\(\int \frac {(f+g x)^2}{(d^2-e^2 x^2)^3} \, dx\) [576]

Optimal result

Integrand size = 22, antiderivative size = 127 \[ \int \frac {(f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx=\frac {\left (d^2 g+e^2 f x\right ) (f+g x)}{4 d^2 e^2 \left (d^2-e^2 x^2\right )^2}+\frac {2 d^2 f g+\left (3 e^2 f^2-d^2 g^2\right ) x}{8 d^4 e^2 \left (d^2-e^2 x^2\right )}+\frac {\left (3 e^2 f^2-d^2 g^2\right ) \text {arctanh}\left (\frac {e x}{d}\right )}{8 d^5 e^3} \]

[Out]

1/4*(e^2*f*x+d^2*g)*(g*x+f)/d^2/e^2/(-e^2*x^2+d^2)^2+1/8*(2*d^2*f*g+(-d^2*g^2+3*e^2*f^2)*x)/d^4/e^2/(-e^2*x^2+
d^2)+1/8*(-d^2*g^2+3*e^2*f^2)*arctanh(e*x/d)/d^5/e^3

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {753, 653, 214} \[ \int \frac {(f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx=\frac {\text {arctanh}\left (\frac {e x}{d}\right ) \left (3 e^2 f^2-d^2 g^2\right )}{8 d^5 e^3}+\frac {(f+g x) \left (d^2 g+e^2 f x\right )}{4 d^2 e^2 \left (d^2-e^2 x^2\right )^2}+\frac {x \left (3 e^2 f^2-d^2 g^2\right )+2 d^2 f g}{8 d^4 e^2 \left (d^2-e^2 x^2\right )} \]

[In]

Int[(f + g*x)^2/(d^2 - e^2*x^2)^3,x]

[Out]

((d^2*g + e^2*f*x)*(f + g*x))/(4*d^2*e^2*(d^2 - e^2*x^2)^2) + (2*d^2*f*g + (3*e^2*f^2 - d^2*g^2)*x)/(8*d^4*e^2
*(d^2 - e^2*x^2)) + ((3*e^2*f^2 - d^2*g^2)*ArcTanh[(e*x)/d])/(8*d^5*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 653

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

Rule 753

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[1/((p + 1)*(-2*a*c)), Int[(d + e*x)^(m - 2)*Simp[a*e^2*(m - 1) -
c*d^2*(2*p + 3) - d*c*e*(m + 2*p + 2)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^
2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (d^2 g+e^2 f x\right ) (f+g x)}{4 d^2 e^2 \left (d^2-e^2 x^2\right )^2}-\frac {\int \frac {-3 e^2 f^2+d^2 g^2-2 e^2 f g x}{\left (d^2-e^2 x^2\right )^2} \, dx}{4 d^2 e^2} \\ & = \frac {\left (d^2 g+e^2 f x\right ) (f+g x)}{4 d^2 e^2 \left (d^2-e^2 x^2\right )^2}+\frac {2 d^2 f g+\left (3 e^2 f^2-d^2 g^2\right ) x}{8 d^4 e^2 \left (d^2-e^2 x^2\right )}-\frac {\left (-\frac {3 e^2 f^2}{d^2}+g^2\right ) \int \frac {1}{d^2-e^2 x^2} \, dx}{8 d^2 e^2} \\ & = \frac {\left (d^2 g+e^2 f x\right ) (f+g x)}{4 d^2 e^2 \left (d^2-e^2 x^2\right )^2}+\frac {2 d^2 f g+\left (3 e^2 f^2-d^2 g^2\right ) x}{8 d^4 e^2 \left (d^2-e^2 x^2\right )}+\frac {\left (3 e^2 f^2-d^2 g^2\right ) \tanh ^{-1}\left (\frac {e x}{d}\right )}{8 d^5 e^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.87 \[ \int \frac {(f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx=\frac {-3 d e^5 f^2 x^3+d^5 e g (4 f+g x)+d^3 e^3 x \left (5 f^2+g^2 x^2\right )+\left (3 e^2 f^2-d^2 g^2\right ) \left (d^2-e^2 x^2\right )^2 \text {arctanh}\left (\frac {e x}{d}\right )}{8 d^5 e^3 \left (d^2-e^2 x^2\right )^2} \]

