\(\int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^2} \, dx\) [92]

Optimal result

Integrand size = 21, antiderivative size = 358 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=-\frac {i b g \arcsin (c x)^2}{2 e^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{e^2 (d+e x)}+\frac {b c (e f-d g) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {b g \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2}+\frac {b g \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2}-\frac {b g \arcsin (c x) \log (d+e x)}{e^2}+\frac {g (a+b \arcsin (c x)) \log (d+e x)}{e^2}-\frac {i b g \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2}-\frac {i b g \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2} \]

[Out]

-1/2*I*b*g*arcsin(c*x)^2/e^2-(-d*g+e*f)*(a+b*arcsin(c*x))/e^2/(e*x+d)-b*g*arcsin(c*x)*ln(e*x+d)/e^2+g*(a+b*arc
sin(c*x))*ln(e*x+d)/e^2+b*g*arcsin(c*x)*ln(1-I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e^2+b*g
*arcsin(c*x)*ln(1-I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e^2-I*b*g*polylog(2,I*e*(I*c*x+(-c
^2*x^2+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e^2-I*b*g*polylog(2,I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2-e
^2)^(1/2)))/e^2+b*c*(-d*g+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)^(1
/2)

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {45, 4837, 12, 6874, 739, 210, 222, 2451, 4825, 4615, 2221, 2317, 2438} \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=-\frac {(e f-d g) (a+b \arcsin (c x))}{e^2 (d+e x)}+\frac {g \log (d+e x) (a+b \arcsin (c x))}{e^2}-\frac {i b g \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2}-\frac {i b g \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2}+\frac {b g \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2}+\frac {b g \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e^2}-\frac {b g \arcsin (c x) \log (d+e x)}{e^2}-\frac {i b g \arcsin (c x)^2}{2 e^2}+\frac {b c (e f-d g) \arctan \left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}} \]

[In]

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

[Out]

((-1/2*I)*b*g*ArcSin[c*x]^2)/e^2 - ((e*f - d*g)*(a + b*ArcSin[c*x]))/(e^2*(d + e*x)) + (b*c*(e*f - d*g)*ArcTan
[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(e^2*Sqrt[c^2*d^2 - e^2]) + (b*g*ArcSin[c*x]*Log[1 -
(I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^2 + (b*g*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(
c*d + Sqrt[c^2*d^2 - e^2])])/e^2 - (b*g*ArcSin[c*x]*Log[d + e*x])/e^2 + (g*(a + b*ArcSin[c*x])*Log[d + e*x])/e
^2 - (I*b*g*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^2 - (I*b*g*PolyLog[2, (I*e*E^(I
*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^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 222

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

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 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2451

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/Sqrt[(f_) + (g_.)*(x_)^2], x_Symbol] :> With[{u = Int
Hide[1/Sqrt[f + g*x^2], x]}, Simp[u*(a + b*Log[c*(d + e*x)^n]), x] - Dist[b*e*n, Int[SimplifyIntegrand[u/(d +
e*x), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && GtQ[f, 0]

Rule 4615

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
Simp[(-I)*((e + f*x)^(m + 1)/(b*f*(m + 1))), x] + (Int[(e + f*x)^m*(E^(I*(c + d*x))/(a - Rt[a^2 - b^2, 2] - I*
b*E^(I*(c + d*x)))), x] + Int[(e + f*x)^m*(E^(I*(c + d*x))/(a + Rt[a^2 - b^2, 2] - I*b*E^(I*(c + d*x)))), x])
/; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && PosQ[a^2 - b^2]

Rule 4825

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Subst[Int[(a + b*x)^n*(Cos[x]/(
c*d + e*Sin[x])), x], x, ArcSin[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]

