Integrand size = 30, antiderivative size = 90 \[ \int \frac {a+b \arcsin (c x)}{\sqrt {d+c d x} (f-c f x)^{3/2}} \, dx=\frac {\sqrt {d+c d x} (a+b \arcsin (c x))}{c d f \sqrt {f-c f x}}+\frac {b \sqrt {1-c^2 x^2} \log (1-c x)}{c f \sqrt {d+c d x} \sqrt {f-c f x}} \] Output:
(c*d*x+d)^(1/2)*(a+b*arcsin(c*x))/c/d/f/(-c*f*x+f)^(1/2)+b*(-c^2*x^2+1)^(1 /2)*ln(-c*x+1)/c/f/(c*d*x+d)^(1/2)/(-c*f*x+f)^(1/2)
Time = 0.80 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.18 \[ \int \frac {a+b \arcsin (c x)}{\sqrt {d+c d x} (f-c f x)^{3/2}} \, dx=\frac {\sqrt {d+c d x} \sqrt {f-c f x} \left (-a \sqrt {1-c^2 x^2}-b \sqrt {1-c^2 x^2} \arcsin (c x)+b (-1+c x) \log (f-c f x)\right )}{c d f^2 (-1+c x) \sqrt {1-c^2 x^2}} \] Input:
Integrate[(a + b*ArcSin[c*x])/(Sqrt[d + c*d*x]*(f - c*f*x)^(3/2)),x]
Output:
(Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(-(a*Sqrt[1 - c^2*x^2]) - b*Sqrt[1 - c^2* x^2]*ArcSin[c*x] + b*(-1 + c*x)*Log[f - c*f*x]))/(c*d*f^2*(-1 + c*x)*Sqrt[ 1 - c^2*x^2])
Time = 0.45 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.90, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5178, 27, 5260, 27, 451, 16}
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.
\(\displaystyle \int \frac {a+b \arcsin (c x)}{\sqrt {c d x+d} (f-c f x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 5178 |
\(\displaystyle \frac {\left (1-c^2 x^2\right )^{3/2} \int \frac {d (c x+1) (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{(c d x+d)^{3/2} (f-c f x)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {d \left (1-c^2 x^2\right )^{3/2} \int \frac {(c x+1) (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{(c d x+d)^{3/2} (f-c f x)^{3/2}}\) |
\(\Big \downarrow \) 5260 |
\(\displaystyle \frac {d \left (1-c^2 x^2\right )^{3/2} \left (\frac {(c x+1) (a+b \arcsin (c x))}{c \sqrt {1-c^2 x^2}}-b c \int \frac {c x+1}{c \left (1-c^2 x^2\right )}dx\right )}{(c d x+d)^{3/2} (f-c f x)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {d \left (1-c^2 x^2\right )^{3/2} \left (\frac {(c x+1) (a+b \arcsin (c x))}{c \sqrt {1-c^2 x^2}}-b \int \frac {c x+1}{1-c^2 x^2}dx\right )}{(c d x+d)^{3/2} (f-c f x)^{3/2}}\) |
\(\Big \downarrow \) 451 |
\(\displaystyle \frac {d \left (1-c^2 x^2\right )^{3/2} \left (\frac {(c x+1) (a+b \arcsin (c x))}{c \sqrt {1-c^2 x^2}}-b \int \frac {1}{1-c x}dx\right )}{(c d x+d)^{3/2} (f-c f x)^{3/2}}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {d \left (1-c^2 x^2\right )^{3/2} \left (\frac {(c x+1) (a+b \arcsin (c x))}{c \sqrt {1-c^2 x^2}}+\frac {b \log (1-c x)}{c}\right )}{(c d x+d)^{3/2} (f-c f x)^{3/2}}\) |
Input:
Int[(a + b*ArcSin[c*x])/(Sqrt[d + c*d*x]*(f - c*f*x)^(3/2)),x]
Output:
(d*(1 - c^2*x^2)^(3/2)*(((1 + c*x)*(a + b*ArcSin[c*x]))/(c*Sqrt[1 - c^2*x^ 2]) + (b*Log[1 - c*x])/c))/((d + c*d*x)^(3/2)*(f - c*f*x)^(3/2))
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_) + (d_.)*(x_))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[c^2/a In t[1/(c - d*x), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(q_), x_Symbol] :> Simp[(d + e*x)^q*((f + g*x)^q/(1 - c^2*x^ 2)^q) Int[(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e _.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[(f + g*x)^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSin[c*x]) u, x] - Simp[b*c Int[1/Sqrt[1 - c^2*x^2] u, x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IG tQ[m, 0] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && (LtQ[m, -2*p - 1] || GtQ[m, 3] )
Result contains complex when optimal does not.
