Integrand size = 29, antiderivative size = 127 \[ \int \frac {\left (1-d^2 x^2\right )^{3/2}}{(1+d x)^2 (e+f x)^2} \, dx=\frac {(d e+f) \sqrt {1-d^2 x^2}}{(d e-f) f (e+f x)}+\frac {d \arcsin (d x)}{f^2}-\frac {d (d e-2 f) (d e+f)^2 \arctan \left (\frac {f+d^2 e x}{\sqrt {d^2 e^2-f^2} \sqrt {1-d^2 x^2}}\right )}{f^2 \left (d^2 e^2-f^2\right )^{3/2}} \] Output:
(d*e+f)*(-d^2*x^2+1)^(1/2)/(d*e-f)/f/(f*x+e)+d*arcsin(d*x)/f^2-d*(d*e-2*f) *(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)^(3/2)
Time = 0.71 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.15 \[ \int \frac {\left (1-d^2 x^2\right )^{3/2}}{(1+d x)^2 (e+f x)^2} \, dx=\frac {\frac {f (d e+f) \sqrt {1-d^2 x^2}}{(d e-f) (e+f x)}+2 d \arctan \left (\frac {d x}{-1+\sqrt {1-d^2 x^2}}\right )+\frac {2 d (d e-2 f) \sqrt {d^2 e^2-f^2} \arctan \left (\frac {\sqrt {d^2 e^2-f^2} x}{e+f x-e \sqrt {1-d^2 x^2}}\right )}{(-d e+f)^2}}{f^2} \] Input:
Integrate[(1 - d^2*x^2)^(3/2)/((1 + d*x)^2*(e + f*x)^2),x]
Output:
((f*(d*e + f)*Sqrt[1 - d^2*x^2])/((d*e - f)*(e + f*x)) + 2*d*ArcTan[(d*x)/ (-1 + Sqrt[1 - d^2*x^2])] + (2*d*(d*e - 2*f)*Sqrt[d^2*e^2 - f^2]*ArcTan[(S qrt[d^2*e^2 - f^2]*x)/(e + f*x - e*Sqrt[1 - d^2*x^2])])/(-(d*e) + f)^2)/f^ 2
Time = 0.37 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.50, 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.
| \(\displaystyle \int \frac {\left (1-d^2 x^2\right )^{3/2}}{(d x+1)^2 (e+f x)^2} \, dx\) |
| \(\Big \downarrow \) 708 |
| \(\displaystyle \int \left (-\frac {2 d (d e+f)}{f^2 \sqrt {1-d^2 x^2} (e+f x)}+\frac {(d e+f)^2}{f^2 \sqrt {1-d^2 x^2} (e+f x)^2}+\frac {d^2}{f^2 \sqrt {1-d^2 x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \arcsin (d x)}{f^2}+\frac {d^2 e (d e+f)^2 \arctan \left (\frac {d^2 e x+f}{\sqrt {1-d^2 x^2} \sqrt {d^2 e^2-f^2}}\right )}{f^2 \left (d^2 e^2-f^2\right )^{3/2}}-\frac {2 d (d e+f) \arctan \left (\frac {d^2 e x+f}{\sqrt {1-d^2 x^2} \sqrt {d^2 e^2-f^2}}\right )}{f^2 \sqrt {d^2 e^2-f^2}}+\frac {\sqrt {1-d^2 x^2} (d e+f)}{f (d e-f) (e+f x)}\) |
Input:
Int[(1 - d^2*x^2)^(3/2)/((1 + d*x)^2*(e + f*x)^2),x]
Output:
((d*e + f)*Sqrt[1 - d^2*x^2])/((d*e - f)*f*(e + f*x)) + (d*ArcSin[d*x])/f^ 2 + (d^2*e*(d*e + f)^2*ArcTan[(f + d^2*e*x)/(Sqrt[d^2*e^2 - f^2]*Sqrt[1 - d^2*x^2])])/(f^2*(d^2*e^2 - f^2)^(3/2)) - (2*d*(d*e + f)*ArcTan[(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. \(1808\) vs. \(2(119)=238\).
