\(\int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^4} \, dx\) [94]

Optimal result

Integrand size = 21, antiderivative size = 257 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^4} \, dx=\frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{6 e \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c \left (c^2 d f-e g\right ) \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {(e f-d g) (a+b \arcsin (c x))}{3 e^2 (d+e x)^3}-\frac {g (a+b \arcsin (c x))}{2 e^2 (d+e x)^2}+\frac {b c^3 \left (e^2 (e f-4 d g)+c^2 d^2 (2 e f+d g)\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{6 e^2 \left (c^2 d^2-e^2\right )^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {45, 4837, 12, 849, 821, 739, 210} \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^4} \, dx=-\frac {(e f-d g) (a+b \arcsin (c x))}{3 e^2 (d+e x)^3}-\frac {g (a+b \arcsin (c x))}{2 e^2 (d+e x)^2}+\frac {b c^3 \arctan \left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right ) \left (c^2 d^2 (d g+2 e f)+e^2 (e f-4 d g)\right )}{6 e^2 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b c \sqrt {1-c^2 x^2} \left (c^2 d f-e g\right )}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {b c \sqrt {1-c^2 x^2} (e f-d g)}{6 e \left (c^2 d^2-e^2\right ) (d+e x)^2} \]

[In]

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

[Out]

(b*c*(e*f - d*g)*Sqrt[1 - c^2*x^2])/(6*e*(c^2*d^2 - e^2)*(d + e*x)^2) + (b*c*(c^2*d*f - e*g)*Sqrt[1 - c^2*x^2]
)/(2*(c^2*d^2 - e^2)^2*(d + e*x)) - ((e*f - d*g)*(a + b*ArcSin[c*x]))/(3*e^2*(d + e*x)^3) - (g*(a + b*ArcSin[c
*x]))/(2*e^2*(d + e*x)^2) + (b*c^3*(e^2*(e*f - 4*d*g) + c^2*d^2*(2*e*f + d*g))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*
d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(6*e^2*(c^2*d^2 - e^2)^(5/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 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 849

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)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rule 4837

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> With[{u = IntHide[Px*(d
+ e*x)^m, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]
] /; FreeQ[{a, b, c, d, e, m}, x] && PolynomialQ[Px, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {(e f-d g) (a+b \arcsin (c x))}{3 e^2 (d+e x)^3}-\frac {g (a+b \arcsin (c x))}{2 e^2 (d+e x)^2}-(b c) \int \frac {-2 e f-d g-3 e g x}{6 e^2 (d+e x)^3 \sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {(e f-d g) (a+b \arcsin (c x))}{3 e^2 (d+e x)^3}-\frac {g (a+b \arcsin (c x))}{2 e^2 (d+e x)^2}-\frac {(b c) \int \frac {-2 e f-d g-3 e g x}{(d+e x)^3 \sqrt {1-c^2 x^2}} \, dx}{6 e^2} \\ & = \frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{6 e \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{3 e^2 (d+e x)^3}-\frac {g (a+b \arcsin (c x))}{2 e^2 (d+e x)^2}-\frac {(b c) \int \frac {2 \left (3 e^2 g-c^2 d (2 e f+d g)\right )+2 c^2 e (e f-d g) x}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{12 e^2 \left (c^2 d^2-e^2\right )} \\ & = \frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{6 e \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c \left (c^2 d f-e g\right ) \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {(e f-d g) (a+b \arcsin (c x))}{3 e^2 (d+e x)^3}-\frac {g (a+b \arcsin (c x))}{2 e^2 (d+e x)^2}+\frac {\left (b c^3 \left (e^2 (e f-4 d g)+c^2 d^2 (2 e f+d g)\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{6 e^2 \left (c^2 d^2-e^2\right )^2} \\ & = \frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{6 e \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c \left (c^2 d f-e g\right ) \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {(e f-d g) (a+b \arcsin (c x))}{3 e^2 (d+e x)^3}-\frac {g (a+b \arcsin (c x))}{2 e^2 (d+e x)^2}-\frac {\left (b c^3 \left (e^2 (e f-4 d g)+c^2 d^2 (2 e f+d g)\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^2 \left (c^2 d^2-e^2\right )^2} \\ & = \frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{6 e \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c \left (c^2 d f-e g\right ) \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {(e f-d g) (a+b \arcsin (c x))}{3 e^2 (d+e x)^3}-\frac {g (a+b \arcsin (c x))}{2 e^2 (d+e x)^2}+\frac {b c^3 \left (e^2 (e f-4 d g)+c^2 d^2 (2 e f+d g)\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^2 \left (c^2 d^2-e^2\right )^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.25 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^4} \, dx=\frac {\frac {a (-2 e f+2 d g)}{(d+e x)^3}-\frac {3 a g}{(d+e x)^2}+\frac {b c e \sqrt {1-c^2 x^2} \left (c^2 d \left (4 d e f-d^2 g+3 e^2 f x\right )-e^2 (2 d g+e (f+3 g x))\right )}{\left (-c^2 d^2+e^2\right )^2 (d+e x)^2}-\frac {b (2 e f+d g+3 e g x) \arcsin (c x)}{(d+e x)^3}+\frac {b c^3 \left (e^2 (e f-4 d g)+c^2 d^2 (2 e f+d g)\right ) \log (d+e x)}{(-c d+e)^2 (c d+e)^2 \sqrt {-c^2 d^2+e^2}}-\frac {b c^3 \left (e^2 (e f-4 d g)+c^2 d^2 (2 e f+d g)\right ) \log \left (e+c^2 d x+\sqrt {-c^2 d^2+e^2} \sqrt {1-c^2 x^2}\right )}{(-c d+e)^2 (c d+e)^2 \sqrt {-c^2 d^2+e^2}}}{6 e^2} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(907\) vs. \(2(237)=474\).

