\(\int \frac {a+b \arcsin (c x)}{(d+e x)^4} \, dx\) [8]

Optimal result

Integrand size = 16, antiderivative size = 191 \[ \int \frac {a+b \arcsin (c x)}{(d+e x)^4} \, dx=\frac {b c \sqrt {1-c^2 x^2}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c^3 d \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {a+b \arcsin (c x)}{3 e (d+e x)^3}+\frac {b c^3 \left (2 c^2 d^2+e^2\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{6 e \left (c^2 d^2-e^2\right )^{5/2}} \]

[Out]

1/3*(-a-b*arcsin(c*x))/e/(e*x+d)^3+1/6*b*c^3*(2*c^2*d^2+e^2)*arctan((c^2*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c^2*x^2+
1)^(1/2))/e/(c^2*d^2-e^2)^(5/2)+1/6*b*c*(-c^2*x^2+1)^(1/2)/(c^2*d^2-e^2)/(e*x+d)^2+1/2*b*c^3*d*(-c^2*x^2+1)^(1
/2)/(c^2*d^2-e^2)^2/(e*x+d)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {4827, 759, 821, 739, 210} \[ \int \frac {a+b \arcsin (c x)}{(d+e x)^4} \, dx=-\frac {a+b \arcsin (c x)}{3 e (d+e x)^3}+\frac {b c^3 \left (2 c^2 d^2+e^2\right ) \arctan \left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right )}{6 e \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b c \sqrt {1-c^2 x^2}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c^3 d \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)} \]

[In]

Int[(a + b*ArcSin[c*x])/(d + e*x)^4,x]

[Out]

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

Rule 210

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

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 759

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)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[c/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*Simp[d*(m + 1)
- e*(m + 2*p + 3)*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] && NeQ[
m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ[p]) || ILtQ
[Simplify[m + 2*p + 3], 0])

Rule 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g
))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e
^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0
] && EqQ[Simplify[m + 2*p + 3], 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]

Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \arcsin (c x)}{3 e (d+e x)^3}+\frac {(b c) \int \frac {1}{(d+e x)^3 \sqrt {1-c^2 x^2}} \, dx}{3 e} \\ & = \frac {b c \sqrt {1-c^2 x^2}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {a+b \arcsin (c x)}{3 e (d+e x)^3}-\frac {\left (b c^3\right ) \int \frac {-2 d+e x}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{6 e \left (c^2 d^2-e^2\right )} \\ & = \frac {b c \sqrt {1-c^2 x^2}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c^3 d \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {a+b \arcsin (c x)}{3 e (d+e x)^3}+\frac {\left (b c^3 \left (2 c^2 d^2+e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{6 e \left (c^2 d^2-e^2\right )^2} \\ & = \frac {b c \sqrt {1-c^2 x^2}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c^3 d \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {a+b \arcsin (c x)}{3 e (d+e x)^3}-\frac {\left (b c^3 \left (2 c^2 d^2+e^2\right )\right ) \text {Subst}\left (\int \frac {1}{-c^2 d^2+e^2-x^2} \, dx,x,\frac {e+c^2 d x}{\sqrt {1-c^2 x^2}}\right )}{6 e \left (c^2 d^2-e^2\right )^2} \\ & = \frac {b c \sqrt {1-c^2 x^2}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c^3 d \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {a+b \arcsin (c x)}{3 e (d+e x)^3}+\frac {b c^3 \left (2 c^2 d^2+e^2\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{6 e \left (c^2 d^2-e^2\right )^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.26 \[ \int \frac {a+b \arcsin (c x)}{(d+e x)^4} \, dx=\frac {1}{6} \left (-\frac {2 a}{e (d+e x)^3}+\frac {b \sqrt {1-c^2 x^2} \left (-c e^2+c^3 d (4 d+3 e x)\right )}{\left (-c^2 d^2+e^2\right )^2 (d+e x)^2}-\frac {2 b \arcsin (c x)}{e (d+e x)^3}+\frac {b c^3 \left (2 c^2 d^2+e^2\right ) \log (d+e x)}{e (-c d+e)^2 (c d+e)^2 \sqrt {-c^2 d^2+e^2}}-\frac {b c^3 \left (2 c^2 d^2+e^2\right ) \log \left (e+c^2 d x+\sqrt {-c^2 d^2+e^2} \sqrt {1-c^2 x^2}\right )}{e (-c d+e)^2 (c d+e)^2 \sqrt {-c^2 d^2+e^2}}\right ) \]

[In]

Integrate[(a + b*ArcSin[c*x])/(d + e*x)^4,x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(553\) vs. \(2(176)=352\).