[In]

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

[Out]

(-3*d*e^5*f^2*x^3 + d^5*e*g*(4*f + g*x) + d^3*e^3*x*(5*f^2 + g^2*x^2) + (3*e^2*f^2 - d^2*g^2)*(d^2 - e^2*x^2)^
2*ArcTanh[(e*x)/d])/(8*d^5*e^3*(d^2 - e^2*x^2)^2)

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.06

method	result	size
norman	\(\frac {\frac {f g}{2 e^{2}}+\frac {\left (d^{2} g^{2}-3 e^{2} f^{2}\right ) x^{3}}{8 d^{4}}+\frac {\left (d^{2} g^{2}+5 e^{2} f^{2}\right ) x}{8 d^{2} e^{2}}}{\left (-e^{2} x^{2}+d^{2}\right )^{2}}+\frac {\left (d^{2} g^{2}-3 e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{16 d^{5} e^{3}}-\frac {\left (d^{2} g^{2}-3 e^{2} f^{2}\right ) \ln \left (e x +d \right )}{16 d^{5} e^{3}}\)	\(135\)
risch	\(\frac {\frac {f g}{2 e^{2}}+\frac {\left (d^{2} g^{2}-3 e^{2} f^{2}\right ) x^{3}}{8 d^{4}}+\frac {\left (d^{2} g^{2}+5 e^{2} f^{2}\right ) x}{8 d^{2} e^{2}}}{\left (-e^{2} x^{2}+d^{2}\right )^{2}}-\frac {\ln \left (-e x -d \right ) g^{2}}{16 d^{3} e^{3}}+\frac {3 \ln \left (-e x -d \right ) f^{2}}{16 d^{5} e}+\frac {\ln \left (e x -d \right ) g^{2}}{16 d^{3} e^{3}}-\frac {3 \ln \left (e x -d \right ) f^{2}}{16 d^{5} e}\)	\(152\)
default	\(\frac {\left (d^{2} g^{2}-3 e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{16 d^{5} e^{3}}-\frac {-d^{2} g^{2}-2 d e f g -e^{2} f^{2}}{16 e^{3} d^{3} \left (-e x +d \right )^{2}}+\frac {-d^{2} g^{2}+2 d e f g +3 e^{2} f^{2}}{16 e^{3} d^{4} \left (-e x +d \right )}+\frac {\left (-d^{2} g^{2}+3 e^{2} f^{2}\right ) \ln \left (e x +d \right )}{16 e^{3} d^{5}}-\frac {-d^{2} g^{2}-2 d e f g +3 e^{2} f^{2}}{16 e^{3} d^{4} \left (e x +d \right )}-\frac {d^{2} g^{2}-2 d e f g +e^{2} f^{2}}{16 e^{3} d^{3} \left (e x +d \right )^{2}}\)	\(216\)
parallelrisch	\(\frac {\ln \left (e x -d \right ) x^{4} d^{2} e^{5} g^{2}-3 \ln \left (e x -d \right ) x^{4} e^{7} f^{2}-\ln \left (e x +d \right ) x^{4} d^{2} e^{5} g^{2}+3 \ln \left (e x +d \right ) x^{4} e^{7} f^{2}-2 \ln \left (e x -d \right ) x^{2} d^{4} e^{3} g^{2}+6 \ln \left (e x -d \right ) x^{2} d^{2} e^{5} f^{2}+2 \ln \left (e x +d \right ) x^{2} d^{4} e^{3} g^{2}-6 \ln \left (e x +d \right ) x^{2} d^{2} e^{5} f^{2}+2 x^{3} d^{3} e^{4} g^{2}-6 x^{3} d \,e^{6} f^{2}+\ln \left (e x -d \right ) d^{6} e \,g^{2}-3 \ln \left (e x -d \right ) d^{4} e^{3} f^{2}-\ln \left (e x +d \right ) d^{6} e \,g^{2}+3 \ln \left (e x +d \right ) d^{4} e^{3} f^{2}+2 x \,d^{5} e^{2} g^{2}+10 x \,d^{3} e^{4} f^{2}+8 d^{5} e^{2} f g}{16 e^{4} d^{5} \left (e^{2} x^{2}-d^{2}\right )^{2}}\)	\(313\)