Time = 1.91 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.59
method | result | size |
default | \(\frac {a \sqrt {c d x +d}}{f d c \sqrt {-c f x +f}}+b \left (\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \arcsin \left (c x \right )}{d \,f^{2} c \left (c^{2} x^{2}-1\right )}-\frac {\arcsin \left (c x \right ) \left (i \sqrt {-c^{2} x^{2}+1}+c x +1\right ) \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}}{\left (c x -1\right ) c d \,f^{2} \left (c x +1\right )}-\frac {2 \sqrt {-f \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{d \,f^{2} c \left (c^{2} x^{2}-1\right )}\right )\) | \(233\) |
parts | \(\frac {a \sqrt {c d x +d}}{f d c \sqrt {-c f x +f}}+b \left (\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \arcsin \left (c x \right )}{d \,f^{2} c \left (c^{2} x^{2}-1\right )}-\frac {\arcsin \left (c x \right ) \left (i \sqrt {-c^{2} x^{2}+1}+c x +1\right ) \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}}{\left (c x -1\right ) c d \,f^{2} \left (c x +1\right )}-\frac {2 \sqrt {-f \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{d \,f^{2} c \left (c^{2} x^{2}-1\right )}\right )\) | \(233\) |
Input:
int((a+b*arcsin(c*x))/(c*d*x+d)^(1/2)/(-c*f*x+f)^(3/2),x,method=_RETURNVER BOSE)
Output:
a/f/d/c/(-c*f*x+f)^(1/2)*(c*d*x+d)^(1/2)+b*(2*I*(-c^2*x^2+1)^(1/2)*(d*(c*x +1))^(1/2)*(-f*(c*x-1))^(1/2)/d/f^2/c/(c^2*x^2-1)*arcsin(c*x)-arcsin(c*x)* (I*(-c^2*x^2+1)^(1/2)+c*x+1)*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)/(c*x-1)/ c/d/f^2/(c*x+1)-2*(-f*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)/ d/f^2/c/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)-I))
Time = 0.18 (sec) , antiderivative size = 354, normalized size of antiderivative = 3.93 \[ \int \frac {a+b \arcsin (c x)}{\sqrt {d+c d x} (f-c f x)^{3/2}} \, dx=\left [\frac {{\left (b c x - b\right )} \sqrt {d f} \log \left (\frac {c^{6} d f x^{6} - 4 \, c^{5} d f x^{5} + 5 \, c^{4} d f x^{4} - 4 \, c^{2} d f x^{2} + 4 \, c d f x - {\left (c^{4} x^{4} - 4 \, c^{3} x^{3} + 6 \, c^{2} x^{2} - 4 \, c x\right )} \sqrt {-c^{2} x^{2} + 1} \sqrt {c d x + d} \sqrt {-c f x + f} \sqrt {d f} - 2 \, d f}{c^{4} x^{4} - 2 \, c^{3} x^{3} + 2 \, c x - 1}\right ) - 2 \, \sqrt {c d x + d} \sqrt {-c f x + f} {\left (b \arcsin \left (c x\right ) + a\right )}}{2 \, {\left (c^{2} d f^{2} x - c d f^{2}\right )}}, \frac {{\left (b c x - b\right )} \sqrt {-d f} \arctan \left (\frac {{\left (c^{2} x^{2} - 2 \, c x + 2\right )} \sqrt {-c^{2} x^{2} + 1} \sqrt {c d x + d} \sqrt {-c f x + f} \sqrt {-d f}}{c^{4} d f x^{4} - 2 \, c^{3} d f x^{3} - c^{2} d f x^{2} + 2 \, c d f x}\right ) - \sqrt {c d x + d} \sqrt {-c f x + f} {\left (b \arcsin \left (c x\right ) + a\right )}}{c^{2} d f^{2} x - c d f^{2}}\right ] \] Input:
integrate((a+b*arcsin(c*x))/(c*d*x+d)^(1/2)/(-c*f*x+f)^(3/2),x, algorithm= "fricas")
Output:
[1/2*((b*c*x - b)*sqrt(d*f)*log((c^6*d*f*x^6 - 4*c^5*d*f*x^5 + 5*c^4*d*f*x ^4 - 4*c^2*d*f*x^2 + 4*c*d*f*x - (c^4*x^4 - 4*c^3*x^3 + 6*c^2*x^2 - 4*c*x) *sqrt(-c^2*x^2 + 1)*sqrt(c*d*x + d)*sqrt(-c*f*x + f)*sqrt(d*f) - 2*d*f)/(c ^4*x^4 - 2*c^3*x^3 + 2*c*x - 1)) - 2*sqrt(c*d*x + d)*sqrt(-c*f*x + f)*(b*a rcsin(c*x) + a))/(c^2*d*f^2*x - c*d*f^2), ((b*c*x - b)*sqrt(-d*f)*arctan(( c^2*x^2 - 2*c*x + 2)*sqrt(-c^2*x^2 + 1)*sqrt(c*d*x + d)*sqrt(-c*f*x + f)*s qrt(-d*f)/(c^4*d*f*x^4 - 2*c^3*d*f*x^3 - c^2*d*f*x^2 + 2*c*d*f*x)) - sqrt( c*d*x + d)*sqrt(-c*f*x + f)*(b*arcsin(c*x) + a))/(c^2*d*f^2*x - c*d*f^2)]
\[ \int \frac {a+b \arcsin (c x)}{\sqrt {d+c d x} (f-c f x)^{3/2}} \, dx=\int \frac {a + b \operatorname {asin}{\left (c x \right )}}{\sqrt {d \left (c x + 1\right )} \left (- f \left (c x - 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a+b*asin(c*x))/(c*d*x+d)**(1/2)/(-c*f*x+f)**(3/2),x)
Output:
Integral((a + b*asin(c*x))/(sqrt(d*(c*x + 1))*(-f*(c*x - 1))**(3/2)), x)
Time = 0.14 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.09 \[ \int \frac {a+b \arcsin (c x)}{\sqrt {d+c d x} (f-c f x)^{3/2}} \, dx=-\frac {\sqrt {-c^{2} d f x^{2} + d f} b \arcsin \left (c x\right )}{c^{2} d f^{2} x - c d f^{2}} - \frac {\sqrt {-c^{2} d f x^{2} + d f} a}{c^{2} d f^{2} x - c d f^{2}} + \frac {b \log \left (c x - 1\right )}{c \sqrt {d} f^{\frac {3}{2}}} \] Input:
integrate((a+b*arcsin(c*x))/(c*d*x+d)^(1/2)/(-c*f*x+f)^(3/2),x, algorithm= "maxima")
Output:
-sqrt(-c^2*d*f*x^2 + d*f)*b*arcsin(c*x)/(c^2*d*f^2*x - c*d*f^2) - sqrt(-c^ 2*d*f*x^2 + d*f)*a/(c^2*d*f^2*x - c*d*f^2) + b*log(c*x - 1)/(c*sqrt(d)*f^( 3/2))
\[ \int \frac {a+b \arcsin (c x)}{\sqrt {d+c d x} (f-c f x)^{3/2}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{\sqrt {c d x + d} {\left (-c f x + f\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*arcsin(c*x))/(c*d*x+d)^(1/2)/(-c*f*x+f)^(3/2),x, algorithm= "giac")
Output:
integrate((b*arcsin(c*x) + a)/(sqrt(c*d*x + d)*(-c*f*x + f)^(3/2)), x)
Timed out. \[ \int \frac {a+b \arcsin (c x)}{\sqrt {d+c d x} (f-c f x)^{3/2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{\sqrt {d+c\,d\,x}\,{\left (f-c\,f\,x\right )}^{3/2}} \,d x \] Input:
int((a + b*asin(c*x))/((d + c*d*x)^(1/2)*(f - c*f*x)^(3/2)),x)
Output:
int((a + b*asin(c*x))/((d + c*d*x)^(1/2)*(f - c*f*x)^(3/2)), x)
\[ \int \frac {a+b \arcsin (c x)}{\sqrt {d+c d x} (f-c f x)^{3/2}} \, dx=\frac {-\sqrt {-c x +1}\, \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {c x +1}\, \sqrt {-c x +1}\, c x -\sqrt {c x +1}\, \sqrt {-c x +1}}d x \right ) b c +\sqrt {c x +1}\, a}{\sqrt {f}\, \sqrt {d}\, \sqrt {-c x +1}\, c f} \] Input:
int((a+b*asin(c*x))/(c*d*x+d)^(1/2)/(-c*f*x+f)^(3/2),x)
Output:
( - sqrt( - c*x + 1)*int(asin(c*x)/(sqrt(c*x + 1)*sqrt( - c*x + 1)*c*x - s qrt(c*x + 1)*sqrt( - c*x + 1)),x)*b*c + sqrt(c*x + 1)*a)/(sqrt(f)*sqrt(d)* sqrt( - c*x + 1)*c*f)