3.1.11 \(\int (d+e x) (a+b \text {ArcSin}(c x))^2 \, dx\) [11]

Optimal. Leaf size=142 \[ -2 b^2 d x-\frac {1}{4} b^2 e x^2+\frac {2 b d \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c}+\frac {b e x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{2 c}-\frac {d^2 (a+b \text {ArcSin}(c x))^2}{2 e}-\frac {e (a+b \text {ArcSin}(c x))^2}{4 c^2}+\frac {(d+e x)^2 (a+b \text {ArcSin}(c x))^2}{2 e} \]

[Out]

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

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Rubi [A]
time = 0.20, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {4827, 4847, 4737, 4767, 8, 4795, 30} \begin {gather*} \frac {2 b d \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c}+\frac {b e x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{2 c}-\frac {e (a+b \text {ArcSin}(c x))^2}{4 c^2}-\frac {d^2 (a+b \text {ArcSin}(c x))^2}{2 e}+\frac {(d+e x)^2 (a+b \text {ArcSin}(c x))^2}{2 e}-2 b^2 d x-\frac {1}{4} b^2 e x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*(a + b*ArcSin[c*x])^2,x]

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c
^2*d + e, 0] && NeQ[n, -1]

Rule 4767

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^(
p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p
], Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*
d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 4795

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[f
*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m
+ 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x],
x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]

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]

Rule 4847

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g},
x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n, 0] && (m == 1 || p > 0 ||
(n == 1 && p > -1) || (m == 2 && p < -2))

Rubi steps

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

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Mathematica [A]
time = 0.18, size = 147, normalized size = 1.04 \begin {gather*} \frac {c \left (2 a^2 c x (2 d+e x)-b^2 c x (8 d+e x)+2 a b (4 d+e x) \sqrt {1-c^2 x^2}\right )+2 b \left (4 a c^2 d x+b c (4 d+e x) \sqrt {1-c^2 x^2}+a e \left (-1+2 c^2 x^2\right )\right ) \text {ArcSin}(c x)+b^2 \left (4 c^2 d x+e \left (-1+2 c^2 x^2\right )\right ) \text {ArcSin}(c x)^2}{4 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)*(a + b*ArcSin[c*x])^2,x]

[Out]

(c*(2*a^2*c*x*(2*d + e*x) - b^2*c*x*(8*d + e*x) + 2*a*b*(4*d + e*x)*Sqrt[1 - c^2*x^2]) + 2*b*(4*a*c^2*d*x + b*
c*(4*d + e*x)*Sqrt[1 - c^2*x^2] + a*e*(-1 + 2*c^2*x^2))*ArcSin[c*x] + b^2*(4*c^2*d*x + e*(-1 + 2*c^2*x^2))*Arc
Sin[c*x]^2)/(4*c^2)

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Maple [A]
time = 0.08, size = 198, normalized size = 1.39

method	result	size
derivativedivides	\(\frac {\frac {a^{2} \left (d \,c^{2} x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b^{2} \left (d c \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {e \left (2 \arcsin \left (c x \right )^{2} c^{2} x^{2}+2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{4}\right )}{c}+\frac {2 a b \left (\arcsin \left (c x \right ) d \,c^{2} x +\frac {\arcsin \left (c x \right ) e \,c^{2} x^{2}}{2}+d c \sqrt {-c^{2} x^{2}+1}-\frac {e \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{2}\right )}{c}}{c}\)	\(198\)
default	\(\frac {\frac {a^{2} \left (d \,c^{2} x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b^{2} \left (d c \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {e \left (2 \arcsin \left (c x \right )^{2} c^{2} x^{2}+2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{4}\right )}{c}+\frac {2 a b \left (\arcsin \left (c x \right ) d \,c^{2} x +\frac {\arcsin \left (c x \right ) e \,c^{2} x^{2}}{2}+d c \sqrt {-c^{2} x^{2}+1}-\frac {e \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{2}\right )}{c}}{c}\)	\(198\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(a+b*arcsin(c*x))^2,x,method=_RETURNVERBOSE)

[Out]

