Integrand size = 29, antiderivative size = 105 \[ \int \frac {\left (1-d^2 x^2\right )^{3/2}}{(1+d x)^2 (e+f x)} \, dx=-\frac {\sqrt {1-d^2 x^2}}{f}-\frac {(d e+2 f) \arcsin (d x)}{f^2}+\frac {(d e+f)^2 \arctan \left (\frac {f+d^2 e x}{\sqrt {d^2 e^2-f^2} \sqrt {1-d^2 x^2}}\right )}{f^2 \sqrt {d^2 e^2-f^2}} \] Output:
-(-d^2*x^2+1)^(1/2)/f-(d*e+2*f)*arcsin(d*x)/f^2+(d*e+f)^2*arctan((d^2*e*x+ f)/(d^2*e^2-f^2)^(1/2)/(-d^2*x^2+1)^(1/2))/f^2/(d^2*e^2-f^2)^(1/2)
Time = 0.53 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.24 \[ \int \frac {\left (1-d^2 x^2\right )^{3/2}}{(1+d x)^2 (e+f x)} \, dx=-\frac {f \sqrt {1-d^2 x^2}+2 (d e+2 f) \arctan \left (\frac {d x}{-1+\sqrt {1-d^2 x^2}}\right )+\frac {2 (d e+f) \sqrt {d^2 e^2-f^2} \arctan \left (\frac {\sqrt {d^2 e^2-f^2} x}{e+f x-e \sqrt {1-d^2 x^2}}\right )}{d e-f}}{f^2} \] Input:
Integrate[(1 - d^2*x^2)^(3/2)/((1 + d*x)^2*(e + f*x)),x]
Output:
-((f*Sqrt[1 - d^2*x^2] + 2*(d*e + 2*f)*ArcTan[(d*x)/(-1 + Sqrt[1 - d^2*x^2 ])] + (2*(d*e + f)*Sqrt[d^2*e^2 - f^2]*ArcTan[(Sqrt[d^2*e^2 - f^2]*x)/(e + f*x - e*Sqrt[1 - d^2*x^2])])/(d*e - f))/f^2)
Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {708, 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.
| \(\displaystyle \int \frac {\left (1-d^2 x^2\right )^{3/2}}{(d x+1)^2 (e+f x)} \, dx\) |
| \(\Big \downarrow \) 708 |
| \(\displaystyle \int \left (-\frac {d (d e+2 f)}{f^2 \sqrt {1-d^2 x^2}}+\frac {(d e+f)^2}{f^2 \sqrt {1-d^2 x^2} (e+f x)}+\frac {d^2 x}{f \sqrt {1-d^2 x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\arcsin (d x) (d e+2 f)}{f^2}+\frac {(d e+f)^2 \arctan \left (\frac {d^2 e x+f}{\sqrt {1-d^2 x^2} \sqrt {d^2 e^2-f^2}}\right )}{f^2 \sqrt {d^2 e^2-f^2}}-\frac {\sqrt {1-d^2 x^2}}{f}\) |
Input:
Int[(1 - d^2*x^2)^(3/2)/((1 + d*x)^2*(e + f*x)),x]
Output:
-(Sqrt[1 - d^2*x^2]/f) - ((d*e + 2*f)*ArcSin[d*x])/f^2 + ((d*e + f)^2*ArcT an[(f + d^2*e*x)/(Sqrt[d^2*e^2 - f^2]*Sqrt[1 - d^2*x^2])])/(f^2*Sqrt[d^2*e ^2 - f^2])
Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2 )^(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + c*x^2], (d + e*x)^m*(f + g*x)^n*(a + c*x^2)^(p + 1/2), x], x] /; FreeQ[{a, c, d, e, f, g, n, p}, x ] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p - 1/2] && ILtQ[m, 0] && ILtQ[n, 0] && !IGtQ[n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(228\) vs. \(2(97)=194\).