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]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = -\frac {(e f-d g) (a+b \arcsin (c x))}{e^2 (d+e x)}+\frac {g (a+b \arcsin (c x)) \log (d+e x)}{e^2}-(b c) \int \frac {-e f \left (1-\frac {d g}{e f}\right )+g (d+e x) \log (d+e x)}{e^2 (d+e x) \sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {(e f-d g) (a+b \arcsin (c x))}{e^2 (d+e x)}+\frac {g (a+b \arcsin (c x)) \log (d+e x)}{e^2}-\frac {(b c) \int \frac {-e f \left (1-\frac {d g}{e f}\right )+g (d+e x) \log (d+e x)}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{e^2} \\ & = -\frac {(e f-d g) (a+b \arcsin (c x))}{e^2 (d+e x)}+\frac {g (a+b \arcsin (c x)) \log (d+e x)}{e^2}-\frac {(b c) \int \left (\frac {-e f+d g}{(d+e x) \sqrt {1-c^2 x^2}}+\frac {g \log (d+e x)}{\sqrt {1-c^2 x^2}}\right ) \, dx}{e^2} \\ & = -\frac {(e f-d g) (a+b \arcsin (c x))}{e^2 (d+e x)}+\frac {g (a+b \arcsin (c x)) \log (d+e x)}{e^2}-\frac {(b c g) \int \frac {\log (d+e x)}{\sqrt {1-c^2 x^2}} \, dx}{e^2}+\frac {(b c (e f-d g)) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{e^2} \\ & = -\frac {(e f-d g) (a+b \arcsin (c x))}{e^2 (d+e x)}-\frac {b g \arcsin (c x) \log (d+e x)}{e^2}+\frac {g (a+b \arcsin (c x)) \log (d+e x)}{e^2}+\frac {(b c g) \int \frac {\arcsin (c x)}{c d+c e x} \, dx}{e}-\frac {(b c (e f-d g)) \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 )}{e^2} \\ & = -\frac {(e f-d g) (a+b \arcsin (c x))}{e^2 (d+e x)}+\frac {b c (e f-d g) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {b g \arcsin (c x) \log (d+e x)}{e^2}+\frac {g (a+b \arcsin (c x)) \log (d+e x)}{e^2}+\frac {(b c g) \text {Subst}\left (\int \frac {x \cos (x)}{c^2 d+c e \sin (x)} \, dx,x,\arcsin (c x)\right )}{e} \\ & = -\frac {i b g \arcsin (c x)^2}{2 e^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{e^2 (d+e x)}+\frac {b c (e f-d g) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {b g \arcsin (c x) \log (d+e x)}{e^2}+\frac {g (a+b \arcsin (c x)) \log (d+e x)}{e^2}+\frac {(b c g) \text {Subst}\left (\int \frac {e^{i x} x}{c^2 d-c \sqrt {c^2 d^2-e^2}-i c e e^{i x}} \, dx,x,\arcsin (c x)\right )}{e}+\frac {(b c g) \text {Subst}\left (\int \frac {e^{i x} x}{c^2 d+c \sqrt {c^2 d^2-e^2}-i c e e^{i x}} \, dx,x,\arcsin (c x)\right )}{e} \\ & = -\frac {i b g \arcsin (c x)^2}{2 e^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{e^2 (d+e x)}+\frac {b c (e f-d g) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {b g \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2}+\frac {b g \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2}-\frac {b g \arcsin (c x) \log (d+e x)}{e^2}+\frac {g (a+b \arcsin (c x)) \log (d+e x)}{e^2}-\frac {(b g) \text {Subst}\left (\int \log \left (1-\frac {i c e e^{i x}}{c^2 d-c \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\arcsin (c x)\right )}{e^2}-\frac {(b g) \text {Subst}\left (\int \log \left (1-\frac {i c e e^{i x}}{c^2 d+c \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\arcsin (c x)\right )}{e^2} \\ & = -\frac {i b g \arcsin (c x)^2}{2 e^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{e^2 (d+e x)}+\frac {b c (e f-d g) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {b g \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2}+\frac {b g \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2}-\frac {b g \arcsin (c x) \log (d+e x)}{e^2}+\frac {g (a+b \arcsin (c x)) \log (d+e x)}{e^2}+\frac {(i b g) \text {Subst}\left (\int \frac {\log \left (1-\frac {i c e x}{c^2 d-c \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{e^2}+\frac {(i b g) \text {Subst}\left (\int \frac {\log \left (1-\frac {i c e x}{c^2 d+c \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{e^2} \\ & = -\frac {i b g \arcsin (c x)^2}{2 e^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{e^2 (d+e x)}+\frac {b c (e f-d g) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {b g \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2}+\frac {b g \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2}-\frac {b g \arcsin (c x) \log (d+e x)}{e^2}+\frac {g (a+b \arcsin (c x)) \log (d+e x)}{e^2}-\frac {i b g \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2}-\frac {i b g \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.93 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\frac {-\frac {1}{2} i b g \arcsin (c x)^2-\frac {(e f-d g) (a+b \arcsin (c x))}{d+e x}+\frac {b c (e f-d g) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d^2-e^2}}+b g \arcsin (c x) \log \left (1+\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+b g \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )-b g \arcsin (c x) \log (d+e x)+g (a+b \arcsin (c x)) \log (d+e x)-i b g \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )-i b g \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2} \]

