3.2.80 \(\int (c e+d e x) (a+b \text {ArcSin}(c+d x)) \, dx\) [180]

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

[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

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Rubi [A]
time = 0.03, 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 = {4889, 12, 4723, 327, 222} \begin {gather*} \frac {e (c+d x)^2 (a+b \text {ArcSin}(c+d x))}{2 d}-\frac {b e \text {ArcSin}(c+d x)}{4 d}+\frac {b e \sqrt {1-(c+d x)^2} (c+d x)}{4 d} \end {gather*}

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 222

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 327

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 4723

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 4889

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 {\text {Subst}\left (\int e x \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {e \text {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) \text {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) \text {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*}

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Mathematica [A]
time = 0.05, size = 59, normalized size = 0.84 \begin {gather*} \frac {e \left (b (c+d x) \sqrt {1-(c+d x)^2}-b \text {ArcSin}(c+d x)+2 (c+d x)^2 (a+b \text {ArcSin}(c+d x))\right )}{4 d} \end {gather*}

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 = 0.01, size = 64, normalized size = 0.91

method	result	size
derivativedivides	\(\frac {\frac {e \left (d x +c \right )^{2} a}{2}+e b \left (\frac {\left (d x +c \right )^{2} \arcsin \left (d x +c \right )}{2}+\frac {\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{4}-\frac {\arcsin \left (d x +c \right )}{4}\right )}{d}\)	\(64\)
default	\(\frac {\frac {e \left (d x +c \right )^{2} a}{2}+e b \left (\frac {\left (d x +c \right )^{2} \arcsin \left (d x +c \right )}{2}+\frac {\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{4}-\frac {\arcsin \left (d x +c \right )}{4}\right )}{d}\)	\(64\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 208 vs. \(2 (65) = 130\).
time = 0.47, size = 208, normalized size = 2.97 \begin {gather*} \frac {1}{2} \, a d x^{2} e + \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 x e + \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} \end {gather*}

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

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Fricas [A]
time = 3.30, size = 92, normalized size = 1.31 \begin {gather*} \frac {{\left (2 \, b d^{2} x^{2} + 4 \, b c d x + 2 \, b c^{2} - b\right )} \arcsin \left (d x + c\right ) e + \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} {\left (b d x + b c\right )} e + 2 \, {\left (a d^{2} x^{2} + 2 \, a c d x\right )} e}{4 \, d} \end {gather*}

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (58) = 116\).
time = 0.12, size = 148, normalized size = 2.11 \begin {gather*} \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 {gather*}

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 = 0.40, size = 77, normalized size = 1.10 \begin {gather*} \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} b e \arcsin \left (d x + c\right )}{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 e \arcsin \left (d x + c\right )}{4 \, d} \end {gather*}

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*e*arcsin(d*x + c)/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*e*arcsin(d*x + c)/d

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

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)

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