1/c*(a^2/c*(d*c^2*x+1/2*c^2*e*x^2)+b^2/c*(d*c*(c*x*arcsin(c*x)^2-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2))+1/4*e
*(2*arcsin(c*x)^2*c^2*x^2+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c*x-arcsin(c*x)^2-c^2*x^2))+2*a*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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

b^2*d*x*arcsin(c*x)^2 + 1/2*a^2*x^2*e - 2*b^2*d*(x - sqrt(-c^2*x^2 + 1)*arcsin(c*x)/c) + a^2*d*x + 1/2*(2*x^2*
arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x/c^2 - arcsin(c*x)/c^3))*a*b*e + 1/2*(x^2*arctan2(c*x, sqrt(c*x + 1)*sqrt
(-c*x + 1))^2 + 2*c*integrate(sqrt(c*x + 1)*sqrt(-c*x + 1)*x^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^2
*x^2 - 1), x))*b^2*e + 2*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*a*b*d/c

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Fricas [A]
time = 2.13, size = 163, normalized size = 1.15 \begin {gather*} \frac {{\left (2 \, a^{2} - b^{2}\right )} c^{2} x^{2} e + 4 \, {\left (a^{2} - 2 \, b^{2}\right )} c^{2} d x + {\left (4 \, b^{2} c^{2} d x + {\left (2 \, b^{2} c^{2} x^{2} - b^{2}\right )} e\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (4 \, a b c^{2} d x + {\left (2 \, a b c^{2} x^{2} - a b\right )} e\right )} \arcsin \left (c x\right ) + 2 \, {\left (a b c x e + 4 \, a b c d + {\left (b^{2} c x e + 4 \, b^{2} c d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{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))^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.18, size = 233, normalized size = 1.64 \begin {gather*} \begin {cases} a^{2} d x + \frac {a^{2} e x^{2}}{2} + 2 a b d x \operatorname {asin}{\left (c x \right )} + a b e x^{2} \operatorname {asin}{\left (c x \right )} + \frac {2 a b d \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {a b e x \sqrt {- c^{2} x^{2} + 1}}{2 c} - \frac {a b e \operatorname {asin}{\left (c x \right )}}{2 c^{2}} + b^{2} d x \operatorname {asin}^{2}{\left (c x \right )} - 2 b^{2} d x + \frac {b^{2} e x^{2} \operatorname {asin}^{2}{\left (c x \right )}}{2} - \frac {b^{2} e x^{2}}{4} + \frac {2 b^{2} d \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{c} + \frac {b^{2} e x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{2 c} - \frac {b^{2} e \operatorname {asin}^{2}{\left (c x \right )}}{4 c^{2}} & \text {for}\: c \neq 0 \\a^{2} \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))**2,x)

[Out]

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

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Giac [A]
time = 0.40, size = 244, normalized size = 1.72 \begin {gather*} b^{2} d x \arcsin \left (c x\right )^{2} + 2 \, a b d x \arcsin \left (c x\right ) + \frac {\sqrt {-c^{2} x^{2} + 1} b^{2} e x \arcsin \left (c x\right )}{2 \, c} + a^{2} d x - 2 \, b^{2} d x + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} e \arcsin \left (c x\right )^{2}}{2 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} a b e x}{2 \, c} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d \arcsin \left (c x\right )}{c} + \frac {{\left (c^{2} x^{2} - 1\right )} a b e \arcsin \left (c x\right )}{c^{2}} + \frac {b^{2} e \arcsin \left (c x\right )^{2}}{4 \, c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b d}{c} + \frac {{\left (c^{2} x^{2} - 1\right )} a^{2} e}{2 \, c^{2}} - \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} e}{4 \, c^{2}} + \frac {a b e \arcsin \left (c x\right )}{2 \, c^{2}} - \frac {b^{2} e}{8 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

b^2*d*x*arcsin(c*x)^2 + 2*a*b*d*x*arcsin(c*x) + 1/2*sqrt(-c^2*x^2 + 1)*b^2*e*x*arcsin(c*x)/c + a^2*d*x - 2*b^2
*d*x + 1/2*(c^2*x^2 - 1)*b^2*e*arcsin(c*x)^2/c^2 + 1/2*sqrt(-c^2*x^2 + 1)*a*b*e*x/c + 2*sqrt(-c^2*x^2 + 1)*b^2
*d*arcsin(c*x)/c + (c^2*x^2 - 1)*a*b*e*arcsin(c*x)/c^2 + 1/4*b^2*e*arcsin(c*x)^2/c^2 + 2*sqrt(-c^2*x^2 + 1)*a*
b*d/c + 1/2*(c^2*x^2 - 1)*a^2*e/c^2 - 1/4*(c^2*x^2 - 1)*b^2*e/c^2 + 1/2*a*b*e*arcsin(c*x)/c^2 - 1/8*b^2*e/c^2

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (d+e\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(c*x))^2*(d + e*x),x)

[Out]

int((a + b*asin(c*x))^2*(d + e*x), x)

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