3.6.67 \(\int \frac {(f+g x)^2}{(d+e x)^3 (d^2-e^2 x^2)^2} \, dx\) [567]

Optimal. Leaf size=178 \[ \frac {(e f+d g)^2}{32 d^5 e^3 (d-e x)}-\frac {(e f-d g)^2}{16 d^2 e^3 (d+e x)^4}-\frac {e^2 f^2-d^2 g^2}{12 d^3 e^3 (d+e x)^3}-\frac {(3 e f-d g) (e f+d g)}{32 d^4 e^3 (d+e x)^2}-\frac {f (e f+d g)}{8 d^5 e^2 (d+e x)}+\frac {(e f+d g) (5 e f+d g) \tanh ^{-1}\left (\frac {e x}{d}\right )}{32 d^6 e^3} \]

[Out]

1/32*(d*g+e*f)^2/d^5/e^3/(-e*x+d)-1/16*(-d*g+e*f)^2/d^2/e^3/(e*x+d)^4+1/12*(d^2*g^2-e^2*f^2)/d^3/e^3/(e*x+d)^3
-1/32*(-d*g+3*e*f)*(d*g+e*f)/d^4/e^3/(e*x+d)^2-1/8*f*(d*g+e*f)/d^5/e^2/(e*x+d)+1/32*(d*g+e*f)*(d*g+5*e*f)*arct
anh(e*x/d)/d^6/e^3

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Rubi [A]
time = 0.13, antiderivative size = 178, 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) (d g+5 e f) \tanh ^{-1}\left (\frac {e x}{d}\right )}{32 d^6 e^3}+\frac {(d g+e f)^2}{32 d^5 e^3 (d-e x)}-\frac {f (d g+e f)}{8 d^5 e^2 (d+e x)}-\frac {(3 e f-d g) (d g+e f)}{32 d^4 e^3 (d+e x)^2}-\frac {(e f-d g)^2}{16 d^2 e^3 (d+e x)^4}-\frac {e^2 f^2-d^2 g^2}{12 d^3 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)^2),x]

[Out]

(e*f + d*g)^2/(32*d^5*e^3*(d - e*x)) - (e*f - d*g)^2/(16*d^2*e^3*(d + e*x)^4) - (e^2*f^2 - d^2*g^2)/(12*d^3*e^
3*(d + e*x)^3) - ((3*e*f - d*g)*(e*f + d*g))/(32*d^4*e^3*(d + e*x)^2) - (f*(e*f + d*g))/(8*d^5*e^2*(d + e*x))
+ ((e*f + d*g)*(5*e*f + d*g)*ArcTanh[(e*x)/d])/(32*d^6*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)^3 \left (d^2-e^2 x^2\right )^2} \, dx &=\int \frac {(f+g x)^2}{(d-e x)^2 (d+e x)^5} \, dx\\ &=\int \left (\frac {(e f+d g)^2}{32 d^5 e^2 (d-e x)^2}+\frac {(-e f+d g)^2}{4 d^2 e^2 (d+e x)^5}+\frac {e^2 f^2-d^2 g^2}{4 d^3 e^2 (d+e x)^4}+\frac {(3 e f-d g) (e f+d g)}{16 d^4 e^2 (d+e x)^3}+\frac {f (e f+d g)}{8 d^5 e (d+e x)^2}+\frac {(e f+d g) (5 e f+d g)}{32 d^5 e^2 \left (d^2-e^2 x^2\right )}\right ) \, dx\\ &=\frac {(e f+d g)^2}{32 d^5 e^3 (d-e x)}-\frac {(e f-d g)^2}{16 d^2 e^3 (d+e x)^4}-\frac {e^2 f^2-d^2 g^2}{12 d^3 e^3 (d+e x)^3}-\frac {(3 e f-d g) (e f+d g)}{32 d^4 e^3 (d+e x)^2}-\frac {f (e f+d g)}{8 d^5 e^2 (d+e x)}+\frac {((e f+d g) (5 e f+d g)) \int \frac {1}{d^2-e^2 x^2} \, dx}{32 d^5 e^2}\\ &=\frac {(e f+d g)^2}{32 d^5 e^3 (d-e x)}-\frac {(e f-d g)^2}{16 d^2 e^3 (d+e x)^4}-\frac {e^2 f^2-d^2 g^2}{12 d^3 e^3 (d+e x)^3}-\frac {(3 e f-d g) (e f+d g)}{32 d^4 e^3 (d+e x)^2}-\frac {f (e f+d g)}{8 d^5 e^2 (d+e x)}+\frac {(e f+d g) (5 e f+d g) \tanh ^{-1}\left (\frac {e x}{d}\right )}{32 d^6 e^3}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 195, normalized size = 1.10 \begin {gather*} \frac {\frac {6 d (e f+d g)^2}{d-e x}-\frac {12 d^4 (e f-d g)^2}{(d+e x)^4}+\frac {16 d^3 \left (-e^2 f^2+d^2 g^2\right )}{(d+e x)^3}+\frac {6 d^2 \left (-3 e^2 f^2-2 d e f g+d^2 g^2\right )}{(d+e x)^2}-\frac {24 d e f (e f+d g)}{d+e x}-3 \left (5 e^2 f^2+6 d e f g+d^2 g^2\right ) \log (d-e x)+3 \left (5 e^2 f^2+6 d e f g+d^2 g^2\right ) \log (d+e x)}{192 d^6 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((6*d*(e*f + d*g)^2)/(d - e*x) - (12*d^4*(e*f - d*g)^2)/(d + e*x)^4 + (16*d^3*(-(e^2*f^2) + d^2*g^2))/(d + e*x
)^3 + (6*d^2*(-3*e^2*f^2 - 2*d*e*f*g + d^2*g^2))/(d + e*x)^2 - (24*d*e*f*(e*f + d*g))/(d + e*x) - 3*(5*e^2*f^2
+ 6*d*e*f*g + d^2*g^2)*Log[d - e*x] + 3*(5*e^2*f^2 + 6*d*e*f*g + d^2*g^2)*Log[d + e*x])/(192*d^6*e^3)