[In]

int((g*x+f)^2/(-e^2*x^2+d^2)^3,x,method=_RETURNVERBOSE)

[Out]

(1/2*f*g/e^2+1/8/d^4*(d^2*g^2-3*e^2*f^2)*x^3+1/8*(d^2*g^2+5*e^2*f^2)/d^2/e^2*x)/(-e^2*x^2+d^2)^2+1/16*(d^2*g^2
-3*e^2*f^2)/d^5/e^3*ln(-e*x+d)-1/16*(d^2*g^2-3*e^2*f^2)/d^5/e^3*ln(e*x+d)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (123) = 246\).

Time = 0.27 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.98 \[ \int \frac {(f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx=\frac {8 \, d^{5} e f g - 2 \, {\left (3 \, d e^{5} f^{2} - d^{3} e^{3} g^{2}\right )} x^{3} + 2 \, {\left (5 \, d^{3} e^{3} f^{2} + d^{5} e g^{2}\right )} x + {\left (3 \, d^{4} e^{2} f^{2} - d^{6} g^{2} + {\left (3 \, e^{6} f^{2} - d^{2} e^{4} g^{2}\right )} x^{4} - 2 \, {\left (3 \, d^{2} e^{4} f^{2} - d^{4} e^{2} g^{2}\right )} x^{2}\right )} \log \left (e x + d\right ) - {\left (3 \, d^{4} e^{2} f^{2} - d^{6} g^{2} + {\left (3 \, e^{6} f^{2} - d^{2} e^{4} g^{2}\right )} x^{4} - 2 \, {\left (3 \, d^{2} e^{4} f^{2} - d^{4} e^{2} g^{2}\right )} x^{2}\right )} \log \left (e x - d\right )}{16 \, {\left (d^{5} e^{7} x^{4} - 2 \, d^{7} e^{5} x^{2} + d^{9} e^{3}\right )}} \]

[In]

integrate((g*x+f)^2/(-e^2*x^2+d^2)^3,x, algorithm="fricas")

[Out]

1/16*(8*d^5*e*f*g - 2*(3*d*e^5*f^2 - d^3*e^3*g^2)*x^3 + 2*(5*d^3*e^3*f^2 + d^5*e*g^2)*x + (3*d^4*e^2*f^2 - d^6
*g^2 + (3*e^6*f^2 - d^2*e^4*g^2)*x^4 - 2*(3*d^2*e^4*f^2 - d^4*e^2*g^2)*x^2)*log(e*x + d) - (3*d^4*e^2*f^2 - d^
6*g^2 + (3*e^6*f^2 - d^2*e^4*g^2)*x^4 - 2*(3*d^2*e^4*f^2 - d^4*e^2*g^2)*x^2)*log(e*x - d))/(d^5*e^7*x^4 - 2*d^
7*e^5*x^2 + d^9*e^3)

Sympy [A] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.13 \[ \int \frac {(f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx=- \frac {- 4 d^{4} f g + x^{3} \left (- d^{2} e^{2} g^{2} + 3 e^{4} f^{2}\right ) + x \left (- d^{4} g^{2} - 5 d^{2} e^{2} f^{2}\right )}{8 d^{8} e^{2} - 16 d^{6} e^{4} x^{2} + 8 d^{4} e^{6} x^{4}} + \frac {\left (d^{2} g^{2} - 3 e^{2} f^{2}\right ) \log {\left (- \frac {d}{e} + x \right )}}{16 d^{5} e^{3}} - \frac {\left (d^{2} g^{2} - 3 e^{2} f^{2}\right ) \log {\left (\frac {d}{e} + x \right )}}{16 d^{5} e^{3}} \]

[In]

integrate((g*x+f)**2/(-e**2*x**2+d**2)**3,x)

[Out]