Time = 3.43 (sec) , antiderivative size = 908, normalized size of antiderivative = 3.53

method	result	size
parts	\(a \left (-\frac {g}{2 e^{2} \left (e x +d \right )^{2}}-\frac {-d g +e f}{3 e^{2} \left (e x +d \right )^{3}}\right )+\frac {b \left (\frac {c^{4} \arcsin \left (c x \right ) d g}{3 e^{2} \left (c e x +d c \right )^{3}}-\frac {c^{4} \arcsin \left (c x \right ) f}{3 e \left (c e x +d c \right )^{3}}-\frac {c^{3} \arcsin \left (c x \right ) g}{2 e^{2} \left (c e x +d c \right )^{2}}+\frac {c^{3} \left (\frac {3 g \left (\frac {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}}}}{\left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {d c e \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 )}{\left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{e^{2}}-\frac {2 c \left (d g -e f \right ) \left (\frac {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}}}}{2 \left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )^{2}}+\frac {3 d c e \left (\frac {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}}}}{\left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {d c e \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 )}{\left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{2 \left (c^{2} d^{2}-e^{2}\right )}+\frac {e^{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 \left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{e^{3}}\right )}{6 e^{2}}\right )}{c}\)	\(908\)
derivativedivides	\(\frac {a \,c^{3} \left (\frac {c \left (d g -e f \right )}{3 e^{2} \left (c e x +d c \right )^{3}}-\frac {g}{2 e^{2} \left (c e x +d c \right )^{2}}\right )+b \,c^{3} \left (\frac {\arcsin \left (c x \right ) c d g}{3 e^{2} \left (c e x +d c \right )^{3}}-\frac {\arcsin \left (c x \right ) c f}{3 e \left (c e x +d c \right )^{3}}-\frac {\arcsin \left (c x \right ) g}{2 e^{2} \left (c e x +d c \right )^{2}}+\frac {\frac {3 g \left (\frac {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}}}}{\left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {d c e \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 )}{\left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{e^{2}}-\frac {2 c \left (d g -e f \right ) \left (\frac {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}}}}{2 \left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )^{2}}+\frac {3 d c e \left (\frac {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}}}}{\left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {d c e \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 )}{\left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{2 \left (c^{2} d^{2}-e^{2}\right )}+\frac {e^{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 \left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{e^{3}}}{6 e^{2}}\right )}{c}\)	\(912\)
default	\(\frac {a \,c^{3} \left (\frac {c \left (d g -e f \right )}{3 e^{2} \left (c e x +d c \right )^{3}}-\frac {g}{2 e^{2} \left (c e x +d c \right )^{2}}\right )+b \,c^{3} \left (\frac {\arcsin \left (c x \right ) c d g}{3 e^{2} \left (c e x +d c \right )^{3}}-\frac {\arcsin \left (c x \right ) c f}{3 e \left (c e x +d c \right )^{3}}-\frac {\arcsin \left (c x \right ) g}{2 e^{2} \left (c e x +d c \right )^{2}}+\frac {\frac {3 g \left (\frac {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}}}}{\left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {d c e \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 )}{\left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{e^{2}}-\frac {2 c \left (d g -e f \right ) \left (\frac {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}}}}{2 \left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )^{2}}+\frac {3 d c e \left (\frac {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}}}}{\left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {d c e \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 )}{\left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{2 \left (c^{2} d^{2}-e^{2}\right )}+\frac {e^{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 \left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{e^{3}}}{6 e^{2}}\right )}{c}\)	\(912\)

