3.6.54 \(\int \frac {(f+g x)^2}{(d+e x)^2 (d^2-e^2 x^2)} \, dx\) [554]

Optimal. Leaf size=87 \[ -\frac {(e f-d g)^2}{4 d e^3 (d+e x)^2}-\frac {(e f-d g) (e f+3 d g)}{4 d^2 e^3 (d+e x)}+\frac {(e f+d g)^2 \tanh ^{-1}\left (\frac {e x}{d}\right )}{4 d^3 e^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.06, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {862, 90, 214} \begin {gather*} \frac {(d g+e f)^2 \tanh ^{-1}\left (\frac {e x}{d}\right )}{4 d^3 e^3}-\frac {(3 d g+e f) (e f-d g)}{4 d^2 e^3 (d+e x)}-\frac {(e f-d g)^2}{4 d e^3 (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(e*f - d*g)^2/(d*e^3*(d + e*x)^2) - ((e*f - d*g)*(e*f + 3*d*g))/(4*d^2*e^3*(d + e*x)) + ((e*f + d*g)^2*Ar
cTanh[(e*x)/d])/(4*d^3*e^3)

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

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 862

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)
^(m + p)*(f + g*x)^n*(a/d + (c/e)*x)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c
*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && EqQ[m + p, 0]))

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.05, size = 87, normalized size = 1.00 \begin {gather*} \frac {\frac {2 d (-e f+d g) \left (2 d^2 g+e^2 f x+d e (2 f+3 g x)\right )}{(d+e x)^2}-(e f+d g)^2 \log (d-e x)+(e f+d g)^2 \log (d+e x)}{8 d^3 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.09, size = 148, normalized size = 1.70

