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]

(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)

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Rubi [A]  time = 0.0404498, antiderivative size = 70, normalized size of antiderivative = 1., 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 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]

Rule 12

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

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 = 0.0678682, 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)

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Maple [A]  time = 0.004, size = 64, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({\frac{e \left ( dx+c \right ) ^{2}a}{2}}+eb \left ({\frac{\arcsin \left ( dx+c \right ) \left ( dx+c \right ) ^{2}}{2}}+{\frac{dx+c}{4}\sqrt{1- \left ( dx+c \right ) ^{2}}}-{\frac{\arcsin \left ( dx+c \right ) }{4}} \right ) \right ) } \end{align*}

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+e*b*(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)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 2.10735, size = 213, normalized size = 3.04 \begin{align*} \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} \end{align*}

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

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Sympy [A]  time = 0.475252, size = 148, normalized size = 2.11 \begin{align*} \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}{\left (c \right )}\right ) & \text{otherwise} \end{cases} \end{align*}

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))

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Giac [A]  time = 1.18343, size = 109, normalized size = 1.56 \begin{align*} \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} \end{align*}

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