3.6.65 \(\int \frac {(f+g x)^2}{(d+e x) (d^2-e^2 x^2)^2} \, dx\) [565]

Optimal. Leaf size=121 \[ \frac {(e f+d g)^2}{8 d^3 e^3 (d-e x)}-\frac {(e f-d g)^2}{8 d^2 e^3 (d+e x)^2}-\frac {e^2 f^2-d^2 g^2}{4 d^3 e^3 (d+e x)}+\frac {(3 e f-d g) (e f+d g) \tanh ^{-1}\left (\frac {e x}{d}\right )}{8 d^4 e^3} \]

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 121, 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 {(3 e f-d g) (d g+e f) \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 {(e f-d g)^2}{8 d^2 e^3 (d+e x)^2}-\frac {e^2 f^2-d^2 g^2}{4 d^3 e^3 (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.08, size = 139, normalized size = 1.15 \begin {gather*} \frac {\frac {2 d (e f+d g)^2}{d-e x}-\frac {2 d^2 (e f-d g)^2}{(d+e x)^2}+\frac {4 d \left (-e^2 f^2+d^2 g^2\right )}{d+e x}+\left (-3 e^2 f^2-2 d e f g+d^2 g^2\right ) \log (d-e x)+\left (3 e^2 f^2+2 d e f g-d^2 g^2\right ) \log (d+e x)}{16 d^4 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*d*(e*f + d*g)^2)/(d - e*x) - (2*d^2*(e*f - d*g)^2)/(d + e*x)^2 + (4*d*(-(e^2*f^2) + d^2*g^2))/(d + e*x) +
(-3*e^2*f^2 - 2*d*e*f*g + d^2*g^2)*Log[d - e*x] + (3*e^2*f^2 + 2*d*e*f*g - d^2*g^2)*Log[d + e*x])/(16*d^4*e^3)

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Maple [A]
time = 0.09, size = 180, normalized size = 1.49