Time = 0.21 (sec) , antiderivative size = 554, normalized size of antiderivative = 2.90

method	result	size
parts	\(-\frac {a}{3 \left (e x +d \right )^{3} e}-\frac {b \,c^{3} \arcsin \left (c x \right )}{3 \left (c e x +d c \right )^{3} e}+\frac {b \,c^{3} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{6 e^{2} \left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )^{2}}+\frac {b \,c^{4} d \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{2 e \left (c^{2} d^{2}-e^{2}\right )^{2} \left (c x +\frac {d c}{e}\right )}-\frac {b \,c^{5} d^{2} \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{2 e^{2} \left (c^{2} d^{2}-e^{2}\right )^{2} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}+\frac {b \,c^{3} \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{6 e^{2} \left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\)	\(554\)
derivativedivides	\(\frac {-\frac {a \,c^{4}}{3 \left (c e x +d c \right )^{3} e}-\frac {b \,c^{4} \arcsin \left (c x \right )}{3 \left (c e x +d c \right )^{3} e}+\frac {b \,c^{4} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{6 e^{2} \left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )^{2}}+\frac {b \,c^{5} d \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{2 e \left (c^{2} d^{2}-e^{2}\right )^{2} \left (c x +\frac {d c}{e}\right )}-\frac {b \,c^{6} d^{2} \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{2 e^{2} \left (c^{2} d^{2}-e^{2}\right )^{2} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}+\frac {b \,c^{4} \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{6 e^{2} \left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}}{c}\)	\(564\)
default	\(\frac {-\frac {a \,c^{4}}{3 \left (c e x +d c \right )^{3} e}-\frac {b \,c^{4} \arcsin \left (c x \right )}{3 \left (c e x +d c \right )^{3} e}+\frac {b \,c^{4} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{6 e^{2} \left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )^{2}}+\frac {b \,c^{5} d \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{2 e \left (c^{2} d^{2}-e^{2}\right )^{2} \left (c x +\frac {d c}{e}\right )}-\frac {b \,c^{6} d^{2} \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{2 e^{2} \left (c^{2} d^{2}-e^{2}\right )^{2} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}+\frac {b \,c^{4} \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{6 e^{2} \left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}}{c}\)	\(564\)

[In]

int((a+b*arcsin(c*x))/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 550 vs. \(2 (173) = 346\).

