\(\int \frac {(f+g x) (a+b \arcsin (c x))}{d+e x} \, dx\) [34]

Optimal result

Integrand size = 21, antiderivative size = 344 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{d+e x} \, dx=\frac {b g \sqrt {1-c^2 x^2}}{c e}-\frac {i b (e f-d g) \arcsin (c x)^2}{2 e^2}+\frac {g x (a+b \arcsin (c x))}{e}+\frac {b (e f-d g) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2}+\frac {b (e f-d g) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2}-\frac {b (e f-d g) \arcsin (c x) \log (d+e x)}{e^2}+\frac {(e f-d g) (a+b \arcsin (c x)) \log (d+e x)}{e^2}-\frac {i b (e f-d g) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2}-\frac {i b (e f-d g) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.94 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{d+e x} \, dx=\frac {\frac {b e g \sqrt {1-c^2 x^2}}{c}-\frac {1}{2} i b (e f-d g) \arcsin (c x)^2+e g x (a+b \arcsin (c x))+b (e f-d g) \arcsin (c x) \log \left (1+\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+b (e f-d g) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )-b (e f-d g) \arcsin (c x) \log (d+e x)+(e f-d g) (a+b \arcsin (c x)) \log (d+e x)-i b (e f-d g) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )-i b (e f-d g) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5252, 27, 7293, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(f+g x) (a+b \arcsin (c x))}{d+e x} \, dx\)

\(\Big \downarrow \) 5252

\(\displaystyle -b c \int \frac {e g x+(e f-d g) \log (d+e x)}{e^2 \sqrt {1-c^2 x^2}}dx+\frac {(e f-d g) \log (d+e x) (a+b \arcsin (c x))}{e^2}+\frac {g x (a+b \arcsin (c x))}{e}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c \int \frac {e g x+(e f-d g) \log (d+e x)}{\sqrt {1-c^2 x^2}}dx}{e^2}+\frac {(e f-d g) \log (d+e x) (a+b \arcsin (c x))}{e^2}+\frac {g x (a+b \arcsin (c x))}{e}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {b c \int \left (\frac {e g x}{\sqrt {1-c^2 x^2}}+\frac {(e f-d g) \log (d+e x)}{\sqrt {1-c^2 x^2}}\right )dx}{e^2}+\frac {(e f-d g) \log (d+e x) (a+b \arcsin (c x))}{e^2}+\frac {g x (a+b \arcsin (c x))}{e}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(e f-d g) \log (d+e x) (a+b \arcsin (c x))}{e^2}+\frac {g x (a+b \arcsin (c x))}{e}-\frac {b c \left (\frac {i (e f-d g) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{c}+\frac {i (e f-d g) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{c}-\frac {\arcsin (c x) (e f-d g) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{c}-\frac {\arcsin (c x) (e f-d g) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{c}+\frac {i \arcsin (c x)^2 (e f-d g)}{2 c}+\frac {\arcsin (c x) (e f-d g) \log (d+e x)}{c}-\frac {e g \sqrt {1-c^2 x^2}}{c^2}\right )}{e^2}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5252

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_ 
Symbol] :> With[{u = IntHide[Px*(d + e*x)^m, x]}, Simp[(a + b*ArcSin[c*x]) 
  u, x] - Simp[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 7293

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1565 vs. \(2 (357 ) = 714\).

Time = 1.82 (sec) , antiderivative size = 1566, normalized size of antiderivative = 4.55

method	result	size
parts	\(\text {Expression too large to display}\)	\(1566\)
derivativedivides	\(\text {Expression too large to display}\)	\(1584\)
default	\(\text {Expression too large to display}\)	\(1584\)

Input:

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

Output:

a*(g/e*x+(-d*g+e*f)/e^2*ln(e*x+d))+I*b*c^2/e^2*d^3*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*b*c^2/e*f/(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)))*d^2+I*b*e*f/(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)))+b*d*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)))+b*d*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)))-b*e*f*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)))-b*e*f*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)))-b*c^2/e^2*d^3*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)))+b*c^2/e*f* 
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)))*d^2+b*c^2/e*f*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)))*d^2-b*c^2/e^2*d^3*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)))+I*b*c^2/e^2*d^3*g/(c^2*d^2-e^2)*dilog((I*d*c+(I*c*x+(-c^...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(f+g x) (a+b \arcsin (c x))}{d+e x} \, dx=\frac {\mathit {asin} \left (c x \right ) b c e g x +\sqrt {-c^{2} x^{2}+1}\, b e g -\left (\int \frac {\mathit {asin} \left (c x \right )}{e x +d}d x \right ) b c d e g +\left (\int \frac {\mathit {asin} \left (c x \right )}{e x +d}d x \right ) b c \,e^{2} f -\mathrm {log}\left (e x +d \right ) a c d g +\mathrm {log}\left (e x +d \right ) a c e f +a c e g x}{c \,e^{2}} \] Input:

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

Output:

(asin(c*x)*b*c*e*g*x + sqrt( - c**2*x**2 + 1)*b*e*g - int(asin(c*x)/(d + e 
*x),x)*b*c*d*e*g + int(asin(c*x)/(d + e*x),x)*b*c*e**2*f - log(d + e*x)*a* 
c*d*g + log(d + e*x)*a*c*e*f + a*c*e*g*x)/(c*e**2)