Time = 1.16 (sec) , antiderivative size = 1809, normalized size of antiderivative = 14.24
Input:
int((-d^2*x^2+1)^(3/2)/(d*x+1)^2/(f*x+e)^2,x,method=_RETURNVERBOSE)
Output:
1/(d*e-f)^2*(1/d/(x+1/d)^2*(-d^2*(x+1/d)^2+2*d*(x+1/d))^(5/2)+3*d*(1/3*(-d ^2*(x+1/d)^2+2*d*(x+1/d))^(3/2)+d*(-1/4*(-2*d^2*(x+1/d)+2*d)/d^2*(-d^2*(x+ 1/d)^2+2*d*(x+1/d))^(1/2)+1/2/(d^2)^(1/2)*arctan((d^2)^(1/2)*x/(-d^2*(x+1/ d)^2+2*d*(x+1/d))^(1/2)))))+1/(d*e-f)^2*(1/(d^2*e^2-f^2)*f^2/(x+e/f)*(-d^2 *(x+e/f)^2+2*d^2*e/f*(x+e/f)-(d^2*e^2-f^2)/f^2)^(5/2)-3*d^2*e*f/(d^2*e^2-f ^2)*(1/3*(-d^2*(x+e/f)^2+2*d^2*e/f*(x+e/f)-(d^2*e^2-f^2)/f^2)^(3/2)+d^2*e/ f*(-1/4*(-2*d^2*(x+e/f)+2*d^2*e/f)/d^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)-1/8*(4*d^2*(d^2*e^2-f^2)/f^2-4*d^4*e^2/f^2)/d^2/(d ^2)^(1/2)*arctan((d^2)^(1/2)*x/(-d^2*(x+e/f)^2+2*d^2*e/f*(x+e/f)-(d^2*e^2- f^2)/f^2)^(1/2)))-(d^2*e^2-f^2)/f^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)+d^2*e/f/(d^2)^(1/2)*arctan((d^2)^(1/2)*x/(-d^2*(x+e/ f)^2+2*d^2*e/f*(x+e/f)-(d^2*e^2-f^2)/f^2)^(1/2))+(d^2*e^2-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))))+4*d^2/(d^2*e^2-f^2)*f^2*(-1/8*(-2*d^2*(x+e/f)+2*d^2*e/f)/d^ 2*(-d^2*(x+e/f)^2+2*d^2*e/f*(x+e/f)-(d^2*e^2-f^2)/f^2)^(3/2)-3/16*(4*d^2*( d^2*e^2-f^2)/f^2-4*d^4*e^2/f^2)/d^2*(-1/4*(-2*d^2*(x+e/f)+2*d^2*e/f)/d^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)-1/8*(4*d^2*(d^2* e^2-f^2)/f^2-4*d^4*e^2/f^2)/d^2/(d^2)^(1/2)*arctan((d^2)^(1/2)*x/(-d^2*(x+ e/f)^2+2*d^2*e/f*(x+e/f)-(d^2*e^2-f^2)/f^2)^(1/2)))))-2*d/(d*e-f)^3*f*(...
Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (119) = 238\).
Time = 0.19 (sec) , antiderivative size = 558, normalized size of antiderivative = 4.39 \[ \int \frac {\left (1-d^2 x^2\right )^{3/2}}{(1+d x)^2 (e+f x)^2} \, dx=\left [\frac {d e^{2} f + e f^{2} + {\left (d^{2} e^{3} - 2 \, d e^{2} f + {\left (d^{2} e^{2} f - 2 \, d e f^{2}\right )} x\right )} \sqrt {-\frac {d e + f}{d e - f}} \log \left (\frac {d^{2} e f x + f^{2} - {\left (d^{2} e^{2} - f^{2}\right )} \sqrt {-d^{2} x^{2} + 1} - {\left (d e f - f^{2} + {\left (d^{3} e^{2} - d^{2} e f\right )} x + \sqrt {-d^{2} x^{2} + 1} {\left (d e f - f^{2}\right )}\right )} \sqrt {-\frac {d e + f}{d e - f}}}{f x + e}\right ) + {\left (d e f^{2} + f^{3}\right )} x - 2 \, {\left (d^{2} e^{3} - d e^{2} f + {\left (d^{2} e^{2} f - d e f^{2}\right )} x\right )} \arctan \left (\frac {\sqrt {-d^{2} x^{2} + 1} - 1}{d x}\right ) + {\left (d e^{2} f + e f^{2}\right )} \sqrt {-d^{2} x^{2} + 1}}{d e^{3} f^{2} - e^{2} f^{3} + {\left (d e^{2} f^{3} - e f^{4}\right )} x}, \frac {d e^{2} f + e f^{2} + 2 \, {\left (d^{2} e^{3} - 2 \, d e^{2} f + {\left (d^{2} e^{2} f - 2 \, d e f^{2}\right )} x\right )} \sqrt {\frac {d e + f}{d e - f}} \arctan \left (-\frac {{\left (f x - \sqrt {-d^{2} x^{2} + 1} e + e\right )} \sqrt {\frac {d e + f}{d e - f}}}{{\left (d e + f\right )} x}\right ) + {\left (d e f^{2} + f^{3}\right )} x - 2 \, {\left (d^{2} e^{3} - d e^{2} f + {\left (d^{2} e^{2} f - d e f^{2}\right )} x\right )} \arctan \left (\frac {\sqrt {-d^{2} x^{2} + 1} - 1}{d x}\right ) + {\left (d e^{2} f + e f^{2}\right )} \sqrt {-d^{2} x^{2} + 1}}{d e^{3} f^{2} - e^{2} f^{3} + {\left (d e^{2} f^{3} - e f^{4}\right )} x}\right ] \] Input:
integrate((-d^2*x^2+1)^(3/2)/(d*x+1)^2/(f*x+e)^2,x, algorithm="fricas")
Output:
[(d*e^2*f + e*f^2 + (d^2*e^3 - 2*d*e^2*f + (d^2*e^2*f - 2*d*e*f^2)*x)*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)) + (d*e*f^2 + f^3)*x - 2*(d^ 2*e^3 - d*e^2*f + (d^2*e^2*f - d*e*f^2)*x)*arctan((sqrt(-d^2*x^2 + 1) - 1) /(d*x)) + (d*e^2*f + e*f^2)*sqrt(-d^2*x^2 + 1))/(d*e^3*f^2 - e^2*f^3 + (d* e^2*f^3 - e*f^4)*x), (d*e^2*f + e*f^2 + 2*(d^2*e^3 - 2*d*e^2*f + (d^2*e^2* f - 2*d*e*f^2)*x)*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)) + (d*e*f^2 + f^3)*x - 2*(d^2*e^3 - d*e^2*f + (d^2*e^2*f - d*e*f^2)*x)*arctan((sqrt(-d^2*x^2 + 1) - 1)/(d*x)) + (d*e^2*f + e*f^2)*sqrt(-d^2*x^2 + 1))/(d*e^3*f^2 - e^2*f^3 + (d*e^2*f^3 - e*f^4)*x)]
\[ \int \frac {\left (1-d^2 x^2\right )^{3/2}}{(1+d x)^2 (e+f x)^2} \, dx=\int \frac {\left (- \left (d x - 1\right ) \left (d x + 1\right )\right )^{\frac {3}{2}}}{\left (e + f x\right )^{2} \left (d x + 1\right )^{2}}\, dx \] Input:
integrate((-d**2*x**2+1)**(3/2)/(d*x+1)**2/(f*x+e)**2,x)
Output:
Integral((-(d*x - 1)*(d*x + 1))**(3/2)/((e + f*x)**2*(d*x + 1)**2), x)
\[ \int \frac {\left (1-d^2 x^2\right )^{3/2}}{(1+d x)^2 (e+f x)^2} \, dx=\int { \frac {{\left (-d^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (d x + 1\right )}^{2} {\left (f x + e\right )}^{2}} \,d x } \] Input:
integrate((-d^2*x^2+1)^(3/2)/(d*x+1)^2/(f*x+e)^2,x, algorithm="maxima")
Output:
integrate((-d^2*x^2 + 1)^(3/2)/((d*x + 1)^2*(f*x + e)^2), x)
Exception generated. \[ \int \frac {\left (1-d^2 x^2\right )^{3/2}}{(1+d x)^2 (e+f x)^2} \, dx=\text {Exception raised: NotImplementedError} \] Input:
integrate((-d^2*x^2+1)^(3/2)/(d*x+1)^2/(f*x+e)^2,x, algorithm="giac")
Output:
Exception raised: NotImplementedError >> unable to parse Giac output: abs( sageVARd)*((-2*sageVARf*sqrt(2*sageVARd*(sageVARd*sageVARx+1)^-1/sageVARd- 1)*sign((sageVARd*sageVARx+1)^-1)*sign(sageVARd)-2*sageVARd*sageVARe*sqrt( 2*sageVARd*(sageVAR
Time = 6.48 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.27 \[ \int \frac {\left (1-d^2 x^2\right )^{3/2}}{(1+d x)^2 (e+f x)^2} \, dx=\frac {d^2\,\mathrm {asinh}\left (x\,\sqrt {-d^2}\right )}{f^2\,\sqrt {-d^2}}-\frac {\frac {\sqrt {1-d^2\,x^2}\,\left (d^4\,e^2+2\,d^3\,e\,f+d^2\,f^2\right )}{f^2\,\left (\frac {d^2\,e^2}{f^2}-1\right )\,\left (x\,\sqrt {-d^2}+\frac {e\,\sqrt {-d^2}}{f}\right )}-\frac {e\,\left (\ln \left (\sqrt {1-\frac {d^2\,e^2}{f^2}}\,\sqrt {1-d^2\,x^2}+\frac {d^2\,e\,x}{f}+1\right )-\ln \left (e+f\,x\right )\right )\,\sqrt {-d^2}\,\left (d^4\,e^2+2\,d^3\,e\,f+d^2\,f^2\right )}{f^3\,{\left (1-\frac {d^2\,e^2}{f^2}\right )}^{3/2}}}{f^2\,\sqrt {-d^2}}+\frac {2\,d\,\left (\ln \left (\sqrt {1-\frac {d^2\,e^2}{f^2}}\,\sqrt {1-d^2\,x^2}+\frac {d^2\,e\,x}{f}+1\right )-\ln \left (e+f\,x\right )\right )\,\left (f+d\,e\right )}{f^3\,\sqrt {1-\frac {d^2\,e^2}{f^2}}} \] Input:
int((1 - d^2*x^2)^(3/2)/((e + f*x)^2*(d*x + 1)^2),x)
Output:
(d^2*asinh(x*(-d^2)^(1/2)))/(f^2*(-d^2)^(1/2)) - (((1 - d^2*x^2)^(1/2)*(d^ 4*e^2 + d^2*f^2 + 2*d^3*e*f))/(f^2*((d^2*e^2)/f^2 - 1)*(x*(-d^2)^(1/2) + ( e*(-d^2)^(1/2))/f)) - (e*(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))*(-d^2)^(1/2)*(d^4*e^2 + d^2*f^2 + 2*d ^3*e*f))/(f^3*(1 - (d^2*e^2)/f^2)^(3/2)))/(f^2*(-d^2)^(1/2)) + (2*d*(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 + d*e))/(f^3*(1 - (d^2*e^2)/f^2)^(1/2))
Time = 0.25 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.83 \[ \int \frac {\left (1-d^2 x^2\right )^{3/2}}{(1+d x)^2 (e+f x)^2} \, dx=\frac {\mathit {asin} \left (d x \right ) d^{3} e^{3}+\mathit {asin} \left (d x \right ) d^{3} e^{2} f x -2 \mathit {asin} \left (d x \right ) d^{2} e^{2} f -2 \mathit {asin} \left (d x \right ) d^{2} e \,f^{2} x +\mathit {asin} \left (d x \right ) d e \,f^{2}+\mathit {asin} \left (d x \right ) d \,f^{3} x -2 \sqrt {d^{2} e^{2}-f^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (d x \right )}{2}\right ) d e +f}{\sqrt {d^{2} e^{2}-f^{2}}}\right ) d^{2} e^{2}-2 \sqrt {d^{2} e^{2}-f^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (d x \right )}{2}\right ) d e +f}{\sqrt {d^{2} e^{2}-f^{2}}}\right ) d^{2} e f x +4 \sqrt {d^{2} e^{2}-f^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (d x \right )}{2}\right ) d e +f}{\sqrt {d^{2} e^{2}-f^{2}}}\right ) d e f +4 \sqrt {d^{2} e^{2}-f^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (d x \right )}{2}\right ) d e +f}{\sqrt {d^{2} e^{2}-f^{2}}}\right ) d \,f^{2} x +\sqrt {-d^{2} x^{2}+1}\, d^{2} e^{2} f -\sqrt {-d^{2} x^{2}+1}\, f^{3}}{f^{2} \left (d^{2} e^{2} f x +d^{2} e^{3}-2 d e \,f^{2} x -2 d \,e^{2} f +f^{3} x +e \,f^{2}\right )} \] Input:
int((-d^2*x^2+1)^(3/2)/(d*x+1)^2/(f*x+e)^2,x)
Output:
(asin(d*x)*d**3*e**3 + asin(d*x)*d**3*e**2*f*x - 2*asin(d*x)*d**2*e**2*f - 2*asin(d*x)*d**2*e*f**2*x + asin(d*x)*d*e*f**2 + asin(d*x)*d*f**3*x - 2*s qrt(d**2*e**2 - f**2)*atan((tan(asin(d*x)/2)*d*e + f)/sqrt(d**2*e**2 - f** 2))*d**2*e**2 - 2*sqrt(d**2*e**2 - f**2)*atan((tan(asin(d*x)/2)*d*e + f)/s qrt(d**2*e**2 - f**2))*d**2*e*f*x + 4*sqrt(d**2*e**2 - f**2)*atan((tan(asi n(d*x)/2)*d*e + f)/sqrt(d**2*e**2 - f**2))*d*e*f + 4*sqrt(d**2*e**2 - f**2 )*atan((tan(asin(d*x)/2)*d*e + f)/sqrt(d**2*e**2 - f**2))*d*f**2*x + sqrt( - d**2*x**2 + 1)*d**2*e**2*f - sqrt( - d**2*x**2 + 1)*f**3)/(f**2*(d**2*e **3 + d**2*e**2*f*x - 2*d*e**2*f - 2*d*e*f**2*x + e*f**2 + f**3*x))