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Maple [A]
time = 0.10, size = 241, normalized size = 1.35

method	result	size
default	\(\frac {\left (d^{2} g^{2}+6 d e f g +5 e^{2} f^{2}\right ) \ln \left (e x +d \right )}{64 e^{3} d^{6}}-\frac {-d^{2} g^{2}+e^{2} f^{2}}{12 e^{3} d^{3} \left (e x +d \right )^{3}}-\frac {-d^{2} g^{2}+2 d e f g +3 e^{2} f^{2}}{32 e^{3} d^{4} \left (e x +d \right )^{2}}-\frac {d^{2} g^{2}-2 d e f g +e^{2} f^{2}}{16 e^{3} d^{2} \left (e x +d \right )^{4}}-\frac {f \left (d g +e f \right )}{8 d^{5} e^{2} \left (e x +d \right )}+\frac {\left (-d^{2} g^{2}-6 d e f g -5 e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{64 e^{3} d^{6}}+\frac {d^{2} g^{2}+2 d e f g +e^{2} f^{2}}{32 e^{3} d^{5} \left (-e x +d \right )}\)	\(241\)
norman	\(\frac {\frac {\left (25 d^{2} g^{2}+54 d e f g -19 e^{2} f^{2}\right ) x^{3}}{96 d^{4}}-\frac {\left (3 d^{2} g^{2}-14 d e f g -33 e^{2} f^{2}\right ) x^{2}}{32 e \,d^{3}}+\frac {3 e \left (3 d^{2} g^{2}+2 d e f g -9 e^{2} f^{2}\right ) x^{4}}{32 d^{5}}+\frac {e^{2} \left (d^{2} g^{2}-4 e^{2} f^{2}\right ) x^{5}}{12 d^{6}}-\frac {\left (d^{2} g^{2}+6 d e f g -27 e^{2} f^{2}\right ) x}{32 d^{2} e^{2}}}{\left (e x +d \right )^{4} \left (-e x +d \right )}-\frac {\left (d^{2} g^{2}+6 d e f g +5 e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{64 e^{3} d^{6}}+\frac {\left (d^{2} g^{2}+6 d e f g +5 e^{2} f^{2}\right ) \ln \left (e x +d \right )}{64 e^{3} d^{6}}\)	\(247\)
risch	\(\frac {\frac {e \left (d^{2} g^{2}+6 d e f g +5 e^{2} f^{2}\right ) x^{4}}{32 d^{5}}+\frac {3 \left (d^{2} g^{2}+6 d e f g +5 e^{2} f^{2}\right ) x^{3}}{32 d^{4}}+\frac {7 \left (d^{2} g^{2}+6 d e f g +5 e^{2} f^{2}\right ) x^{2}}{96 d^{3} e}+\frac {\left (7 d^{2} g^{2}-6 d e f g -5 e^{2} f^{2}\right ) x}{32 d^{2} e^{2}}+\frac {d^{2} g^{2}-4 e^{2} f^{2}}{12 e^{3} d}}{\left (e x +d \right )^{3} \left (-e^{2} x^{2}+d^{2}\right )}-\frac {\ln \left (-e x +d \right ) g^{2}}{64 e^{3} d^{4}}-\frac {3 \ln \left (-e x +d \right ) f g}{32 e^{2} d^{5}}-\frac {5 \ln \left (-e x +d \right ) f^{2}}{64 e \,d^{6}}+\frac {\ln \left (e x +d \right ) g^{2}}{64 e^{3} d^{4}}+\frac {3 \ln \left (e x +d \right ) f g}{32 e^{2} d^{5}}+\frac {5 \ln \left (e x +d \right ) f^{2}}{64 e \,d^{6}}\)	\(278\)