[In]

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

[Out]

a*(-1/2*g/e^2/(e*x+d)^2-1/3*(-d*g+e*f)/e^2/(e*x+d)^3)+b/c*(1/3*c^4*arcsin(c*x)/e^2/(c*e*x+c*d)^3*d*g-1/3*c^4*a
rcsin(c*x)/e/(c*e*x+c*d)^3*f-1/2*c^3*arcsin(c*x)*g/e^2/(c*e*x+c*d)^2+1/6*c^3/e^2*(3*g/e^2*(1/(c^2*d^2-e^2)*e^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)-d*c*e/(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)))-2*c*(d*g-e*f)/e^3*(1/2/(c^2*d^2-e^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)+3/2*d*c*e/(c^2*d^2-e^2)*(1/(c^2*d^2-e^2)*e^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)-d*c*e/(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)))+1/2/(c^2*d^2-e^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. 946 vs. \(2 (237) = 474\).

Time = 16.15 (sec) , antiderivative size = 1920, normalized size of antiderivative = 7.47 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[-1/12*(6*(a*c^6*d^6*e - 3*a*c^4*d^4*e^3 + 3*a*c^2*d^2*e^5 - a*e^7)*g*x - sqrt(-c^2*d^2 + e^2)*(((2*b*c^5*d^2*
e^4 + b*c^3*e^6)*f + (b*c^5*d^3*e^3 - 4*b*c^3*d*e^5)*g)*x^3 + 3*((2*b*c^5*d^3*e^3 + b*c^3*d*e^5)*f + (b*c^5*d^
4*e^2 - 4*b*c^3*d^2*e^4)*g)*x^2 + (2*b*c^5*d^5*e + b*c^3*d^3*e^3)*f + (b*c^5*d^6 - 4*b*c^3*d^4*e^2)*g + 3*((2*
b*c^5*d^4*e^2 + b*c^3*d^2*e^4)*f + (b*c^5*d^5*e - 4*b*c^3*d^3*e^3)*g)*x)*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)*sqrt(-c^2*x^2 + 1) + 2*e^2)/(e^2*x^2 + 2*d*e*x + d^2)
) + 4*(a*c^6*d^6*e - 3*a*c^4*d^4*e^3 + 3*a*c^2*d^2*e^5 - a*e^7)*f + 2*(a*c^6*d^7 - 3*a*c^4*d^5*e^2 + 3*a*c^2*d
^3*e^4 - a*d*e^6)*g + 2*(3*(b*c^6*d^6*e - 3*b*c^4*d^4*e^3 + 3*b*c^2*d^2*e^5 - b*e^7)*g*x + 2*(b*c^6*d^6*e - 3*
b*c^4*d^4*e^3 + 3*b*c^2*d^2*e^5 - b*e^7)*f + (b*c^6*d^7 - 3*b*c^4*d^5*e^2 + 3*b*c^2*d^3*e^4 - b*d*e^6)*g)*arcs
in(c*x) - 2*sqrt(-c^2*x^2 + 1)*(3*((b*c^5*d^3*e^4 - b*c^3*d*e^6)*f - (b*c^3*d^2*e^5 - b*c*e^7)*g)*x^2 + (4*b*c
