\(\int \frac {(f+g x)^2}{(d^2-e^2 x^2)^2} \, dx\) [564]

Optimal result

Integrand size = 22, antiderivative size = 74 \[ \int \frac {(f+g x)^2}{\left (d^2-e^2 x^2\right )^2} \, dx=\frac {\left (d^2 g+e^2 f x\right ) (f+g x)}{2 d^2 e^2 \left (d^2-e^2 x^2\right )}+\frac {(e f-d g) (e f+d g) \text {arctanh}\left (\frac {e x}{d}\right )}{2 d^3 e^3} \]

[Out]

1/2*(e^2*f*x+d^2*g)*(g*x+f)/d^2/e^2/(-e^2*x^2+d^2)+1/2*(-d*g+e*f)*(d*g+e*f)*arctanh(e*x/d)/d^3/e^3

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {737, 214} \[ \int \frac {(f+g x)^2}{\left (d^2-e^2 x^2\right )^2} \, dx=\frac {\text {arctanh}\left (\frac {e x}{d}\right ) (e f-d g) (d g+e f)}{2 d^3 e^3}+\frac {(f+g x) \left (d^2 g+e^2 f x\right )}{2 d^2 e^2 \left (d^2-e^2 x^2\right )} \]

[In]

Int[(f + g*x)^2/(d^2 - e^2*x^2)^2,x]

[Out]

((d^2*g + e^2*f*x)*(f + g*x))/(2*d^2*e^2*(d^2 - e^2*x^2)) + ((e*f - d*g)*(e*f + d*g)*ArcTanh[(e*x)/d])/(2*d^3*
e^3)

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 737

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(a*e - c*d*x)*((a
+ c*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Dist[(2*p + 3)*((c*d^2 + a*e^2)/(2*a*c*(p + 1))), Int[(d + e*x)^(m -
2)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2, 0] && Lt
Q[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (d^2 g+e^2 f x\right ) (f+g x)}{2 d^2 e^2 \left (d^2-e^2 x^2\right )}-\frac {1}{2} \left (-\frac {f^2}{d^2}+\frac {g^2}{e^2}\right ) \int \frac {1}{d^2-e^2 x^2} \, dx \\ & = \frac {\left (d^2 g+e^2 f x\right ) (f+g x)}{2 d^2 e^2 \left (d^2-e^2 x^2\right )}+\frac {(e f-d g) (e f+d g) \tanh ^{-1}\left (\frac {e x}{d}\right )}{2 d^3 e^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.15 \[ \int \frac {(f+g x)^2}{\left (d^2-e^2 x^2\right )^2} \, dx=\frac {-2 d^2 f g-e^2 f^2 x-d^2 g^2 x}{2 d^2 e^2 \left (-d^2+e^2 x^2\right )}-\frac {\left (-e^2 f^2+d^2 g^2\right ) \text {arctanh}\left (\frac {e x}{d}\right )}{2 d^3 e^3} \]

[In]

Integrate[(f + g*x)^2/(d^2 - e^2*x^2)^2,x]

[Out]

(-2*d^2*f*g - e^2*f^2*x - d^2*g^2*x)/(2*d^2*e^2*(-d^2 + e^2*x^2)) - ((-(e^2*f^2) + d^2*g^2)*ArcTanh[(e*x)/d])/
(2*d^3*e^3)