[In]

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

[Out]

((-1/2*I)*b*g*ArcSin[c*x]^2 - ((e*f - d*g)*(a + b*ArcSin[c*x]))/(d + e*x) + (b*c*(e*f - d*g)*ArcTan[(e + c^2*d
*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d^2 - e^2] + b*g*ArcSin[c*x]*Log[1 + (I*e*E^(I*ArcSin[c
*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] + b*g*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 -
e^2])] - b*g*ArcSin[c*x]*Log[d + e*x] + g*(a + b*ArcSin[c*x])*Log[d + e*x] - I*b*g*PolyLog[2, (I*e*E^(I*ArcSin
[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] - I*b*g*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/
e^2

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 953 vs. \(2 (367 ) = 734\).

Time = 2.27 (sec) , antiderivative size = 954, normalized size of antiderivative = 2.66

method	result	size
derivativedivides	\(\frac {a c \left (\frac {g \ln \left (c e x +d c \right )}{e^{2}}+\frac {c \left (d g -e f \right )}{e^{2} \left (c e x +d c \right )}\right )+b c \left (-\frac {i g \arcsin \left (c x \right )^{2}}{2 e^{2}}+\frac {\left (d g -e f \right ) \arcsin \left (c x \right ) c}{e^{2} \left (c e x +d c \right )}+\frac {i g \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {i g \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}}{e^{2} \left (c^{2} d^{2}-e^{2}\right )}-\frac {2 i c f \,\operatorname {arctanh}\left (\frac {2 i e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 d c}{2 \sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}+\frac {g \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}}{e^{2} \left (c^{2} d^{2}-e^{2}\right )}+\frac {g \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}}{e^{2} \left (c^{2} d^{2}-e^{2}\right )}-\frac {g \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {g \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{c^{2} d^{2}-e^{2}}+\frac {2 i c d g \,\operatorname {arctanh}\left (\frac {2 i e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 d c}{2 \sqrt {c^{2} d^{2}-e^{2}}}\right )}{e^{2} \sqrt {c^{2} d^{2}-e^{2}}}+\frac {i g \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {i g \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}}{e^{2} \left (c^{2} d^{2}-e^{2}\right )}\right )}{c}\)	\(954\)
default	\(\frac {a c \left (\frac {g \ln \left (c e x +d c \right )}{e^{2}}+\frac {c \left (d g -e f \right )}{e^{2} \left (c e x +d c \right )}\right )+b c \left (-\frac {i g \arcsin \left (c x \right )^{2}}{2 e^{2}}+\frac {\left (d g -e f \right ) \arcsin \left (c x \right ) c}{e^{2} \left (c e x +d c \right )}+\frac {i g \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {i g \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}}{e^{2} \left (c^{2} d^{2}-e^{2}\right )}-\frac {2 i c f \,\operatorname {arctanh}\left (\frac {2 i e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 d c}{2 \sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}+\frac {g \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}}{e^{2} \left (c^{2} d^{2}-e^{2}\right )}+\frac {g \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}}{e^{2} \left (c^{2} d^{2}-e^{2}\right )}-\frac {g \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {g \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{c^{2} d^{2}-e^{2}}+\frac {2 i c d g \,\operatorname {arctanh}\left (\frac {2 i e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 d c}{2 \sqrt {c^{2} d^{2}-e^{2}}}\right )}{e^{2} \sqrt {c^{2} d^{2}-e^{2}}}+\frac {i g \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {i g \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}}{e^{2} \left (c^{2} d^{2}-e^{2}\right )}\right )}{c}\)	\(954\)
parts	\(a \left (-\frac {-d g +e f}{e^{2} \left (e x +d \right )}+\frac {g \ln \left (e x +d \right )}{e^{2}}\right )+\frac {b \left (-\frac {i c \arcsin \left (c x \right )^{2} g}{2 e^{2}}+\frac {\left (d g -e f \right ) \arcsin \left (c x \right ) c^{2}}{e^{2} \left (c e x +d c \right )}-\frac {i c^{3} g \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right ) d^{2}}{e^{2} \left (c^{2} d^{2}-e^{2}\right )}-\frac {i c^{3} g \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right ) d^{2}}{e^{2} \left (c^{2} d^{2}-e^{2}\right )}+\frac {c^{3} g \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right ) d^{2}}{e^{2} \left (c^{2} d^{2}-e^{2}\right )}+\frac {c^{3} g \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right ) d^{2}}{e^{2} \left (c^{2} d^{2}-e^{2}\right )}-\frac {c g \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {c g \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{c^{2} d^{2}-e^{2}}+\frac {i c g \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{c^{2} d^{2}-e^{2}}+\frac {i c g \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{c^{2} d^{2}-e^{2}}+\frac {2 c^{2} f \arctan \left (\frac {2 \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +2 i d c}{2 \sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}-\frac {2 c^{2} d g \arctan \left (\frac {2 \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +2 i d c}{2 \sqrt {c^{2} d^{2}-e^{2}}}\right )}{e^{2} \sqrt {c^{2} d^{2}-e^{2}}}\right )}{c}\)	\(954\)