-(-4*d**4*f*g + x**3*(-d**2*e**2*g**2 + 3*e**4*f**2) + x*(-d**4*g**2 - 5*d**2*e**2*f**2))/(8*d**8*e**2 - 16*d*
*6*e**4*x**2 + 8*d**4*e**6*x**4) + (d**2*g**2 - 3*e**2*f**2)*log(-d/e + x)/(16*d**5*e**3) - (d**2*g**2 - 3*e**
2*f**2)*log(d/e + x)/(16*d**5*e**3)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.20 \[ \int \frac {(f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx=\frac {4 \, d^{4} f g - {\left (3 \, e^{4} f^{2} - d^{2} e^{2} g^{2}\right )} x^{3} + {\left (5 \, d^{2} e^{2} f^{2} + d^{4} g^{2}\right )} x}{8 \, {\left (d^{4} e^{6} x^{4} - 2 \, d^{6} e^{4} x^{2} + d^{8} e^{2}\right )}} + \frac {{\left (3 \, e^{2} f^{2} - d^{2} g^{2}\right )} \log \left (e x + d\right )}{16 \, d^{5} e^{3}} - \frac {{\left (3 \, e^{2} f^{2} - d^{2} g^{2}\right )} \log \left (e x - d\right )}{16 \, d^{5} e^{3}} \]

[In]

integrate((g*x+f)^2/(-e^2*x^2+d^2)^3,x, algorithm="maxima")

[Out]

1/8*(4*d^4*f*g - (3*e^4*f^2 - d^2*e^2*g^2)*x^3 + (5*d^2*e^2*f^2 + d^4*g^2)*x)/(d^4*e^6*x^4 - 2*d^6*e^4*x^2 + d
^8*e^2) + 1/16*(3*e^2*f^2 - d^2*g^2)*log(e*x + d)/(d^5*e^3) - 1/16*(3*e^2*f^2 - d^2*g^2)*log(e*x - d)/(d^5*e^3
)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.15 \[ \int \frac {(f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx=-\frac {3 \, e^{4} f^{2} x^{3} - d^{2} e^{2} g^{2} x^{3} - 5 \, d^{2} e^{2} f^{2} x - d^{4} g^{2} x - 4 \, d^{4} f g}{8 \, {\left (e^{2} x^{2} - d^{2}\right )}^{2} d^{4} e^{2}} + \frac {{\left (3 \, e^{3} f^{2} - d^{2} e g^{2}\right )} \log \left ({\left | e x + d \right |}\right )}{16 \, d^{5} e^{4}} - \frac {{\left (3 \, e^{3} f^{2} - d^{2} e g^{2}\right )} \log \left ({\left | e x - d \right |}\right )}{16 \, d^{5} e^{4}} \]

[In]

integrate((g*x+f)^2/(-e^2*x^2+d^2)^3,x, algorithm="giac")

[Out]

-1/8*(3*e^4*f^2*x^3 - d^2*e^2*g^2*x^3 - 5*d^2*e^2*f^2*x - d^4*g^2*x - 4*d^4*f*g)/((e^2*x^2 - d^2)^2*d^4*e^2) +
1/16*(3*e^3*f^2 - d^2*e*g^2)*log(abs(e*x + d))/(d^5*e^4) - 1/16*(3*e^3*f^2 - d^2*e*g^2)*log(abs(e*x - d))/(d^
5*e^4)

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.90 \[ \int \frac {(f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx=\frac {\frac {x^3\,\left (d^2\,g^2-3\,e^2\,f^2\right )}{8\,d^4}+\frac {f\,g}{2\,e^2}+\frac {x\,\left (d^2\,g^2+5\,e^2\,f^2\right )}{8\,d^2\,e^2}}{d^4-2\,d^2\,e^2\,x^2+e^4\,x^4}-\frac {\mathrm {atanh}\left (\frac {e\,x}{d}\right )\,\left (d^2\,g^2-3\,e^2\,f^2\right )}{8\,d^5\,e^3} \]

[In]

int((f + g*x)^2/(d^2 - e^2*x^2)^3,x)

[Out]

((x^3*(d^2*g^2 - 3*e^2*f^2))/(8*d^4) + (f*g)/(2*e^2) + (x*(d^2*g^2 + 5*e^2*f^2))/(8*d^2*e^2))/(d^4 + e^4*x^4 -
2*d^2*e^2*x^2) - (atanh((e*x)/d)*(d^2*g^2 - 3*e^2*f^2))/(8*d^5*e^3)