Integrand size = 21, antiderivative size = 358 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=-\frac {i b g \arcsin (c x)^2}{2 e^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{e^2 (d+e x)}+\frac {b c (e f-d g) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {b g \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2}+\frac {b g \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2}-\frac {b g \arcsin (c x) \log (d+e x)}{e^2}+\frac {g (a+b \arcsin (c x)) \log (d+e x)}{e^2}-\frac {i b g \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2}-\frac {i b g \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2} \] Output:
-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*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)+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-b*g*arcsin(c*x)*ln(e*x+d)/e^2+g*(a+b *arcsin(c*x))*ln(e*x+d)/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
Time = 0.32 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.93 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\frac {-\frac {1}{2} i b g \arcsin (c x)^2-\frac {(e f-d g) (a+b \arcsin (c x))}{d+e x}+\frac {b c (e f-d g) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d^2-e^2}}+b g \arcsin (c x) \log \left (1+\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+b g \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )-b g \arcsin (c x) \log (d+e x)+g (a+b \arcsin (c x)) \log (d+e x)-i b g \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )-i b g \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2} \] Input:
Integrate[((f + g*x)*(a + b*ArcSin[c*x]))/(d + e*x)^2,x]
Output:
((-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*Ar cSin[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
Time = 1.80 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5252, 25, 27, 7292, 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.
| \(\displaystyle \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 5252 |
| \(\displaystyle -b c \int -\frac {e f-d g-g (d+e x) \log (d+e x)}{e^2 (d+e x) \sqrt {1-c^2 x^2}}dx-\frac {(e f-d g) (a+b \arcsin (c x))}{e^2 (d+e x)}+\frac {g \log (d+e x) (a+b \arcsin (c x))}{e^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle b c \int \frac {e f-d g-g (d+e x) \log (d+e x)}{e^2 (d+e x) \sqrt {1-c^2 x^2}}dx-\frac {(e f-d g) (a+b \arcsin (c x))}{e^2 (d+e x)}+\frac {g \log (d+e x) (a+b \arcsin (c x))}{e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c \int \frac {e f-d g-g (d+e x) \log (d+e x)}{(d+e x) \sqrt {1-c^2 x^2}}dx}{e^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{e^2 (d+e x)}+\frac {g \log (d+e x) (a+b \arcsin (c x))}{e^2}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {b c \int \frac {e f \left (1-\frac {d g}{e f}\right )-g (d+e x) \log (d+e x)}{(d+e x) \sqrt {1-c^2 x^2}}dx}{e^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{e^2 (d+e x)}+\frac {g \log (d+e x) (a+b \arcsin (c x))}{e^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {b c \int \left (\frac {e f-d g}{(d+e x) \sqrt {1-c^2 x^2}}-\frac {g \log (d+e x)}{\sqrt {1-c^2 x^2}}\right )dx}{e^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{e^2 (d+e x)}+\frac {g \log (d+e x) (a+b \arcsin (c x))}{e^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(e f-d g) (a+b \arcsin (c x))}{e^2 (d+e x)}+\frac {g \log (d+e x) (a+b \arcsin (c x))}{e^2}+\frac {b c \left (-\frac {i g \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{c}-\frac {i g \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{c}+\frac {g \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{c}+\frac {g \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{c}-\frac {g \arcsin (c x) \log (d+e x)}{c}-\frac {i g \arcsin (c x)^2}{2 c}+\frac {(e f-d g) \arctan \left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right )}{\sqrt {c^2 d^2-e^2}}\right )}{e^2}\) |
Input:
Int[((f + g*x)*(a + b*ArcSin[c*x]))/(d + e*x)^2,x]
Output:
-(((e*f - d*g)*(a + b*ArcSin[c*x]))/(e^2*(d + e*x))) + (g*(a + b*ArcSin[c* x])*Log[d + e*x])/e^2 + (b*c*(((-1/2*I)*g*ArcSin[c*x]^2)/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] + (g*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2* d^2 - e^2])])/c + (g*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sq rt[c^2*d^2 - e^2])])/c - (g*ArcSin[c*x]*Log[d + e*x])/c - (I*g*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/c - (I*g*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/c))/e^2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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 = 1.76 (sec) , antiderivative size = 954, normalized size of antiderivative = 2.66
| method | result | size |
