3.1.3 \(\int (d+e x) (a+b \text {ArcSin}(c x)) \, dx\) [3]

Optimal. Leaf size=98 \[ \frac {3 b d \sqrt {1-c^2 x^2}}{4 c}+\frac {b (d+e x) \sqrt {1-c^2 x^2}}{4 c}-\frac {b \left (2 d^2+\frac {e^2}{c^2}\right ) \text {ArcSin}(c x)}{4 e}+\frac {(d+e x)^2 (a+b \text {ArcSin}(c x))}{2 e} \]

[Out]

-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

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Rubi [A]
time = 0.04, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4827, 757, 655, 222} \begin {gather*} \frac {(d+e x)^2 (a+b \text {ArcSin}(c x))}{2 e}-\frac {b \text {ArcSin}(c x) \left (\frac {e^2}{c^2}+2 d^2\right )}{4 e}+\frac {b \sqrt {1-c^2 x^2} (d+e x)}{4 c}+\frac {3 b d \sqrt {1-c^2 x^2}}{4 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*(a + b*ArcSin[c*x]),x]

[Out]

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

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 655

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[e*((a + c*x^2)^(p + 1)/(2*c*(p + 1))),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 757

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*((a + c*x^2)^(p
+ 1)/(c*(m + 2*p + 1))), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - a*e^
2*(m - 1) + 2*c*d*e*(m + p)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2,
0] && If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, 0, c, d, e, m
, p, x]

Rule 4827

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

Rubi steps

\begin {align*} \int (d+e x) \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {(d+e x)^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e}-\frac {(b c) \int \frac {(d+e x)^2}{\sqrt {1-c^2 x^2}} \, dx}{2 e}\\ &=\frac {b (d+e x) \sqrt {1-c^2 x^2}}{4 c}+\frac {(d+e x)^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e}+\frac {b \int \frac {-2 c^2 d^2-e^2-3 c^2 d e x}{\sqrt {1-c^2 x^2}} \, dx}{4 c e}\\ &=\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 {(d+e x)^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e}-\frac {\left (b \left (2 c^2 d^2+e^2\right )\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 c e}\\ &=\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 ) \sin ^{-1}(c x)}{4 e}+\frac {(d+e x)^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 92, normalized size = 0.94 \begin {gather*} 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 \text {ArcSin}(c x)}{4 c^2}+b d x \text {ArcSin}(c x)+\frac {1}{2} b e x^2 \text {ArcSin}(c x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.01, size = 97, normalized size = 0.99

method	result	size
derivativedivides	\(\frac {\frac {a \left (d \,c^{2} x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b \left (\arcsin \left (c x \right ) d \,c^{2} x +\frac {\arcsin \left (c x \right ) e \,c^{2} x^{2}}{2}+d c \sqrt {-c^{2} x^{2}+1}-\frac {e \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{2}\right )}{c}}{c}\)	\(97\)
default	\(\frac {\frac {a \left (d \,c^{2} x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b \left (\arcsin \left (c x \right ) d \,c^{2} x +\frac {\arcsin \left (c x \right ) e \,c^{2} x^{2}}{2}+d c \sqrt {-c^{2} x^{2}+1}-\frac {e \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{2}\right )}{c}}{c}\)	\(97\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.48, size = 83, normalized size = 0.85 \begin {gather*} \frac {1}{2} \, a x^{2} e + a d x + \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 + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.89, size = 80, normalized size = 0.82 \begin {gather*} \frac {2 \, a c^{2} x^{2} e + 4 \, a c^{2} d x + {\left (4 \, b c^{2} d x + {\left (2 \, b c^{2} x^{2} - b\right )} e\right )} \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1} {\left (b c x e + 4 \, b c d\right )}}{4 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.12, size = 99, normalized size = 1.01 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*sq
rt(-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))

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Giac [A]
time = 0.38, size = 98, normalized size = 1.00 \begin {gather*} 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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.41, size = 77, normalized size = 0.79 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(c*x))*(d + e*x),x)

[Out]

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

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