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

Optimal. Leaf size=113 \[ \frac {(d g+e f)^2 \tanh ^{-1}\left (\frac {e x}{d}\right )}{8 d^4 e^3}-\frac {(d g+e f)^2}{8 d^3 e^3 (d+e x)}-\frac {(3 d g+e f) (e f-d g)}{8 d^2 e^3 (d+e x)^2}-\frac {(e f-d g)^2}{6 d e^3 (d+e x)^3} \]

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 113, 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 = {848, 88, 208} \begin {gather*} -\frac {(3 d g+e f) (e f-d g)}{8 d^2 e^3 (d+e x)^2}-\frac {(d g+e f)^2}{8 d^3 e^3 (d+e x)}+\frac {(d g+e f)^2 \tanh ^{-1}\left (\frac {e x}{d}\right )}{8 d^4 e^3}-\frac {(e f-d g)^2}{6 d e^3 (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(e*f - d*g)^2/(6*d*e^3*(d + e*x)^3) - ((e*f - d*g)*(e*f + 3*d*g))/(8*d^2*e^3*(d + e*x)^2) - (e*f + d*g)^2/(8*
d^3*e^3*(d + e*x)) + ((e*f + d*g)^2*ArcTanh[(e*x)/d])/(8*d^4*e^3)

Rule 88

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 208

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

Rule 848

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*x)/e)^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)^3 \left (d^2-e^2 x^2\right )} \, dx &=\int \frac {(f+g x)^2}{(d-e x) (d+e x)^4} \, dx\\ &=\int \left (\frac {(-e f+d g)^2}{2 d e^2 (d+e x)^4}+\frac {(e f-d g) (e f+3 d g)}{4 d^2 e^2 (d+e x)^3}+\frac {(e f+d g)^2}{8 d^3 e^2 (d+e x)^2}+\frac {(e f+d g)^2}{8 d^3 e^2 \left (d^2-e^2 x^2\right )}\right ) \, dx\\ &=-\frac {(e f-d g)^2}{6 d e^3 (d+e x)^3}-\frac {(e f-d g) (e f+3 d g)}{8 d^2 e^3 (d+e x)^2}-\frac {(e f+d g)^2}{8 d^3 e^3 (d+e x)}+\frac {(e f+d g)^2 \int \frac {1}{d^2-e^2 x^2} \, dx}{8 d^3 e^2}\\ &=-\frac {(e f-d g)^2}{6 d e^3 (d+e x)^3}-\frac {(e f-d g) (e f+3 d g)}{8 d^2 e^3 (d+e x)^2}-\frac {(e f+d g)^2}{8 d^3 e^3 (d+e x)}+\frac {(e f+d g)^2 \tanh ^{-1}\left (\frac {e x}{d}\right )}{8 d^4 e^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 122, normalized size = 1.08 \begin {gather*} \frac {-\frac {8 d^3 (e f-d g)^2}{(d+e x)^3}+\frac {6 d^2 \left (3 d^2 g^2-2 d e f g-e^2 f^2\right )}{(d+e x)^2}-\frac {6 d (d g+e f)^2}{d+e x}-3 (d g+e f)^2 \log (d-e x)+3 (d g+e f)^2 \log (d+e x)}{48 d^4 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-8*d^3*(e*f - d*g)^2)/(d + e*x)^3 + (6*d^2*(-(e^2*f^2) - 2*d*e*f*g + 3*d^2*g^2))/(d + e*x)^2 - (6*d*(e*f + d
*g)^2)/(d + e*x) - 3*(e*f + d*g)^2*Log[d - e*x] + 3*(e*f + d*g)^2*Log[d + e*x])/(48*d^4*e^3)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(f+g x)^2}{(d+e x)^3 \left (d^2-e^2 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.39, size = 400, normalized size = 3.54 \begin {gather*} -\frac {20 \, d^{3} e^{2} f^{2} + 8 \, d^{4} e f g - 4 \, d^{5} g^{2} + 6 \, {\left (d e^{4} f^{2} + 2 \, d^{2} e^{3} f g + d^{3} e^{2} g^{2}\right )} x^{2} + 6 \, {\left (3 \, d^{2} e^{3} f^{2} + 6 \, d^{3} e^{2} f g - d^{4} e g^{2}\right )} x - 3 \, {\left (d^{3} e^{2} f^{2} + 2 \, d^{4} e f g + d^{5} g^{2} + {\left (e^{5} f^{2} + 2 \, d e^{4} f g + d^{2} e^{3} g^{2}\right )} x^{3} + 3 \, {\left (d e^{4} f^{2} + 2 \, d^{2} e^{3} f g + d^{3} e^{2} g^{2}\right )} x^{2} + 3 \, {\left (d^{2} e^{3} f^{2} + 2 \, d^{3} e^{2} f g + d^{4} e g^{2}\right )} x\right )} \log \left (e x + d\right ) + 3 \, {\left (d^{3} e^{2} f^{2} + 2 \, d^{4} e f g + d^{5} g^{2} + {\left (e^{5} f^{2} + 2 \, d e^{4} f g + d^{2} e^{3} g^{2}\right )} x^{3} + 3 \, {\left (d e^{4} f^{2} + 2 \, d^{2} e^{3} f g + d^{3} e^{2} g^{2}\right )} x^{2} + 3 \, {\left (d^{2} e^{3} f^{2} + 2 \, d^{3} e^{2} f g + d^{4} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{48 \, {\left (d^{4} e^{6} x^{3} + 3 \, d^{5} e^{5} x^{2} + 3 \, d^{6} e^{4} x + d^{7} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(20*d^3*e^2*f^2 + 8*d^4*e*f*g - 4*d^5*g^2 + 6*(d*e^4*f^2 + 2*d^2*e^3*f*g + d^3*e^2*g^2)*x^2 + 6*(3*d^2*e
^3*f^2 + 6*d^3*e^2*f*g - d^4*e*g^2)*x - 3*(d^3*e^2*f^2 + 2*d^4*e*f*g + d^5*g^2 + (e^5*f^2 + 2*d*e^4*f*g + d^2*
e^3*g^2)*x^3 + 3*(d*e^4*f^2 + 2*d^2*e^3*f*g + d^3*e^2*g^2)*x^2 + 3*(d^2*e^3*f^2 + 2*d^3*e^2*f*g + d^4*e*g^2)*x
)*log(e*x + d) + 3*(d^3*e^2*f^2 + 2*d^4*e*f*g + d^5*g^2 + (e^5*f^2 + 2*d*e^4*f*g + d^2*e^3*g^2)*x^3 + 3*(d*e^4
*f^2 + 2*d^2*e^3*f*g + d^3*e^2*g^2)*x^2 + 3*(d^2*e^3*f^2 + 2*d^3*e^2*f*g + d^4*e*g^2)*x)*log(e*x - d))/(d^4*e^
6*x^3 + 3*d^5*e^5*x^2 + 3*d^6*e^4*x + d^7*e^3)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: -(2*exp(2)^2*d*g*f-3*exp(2)^2*exp(1)*f^2
-3*exp(2)*d^2*exp(1)*g^2+6*exp(2)*d*exp(1)^2*g*f-exp(2)*exp(1)^3*f^2-d^2*exp(1)^3*g^2)/(2*exp(2)^3*d^4-6*exp(2
)^2*d^4*exp(1)^2+6*exp(2)*d^4*exp(1)^4-2*d^4*exp(1)^6)*ln(abs(-x^2*exp(2)+d^2))-(-exp(2)^3*f^2-exp(2)^2*d^2*g^
2+6*exp(2)^2*d*exp(1)*g*f-3*exp(2)^2*exp(1)^2*f^2-3*exp(2)*d^2*exp(1)^2*g^2+2*exp(2)*d*exp(1)^3*g*f)*1/2/(exp(
2)^3*d^3-3*exp(2)^2*d^3*exp(1)^2+3*exp(2)*d^3*exp(1)^4-d^3*exp(1)^6)/exp(1)/abs(d)*ln(abs(-2*x*exp(2)-2*exp(1)
*abs(d))/abs(-2*x*exp(2)+2*exp(1)*abs(d)))-(-2*exp(2)^2*d*exp(1)*g*f+3*exp(2)^2*exp(1)^2*f^2+3*exp(2)*d^2*exp(
1)^2*g^2-6*exp(2)*d*exp(1)^3*g*f+exp(2)*exp(1)^4*f^2+d^2*exp(1)^4*g^2)/(exp(2)^3*d^4*exp(1)-3*exp(2)^2*d^4*exp
(1)^3+3*exp(2)*d^4*exp(1)^5-d^4*exp(1)^7)*ln(abs(x*exp(1)+d))-(-exp(2)^2*d^4*g^2+6*exp(2)^2*d^3*exp(1)*g*f-5*e
xp(2)^2*d^2*exp(1)^2*f^2-2*exp(2)*d^4*exp(1)^2*g^2-4*exp(2)*d^3*exp(1)^3*g*f+6*exp(2)*d^2*exp(1)^4*f^2+3*d^4*e
xp(1)^4*g^2-2*d^3*exp(1)^5*g*f-d^2*exp(1)^6*f^2+(4*exp(2)^2*d^2*exp(1)^2*g*f-4*exp(2)^2*d*exp(1)^3*f^2-4*exp(2
)*d^3*exp(1)^3*g^2+4*exp(2)*d*exp(1)^5*f^2+4*d^3*exp(1)^5*g^2-4*d^2*exp(1)^6*g*f)*x)/2/d^4/exp(1)/(exp(2)-exp(
1)^2)^3/(x*exp(1)+d)^2