Time = 0.86 (sec) , antiderivative size = 1125, normalized size of antiderivative = 5.89 \[ \int \frac {a+b \arcsin (c x)}{(d+e x)^4} \, dx=\left [-\frac {4 \, a c^{6} d^{6} - 12 \, a c^{4} d^{4} e^{2} + 12 \, a c^{2} d^{2} e^{4} - 4 \, a e^{6} + {\left (2 \, b c^{5} d^{5} + b c^{3} d^{3} e^{2} + {\left (2 \, b c^{5} d^{2} e^{3} + b c^{3} e^{5}\right )} x^{3} + 3 \, {\left (2 \, b c^{5} d^{3} e^{2} + b c^{3} d e^{4}\right )} x^{2} + 3 \, {\left (2 \, b c^{5} d^{4} e + b c^{3} d^{2} e^{3}\right )} x\right )} \sqrt {-c^{2} d^{2} + e^{2}} \log \left (\frac {2 \, c^{2} d e x - c^{2} d^{2} + {\left (2 \, c^{4} d^{2} - c^{2} e^{2}\right )} x^{2} - 2 \, \sqrt {-c^{2} d^{2} + e^{2}} {\left (c^{2} d x + e\right )} \sqrt {-c^{2} x^{2} + 1} + 2 \, e^{2}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 4 \, {\left (b c^{6} d^{6} - 3 \, b c^{4} d^{4} e^{2} + 3 \, b c^{2} d^{2} e^{4} - b e^{6}\right )} \arcsin \left (c x\right ) - 2 \, {\left (4 \, b c^{5} d^{5} e - 5 \, b c^{3} d^{3} e^{3} + b c d e^{5} + 3 \, {\left (b c^{5} d^{3} e^{3} - b c^{3} d e^{5}\right )} x^{2} + {\left (7 \, b c^{5} d^{4} e^{2} - 8 \, b c^{3} d^{2} e^{4} + b c e^{6}\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{12 \, {\left (c^{6} d^{9} e - 3 \, c^{4} d^{7} e^{3} + 3 \, c^{2} d^{5} e^{5} - d^{3} e^{7} + {\left (c^{6} d^{6} e^{4} - 3 \, c^{4} d^{4} e^{6} + 3 \, c^{2} d^{2} e^{8} - e^{10}\right )} x^{3} + 3 \, {\left (c^{6} d^{7} e^{3} - 3 \, c^{4} d^{5} e^{5} + 3 \, c^{2} d^{3} e^{7} - d e^{9}\right )} x^{2} + 3 \, {\left (c^{6} d^{8} e^{2} - 3 \, c^{4} d^{6} e^{4} + 3 \, c^{2} d^{4} e^{6} - d^{2} e^{8}\right )} x\right )}}, -\frac {2 \, a c^{6} d^{6} - 6 \, a c^{4} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} - 2 \, a e^{6} - {\left (2 \, b c^{5} d^{5} + b c^{3} d^{3} e^{2} + {\left (2 \, b c^{5} d^{2} e^{3} + b c^{3} e^{5}\right )} x^{3} + 3 \, {\left (2 \, b c^{5} d^{3} e^{2} + b c^{3} d e^{4}\right )} x^{2} + 3 \, {\left (2 \, b c^{5} d^{4} e + b c^{3} d^{2} e^{3}\right )} x\right )} \sqrt {c^{2} d^{2} - e^{2}} \arctan \left (\frac {\sqrt {c^{2} d^{2} - e^{2}} {\left (c^{2} d x + e\right )} \sqrt {-c^{2} x^{2} + 1}}{c^{2} d^{2} - {\left (c^{4} d^{2} - c^{2} e^{2}\right )} x^{2} - e^{2}}\right ) + 2 \, {\left (b c^{6} d^{6} - 3 \, b c^{4} d^{4} e^{2} + 3 \, b c^{2} d^{2} e^{4} - b e^{6}\right )} \arcsin \left (c x\right ) - {\left (4 \, b c^{5} d^{5} e - 5 \, b c^{3} d^{3} e^{3} + b c d e^{5} + 3 \, {\left (b c^{5} d^{3} e^{3} - b c^{3} d e^{5}\right )} x^{2} + {\left (7 \, b c^{5} d^{4} e^{2} - 8 \, b c^{3} d^{2} e^{4} + b c e^{6}\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{6 \, {\left (c^{6} d^{9} e - 3 \, c^{4} d^{7} e^{3} + 3 \, c^{2} d^{5} e^{5} - d^{3} e^{7} + {\left (c^{6} d^{6} e^{4} - 3 \, c^{4} d^{4} e^{6} + 3 \, c^{2} d^{2} e^{8} - e^{10}\right )} x^{3} + 3 \, {\left (c^{6} d^{7} e^{3} - 3 \, c^{4} d^{5} e^{5} + 3 \, c^{2} d^{3} e^{7} - d e^{9}\right )} x^{2} + 3 \, {\left (c^{6} d^{8} e^{2} - 3 \, c^{4} d^{6} e^{4} + 3 \, c^{2} d^{4} e^{6} - d^{2} e^{8}\right )} x\right )}}\right ] \]

[In]

integrate((a+b*arcsin(c*x))/(e*x+d)^4,x, algorithm="fricas")

[Out]