method	result	size
default	\(-\frac {-d^{2} g^{2}+e^{2} f^{2}}{4 e^{3} d^{3} \left (e x +d \right )}+\frac {\left (-d^{2} g^{2}+2 d e f g +3 e^{2} f^{2}\right ) \ln \left (e x +d \right )}{16 e^{3} d^{4}}-\frac {d^{2} g^{2}-2 d e f g +e^{2} f^{2}}{8 e^{3} d^{2} \left (e x +d \right )^{2}}+\frac {\left (d^{2} g^{2}-2 d e f g -3 e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{16 e^{3} d^{4}}+\frac {d^{2} g^{2}+2 d e f g +e^{2} f^{2}}{8 e^{3} d^{3} \left (-e x +d \right )}\)	\(180\)
norman	\(\frac {-\frac {-d^{2} g^{2}-2 d e f g +e^{2} f^{2}}{4 d \,e^{3}}-\frac {\left (-3 d^{2} g^{2}-2 d e f g -3 e^{2} f^{2}\right ) x}{8 e^{2} d^{2}}+\frac {\left (-d^{2} g^{2}+2 d e f g +3 e^{2} f^{2}\right ) x^{2}}{8 e \,d^{3}}}{\left (e x +d \right )^{2} \left (-e x +d \right )}+\frac {\left (d^{2} g^{2}-2 d e f g -3 e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{16 e^{3} d^{4}}-\frac {\left (d^{2} g^{2}-2 d e f g -3 e^{2} f^{2}\right ) \ln \left (e x +d \right )}{16 e^{3} d^{4}}\)	\(188\)
risch	\(\frac {-\frac {\left (d^{2} g^{2}-2 d e f g -3 e^{2} f^{2}\right ) x^{2}}{8 e \,d^{3}}+\frac {\left (3 d^{2} g^{2}+2 d e f g +3 e^{2} f^{2}\right ) x}{8 e^{2} d^{2}}+\frac {d^{2} g^{2}+2 d e f g -e^{2} f^{2}}{4 d \,e^{3}}}{\left (e x +d \right ) \left (-e^{2} x^{2}+d^{2}\right )}+\frac {\ln \left (e x -d \right ) g^{2}}{16 e^{3} d^{2}}-\frac {\ln \left (e x -d \right ) f g}{8 e^{2} d^{3}}-\frac {3 \ln \left (e x -d \right ) f^{2}}{16 e \,d^{4}}-\frac {\ln \left (-e x -d \right ) g^{2}}{16 e^{3} d^{2}}+\frac {\ln \left (-e x -d \right ) f g}{8 e^{2} d^{3}}+\frac {3 \ln \left (-e x -d \right ) f^{2}}{16 e \,d^{4}}\)	\(235\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 200, normalized size = 1.65 \begin {gather*} -\frac {2 \, d^{4} g^{2} + 4 \, d^{3} f g e - 2 \, d^{2} f^{2} e^{2} - {\left (d^{2} g^{2} e^{2} - 2 \, d f g e^{3} - 3 \, f^{2} e^{4}\right )} x^{2} + {\left (3 \, d^{3} g^{2} e + 2 \, d^{2} f g e^{2} + 3 \, d f^{2} e^{3}\right )} x}{8 \, {\left (d^{3} x^{3} e^{6} + d^{4} x^{2} e^{5} - d^{5} x e^{4} - d^{6} e^{3}\right )}} - \frac {{\left (d^{2} g^{2} - 2 \, d f g e - 3 \, f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left (x e + d\right )}{16 \, d^{4}} + \frac {{\left (d^{2} g^{2} - 2 \, d f g e - 3 \, f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left (x e - d\right )}{16 \, d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 399 vs. \(2 (116) = 232\).
time = 2.07, size = 399, normalized size = 3.30 \begin {gather*} -\frac {4 \, d^{5} g^{2} + 6 \, d f^{2} x^{2} e^{4} + 2 \, {\left (2 \, d^{2} f g x^{2} + 3 \, d^{2} f^{2} x\right )} e^{3} - 2 \, {\left (d^{3} g^{2} x^{2} - 2 \, d^{3} f g x + 2 \, d^{3} f^{2}\right )} e^{2} + 2 \, {\left (3 \, d^{4} g^{2} x + 4 \, d^{4} f g\right )} e - {\left (d^{5} g^{2} + 3 \, f^{2} x^{3} e^{5} + {\left (2 \, d f g x^{3} + 3 \, d f^{2} x^{2}\right )} e^{4} - {\left (d^{2} g^{2} x^{3} - 2 \, d^{2} f g x^{2} + 3 \, d^{2} f^{2} x\right )} e^{3} - {\left (d^{3} g^{2} x^{2} + 2 \, d^{3} f g x + 3 \, d^{3} f^{2}\right )} e^{2} + {\left (d^{4} g^{2} x - 2 \, d^{4} f g\right )} e\right )} \log \left (x e + d\right ) + {\left (d^{5} g^{2} + 3 \, f^{2} x^{3} e^{5} + {\left (2 \, d f g x^{3} + 3 \, d f^{2} x^{2}\right )} e^{4} - {\left (d^{2} g^{2} x^{3} - 2 \, d^{2} f g x^{2} + 3 \, d^{2} f^{2} x\right )} e^{3} - {\left (d^{3} g^{2} x^{2} + 2 \, d^{3} f g x + 3 \, d^{3} f^{2}\right )} e^{2} + {\left (d^{4} g^{2} x - 2 \, d^{4} f g\right )} e\right )} \log \left (x e - d\right )}{16 \, {\left (d^{4} x^{3} e^{6} + d^{5} x^{2} e^{5} - d^{6} x e^{4} - 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)/(-e^2*x^2+d^2)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (105) = 210\).
time = 0.63, size = 279, normalized size = 2.31 \begin {gather*} \frac {- 2 d^{4} g^{2} - 4 d^{3} e f g + 2 d^{2} e^{2} f^{2} + x^{2} \left (d^{2} e^{2} g^{2} - 2 d e^{3} f g - 3 e^{4} f^{2}\right ) + x \left (- 3 d^{3} e g^{2} - 2 d^{2} e^{2} f g - 3 d e^{3} f^{2}\right )}{- 8 d^{6} e^{3} - 8 d^{5} e^{4} x + 8 d^{4} e^{5} x^{2} + 8 d^{3} e^{6} x^{3}} + \frac {\left (d g - 3 e f\right ) \left (d g + e f\right ) \log {\left (- \frac {d \left (d g - 3 e f\right ) \left (d g + e f\right )}{e \left (d^{2} g^{2} - 2 d e f g - 3 e^{2} f^{2}\right )} + x \right )}}{16 d^{4} e^{3}} - \frac {\left (d g - 3 e f\right ) \left (d g + e f\right ) \log {\left (\frac {d \left (d g - 3 e f\right ) \left (d g + e f\right )}{e \left (d^{2} g^{2} - 2 d e f g - 3 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)/(-e**2*x**2+d**2)**2,x)

[Out]

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

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Giac [A]
time = 1.99, size = 194, normalized size = 1.60 \begin {gather*} -\frac {{\left (d^{2} g^{2} - 2 \, d f g e - 3 \, f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right )}{16 \, d^{4}} + \frac {{\left (d^{2} g^{2} - 2 \, d f g e - 3 \, f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e - d \right |}\right )}{16 \, d^{4}} - \frac {{\left (2 \, d^{5} g^{2} + 4 \, d^{4} f g e - 2 \, d^{3} f^{2} e^{2} - {\left (d^{3} g^{2} e^{2} - 2 \, d^{2} f g e^{3} - 3 \, d f^{2} e^{4}\right )} x^{2} + {\left (3 \, d^{4} g^{2} e + 2 \, d^{3} f g e^{2} + 3 \, d^{2} f^{2} e^{3}\right )} x\right )} e^{\left (-3\right )}}{8 \, {\left (x e + d\right )}^{2} {\left (x e - d\right )} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.15, size = 198, normalized size = 1.64 \begin {gather*} \frac {\frac {d^2\,g^2+2\,d\,e\,f\,g-e^2\,f^2}{4\,d\,e^3}+\frac {x\,\left (3\,d^2\,g^2+2\,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+3\,e^2\,f^2\right )}{8\,d^3\,e}}{d^3+d^2\,e\,x-d\,e^2\,x^2-e^3\,x^3}+\frac {\mathrm {atanh}\left (\frac {e\,x\,\left (d\,g+e\,f\right )\,\left (d\,g-3\,e\,f\right )}{d\,\left (-d^2\,g^2+2\,d\,e\,f\,g+3\,e^2\,f^2\right )}\right )\,\left (d\,g+e\,f\right )\,\left (d\,g-3\,e\,f\right )}{8\,d^4\,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)^2*(d + e*x)),x)

[Out]

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

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