[In]

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

[Out]

1/c*(a*c*(g/e^2*ln(c*e*x+c*d)+c*(d*g-e*f)/e^2/(c*e*x+c*d))+b*c*(-1/2*I*g*arcsin(c*x)^2/e^2+(d*g-e*f)*arcsin(c*
x)*c/e^2/(c*e*x+c*d)+I*g/(c^2*d^2-e^2)*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+
(-c^2*d^2+e^2)^(1/2)))-I/e^2*g/(c^2*d^2-e^2)*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(
I*d*c-(-c^2*d^2+e^2)^(1/2)))*c^2*d^2-2*I/e*c*f/(c^2*d^2-e^2)^(1/2)*arctanh(1/2*(2*I*e*(I*c*x+(-c^2*x^2+1)^(1/2
))-2*d*c)/(c^2*d^2-e^2)^(1/2))+1/e^2*g*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*
d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))*c^2*d^2+1/e^2*g*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+(I*c*x+(-c^2
*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))*c^2*d^2-g*arcsin(c*x)/(c^2*d^2-e^2)*ln((I
*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))-g*arcsin(c*x)/(c^2*d^2-e
^2)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))+2*I/e^2*c*d*g/(
c^2*d^2-e^2)^(1/2)*arctanh(1/2*(2*I*e*(I*c*x+(-c^2*x^2+1)^(1/2))-2*d*c)/(c^2*d^2-e^2)^(1/2))+I*g/(c^2*d^2-e^2)
*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))-I/e^2*g/(c^2*d^
2-e^2)*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))*c^2*d^2))

Fricas [F]

\[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{2}} \,d x } \]

[In]

integrate((g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^2,x, algorithm="fricas")

[Out]

integral((a*g*x + a*f + (b*g*x + b*f)*arcsin(c*x))/(e^2*x^2 + 2*d*e*x + d^2), x)

Sympy [F]

\[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x\right )}{\left (d + e x\right )^{2}}\, dx \]

[In]

integrate((g*x+f)*(a+b*asin(c*x))/(e*x+d)**2,x)

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume((e-c*d)*(e+c*d)>0)', see `assu
me?` for mor

Giac [F]

\[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

integrate((g*x + f)*(b*arcsin(c*x) + a)/(e*x + d)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\int \frac {\left (f+g\,x\right )\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d+e\,x\right )}^2} \,d x \]

[In]

int(((f + g*x)*(a + b*asin(c*x)))/(d + e*x)^2,x)

[Out]

int(((f + g*x)*(a + b*asin(c*x)))/(d + e*x)^2, x)