^5*d^5*e^2 - 5*b*c^3*d^3*e^4 + b*c*d*e^6)*f - (b*c^5*d^6*e + b*c^3*d^4*e^3 - 2*b*c*d^2*e^5)*g + ((7*b*c^5*d^4*
e^3 - 8*b*c^3*d^2*e^5 + b*c*e^7)*f - (b*c^5*d^5*e^2 + 4*b*c^3*d^3*e^4 - 5*b*c*d*e^6)*g)*x))/(c^6*d^9*e^2 - 3*c
^4*d^7*e^4 + 3*c^2*d^5*e^6 - d^3*e^8 + (c^6*d^6*e^5 - 3*c^4*d^4*e^7 + 3*c^2*d^2*e^9 - e^11)*x^3 + 3*(c^6*d^7*e
^4 - 3*c^4*d^5*e^6 + 3*c^2*d^3*e^8 - d*e^10)*x^2 + 3*(c^6*d^8*e^3 - 3*c^4*d^6*e^5 + 3*c^2*d^4*e^7 - d^2*e^9)*x
), -1/6*(3*(a*c^6*d^6*e - 3*a*c^4*d^4*e^3 + 3*a*c^2*d^2*e^5 - a*e^7)*g*x - sqrt(c^2*d^2 - e^2)*(((2*b*c^5*d^2*
e^4 + b*c^3*e^6)*f + (b*c^5*d^3*e^3 - 4*b*c^3*d*e^5)*g)*x^3 + 3*((2*b*c^5*d^3*e^3 + b*c^3*d*e^5)*f + (b*c^5*d^
4*e^2 - 4*b*c^3*d^2*e^4)*g)*x^2 + (2*b*c^5*d^5*e + b*c^3*d^3*e^3)*f + (b*c^5*d^6 - 4*b*c^3*d^4*e^2)*g + 3*((2*
b*c^5*d^4*e^2 + b*c^3*d^2*e^4)*f + (b*c^5*d^5*e - 4*b*c^3*d^3*e^3)*g)*x)*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*(a*c^6*d^6*e - 3*a*c^4*d^4*e^3 + 3*a*c^2
*d^2*e^5 - a*e^7)*f + (a*c^6*d^7 - 3*a*c^4*d^5*e^2 + 3*a*c^2*d^3*e^4 - a*d*e^6)*g + (3*(b*c^6*d^6*e - 3*b*c^4*
d^4*e^3 + 3*b*c^2*d^2*e^5 - b*e^7)*g*x + 2*(b*c^6*d^6*e - 3*b*c^4*d^4*e^3 + 3*b*c^2*d^2*e^5 - b*e^7)*f + (b*c^
6*d^7 - 3*b*c^4*d^5*e^2 + 3*b*c^2*d^3*e^4 - b*d*e^6)*g)*arcsin(c*x) - sqrt(-c^2*x^2 + 1)*(3*((b*c^5*d^3*e^4 -
b*c^3*d*e^6)*f - (b*c^3*d^2*e^5 - b*c*e^7)*g)*x^2 + (4*b*c^5*d^5*e^2 - 5*b*c^3*d^3*e^4 + b*c*d*e^6)*f - (b*c^5
*d^6*e + b*c^3*d^4*e^3 - 2*b*c*d^2*e^5)*g + ((7*b*c^5*d^4*e^3 - 8*b*c^3*d^2*e^5 + b*c*e^7)*f - (b*c^5*d^5*e^2
+ 4*b*c^3*d^3*e^4 - 5*b*c*d*e^6)*g)*x))/(c^6*d^9*e^2 - 3*c^4*d^7*e^4 + 3*c^2*d^5*e^6 - d^3*e^8 + (c^6*d^6*e^5
- 3*c^4*d^4*e^7 + 3*c^2*d^2*e^9 - e^11)*x^3 + 3*(c^6*d^7*e^4 - 3*c^4*d^5*e^6 + 3*c^2*d^3*e^8 - d*e^10)*x^2 + 3
*(c^6*d^8*e^3 - 3*c^4*d^6*e^5 + 3*c^2*d^4*e^7 - d^2*e^9)*x)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(((f + g*x)*(a + b*asin(c*x)))/(d + e*x)^4,x)

[Out]

int(((f + g*x)*(a + b*asin(c*x)))/(d + e*x)^4, x)