3.187 \(\int \frac{a+b \sin ^{-1}(c+d x)}{(c e+d e x)^6} \, dx\)

Optimal. Leaf size=121 \[ -\frac{a+b \sin ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}-\frac{3 b \sqrt{1-(c+d x)^2}}{40 d e^6 (c+d x)^2}-\frac{b \sqrt{1-(c+d x)^2}}{20 d e^6 (c+d x)^4}-\frac{3 b \tanh ^{-1}\left (\sqrt{1-(c+d x)^2}\right )}{40 d e^6} \]

[Out]

-(b*Sqrt[1 - (c + d*x)^2])/(20*d*e^6*(c + d*x)^4) - (3*b*Sqrt[1 - (c + d*x)^2])/(40*d*e^6*(c + d*x)^2) - (a +
b*ArcSin[c + d*x])/(5*d*e^6*(c + d*x)^5) - (3*b*ArcTanh[Sqrt[1 - (c + d*x)^2]])/(40*d*e^6)

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Rubi [A]  time = 0.0925666, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4805, 12, 4627, 266, 51, 63, 206} \[ -\frac{a+b \sin ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}-\frac{3 b \sqrt{1-(c+d x)^2}}{40 d e^6 (c+d x)^2}-\frac{b \sqrt{1-(c+d x)^2}}{20 d e^6 (c+d x)^4}-\frac{3 b \tanh ^{-1}\left (\sqrt{1-(c+d x)^2}\right )}{40 d e^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^6,x]

[Out]

-(b*Sqrt[1 - (c + d*x)^2])/(20*d*e^6*(c + d*x)^4) - (3*b*Sqrt[1 - (c + d*x)^2])/(40*d*e^6*(c + d*x)^2) - (a +
b*ArcSin[c + d*x])/(5*d*e^6*(c + d*x)^5) - (3*b*ArcTanh[Sqrt[1 - (c + d*x)^2]])/(40*d*e^6)

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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{a+b \sin ^{-1}(c+d x)}{(c e+d e x)^6} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b \sin ^{-1}(x)}{e^6 x^6} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{a+b \sin ^{-1}(x)}{x^6} \, dx,x,c+d x\right )}{d e^6}\\ &=-\frac{a+b \sin ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac{b \operatorname{Subst}\left (\int \frac{1}{x^5 \sqrt{1-x^2}} \, dx,x,c+d x\right )}{5 d e^6}\\ &=-\frac{a+b \sin ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x^3} \, dx,x,(c+d x)^2\right )}{10 d e^6}\\ &=-\frac{b \sqrt{1-(c+d x)^2}}{20 d e^6 (c+d x)^4}-\frac{a+b \sin ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x^2} \, dx,x,(c+d x)^2\right )}{40 d e^6}\\ &=-\frac{b \sqrt{1-(c+d x)^2}}{20 d e^6 (c+d x)^4}-\frac{3 b \sqrt{1-(c+d x)^2}}{40 d e^6 (c+d x)^2}-\frac{a+b \sin ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x} \, dx,x,(c+d x)^2\right )}{80 d e^6}\\ &=-\frac{b \sqrt{1-(c+d x)^2}}{20 d e^6 (c+d x)^4}-\frac{3 b \sqrt{1-(c+d x)^2}}{40 d e^6 (c+d x)^2}-\frac{a+b \sin ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-(c+d x)^2}\right )}{40 d e^6}\\ &=-\frac{b \sqrt{1-(c+d x)^2}}{20 d e^6 (c+d x)^4}-\frac{3 b \sqrt{1-(c+d x)^2}}{40 d e^6 (c+d x)^2}-\frac{a+b \sin ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}-\frac{3 b \tanh ^{-1}\left (\sqrt{1-(c+d x)^2}\right )}{40 d e^6}\\ \end{align*}

Mathematica [C]  time = 0.0392163, size = 68, normalized size = 0.56 \[ \frac{-\frac{1}{5} b \sqrt{1-(c+d x)^2} \text{Hypergeometric2F1}\left (\frac{1}{2},3,\frac{3}{2},1-(c+d x)^2\right )-\frac{a+b \sin ^{-1}(c+d x)}{5 (c+d x)^5}}{d e^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^6,x]

