3.180 \(\int (c e+d e x) (a+b \sin ^{-1}(c+d x)) \, dx\)

Optimal. Leaf size=70 \[ \frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )}{2 d}+\frac {b e \sqrt {1-(c+d x)^2} (c+d x)}{4 d}-\frac {b e \sin ^{-1}(c+d x)}{4 d} \]

[Out]

-1/4*b*e*arcsin(d*x+c)/d+1/2*e*(d*x+c)^2*(a+b*arcsin(d*x+c))/d+1/4*b*e*(d*x+c)*(1-(d*x+c)^2)^(1/2)/d

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {4805, 12, 4627, 321, 216} \[ \frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )}{2 d}+\frac {b e \sqrt {1-(c+d x)^2} (c+d x)}{4 d}-\frac {b e \sin ^{-1}(c+d x)}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)*(a + b*ArcSin[c + d*x]),x]

[Out]

(b*e*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(4*d) - (b*e*ArcSin[c + d*x])/(4*d) + (e*(c + d*x)^2*(a + b*ArcSin[c + d
*x]))/(2*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 216

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]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 4627

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcSi
n[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSin[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 4805

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]

Rubi steps

\begin {align*} \int (c e+d e x) \left (a+b \sin ^{-1}(c+d x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int e x \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {e \operatorname {Subst}\left (\int x \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )}{2 d}-\frac {(b e) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{2 d}\\ &=\frac {b e (c+d x) \sqrt {1-(c+d x)^2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )}{2 d}-\frac {(b e) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{4 d}\\ &=\frac {b e (c+d x) \sqrt {1-(c+d x)^2}}{4 d}-\frac {b e \sin ^{-1}(c+d x)}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )}{2 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.09, size = 59, normalized size = 0.84 \[ \frac {e \left (2 (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )+b \sqrt {1-(c+d x)^2} (c+d x)-b \sin ^{-1}(c+d x)\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)*(a + b*ArcSin[c + d*x]),x]

[Out]

(e*(b*(c + d*x)*Sqrt[1 - (c + d*x)^2] - b*ArcSin[c + d*x] + 2*(c + d*x)^2*(a + b*ArcSin[c + d*x])))/(4*d)

________________________________________________________________________________________

fricas [A]  time = 0.46, size = 93, normalized size = 1.33 \[ \frac {2 \, a d^{2} e x^{2} + 4 \, a c d e x + {\left (2 \, b d^{2} e x^{2} + 4 \, b c d e x + {\left (2 \, b c^{2} - b\right )} e\right )} \arcsin \left (d x + c\right ) + {\left (b d e x + b c e\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)*(a+b*arcsin(d*x+c)),x, algorithm="fricas")

[Out]

1/4*(2*a*d^2*e*x^2 + 4*a*c*d*e*x + (2*b*d^2*e*x^2 + 4*b*c*d*e*x + (2*b*c^2 - b)*e)*arcsin(d*x + c) + (b*d*e*x
+ b*c*e)*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1))/d

________________________________________________________________________________________

giac [A]  time = 0.64, size = 81, normalized size = 1.16 \[ \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} b \arcsin \left (d x + c\right ) e}{2 \, d} + \frac {\sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} b e}{4 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} a e}{2 \, d} + \frac {b \arcsin \left (d x + c\right ) e}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)*(a+b*arcsin(d*x+c)),x, algorithm="giac")

[Out]

1/2*((d*x + c)^2 - 1)*b*arcsin(d*x + c)*e/d + 1/4*sqrt(-(d*x + c)^2 + 1)*(d*x + c)*b*e/d + 1/2*((d*x + c)^2 -
1)*a*e/d + 1/4*b*arcsin(d*x + c)*e/d

________________________________________________________________________________________

maple [A]  time = 0.01, size = 64, normalized size = 0.91 \[ \frac {\frac {e \left (d x +c \right )^{2} a}{2}+b e \left (\frac {\arcsin \left (d x +c \right ) \left (d x +c \right )^{2}}{2}+\frac {\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{4}-\frac {\arcsin \left (d x +c \right )}{4}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)*(a+b*arcsin(d*x+c)),x)

[Out]

1/d*(1/2*e*(d*x+c)^2*a+b*e*(1/2*arcsin(d*x+c)*(d*x+c)^2+1/4*(d*x+c)*(1-(d*x+c)^2)^(1/2)-1/4*arcsin(d*x+c)))

________________________________________________________________________________________

maxima [B]  time = 0.42, size = 204, normalized size = 2.91 \[ \frac {1}{2} \, a d e x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \arcsin \left (d x + c\right ) + d {\left (\frac {3 \, c^{2} \arcsin \left (-\frac {d^{2} x + c d}{\sqrt {c^{2} d^{2} - {\left (c^{2} - 1\right )} d^{2}}}\right )}{d^{3}} + \frac {\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} x}{d^{2}} - \frac {{\left (c^{2} - 1\right )} \arcsin \left (-\frac {d^{2} x + c d}{\sqrt {c^{2} d^{2} - {\left (c^{2} - 1\right )} d^{2}}}\right )}{d^{3}} - \frac {3 \, \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} c}{d^{3}}\right )}\right )} b d e + a c e x + \frac {{\left ({\left (d x + c\right )} \arcsin \left (d x + c\right ) + \sqrt {-{\left (d x + c\right )}^{2} + 1}\right )} b c e}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)*(a+b*arcsin(d*x+c)),x, algorithm="maxima")

[Out]

1/2*a*d*e*x^2 + 1/4*(2*x^2*arcsin(d*x + c) + d*(3*c^2*arcsin(-(d^2*x + c*d)/sqrt(c^2*d^2 - (c^2 - 1)*d^2))/d^3
+ sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*x/d^2 - (c^2 - 1)*arcsin(-(d^2*x + c*d)/sqrt(c^2*d^2 - (c^2 - 1)*d^2))/d
^3 - 3*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*c/d^3))*b*d*e + a*c*e*x + ((d*x + c)*arcsin(d*x + c) + sqrt(-(d*x +
c)^2 + 1))*b*c*e/d

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (c\,e+d\,e\,x\right )\,\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)*(a + b*asin(c + d*x)),x)

[Out]

int((c*e + d*e*x)*(a + b*asin(c + d*x)), x)

________________________________________________________________________________________

sympy [A]  time = 0.31, size = 148, normalized size = 2.11 \[ \begin {cases} a c e x + \frac {a d e x^{2}}{2} + \frac {b c^{2} e \operatorname {asin}{\left (c + d x \right )}}{2 d} + b c e x \operatorname {asin}{\left (c + d x \right )} + \frac {b c e \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{4 d} + \frac {b d e x^{2} \operatorname {asin}{\left (c + d x \right )}}{2} + \frac {b e x \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{4} - \frac {b e \operatorname {asin}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\c e x \left (a + b \operatorname {asin}{\relax (c )}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)*(a+b*asin(d*x+c)),x)

[Out]

Piecewise((a*c*e*x + a*d*e*x**2/2 + b*c**2*e*asin(c + d*x)/(2*d) + b*c*e*x*asin(c + d*x) + b*c*e*sqrt(-c**2 -
2*c*d*x - d**2*x**2 + 1)/(4*d) + b*d*e*x**2*asin(c + d*x)/2 + b*e*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/4 -
b*e*asin(c + d*x)/(4*d), Ne(d, 0)), (c*e*x*(a + b*asin(c)), True))

________________________________________________________________________________________