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.47

method	result	size
norman	\(\frac {\frac {f g}{e^{2}}+\frac {\left (d^{2} g^{2}+e^{2} f^{2}\right ) x}{2 d^{2} e^{2}}}{-e^{2} x^{2}+d^{2}}+\frac {\left (d^{2} g^{2}-e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{4 e^{3} d^{3}}-\frac {\left (d^{2} g^{2}-e^{2} f^{2}\right ) \ln \left (e x +d \right )}{4 e^{3} d^{3}}\)	\(109\)
risch	\(\frac {\frac {f g}{e^{2}}+\frac {\left (d^{2} g^{2}+e^{2} f^{2}\right ) x}{2 d^{2} e^{2}}}{-e^{2} x^{2}+d^{2}}+\frac {\ln \left (e x -d \right ) g^{2}}{4 e^{3} d}-\frac {\ln \left (e x -d \right ) f^{2}}{4 e \,d^{3}}-\frac {\ln \left (-e x -d \right ) g^{2}}{4 e^{3} d}+\frac {\ln \left (-e x -d \right ) f^{2}}{4 e \,d^{3}}\)	\(126\)
default	\(\frac {\left (d^{2} g^{2}-e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{4 e^{3} d^{3}}+\frac {d^{2} g^{2}+2 d e f g +e^{2} f^{2}}{4 d^{2} e^{3} \left (-e x +d \right )}+\frac {\left (-d^{2} g^{2}+e^{2} f^{2}\right ) \ln \left (e x +d \right )}{4 e^{3} d^{3}}-\frac {d^{2} g^{2}-2 d e f g +e^{2} f^{2}}{4 d^{2} e^{3} \left (e x +d \right )}\)	\(136\)
parallelrisch	\(\frac {\ln \left (e x -d \right ) x^{2} d^{2} e^{2} g^{2}-\ln \left (e x -d \right ) x^{2} e^{4} f^{2}-\ln \left (e x +d \right ) x^{2} d^{2} e^{2} g^{2}+\ln \left (e x +d \right ) x^{2} e^{4} f^{2}-\ln \left (e x -d \right ) d^{4} g^{2}+\ln \left (e x -d \right ) d^{2} e^{2} f^{2}+\ln \left (e x +d \right ) d^{4} g^{2}-\ln \left (e x +d \right ) d^{2} e^{2} f^{2}-2 x \,d^{3} e \,g^{2}-2 x d \,e^{3} f^{2}-4 f g e \,d^{3}}{4 d^{3} e^{3} \left (e^{2} x^{2}-d^{2}\right )}\)	\(193\)

[In]

int((g*x+f)^2/(-e^2*x^2+d^2)^2,x,method=_RETURNVERBOSE)

[Out]

(f*g/e^2+1/2*(d^2*g^2+e^2*f^2)/d^2/e^2*x)/(-e^2*x^2+d^2)+1/4/e^3*(d^2*g^2-e^2*f^2)/d^3*ln(-e*x+d)-1/4/e^3*(d^2
*g^2-e^2*f^2)/d^3*ln(e*x+d)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (71) = 142\).

Time = 0.26 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.09 \[ \int \frac {(f+g x)^2}{\left (d^2-e^2 x^2\right )^2} \, dx=-\frac {4 \, d^{3} e f g + 2 \, {\left (d e^{3} f^{2} + d^{3} e g^{2}\right )} x + {\left (d^{2} e^{2} f^{2} - d^{4} g^{2} - {\left (e^{4} f^{2} - d^{2} e^{2} g^{2}\right )} x^{2}\right )} \log \left (e x + d\right ) - {\left (d^{2} e^{2} f^{2} - d^{4} g^{2} - {\left (e^{4} f^{2} - d^{2} e^{2} g^{2}\right )} x^{2}\right )} \log \left (e x - d\right )}{4 \, {\left (d^{3} e^{5} x^{2} - d^{5} e^{3}\right )}} \]

[In]

integrate((g*x+f)^2/(-e^2*x^2+d^2)^2,x, algorithm="fricas")

[Out]

-1/4*(4*d^3*e*f*g + 2*(d*e^3*f^2 + d^3*e*g^2)*x + (d^2*e^2*f^2 - d^4*g^2 - (e^4*f^2 - d^2*e^2*g^2)*x^2)*log(e*
x + d) - (d^2*e^2*f^2 - d^4*g^2 - (e^4*f^2 - d^2*e^2*g^2)*x^2)*log(e*x - d))/(d^3*e^5*x^2 - d^5*e^3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (66) = 132\).

Time = 0.33 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.11 \[ \int \frac {(f+g x)^2}{\left (d^2-e^2 x^2\right )^2} \, dx=\frac {- 2 d^{2} f g + x \left (- d^{2} g^{2} - e^{2} f^{2}\right )}{- 2 d^{4} e^{2} + 2 d^{2} e^{4} x^{2}} + \frac {\left (d g - e f\right ) \left (d g + e f\right ) \log {\left (- \frac {d \left (d g - e f\right ) \left (d g + e f\right )}{e \left (d^{2} g^{2} - e^{2} f^{2}\right )} + x \right )}}{4 d^{3} e^{3}} - \frac {\left (d g - e f\right ) \left (d g + e f\right ) \log {\left (\frac {d \left (d g - e f\right ) \left (d g + e f\right )}{e \left (d^{2} g^{2} - e^{2} f^{2}\right )} + x \right )}}{4 d^{3} e^{3}} \]