[Out]

(-(a + b*ArcSin[c + d*x])/(5*(c + d*x)^5) - (b*Sqrt[1 - (c + d*x)^2]*Hypergeometric2F1[1/2, 3, 3/2, 1 - (c + d
*x)^2])/5)/(d*e^6)

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Maple [A]  time = 0.004, size = 100, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( -{\frac{a}{5\,{e}^{6} \left ( dx+c \right ) ^{5}}}+{\frac{b}{{e}^{6}} \left ( -{\frac{\arcsin \left ( dx+c \right ) }{5\, \left ( dx+c \right ) ^{5}}}-{\frac{1}{20\, \left ( dx+c \right ) ^{4}}\sqrt{1- \left ( dx+c \right ) ^{2}}}-{\frac{3}{40\, \left ( dx+c \right ) ^{2}}\sqrt{1- \left ( dx+c \right ) ^{2}}}-{\frac{3}{40}{\it Artanh} \left ({\frac{1}{\sqrt{1- \left ( dx+c \right ) ^{2}}}} \right ) } \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)^6,x)

[Out]

1/d*(-1/5*a/e^6/(d*x+c)^5+b/e^6*(-1/5/(d*x+c)^5*arcsin(d*x+c)-1/20/(d*x+c)^4*(1-(d*x+c)^2)^(1/2)-3/40/(d*x+c)^
2*(1-(d*x+c)^2)^(1/2)-3/40*arctanh(1/(1-(d*x+c)^2)^(1/2))))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left ({\left (d^{6} e^{6} x^{5} + 5 \, c d^{5} e^{6} x^{4} + 10 \, c^{2} d^{4} e^{6} x^{3} + 10 \, c^{3} d^{3} e^{6} x^{2} + 5 \, c^{4} d^{2} e^{6} x + c^{5} d e^{6}\right )} \int \frac{e^{\left (\frac{1}{2} \, \log \left (d x + c + 1\right ) + \frac{1}{2} \, \log \left (-d x - c + 1\right )\right )}}{d^{9} e^{6} x^{9} + 9 \, c d^{8} e^{6} x^{8} +{\left (36 \, c^{2} - 1\right )} d^{7} e^{6} x^{7} + 7 \,{\left (12 \, c^{3} - c\right )} d^{6} e^{6} x^{6} + 21 \,{\left (6 \, c^{4} - c^{2}\right )} d^{5} e^{6} x^{5} + 7 \,{\left (18 \, c^{5} - 5 \, c^{3}\right )} d^{4} e^{6} x^{4} + 7 \,{\left (12 \, c^{6} - 5 \, c^{4}\right )} d^{3} e^{6} x^{3} + 3 \,{\left (12 \, c^{7} - 7 \, c^{5}\right )} d^{2} e^{6} x^{2} +{\left (9 \, c^{8} - 7 \, c^{6}\right )} d e^{6} x +{\left (c^{9} - c^{7}\right )} e^{6} -{\left (d^{7} e^{6} x^{7} + 7 \, c d^{6} e^{6} x^{6} +{\left (21 \, c^{2} - 1\right )} d^{5} e^{6} x^{5} + 5 \,{\left (7 \, c^{3} - c\right )} d^{4} e^{6} x^{4} + 5 \,{\left (7 \, c^{4} - 2 \, c^{2}\right )} d^{3} e^{6} x^{3} +{\left (21 \, c^{5} - 10 \, c^{3}\right )} d^{2} e^{6} x^{2} +{\left (7 \, c^{6} - 5 \, c^{4}\right )} d e^{6} x +{\left (c^{7} - c^{5}\right )} e^{6}\right )}{\left (d x + c + 1\right )}{\left (d x + c - 1\right )}}\,{d x} + \arctan \left (d x + c, \sqrt{d x + c + 1} \sqrt{-d x - c + 1}\right )\right )} b}{5 \,{\left (d^{6} e^{6} x^{5} + 5 \, c d^{5} e^{6} x^{4} + 10 \, c^{2} d^{4} e^{6} x^{3} + 10 \, c^{3} d^{3} e^{6} x^{2} + 5 \, c^{4} d^{2} e^{6} x + c^{5} d e^{6}\right )}} - \frac{a}{5 \,{\left (d^{6} e^{6} x^{5} + 5 \, c d^{5} e^{6} x^{4} + 10 \, c^{2} d^{4} e^{6} x^{3} + 10 \, c^{3} d^{3} e^{6} x^{2} + 5 \, c^{4} d^{2} e^{6} x + c^{5} d e^{6}\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)^6,x, algorithm="maxima")