Time = 1.27 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.18
| method | result | size |
| risch | \(\frac {d^{2} x^{2}-1}{f \sqrt {-d^{2} x^{2}+1}}-\frac {\frac {d \left (d e +2 f \right ) \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right )}{f \sqrt {d^{2}}}+\frac {\left (d^{2} e^{2}+2 d e f +f^{2}\right ) \ln \left (\frac {-\frac {2 \left (d^{2} e^{2}-f^{2}\right )}{f^{2}}+\frac {2 d^{2} e \left (x +\frac {e}{f}\right )}{f}+2 \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, \sqrt {-d^{2} \left (x +\frac {e}{f}\right )^{2}+\frac {2 d^{2} e \left (x +\frac {e}{f}\right )}{f}-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}}{x +\frac {e}{f}}\right )}{f^{2} \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}}}{f}\) | \(229\) |
| default | \(\frac {\frac {\left (-d^{2} \left (x +\frac {1}{d}\right )^{2}+2 d \left (x +\frac {1}{d}\right )\right )^{\frac {5}{2}}}{d \left (x +\frac {1}{d}\right )^{2}}+3 d \left (\frac {\left (-d^{2} \left (x +\frac {1}{d}\right )^{2}+2 d \left (x +\frac {1}{d}\right )\right )^{\frac {3}{2}}}{3}+d \left (-\frac {\left (-2 d^{2} \left (x +\frac {1}{d}\right )+2 d \right ) \sqrt {-d^{2} \left (x +\frac {1}{d}\right )^{2}+2 d \left (x +\frac {1}{d}\right )}}{4 d^{2}}+\frac {\arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} \left (x +\frac {1}{d}\right )^{2}+2 d \left (x +\frac {1}{d}\right )}}\right )}{2 \sqrt {d^{2}}}\right )\right )}{d \left (d e -f \right )}+\frac {f \left (\frac {\left (-d^{2} \left (x +\frac {e}{f}\right )^{2}+\frac {2 d^{2} e \left (x +\frac {e}{f}\right )}{f}-\frac {d^{2} e^{2}-f^{2}}{f^{2}}\right )^{\frac {3}{2}}}{3}+\frac {d^{2} e \left (-\frac {\left (-2 d^{2} \left (x +\frac {e}{f}\right )+\frac {2 d^{2} e}{f}\right ) \sqrt {-d^{2} \left (x +\frac {e}{f}\right )^{2}+\frac {2 d^{2} e \left (x +\frac {e}{f}\right )}{f}-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}}{4 d^{2}}-\frac {\left (\frac {4 d^{2} \left (d^{2} e^{2}-f^{2}\right )}{f^{2}}-\frac {4 d^{4} e^{2}}{f^{2}}\right ) \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} \left (x +\frac {e}{f}\right )^{2}+\frac {2 d^{2} e \left (x +\frac {e}{f}\right )}{f}-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}}\right )}{8 d^{2} \sqrt {d^{2}}}\right )}{f}-\frac {\left (d^{2} e^{2}-f^{2}\right ) \left (\sqrt {-d^{2} \left (x +\frac {e}{f}\right )^{2}+\frac {2 d^{2} e \left (x +\frac {e}{f}\right )}{f}-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}+\frac {d^{2} e \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} \left (x +\frac {e}{f}\right )^{2}+\frac {2 d^{2} e \left (x +\frac {e}{f}\right )}{f}-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}}\right )}{f \sqrt {d^{2}}}+\frac {\left (d^{2} e^{2}-f^{2}\right ) \ln \left (\frac {-\frac {2 \left (d^{2} e^{2}-f^{2}\right )}{f^{2}}+\frac {2 d^{2} e \left (x +\frac {e}{f}\right )}{f}+2 \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, \sqrt {-d^{2} \left (x +\frac {e}{f}\right )^{2}+\frac {2 d^{2} e \left (x +\frac {e}{f}\right )}{f}-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}}{x +\frac {e}{f}}\right )}{f^{2} \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}}\right )}{f^{2}}\right )}{\left (d e -f \right )^{2}}-\frac {f \left (\frac {\left (-d^{2} \left (x +\frac {1}{d}\right )^{2}+2 d \left (x +\frac {1}{d}\right )\right )^{\frac {3}{2}}}{3}+d \left (-\frac {\left (-2 d^{2} \left (x +\frac {1}{d}\right )+2 d \right ) \sqrt {-d^{2} \left (x +\frac {1}{d}\right )^{2}+2 d \left (x +\frac {1}{d}\right )}}{4 d^{2}}+\frac {\arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} \left (x +\frac {1}{d}\right )^{2}+2 d \left (x +\frac {1}{d}\right )}}\right )}{2 \sqrt {d^{2}}}\right )\right )}{\left (d e -f \right )^{2}}\) | \(835\) |
Input:
int((-d^2*x^2+1)^(3/2)/(d*x+1)^2/(f*x+e),x,method=_RETURNVERBOSE)
Output:
1/f*(d^2*x^2-1)/(-d^2*x^2+1)^(1/2)-1/f*(d*(d*e+2*f)/f/(d^2)^(1/2)*arctan(( d^2)^(1/2)*x/(-d^2*x^2+1)^(1/2))+(d^2*e^2+2*d*e*f+f^2)/f^2/(-(d^2*e^2-f^2) /f^2)^(1/2)*ln((-2*(d^2*e^2-f^2)/f^2+2*d^2*e/f*(x+e/f)+2*(-(d^2*e^2-f^2)/f ^2)^(1/2)*(-d^2*(x+e/f)^2+2*d^2*e/f*(x+e/f)-(d^2*e^2-f^2)/f^2)^(1/2))/(x+e /f)))
Time = 0.16 (sec) , antiderivative size = 317, normalized size of antiderivative = 3.02 \[ \int \frac {\left (1-d^2 x^2\right )^{3/2}}{(1+d x)^2 (e+f x)} \, dx=\left [\frac {{\left (d e + f\right )} \sqrt {-\frac {d e + f}{d e - f}} \log \left (\frac {d^{2} e f x + f^{2} - {\left (d^{2} e^{2} - f^{2}\right )} \sqrt {-d^{2} x^{2} + 1} + {\left (d e f - f^{2} + {\left (d^{3} e^{2} - d^{2} e f\right )} x + \sqrt {-d^{2} x^{2} + 1} {\left (d e f - f^{2}\right )}\right )} \sqrt {-\frac {d e + f}{d e - f}}}{f x + e}\right ) + 2 \, {\left (d e + 2 \, f\right )} \arctan \left (\frac {\sqrt {-d^{2} x^{2} + 1} - 1}{d x}\right ) - \sqrt {-d^{2} x^{2} + 1} f}{f^{2}}, -\frac {2 \, {\left (d e + f\right )} \sqrt {\frac {d e + f}{d e - f}} \arctan \left (-\frac {{\left (f x - \sqrt {-d^{2} x^{2} + 1} e + e\right )} \sqrt {\frac {d e + f}{d e - f}}}{{\left (d e + f\right )} x}\right ) - 2 \, {\left (d e + 2 \, f\right )} \arctan \left (\frac {\sqrt {-d^{2} x^{2} + 1} - 1}{d x}\right ) + \sqrt {-d^{2} x^{2} + 1} f}{f^{2}}\right ] \] Input:
integrate((-d^2*x^2+1)^(3/2)/(d*x+1)^2/(f*x+e),x, algorithm="fricas")
Output:
[((d*e + f)*sqrt(-(d*e + f)/(d*e - f))*log((d^2*e*f*x + f^2 - (d^2*e^2 - f ^2)*sqrt(-d^2*x^2 + 1) + (d*e*f - f^2 + (d^3*e^2 - d^2*e*f)*x + sqrt(-d^2* x^2 + 1)*(d*e*f - f^2))*sqrt(-(d*e + f)/(d*e - f)))/(f*x + e)) + 2*(d*e + 2*f)*arctan((sqrt(-d^2*x^2 + 1) - 1)/(d*x)) - sqrt(-d^2*x^2 + 1)*f)/f^2, - (2*(d*e + f)*sqrt((d*e + f)/(d*e - f))*arctan(-(f*x - sqrt(-d^2*x^2 + 1)*e + e)*sqrt((d*e + f)/(d*e - f))/((d*e + f)*x)) - 2*(d*e + 2*f)*arctan((sqr t(-d^2*x^2 + 1) - 1)/(d*x)) + sqrt(-d^2*x^2 + 1)*f)/f^2]
\[ \int \frac {\left (1-d^2 x^2\right )^{3/2}}{(1+d x)^2 (e+f x)} \, dx=\int \frac {\left (- \left (d x - 1\right ) \left (d x + 1\right )\right )^{\frac {3}{2}}}{\left (e + f x\right ) \left (d x + 1\right )^{2}}\, dx \] Input:
integrate((-d**2*x**2+1)**(3/2)/(d*x+1)**2/(f*x+e),x)
Output:
Integral((-(d*x - 1)*(d*x + 1))**(3/2)/((e + f*x)*(d*x + 1)**2), x)
\[ \int \frac {\left (1-d^2 x^2\right )^{3/2}}{(1+d x)^2 (e+f x)} \, dx=\int { \frac {{\left (-d^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (d x + 1\right )}^{2} {\left (f x + e\right )}} \,d x } \] Input:
integrate((-d^2*x^2+1)^(3/2)/(d*x+1)^2/(f*x+e),x, algorithm="maxima")
Output:
integrate((-d^2*x^2 + 1)^(3/2)/((d*x + 1)^2*(f*x + e)), x)
Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (97) = 194\).
Time = 0.19 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.99 \[ \int \frac {\left (1-d^2 x^2\right )^{3/2}}{(1+d x)^2 (e+f x)} \, dx=-{\left (\frac {{\left (d x + 1\right )} \sqrt {\frac {2}{d x + 1} - 1} \mathrm {sgn}\left (\frac {1}{d x + 1}\right ) \mathrm {sgn}\left (d\right )}{d f} - \frac {2 \, {\left (d e \mathrm {sgn}\left (\frac {1}{d x + 1}\right ) \mathrm {sgn}\left (d\right ) + 2 \, f \mathrm {sgn}\left (\frac {1}{d x + 1}\right ) \mathrm {sgn}\left (d\right )\right )} \arctan \left (\sqrt {\frac {2}{d x + 1} - 1}\right )}{d f^{2}} + \frac {2 \, {\left (d^{2} e^{2} \mathrm {sgn}\left (\frac {1}{d x + 1}\right ) \mathrm {sgn}\left (d\right ) + 2 \, d e f \mathrm {sgn}\left (\frac {1}{d x + 1}\right ) \mathrm {sgn}\left (d\right ) + f^{2} \mathrm {sgn}\left (\frac {1}{d x + 1}\right ) \mathrm {sgn}\left (d\right )\right )} \arctan \left (\frac {d e \sqrt {\frac {2}{d x + 1} - 1} - f \sqrt {\frac {2}{d x + 1} - 1}}{\sqrt {d^{2} e^{2} - f^{2}}}\right )}{\sqrt {d^{2} e^{2} - f^{2}} d f^{2}}\right )} {\left | d \right |} \] Input:
integrate((-d^2*x^2+1)^(3/2)/(d*x+1)^2/(f*x+e),x, algorithm="giac")
Output:
-((d*x + 1)*sqrt(2/(d*x + 1) - 1)*sgn(1/(d*x + 1))*sgn(d)/(d*f) - 2*(d*e*s gn(1/(d*x + 1))*sgn(d) + 2*f*sgn(1/(d*x + 1))*sgn(d))*arctan(sqrt(2/(d*x + 1) - 1))/(d*f^2) + 2*(d^2*e^2*sgn(1/(d*x + 1))*sgn(d) + 2*d*e*f*sgn(1/(d* x + 1))*sgn(d) + f^2*sgn(1/(d*x + 1))*sgn(d))*arctan((d*e*sqrt(2/(d*x + 1) - 1) - f*sqrt(2/(d*x + 1) - 1))/sqrt(d^2*e^2 - f^2))/(sqrt(d^2*e^2 - f^2) *d*f^2))*abs(d)
Time = 0.08 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.39 \[ \int \frac {\left (1-d^2 x^2\right )^{3/2}}{(1+d x)^2 (e+f x)} \, dx=\frac {\mathrm {asinh}\left (x\,\sqrt {-d^2}\right )\,\left (2\,d\,\sqrt {-d^2}+\frac {d^2\,e\,\sqrt {-d^2}}{f}\right )}{d^2\,f}-\frac {\sqrt {1-d^2\,x^2}}{f}-\frac {\left (\ln \left (\sqrt {1-\frac {d^2\,e^2}{f^2}}\,\sqrt {1-d^2\,x^2}+\frac {d^2\,e\,x}{f}+1\right )-\ln \left (e+f\,x\right )\right )\,\left (d^2\,e^2+2\,d\,e\,f+f^2\right )}{f^3\,\sqrt {1-\frac {d^2\,e^2}{f^2}}} \] Input:
int((1 - d^2*x^2)^(3/2)/((e + f*x)*(d*x + 1)^2),x)
Output:
(asinh(x*(-d^2)^(1/2))*(2*d*(-d^2)^(1/2) + (d^2*e*(-d^2)^(1/2))/f))/(d^2*f ) - (1 - d^2*x^2)^(1/2)/f - ((log((1 - (d^2*e^2)/f^2)^(1/2)*(1 - d^2*x^2)^ (1/2) + (d^2*e*x)/f + 1) - log(e + f*x))*(f^2 + d^2*e^2 + 2*d*e*f))/(f^3*( 1 - (d^2*e^2)/f^2)^(1/2))
Time = 0.26 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.70 \[ \int \frac {\left (1-d^2 x^2\right )^{3/2}}{(1+d x)^2 (e+f x)} \, dx=\frac {-\mathit {asin} \left (d x \right ) d^{2} e^{2}-\mathit {asin} \left (d x \right ) d e f +2 \mathit {asin} \left (d x \right ) f^{2}+2 \sqrt {d^{2} e^{2}-f^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (d x \right )}{2}\right ) d e +f}{\sqrt {d^{2} e^{2}-f^{2}}}\right ) d e +2 \sqrt {d^{2} e^{2}-f^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (d x \right )}{2}\right ) d e +f}{\sqrt {d^{2} e^{2}-f^{2}}}\right ) f -\sqrt {-d^{2} x^{2}+1}\, d e f +\sqrt {-d^{2} x^{2}+1}\, f^{2}+d e f -f^{2}}{f^{2} \left (d e -f \right )} \] Input:
int((-d^2*x^2+1)^(3/2)/(d*x+1)^2/(f*x+e),x)
Output:
( - asin(d*x)*d**2*e**2 - asin(d*x)*d*e*f + 2*asin(d*x)*f**2 + 2*sqrt(d**2 *e**2 - f**2)*atan((tan(asin(d*x)/2)*d*e + f)/sqrt(d**2*e**2 - f**2))*d*e + 2*sqrt(d**2*e**2 - f**2)*atan((tan(asin(d*x)/2)*d*e + f)/sqrt(d**2*e**2 - f**2))*f - sqrt( - d**2*x**2 + 1)*d*e*f + sqrt( - d**2*x**2 + 1)*f**2 + d*e*f - f**2)/(f**2*(d*e - f))