[In]

integrate((g*x+f)**2/(-e**2*x**2+d**2)**2,x)

[Out]

(-2*d**2*f*g + x*(-d**2*g**2 - e**2*f**2))/(-2*d**4*e**2 + 2*d**2*e**4*x**2) + (d*g - e*f)*(d*g + e*f)*log(-d*
(d*g - e*f)*(d*g + e*f)/(e*(d**2*g**2 - e**2*f**2)) + x)/(4*d**3*e**3) - (d*g - e*f)*(d*g + e*f)*log(d*(d*g -
e*f)*(d*g + e*f)/(e*(d**2*g**2 - e**2*f**2)) + x)/(4*d**3*e**3)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.50 \[ \int \frac {(f+g x)^2}{\left (d^2-e^2 x^2\right )^2} \, dx=-\frac {2 \, d^{2} f g + {\left (e^{2} f^{2} + d^{2} g^{2}\right )} x}{2 \, {\left (d^{2} e^{4} x^{2} - d^{4} e^{2}\right )}} + \frac {{\left (e^{2} f^{2} - d^{2} g^{2}\right )} \log \left (e x + d\right )}{4 \, d^{3} e^{3}} - \frac {{\left (e^{2} f^{2} - d^{2} g^{2}\right )} \log \left (e x - d\right )}{4 \, d^{3} e^{3}} \]

[In]

integrate((g*x+f)^2/(-e^2*x^2+d^2)^2,x, algorithm="maxima")

[Out]

-1/2*(2*d^2*f*g + (e^2*f^2 + d^2*g^2)*x)/(d^2*e^4*x^2 - d^4*e^2) + 1/4*(e^2*f^2 - d^2*g^2)*log(e*x + d)/(d^3*e
^3) - 1/4*(e^2*f^2 - d^2*g^2)*log(e*x - d)/(d^3*e^3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.54 \[ \int \frac {(f+g x)^2}{\left (d^2-e^2 x^2\right )^2} \, dx=-\frac {e^{2} f^{2} x + d^{2} g^{2} x + 2 \, d^{2} f g}{2 \, {\left (e^{2} x^{2} - d^{2}\right )} d^{2} e^{2}} + \frac {{\left (e^{3} f^{2} - d^{2} e g^{2}\right )} \log \left ({\left | e x + d \right |}\right )}{4 \, d^{3} e^{4}} - \frac {{\left (e^{3} f^{2} - d^{2} e g^{2}\right )} \log \left ({\left | e x - d \right |}\right )}{4 \, d^{3} e^{4}} \]

[In]

integrate((g*x+f)^2/(-e^2*x^2+d^2)^2,x, algorithm="giac")

[Out]

-1/2*(e^2*f^2*x + d^2*g^2*x + 2*d^2*f*g)/((e^2*x^2 - d^2)*d^2*e^2) + 1/4*(e^3*f^2 - d^2*e*g^2)*log(abs(e*x + d
))/(d^3*e^4) - 1/4*(e^3*f^2 - d^2*e*g^2)*log(abs(e*x - d))/(d^3*e^4)

Mupad [B] (verification not implemented)

Time = 12.05 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.55 \[ \int \frac {(f+g x)^2}{\left (d^2-e^2 x^2\right )^2} \, dx=\frac {\frac {f\,g}{e^2}+\frac {x\,\left (d^2\,g^2+e^2\,f^2\right )}{2\,d^2\,e^2}}{d^2-e^2\,x^2}-\frac {2\,\mathrm {atanh}\left (\frac {4\,e\,x\,\left (\frac {d^2\,g^2}{4}-\frac {e^2\,f^2}{4}\right )}{d\,\left (d^2\,g^2-e^2\,f^2\right )}\right )\,\left (\frac {d^2\,g^2}{4}-\frac {e^2\,f^2}{4}\right )}{d^3\,e^3} \]

[In]

int((f + g*x)^2/(d^2 - e^2*x^2)^2,x)

[Out]

((f*g)/e^2 + (x*(d^2*g^2 + e^2*f^2))/(2*d^2*e^2))/(d^2 - e^2*x^2) - (2*atanh((4*e*x*((d^2*g^2)/4 - (e^2*f^2)/4
))/(d*(d^2*g^2 - e^2*f^2)))*((d^2*g^2)/4 - (e^2*f^2)/4))/(d^3*e^3)