[Out]

-1/5*(5*(d^6*e^6*x^5 + 5*c*d^5*e^6*x^4 + 10*c^2*d^4*e^6*x^3 + 10*c^3*d^3*e^6*x^2 + 5*c^4*d^2*e^6*x + c^5*d*e^6
)*integrate(1/5*e^(1/2*log(d*x + c + 1) + 1/2*log(-d*x - c + 1))/(d^9*e^6*x^9 + 9*c*d^8*e^6*x^8 + (36*c^2 - 1)
*d^7*e^6*x^7 + 7*(12*c^3 - c)*d^6*e^6*x^6 + 21*(6*c^4 - c^2)*d^5*e^6*x^5 + 7*(18*c^5 - 5*c^3)*d^4*e^6*x^4 + 7*
(12*c^6 - 5*c^4)*d^3*e^6*x^3 + 3*(12*c^7 - 7*c^5)*d^2*e^6*x^2 + (9*c^8 - 7*c^6)*d*e^6*x + (c^9 - c^7)*e^6 + (d
^7*e^6*x^7 + 7*c*d^6*e^6*x^6 + (21*c^2 - 1)*d^5*e^6*x^5 + 5*(7*c^3 - c)*d^4*e^6*x^4 + 5*(7*c^4 - 2*c^2)*d^3*e^
6*x^3 + (21*c^5 - 10*c^3)*d^2*e^6*x^2 + (7*c^6 - 5*c^4)*d*e^6*x + (c^7 - c^5)*e^6)*e^(log(d*x + c + 1) + log(-
d*x - c + 1))), x) + arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))*b/(d^6*e^6*x^5 + 5*c*d^5*e^6*x^4
+ 10*c^2*d^4*e^6*x^3 + 10*c^3*d^3*e^6*x^2 + 5*c^4*d^2*e^6*x + c^5*d*e^6) - 1/5*a/(d^6*e^6*x^5 + 5*c*d^5*e^6*x^
4 + 10*c^2*d^4*e^6*x^3 + 10*c^3*d^3*e^6*x^2 + 5*c^4*d^2*e^6*x + c^5*d*e^6)

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Fricas [B]  time = 3.22165, size = 705, normalized size = 5.83 \begin{align*} -\frac{16 \, b \arcsin \left (d x + c\right ) + 3 \,{\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x + b c^{5}\right )} \log \left (\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} + 1\right ) - 3 \,{\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x + b c^{5}\right )} \log \left (\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} - 1\right ) + 2 \,{\left (3 \, b d^{3} x^{3} + 9 \, b c d^{2} x^{2} + 3 \, b c^{3} +{\left (9 \, b c^{2} + 2 \, b\right )} d x + 2 \, b c\right )} \sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} + 16 \, a}{80 \,{\left (d^{6} e^{6} x^{5} + 5 \, c d^{5} e^{6} x^{4} + 10 \, c^{2} d^{4} e^{6} x^{3} + 10 \, c^{3} d^{3} e^{6} x^{2} + 5 \, c^{4} d^{2} e^{6} x + c^{5} d e^{6}\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)^6,x, algorithm="fricas")

[Out]