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)^2,x,method=_RETURNVERBOSE)

[Out]

1/64/e^3*(d^2*g^2+6*d*e*f*g+5*e^2*f^2)/d^6*ln(e*x+d)-1/12*(-d^2*g^2+e^2*f^2)/e^3/d^3/(e*x+d)^3-1/32*(-d^2*g^2+
2*d*e*f*g+3*e^2*f^2)/e^3/d^4/(e*x+d)^2-1/16*(d^2*g^2-2*d*e*f*g+e^2*f^2)/e^3/d^2/(e*x+d)^4-1/8*f*(d*g+e*f)/d^5/
e^2/(e*x+d)+1/64*(-d^2*g^2-6*d*e*f*g-5*e^2*f^2)/e^3/d^6*ln(-e*x+d)+1/32*(d^2*g^2+2*d*e*f*g+e^2*f^2)/e^3/d^5/(-
e*x+d)

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Maxima [A]
time = 0.31, size = 281, normalized size = 1.58 \begin {gather*} -\frac {8 \, d^{6} g^{2} - 32 \, d^{4} f^{2} e^{2} + 3 \, {\left (d^{2} g^{2} e^{4} + 6 \, d f g e^{5} + 5 \, f^{2} e^{6}\right )} x^{4} + 9 \, {\left (d^{3} g^{2} e^{3} + 6 \, d^{2} f g e^{4} + 5 \, d f^{2} e^{5}\right )} x^{3} + 7 \, {\left (d^{4} g^{2} e^{2} + 6 \, d^{3} f g e^{3} + 5 \, d^{2} f^{2} e^{4}\right )} x^{2} + 3 \, {\left (7 \, d^{5} g^{2} e - 6 \, d^{4} f g e^{2} - 5 \, d^{3} f^{2} e^{3}\right )} x}{96 \, {\left (d^{5} x^{5} e^{8} + 3 \, d^{6} x^{4} e^{7} + 2 \, d^{7} x^{3} e^{6} - 2 \, d^{8} x^{2} e^{5} - 3 \, d^{9} x e^{4} - d^{10} e^{3}\right )}} + \frac {{\left (d^{2} g^{2} + 6 \, d f g e + 5 \, f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left (x e + d\right )}{64 \, d^{6}} - \frac {{\left (d^{2} g^{2} + 6 \, d f g e + 5 \, f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left (x e - d\right )}{64 \, d^{6}} \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)^2,x, algorithm="maxima")

[Out]

-1/96*(8*d^6*g^2 - 32*d^4*f^2*e^2 + 3*(d^2*g^2*e^4 + 6*d*f*g*e^5 + 5*f^2*e^6)*x^4 + 9*(d^3*g^2*e^3 + 6*d^2*f*g
*e^4 + 5*d*f^2*e^5)*x^3 + 7*(d^4*g^2*e^2 + 6*d^3*f*g*e^3 + 5*d^2*f^2*e^4)*x^2 + 3*(7*d^5*g^2*e - 6*d^4*f*g*e^2
- 5*d^3*f^2*e^3)*x)/(d^5*x^5*e^8 + 3*d^6*x^4*e^7 + 2*d^7*x^3*e^6 - 2*d^8*x^2*e^5 - 3*d^9*x*e^4 - d^10*e^3) +
1/64*(d^2*g^2 + 6*d*f*g*e + 5*f^2*e^2)*e^(-3)*log(x*e + d)/d^6 - 1/64*(d^2*g^2 + 6*d*f*g*e + 5*f^2*e^2)*e^(-3)
*log(x*e - d)/d^6

