3.184 \(\int \frac{a+b \sin ^{-1}(c+d x)}{(c e+d e x)^3} \, dx\)

Optimal. Leaf size=61 \[ -\frac{a+b \sin ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}-\frac{b \sqrt{1-(c+d x)^2}}{2 d e^3 (c+d x)} \]

[Out]

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

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Rubi [A]  time = 0.0530201, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {4805, 12, 4627, 264} \[ -\frac{a+b \sin ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}-\frac{b \sqrt{1-(c+d x)^2}}{2 d e^3 (c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^3,x]

[Out]

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

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 264

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

Rubi steps

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

Mathematica [A]  time = 0.0535338, size = 49, normalized size = 0.8 \[ -\frac{a+b (c+d x) \sqrt{1-(c+d x)^2}+b \sin ^{-1}(c+d x)}{2 d e^3 (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^3,x]

[Out]

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

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Maple [A]  time = 0.006, size = 62, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( -{\frac{a}{2\,{e}^{3} \left ( dx+c \right ) ^{2}}}+{\frac{b}{{e}^{3}} \left ( -{\frac{\arcsin \left ( dx+c \right ) }{2\, \left ( dx+c \right ) ^{2}}}-{\frac{1}{2\,dx+2\,c}\sqrt{1- \left ( dx+c \right ) ^{2}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(d*x+c))/(d*e*x+c*e)^3,x)

[Out]

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

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Maxima [B]  time = 1.46822, size = 162, normalized size = 2.66 \begin{align*} -\frac{1}{2} \, b{\left (\frac{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} d}{d^{3} e^{3} x + c d^{2} e^{3}} + \frac{\arcsin \left (d x + c\right )}{d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}}\right )} - \frac{a}{2 \,{\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(d*x+c))/(d*e*x+c*e)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.15265, size = 213, normalized size = 3.49 \begin{align*} \frac{a d^{2} x^{2} + 2 \, a c d x - b c^{2} \arcsin \left (d x + c\right ) -{\left (b c^{2} d x + b c^{3}\right )} \sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{2 \,{\left (c^{2} d^{3} e^{3} x^{2} + 2 \, c^{3} d^{2} e^{3} x + c^{4} d e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(d*x+c))/(d*e*x+c*e)^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac{b \operatorname{asin}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx}{e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(d*x+c))/(d*e*x+c*e)**3,x)

[Out]

(Integral(a/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral(b*asin(c + d*x)/(c**3 + 3*c**2*d*x
+ 3*c*d**2*x**2 + d**3*x**3), x))/e**3

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Giac [B]  time = 1.28168, size = 301, normalized size = 4.93 \begin{align*} -\frac{b \arcsin \left (d x + c\right ) e^{\left (-3\right )}}{4 \, d} - \frac{{\left (d x + c\right )}^{2} b \arcsin \left (d x + c\right ) e^{\left (-3\right )}}{8 \, d{\left (\sqrt{-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2}} - \frac{b{\left (\sqrt{-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2} \arcsin \left (d x + c\right ) e^{\left (-3\right )}}{8 \,{\left (d x + c\right )}^{2} d} - \frac{a e^{\left (-3\right )}}{4 \, d} - \frac{{\left (d x + c\right )}^{2} a e^{\left (-3\right )}}{8 \, d{\left (\sqrt{-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2}} + \frac{{\left (d x + c\right )} b e^{\left (-3\right )}}{4 \, d{\left (\sqrt{-{\left (d x + c\right )}^{2} + 1} + 1\right )}} - \frac{b{\left (\sqrt{-{\left (d x + c\right )}^{2} + 1} + 1\right )} e^{\left (-3\right )}}{4 \,{\left (d x + c\right )} d} - \frac{a{\left (\sqrt{-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2} e^{\left (-3\right )}}{8 \,{\left (d x + c\right )}^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(d*x+c))/(d*e*x+c*e)^3,x, algorithm="giac")

[Out]

-1/4*b*arcsin(d*x + c)*e^(-3)/d - 1/8*(d*x + c)^2*b*arcsin(d*x + c)*e^(-3)/(d*(sqrt(-(d*x + c)^2 + 1) + 1)^2)
- 1/8*b*(sqrt(-(d*x + c)^2 + 1) + 1)^2*arcsin(d*x + c)*e^(-3)/((d*x + c)^2*d) - 1/4*a*e^(-3)/d - 1/8*(d*x + c)
^2*a*e^(-3)/(d*(sqrt(-(d*x + c)^2 + 1) + 1)^2) + 1/4*(d*x + c)*b*e^(-3)/(d*(sqrt(-(d*x + c)^2 + 1) + 1)) - 1/4
*b*(sqrt(-(d*x + c)^2 + 1) + 1)*e^(-3)/((d*x + c)*d) - 1/8*a*(sqrt(-(d*x + c)^2 + 1) + 1)^2*e^(-3)/((d*x + c)^
2*d)