\) | \(97\) |
a*(1/2*e*x^2+d*x)+b/c*(1/2*c*arcsin(c*x)*x^2*e+arcsin(c*x)*d*c*x-1/2/c*(e* (-1/2*c*x*(-c^2*x^2+1)^(1/2)+1/2*arcsin(c*x))-2*d*c*(-c^2*x^2+1)^(1/2)))
Time = 0.25 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.78 \[ \int (d+e x) (a+b \arcsin (c x)) \, dx=\frac {2 \, a c^{2} e x^{2} + 4 \, a c^{2} d x + {\left (2 \, b c^{2} e x^{2} + 4 \, b c^{2} d x - b e\right )} \arcsin \left (c x\right ) + {\left (b c e x + 4 \, b c d\right )} \sqrt {-c^{2} x^{2} + 1}}{4 \, c^{2}} \]
1/4*(2*a*c^2*e*x^2 + 4*a*c^2*d*x + (2*b*c^2*e*x^2 + 4*b*c^2*d*x - b*e)*arc sin(c*x) + (b*c*e*x + 4*b*c*d)*sqrt(-c^2*x^2 + 1))/c^2
Time = 0.17 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.01 \[ \int (d+e x) (a+b \arcsin (c x)) \, dx=\begin {cases} a d x + \frac {a e x^{2}}{2} + b d x \operatorname {asin}{\left (c x \right )} + \frac {b e x^{2} \operatorname {asin}{\left (c x \right )}}{2} + \frac {b d \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {b e x \sqrt {- c^{2} x^{2} + 1}}{4 c} - \frac {b e \operatorname {asin}{\left (c x \right )}}{4 c^{2}} & \text {for}\: c \neq 0 \\a \left (d x + \frac {e x^{2}}{2}\right ) & \text {otherwise} \end {cases} \]
Piecewise((a*d*x + a*e*x**2/2 + b*d*x*asin(c*x) + b*e*x**2*asin(c*x)/2 + b *d*sqrt(-c**2*x**2 + 1)/c + b*e*x*sqrt(-c**2*x**2 + 1)/(4*c) - b*e*asin(c* x)/(4*c**2), Ne(c, 0)), (a*(d*x + e*x**2/2), True))
Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.83 \[ \int (d+e x) (a+b \arcsin (c x)) \, dx=\frac {1}{2} \, a e x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b e + a d x + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d}{c} \]
1/2*a*e*x^2 + 1/4*(2*x^2*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x/c^2 - arcsi n(c*x)/c^3))*b*e + a*d*x + (c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*b*d/c
Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00 \[ \int (d+e x) (a+b \arcsin (c x)) \, dx=b d x \arcsin \left (c x\right ) + a d x + \frac {\sqrt {-c^{2} x^{2} + 1} b e x}{4 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} b e \arcsin \left (c x\right )}{2 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d}{c} + \frac {{\left (c^{2} x^{2} - 1\right )} a e}{2 \, c^{2}} + \frac {b e \arcsin \left (c x\right )}{4 \, c^{2}} \]
b*d*x*arcsin(c*x) + a*d*x + 1/4*sqrt(-c^2*x^2 + 1)*b*e*x/c + 1/2*(c^2*x^2 - 1)*b*e*arcsin(c*x)/c^2 + sqrt(-c^2*x^2 + 1)*b*d/c + 1/2*(c^2*x^2 - 1)*a* e/c^2 + 1/4*b*e*arcsin(c*x)/c^2
Time = 0.42 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79 \[ \int (d+e x) (a+b \arcsin (c x)) \, dx=\frac {a\,x\,\left (2\,d+e\,x\right )}{2}+\frac {b\,e\,\left (\frac {\mathrm {asin}\left (c\,x\right )\,\left (2\,c^2\,x^2-1\right )}{4}+\frac {c\,x\,\sqrt {1-c^2\,x^2}}{4}\right )}{c^2}+\frac {b\,d\,\left (\sqrt {1-c^2\,x^2}+c\,x\,\mathrm {asin}\left (c\,x\right )\right )}{c} \]