________________________________________________________________________________________

maple [B]  time = 0.01, size = 259, normalized size = 2.29 \begin {gather*} -\frac {d \,g^{2}}{6 \left (e x +d \right )^{3} e^{3}}-\frac {f^{2}}{6 \left (e x +d \right )^{3} d e}+\frac {f g}{3 \left (e x +d \right )^{3} e^{2}}-\frac {f g}{4 \left (e x +d \right )^{2} d \,e^{2}}-\frac {f^{2}}{8 \left (e x +d \right )^{2} d^{2} e}+\frac {3 g^{2}}{8 \left (e x +d \right )^{2} e^{3}}-\frac {g^{2}}{8 \left (e x +d \right ) d \,e^{3}}-\frac {f g}{4 \left (e x +d \right ) d^{2} e^{2}}-\frac {g^{2} \ln \left (e x -d \right )}{16 d^{2} e^{3}}+\frac {g^{2} \ln \left (e x +d \right )}{16 d^{2} e^{3}}-\frac {f^{2}}{8 \left (e x +d \right ) d^{3} e}-\frac {f g \ln \left (e x -d \right )}{8 d^{3} e^{2}}+\frac {f g \ln \left (e x +d \right )}{8 d^{3} e^{2}}-\frac {f^{2} \ln \left (e x -d \right )}{16 d^{4} e}+\frac {f^{2} \ln \left (e x +d \right )}{16 d^{4} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.48, size = 206, normalized size = 1.82 \begin {gather*} -\frac {10 \, d^{2} e^{2} f^{2} + 4 \, d^{3} e f g - 2 \, d^{4} g^{2} + 3 \, {\left (e^{4} f^{2} + 2 \, d e^{3} f g + d^{2} e^{2} g^{2}\right )} x^{2} + 3 \, {\left (3 \, d e^{3} f^{2} + 6 \, d^{2} e^{2} f g - d^{3} e g^{2}\right )} x}{24 \, {\left (d^{3} e^{6} x^{3} + 3 \, d^{4} e^{5} x^{2} + 3 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} + \frac {{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x + d\right )}{16 \, d^{4} e^{3}} - \frac {{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{16 \, d^{4} e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(10*d^2*e^2*f^2 + 4*d^3*e*f*g - 2*d^4*g^2 + 3*(e^4*f^2 + 2*d*e^3*f*g + d^2*e^2*g^2)*x^2 + 3*(3*d*e^3*f^2
+ 6*d^2*e^2*f*g - d^3*e*g^2)*x)/(d^3*e^6*x^3 + 3*d^4*e^5*x^2 + 3*d^5*e^4*x + d^6*e^3) + 1/16*(e^2*f^2 + 2*d*e
*f*g + d^2*g^2)*log(e*x + d)/(d^4*e^3) - 1/16*(e^2*f^2 + 2*d*e*f*g + d^2*g^2)*log(e*x - d)/(d^4*e^3)

________________________________________________________________________________________

mupad [B]  time = 2.65, size = 152, normalized size = 1.35 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {e\,x}{d}\right )\,{\left (d\,g+e\,f\right )}^2}{8\,d^4\,e^3}-\frac {\frac {-d^2\,g^2+2\,d\,e\,f\,g+5\,e^2\,f^2}{12\,d\,e^3}+\frac {x\,\left (-d^2\,g^2+6\,d\,e\,f\,g+3\,e^2\,f^2\right )}{8\,d^2\,e^2}+\frac {x^2\,\left (d^2\,g^2+2\,d\,e\,f\,g+e^2\,f^2\right )}{8\,d^3\,e}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^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)^3),x)

[Out]

(atanh((e*x)/d)*(d*g + e*f)^2)/(8*d^4*e^3) - ((5*e^2*f^2 - d^2*g^2 + 2*d*e*f*g)/(12*d*e^3) + (x*(3*e^2*f^2 - d
^2*g^2 + 6*d*e*f*g))/(8*d^2*e^2) + (x^2*(d^2*g^2 + e^2*f^2 + 2*d*e*f*g))/(8*d^3*e))/(d^3 + e^3*x^3 + 3*d*e^2*x
^2 + 3*d^2*e*x)

________________________________________________________________________________________

sympy [B]  time = 1.42, size = 248, normalized size = 2.19 \begin {gather*} - \frac {- 2 d^{4} g^{2} + 4 d^{3} e f g + 10 d^{2} e^{2} f^{2} + x^{2} \left (3 d^{2} e^{2} g^{2} + 6 d e^{3} f g + 3 e^{4} f^{2}\right ) + x \left (- 3 d^{3} e g^{2} + 18 d^{2} e^{2} f g + 9 d e^{3} f^{2}\right )}{24 d^{6} e^{3} + 72 d^{5} e^{4} x + 72 d^{4} e^{5} x^{2} + 24 d^{3} e^{6} x^{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 )}}{16 d^{4} 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 )}}{16 d^{4} e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-2*d**4*g**2 + 4*d**3*e*f*g + 10*d**2*e**2*f**2 + x**2*(3*d**2*e**2*g**2 + 6*d*e**3*f*g + 3*e**4*f**2) + x*(
-3*d**3*e*g**2 + 18*d**2*e**2*f*g + 9*d*e**3*f**2))/(24*d**6*e**3 + 72*d**5*e**4*x + 72*d**4*e**5*x**2 + 24*d*
*3*e**6*x**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)/(16*d**4*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)/(16*d**4*e**3)

________________________________________________________________________________________