[-1/12*(4*a*c^6*d^6 - 12*a*c^4*d^4*e^2 + 12*a*c^2*d^2*e^4 - 4*a*e^6 + (2*b*c^5*d^5 + b*c^3*d^3*e^2 + (2*b*c^5*
d^2*e^3 + b*c^3*e^5)*x^3 + 3*(2*b*c^5*d^3*e^2 + b*c^3*d*e^4)*x^2 + 3*(2*b*c^5*d^4*e + b*c^3*d^2*e^3)*x)*sqrt(-
c^2*d^2 + e^2)*log((2*c^2*d*e*x - c^2*d^2 + (2*c^4*d^2 - c^2*e^2)*x^2 - 2*sqrt(-c^2*d^2 + e^2)*(c^2*d*x + e)*s
qrt(-c^2*x^2 + 1) + 2*e^2)/(e^2*x^2 + 2*d*e*x + d^2)) + 4*(b*c^6*d^6 - 3*b*c^4*d^4*e^2 + 3*b*c^2*d^2*e^4 - b*e
^6)*arcsin(c*x) - 2*(4*b*c^5*d^5*e - 5*b*c^3*d^3*e^3 + b*c*d*e^5 + 3*(b*c^5*d^3*e^3 - b*c^3*d*e^5)*x^2 + (7*b*
c^5*d^4*e^2 - 8*b*c^3*d^2*e^4 + b*c*e^6)*x)*sqrt(-c^2*x^2 + 1))/(c^6*d^9*e - 3*c^4*d^7*e^3 + 3*c^2*d^5*e^5 - d
^3*e^7 + (c^6*d^6*e^4 - 3*c^4*d^4*e^6 + 3*c^2*d^2*e^8 - e^10)*x^3 + 3*(c^6*d^7*e^3 - 3*c^4*d^5*e^5 + 3*c^2*d^3
*e^7 - d*e^9)*x^2 + 3*(c^6*d^8*e^2 - 3*c^4*d^6*e^4 + 3*c^2*d^4*e^6 - d^2*e^8)*x), -1/6*(2*a*c^6*d^6 - 6*a*c^4*
d^4*e^2 + 6*a*c^2*d^2*e^4 - 2*a*e^6 - (2*b*c^5*d^5 + b*c^3*d^3*e^2 + (2*b*c^5*d^2*e^3 + b*c^3*e^5)*x^3 + 3*(2*
b*c^5*d^3*e^2 + b*c^3*d*e^4)*x^2 + 3*(2*b*c^5*d^4*e + b*c^3*d^2*e^3)*x)*sqrt(c^2*d^2 - e^2)*arctan(sqrt(c^2*d^
2 - e^2)*(c^2*d*x + e)*sqrt(-c^2*x^2 + 1)/(c^2*d^2 - (c^4*d^2 - c^2*e^2)*x^2 - e^2)) + 2*(b*c^6*d^6 - 3*b*c^4*
d^4*e^2 + 3*b*c^2*d^2*e^4 - b*e^6)*arcsin(c*x) - (4*b*c^5*d^5*e - 5*b*c^3*d^3*e^3 + b*c*d*e^5 + 3*(b*c^5*d^3*e
^3 - b*c^3*d*e^5)*x^2 + (7*b*c^5*d^4*e^2 - 8*b*c^3*d^2*e^4 + b*c*e^6)*x)*sqrt(-c^2*x^2 + 1))/(c^6*d^9*e - 3*c^
4*d^7*e^3 + 3*c^2*d^5*e^5 - d^3*e^7 + (c^6*d^6*e^4 - 3*c^4*d^4*e^6 + 3*c^2*d^2*e^8 - e^10)*x^3 + 3*(c^6*d^7*e^
3 - 3*c^4*d^5*e^5 + 3*c^2*d^3*e^7 - d*e^9)*x^2 + 3*(c^6*d^8*e^2 - 3*c^4*d^6*e^4 + 3*c^2*d^4*e^6 - d^2*e^8)*x)]

Sympy [F]

\[ \int \frac {a+b \arcsin (c x)}{(d+e x)^4} \, dx=\int \frac {a + b \operatorname {asin}{\left (c x \right )}}{\left (d + e x\right )^{4}}\, dx \]

[In]

integrate((a+b*asin(c*x))/(e*x+d)**4,x)

[Out]

Integral((a + b*asin(c*x))/(d + e*x)**4, x)

Maxima [F]

\[ \int \frac {a+b \arcsin (c x)}{(d+e x)^4} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (e x + d\right )}^{4}} \,d x } \]

[In]

integrate((a+b*arcsin(c*x))/(e*x+d)^4,x, algorithm="maxima")

[Out]

-1/3*(3*(c*e^4*x^3 + 3*c*d*e^3*x^2 + 3*c*d^2*e^2*x + c*d^3*e)*integrate(1/3*e^(1/2*log(c*x + 1) + 1/2*log(-c*x
+ 1))/(c^4*e^4*x^7 + 3*c^4*d*e^3*x^6 - 3*c^2*d^2*e^2*x^3 - c^2*d^3*e*x^2 + (3*c^4*d^2*e^2 - c^2*e^4)*x^5 + (c
^4*d^3*e - 3*c^2*d*e^3)*x^4 + (c^2*e^4*x^5 + 3*c^2*d*e^3*x^4 - 3*d^2*e^2*x - d^3*e + (3*c^2*d^2*e^2 - e^4)*x^3
+ (c^2*d^3*e - 3*d*e^3)*x^2)*e^(log(c*x + 1) + log(-c*x + 1))), x) + arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1
)))*b/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e) - 1/3*a/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e)

Giac [F]

\[ \int \frac {a+b \arcsin (c x)}{(d+e x)^4} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (e x + d\right )}^{4}} \,d x } \]

[In]

integrate((a+b*arcsin(c*x))/(e*x+d)^4,x, algorithm="giac")

[Out]

integrate((b*arcsin(c*x) + a)/(e*x + d)^4, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \arcsin (c x)}{(d+e x)^4} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{{\left (d+e\,x\right )}^4} \,d x \]

[In]

int((a + b*asin(c*x))/(d + e*x)^4,x)

[Out]

int((a + b*asin(c*x))/(d + e*x)^4, x)