| derivativedivides | \(\frac {a c \left (\frac {c \left (d g -e f \right )}{e^{2} \left (c x e +c d \right )}+\frac {g \ln \left (c x e +c d \right )}{e^{2}}\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 x e +c d \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} \left (c^{2} d^{2}-e^{2}\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 {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 {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 {2 i c f \,\operatorname {arctanh}\left (\frac {2 i e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 c d}{2 \sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \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} \left (c^{2} d^{2}-e^{2}\right )}+\frac {2 i d c g \,\operatorname {arctanh}\left (\frac {2 i e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 c d}{2 \sqrt {c^{2} d^{2}-e^{2}}}\right )}{e^{2} \sqrt {c^{2} d^{2}-e^{2}}}\right )}{c}\) | \(954\) |
| default | \(\frac {a c \left (\frac {c \left (d g -e f \right )}{e^{2} \left (c x e +c d \right )}+\frac {g \ln \left (c x e +c d \right )}{e^{2}}\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 x e +c d \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} \left (c^{2} d^{2}-e^{2}\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 {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 {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 {2 i c f \,\operatorname {arctanh}\left (\frac {2 i e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 c d}{2 \sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \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} \left (c^{2} d^{2}-e^{2}\right )}+\frac {2 i d c g \,\operatorname {arctanh}\left (\frac {2 i e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 c d}{2 \sqrt {c^{2} d^{2}-e^{2}}}\right )}{e^{2} \sqrt {c^{2} d^{2}-e^{2}}}\right )}{c}\) | \(954\) |
| parts | \(a \left (\frac {g \ln \left (e x +d \right )}{e^{2}}-\frac {-d g +e f}{e^{2} \left (e x +d \right )}\right )+\frac {b \left (-\frac {i c g \arcsin \left (c x \right )^{2}}{2 e^{2}}+\frac {\left (d g -e f \right ) c^{2} \arcsin \left (c x \right )}{e^{2} \left (c x e +c d \right )}+\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 {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 {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^{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 {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}\) | \(954\) |
Input:
int((g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^2,x,method=_RETURNVERBOSE)
Output:
1/c*(a*c*(c*(d*g-e*f)/e^2/(c*e*x+c*d)+g/e^2*ln(c*e*x+c*d))+b*c*(-1/2*I*g*a rcsin(c*x)^2/e^2+(d*g-e*f)*arcsin(c*x)*c/e^2/(c*e*x+c*d)-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+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)))+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)))+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)))-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*c*d)/(c^2*d^2-e^2)^(1/2))-I/e^2*g/(c^2*d^2-e^2)*d ilog((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^2*d*c*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*c*d)/(c^2*d^2-e^2)^(1/2))))
\[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate((g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^2,x, algorithm="fricas")
Output:
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)
\[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x\right )}{\left (d + e x\right )^{2}}\, dx \] Input:
integrate((g*x+f)*(a+b*asin(c*x))/(e*x+d)**2,x)
Output:
Integral((a + b*asin(c*x))*(f + g*x)/(d + e*x)**2, x)
Exception generated. \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^2,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume((e-c*d)*(e+c*d)>0)', see `assume ?` for mor
Exception generated. \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^2,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
Timed out. \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\int \frac {\left (f+g\,x\right )\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d+e\,x\right )}^2} \,d x \] Input:
int(((f + g*x)*(a + b*asin(c*x)))/(d + e*x)^2,x)
Output:
int(((f + g*x)*(a + b*asin(c*x)))/(d + e*x)^2, x)
\[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\frac {\left (\int \frac {\mathit {asin} \left (c x \right )}{e^{2} x^{2}+2 d e x +d^{2}}d x \right ) b \,d^{2} e^{2} f +\left (\int \frac {\mathit {asin} \left (c x \right )}{e^{2} x^{2}+2 d e x +d^{2}}d x \right ) b d \,e^{3} f x +\left (\int \frac {\mathit {asin} \left (c x \right ) x}{e^{2} x^{2}+2 d e x +d^{2}}d x \right ) b \,d^{2} e^{2} g +\left (\int \frac {\mathit {asin} \left (c x \right ) x}{e^{2} x^{2}+2 d e x +d^{2}}d x \right ) b d \,e^{3} g x +\mathrm {log}\left (e x +d \right ) a \,d^{2} g +\mathrm {log}\left (e x +d \right ) a d e g x -a d e g x +a \,e^{2} f x}{d \,e^{2} \left (e x +d \right )} \] Input:
int((g*x+f)*(a+b*asin(c*x))/(e*x+d)^2,x)
Output:
(int(asin(c*x)/(d**2 + 2*d*e*x + e**2*x**2),x)*b*d**2*e**2*f + int(asin(c* x)/(d**2 + 2*d*e*x + e**2*x**2),x)*b*d*e**3*f*x + int((asin(c*x)*x)/(d**2 + 2*d*e*x + e**2*x**2),x)*b*d**2*e**2*g + int((asin(c*x)*x)/(d**2 + 2*d*e* x + e**2*x**2),x)*b*d*e**3*g*x + log(d + e*x)*a*d**2*g + log(d + e*x)*a*d* e*g*x - a*d*e*g*x + a*e**2*f*x)/(d*e**2*(d + e*x))