-1/80*(16*b*arcsin(d*x + c) + 3*(b*d^5*x^5 + 5*b*c*d^4*x^4 + 10*b*c^2*d^3*x^3 + 10*b*c^3*d^2*x^2 + 5*b*c^4*d*x
+ b*c^5)*log(sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1) + 1) - 3*(b*d^5*x^5 + 5*b*c*d^4*x^4 + 10*b*c^2*d^3*x^3 + 10*b
*c^3*d^2*x^2 + 5*b*c^4*d*x + b*c^5)*log(sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1) - 1) + 2*(3*b*d^3*x^3 + 9*b*c*d^2*x
^2 + 3*b*c^3 + (9*b*c^2 + 2*b)*d*x + 2*b*c)*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1) + 16*a)/(d^6*e^6*x^5 + 5*c*d^5*
e^6*x^4 + 10*c^2*d^4*e^6*x^3 + 10*c^3*d^3*e^6*x^2 + 5*c^4*d^2*e^6*x + c^5*d*e^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a}{c^{6} + 6 c^{5} d x + 15 c^{4} d^{2} x^{2} + 20 c^{3} d^{3} x^{3} + 15 c^{2} d^{4} x^{4} + 6 c d^{5} x^{5} + d^{6} x^{6}}\, dx + \int \frac{b \operatorname{asin}{\left (c + d x \right )}}{c^{6} + 6 c^{5} d x + 15 c^{4} d^{2} x^{2} + 20 c^{3} d^{3} x^{3} + 15 c^{2} d^{4} x^{4} + 6 c d^{5} x^{5} + d^{6} x^{6}}\, dx}{e^{6}} \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)**6,x)

[Out]

(Integral(a/(c**6 + 6*c**5*d*x + 15*c**4*d**2*x**2 + 20*c**3*d**3*x**3 + 15*c**2*d**4*x**4 + 6*c*d**5*x**5 + d
**6*x**6), x) + Integral(b*asin(c + d*x)/(c**6 + 6*c**5*d*x + 15*c**4*d**2*x**2 + 20*c**3*d**3*x**3 + 15*c**2*
d**4*x**4 + 6*c*d**5*x**5 + d**6*x**6), x))/e**6

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Giac [B]  time = 1.81915, size = 783, normalized size = 6.47 \begin{align*} \text{result too large to display} \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)^6,x, algorithm="giac")

[Out]

-1/160*(d*x + c)^5*b*arcsin(d*x + c)*e^(-6)/(d*(sqrt(-(d*x + c)^2 + 1) + 1)^5) - 1/32*(d*x + c)^3*b*arcsin(d*x
+ c)*e^(-6)/(d*(sqrt(-(d*x + c)^2 + 1) + 1)^3) - 1/16*(d*x + c)*b*arcsin(d*x + c)*e^(-6)/(d*(sqrt(-(d*x + c)^
2 + 1) + 1)) - 1/16*b*(sqrt(-(d*x + c)^2 + 1) + 1)*arcsin(d*x + c)*e^(-6)/((d*x + c)*d) - 1/32*b*(sqrt(-(d*x +
c)^2 + 1) + 1)^3*arcsin(d*x + c)*e^(-6)/((d*x + c)^3*d) - 1/160*b*(sqrt(-(d*x + c)^2 + 1) + 1)^5*arcsin(d*x +
c)*e^(-6)/((d*x + c)^5*d) - 3/40*b*e^(-6)*log(sqrt(-(d*x + c)^2 + 1) + 1)/d + 3/40*b*e^(-6)*log(abs(d*x + c))
/d - 1/160*(d*x + c)^5*a*e^(-6)/(d*(sqrt(-(d*x + c)^2 + 1) + 1)^5) + 1/320*(d*x + c)^4*b*e^(-6)/(d*(sqrt(-(d*x
+ c)^2 + 1) + 1)^4) - 1/32*(d*x + c)^3*a*e^(-6)/(d*(sqrt(-(d*x + c)^2 + 1) + 1)^3) + 1/40*(d*x + c)^2*b*e^(-6
)/(d*(sqrt(-(d*x + c)^2 + 1) + 1)^2) - 1/16*(d*x + c)*a*e^(-6)/(d*(sqrt(-(d*x + c)^2 + 1) + 1)) - 1/16*a*(sqrt
(-(d*x + c)^2 + 1) + 1)*e^(-6)/((d*x + c)*d) - 1/40*b*(sqrt(-(d*x + c)^2 + 1) + 1)^2*e^(-6)/((d*x + c)^2*d) -
1/32*a*(sqrt(-(d*x + c)^2 + 1) + 1)^3*e^(-6)/((d*x + c)^3*d) - 1/320*b*(sqrt(-(d*x + c)^2 + 1) + 1)^4*e^(-6)/(
(d*x + c)^4*d) - 1/160*a*(sqrt(-(d*x + c)^2 + 1) + 1)^5*e^(-6)/((d*x + c)^5*d)