method	result	size
norman	\(\frac {\frac {d^{2} g^{2}-e^{2} f^{2}}{2 d \,e^{3}}+\frac {\left (3 d^{2} g^{2}-2 d e f g -e^{2} f^{2}\right ) x}{4 d^{2} e^{2}}}{\left (e x +d \right )^{2}}-\frac {\left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{8 e^{3} d^{3}}+\frac {\left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right ) \ln \left (e x +d \right )}{8 e^{3} d^{3}}\)	\(138\)
default	\(-\frac {-3 d^{2} g^{2}+2 d e f g +e^{2} f^{2}}{4 d^{2} e^{3} \left (e x +d \right )}-\frac {d^{2} g^{2}-2 d e f g +e^{2} f^{2}}{4 e^{3} d \left (e x +d \right )^{2}}+\frac {\left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right ) \ln \left (e x +d \right )}{8 e^{3} d^{3}}+\frac {\left (-d^{2} g^{2}-2 d e f g -e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{8 e^{3} d^{3}}\)	\(148\)
risch	\(\frac {\frac {d^{2} g^{2}-e^{2} f^{2}}{2 d \,e^{3}}+\frac {\left (3 d^{2} g^{2}-2 d e f g -e^{2} f^{2}\right ) x}{4 d^{2} e^{2}}}{\left (e x +d \right )^{2}}-\frac {\ln \left (-e x +d \right ) g^{2}}{8 e^{3} d}-\frac {\ln \left (-e x +d \right ) f g}{4 e^{2} d^{2}}-\frac {\ln \left (-e x +d \right ) f^{2}}{8 e \,d^{3}}+\frac {\ln \left (e x +d \right ) g^{2}}{8 e^{3} d}+\frac {\ln \left (e x +d \right ) f g}{4 e^{2} d^{2}}+\frac {\ln \left (e x +d \right ) f^{2}}{8 e \,d^{3}}\)	\(170\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 145, normalized size = 1.67 \begin {gather*} \frac {2 \, d^{3} g^{2} - 2 \, d f^{2} e^{2} + {\left (3 \, d^{2} g^{2} e - 2 \, d f g e^{2} - f^{2} e^{3}\right )} x}{4 \, {\left (d^{2} x^{2} e^{5} + 2 \, d^{3} x e^{4} + d^{4} e^{3}\right )}} + \frac {{\left (d^{2} g^{2} + 2 \, d f g e + f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left (x e + d\right )}{8 \, d^{3}} - \frac {{\left (d^{2} g^{2} + 2 \, d f g e + f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left (x e - d\right )}{8 \, d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 261 vs. \(2 (85) = 170\).
time = 2.93, size = 261, normalized size = 3.00 \begin {gather*} \frac {6 \, d^{3} g^{2} x e + 4 \, d^{4} g^{2} - 2 \, d f^{2} x e^{3} - 4 \, {\left (d^{2} f g x + d^{2} f^{2}\right )} e^{2} + {\left (d^{4} g^{2} + f^{2} x^{2} e^{4} + 2 \, {\left (d f g x^{2} + d f^{2} x\right )} e^{3} + {\left (d^{2} g^{2} x^{2} + 4 \, d^{2} f g x + d^{2} f^{2}\right )} e^{2} + 2 \, {\left (d^{3} g^{2} x + d^{3} f g\right )} e\right )} \log \left (x e + d\right ) - {\left (d^{4} g^{2} + f^{2} x^{2} e^{4} + 2 \, {\left (d f g x^{2} + d f^{2} x\right )} e^{3} + {\left (d^{2} g^{2} x^{2} + 4 \, d^{2} f g x + d^{2} f^{2}\right )} e^{2} + 2 \, {\left (d^{3} g^{2} x + d^{3} f g\right )} e\right )} \log \left (x e - d\right )}{8 \, {\left (d^{3} x^{2} e^{5} + 2 \, d^{4} x e^{4} + d^{5} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (75) = 150\).
time = 0.49, size = 185, normalized size = 2.13 \begin {gather*} - \frac {- 2 d^{3} g^{2} + 2 d e^{2} f^{2} + x \left (- 3 d^{2} e g^{2} + 2 d e^{2} f g + e^{3} f^{2}\right )}{4 d^{4} e^{3} + 8 d^{3} e^{4} x + 4 d^{2} e^{5} x^{2}} - \frac {\left (d g + e f\right )^{2} \log {\left (- \frac {d \left (d g + e f\right )^{2}}{e \left (d^{2} g^{2} + 2 d e f g + e^{2} f^{2}\right )} + x \right )}}{8 d^{3} e^{3}} + \frac {\left (d g + e f\right )^{2} \log {\left (\frac {d \left (d g + e f\right )^{2}}{e \left (d^{2} g^{2} + 2 d e f g + e^{2} f^{2}\right )} + x \right )}}{8 d^{3} e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 1.28, size = 151, normalized size = 1.74 \begin {gather*} \frac {{\left (\frac {3 \, d^{2} g^{2} e^{3}}{x e + d} - \frac {d^{3} g^{2} e^{3}}{{\left (x e + d\right )}^{2}} - \frac {2 \, d f g e^{4}}{x e + d} + \frac {2 \, d^{2} f g e^{4}}{{\left (x e + d\right )}^{2}} - \frac {f^{2} e^{5}}{x e + d} - \frac {d f^{2} e^{5}}{{\left (x e + d\right )}^{2}}\right )} e^{\left (-6\right )}}{4 \, d^{2}} - \frac {{\left (d^{2} g^{2} + 2 \, d f g e + f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | -\frac {2 \, d}{x e + d} + 1 \right |}\right )}{8 \, d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.13, size = 100, normalized size = 1.15 \begin {gather*} \frac {\frac {d^2\,g^2-e^2\,f^2}{2\,d\,e^3}-\frac {x\,\left (-3\,d^2\,g^2+2\,d\,e\,f\,g+e^2\,f^2\right )}{4\,d^2\,e^2}}{d^2+2\,d\,e\,x+e^2\,x^2}+\frac {\mathrm {atanh}\left (\frac {e\,x}{d}\right )\,{\left (d\,g+e\,f\right )}^2}{4\,d^3\,e^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________