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 638 vs. \(2 (172) = 344\).
time = 3.53, size = 638, normalized size = 3.58 \begin {gather*} -\frac {42 \, d^{6} g^{2} x e + 16 \, d^{7} g^{2} + 30 \, d f^{2} x^{4} e^{6} + 18 \, {\left (2 \, d^{2} f g x^{4} + 5 \, d^{2} f^{2} x^{3}\right )} e^{5} + 2 \, {\left (3 \, d^{3} g^{2} x^{4} + 54 \, d^{3} f g x^{3} + 35 \, d^{3} f^{2} x^{2}\right )} e^{4} + 6 \, {\left (3 \, d^{4} g^{2} x^{3} + 14 \, d^{4} f g x^{2} - 5 \, d^{4} f^{2} x\right )} e^{3} + 2 \, {\left (7 \, d^{5} g^{2} x^{2} - 18 \, d^{5} f g x - 32 \, d^{5} f^{2}\right )} e^{2} + 3 \, {\left (d^{7} g^{2} - 5 \, f^{2} x^{5} e^{7} - 3 \, {\left (2 \, d f g x^{5} + 5 \, d f^{2} x^{4}\right )} e^{6} - {\left (d^{2} g^{2} x^{5} + 18 \, d^{2} f g x^{4} + 10 \, d^{2} f^{2} x^{3}\right )} e^{5} - {\left (3 \, d^{3} g^{2} x^{4} + 12 \, d^{3} f g x^{3} - 10 \, d^{3} f^{2} x^{2}\right )} e^{4} - {\left (2 \, d^{4} g^{2} x^{3} - 12 \, d^{4} f g x^{2} - 15 \, d^{4} f^{2} x\right )} e^{3} + {\left (2 \, d^{5} g^{2} x^{2} + 18 \, d^{5} f g x + 5 \, d^{5} f^{2}\right )} e^{2} + 3 \, {\left (d^{6} g^{2} x + 2 \, d^{6} f g\right )} e\right )} \log \left (x e + d\right ) - 3 \, {\left (d^{7} g^{2} - 5 \, f^{2} x^{5} e^{7} - 3 \, {\left (2 \, d f g x^{5} + 5 \, d f^{2} x^{4}\right )} e^{6} - {\left (d^{2} g^{2} x^{5} + 18 \, d^{2} f g x^{4} + 10 \, d^{2} f^{2} x^{3}\right )} e^{5} - {\left (3 \, d^{3} g^{2} x^{4} + 12 \, d^{3} f g x^{3} - 10 \, d^{3} f^{2} x^{2}\right )} e^{4} - {\left (2 \, d^{4} g^{2} x^{3} - 12 \, d^{4} f g x^{2} - 15 \, d^{4} f^{2} x\right )} e^{3} + {\left (2 \, d^{5} g^{2} x^{2} + 18 \, d^{5} f g x + 5 \, d^{5} f^{2}\right )} e^{2} + 3 \, {\left (d^{6} g^{2} x + 2 \, d^{6} f g\right )} e\right )} \log \left (x e - d\right )}{192 \, {\left (d^{6} x^{5} e^{8} + 3 \, d^{7} x^{4} e^{7} + 2 \, d^{8} x^{3} e^{6} - 2 \, d^{9} x^{2} e^{5} - 3 \, d^{10} x e^{4} - d^{11} 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)^2,x, algorithm="fricas")

[Out]

-1/192*(42*d^6*g^2*x*e + 16*d^7*g^2 + 30*d*f^2*x^4*e^6 + 18*(2*d^2*f*g*x^4 + 5*d^2*f^2*x^3)*e^5 + 2*(3*d^3*g^2
*x^4 + 54*d^3*f*g*x^3 + 35*d^3*f^2*x^2)*e^4 + 6*(3*d^4*g^2*x^3 + 14*d^4*f*g*x^2 - 5*d^4*f^2*x)*e^3 + 2*(7*d^5*
g^2*x^2 - 18*d^5*f*g*x - 32*d^5*f^2)*e^2 + 3*(d^7*g^2 - 5*f^2*x^5*e^7 - 3*(2*d*f*g*x^5 + 5*d*f^2*x^4)*e^6 - (d
^2*g^2*x^5 + 18*d^2*f*g*x^4 + 10*d^2*f^2*x^3)*e^5 - (3*d^3*g^2*x^4 + 12*d^3*f*g*x^3 - 10*d^3*f^2*x^2)*e^4 - (2
*d^4*g^2*x^3 - 12*d^4*f*g*x^2 - 15*d^4*f^2*x)*e^3 + (2*d^5*g^2*x^2 + 18*d^5*f*g*x + 5*d^5*f^2)*e^2 + 3*(d^6*g^
2*x + 2*d^6*f*g)*e)*log(x*e + d) - 3*(d^7*g^2 - 5*f^2*x^5*e^7 - 3*(2*d*f*g*x^5 + 5*d*f^2*x^4)*e^6 - (d^2*g^2*x
^5 + 18*d^2*f*g*x^4 + 10*d^2*f^2*x^3)*e^5 - (3*d^3*g^2*x^4 + 12*d^3*f*g*x^3 - 10*d^3*f^2*x^2)*e^4 - (2*d^4*g^2
*x^3 - 12*d^4*f*g*x^2 - 15*d^4*f^2*x)*e^3 + (2*d^5*g^2*x^2 + 18*d^5*f*g*x + 5*d^5*f^2)*e^2 + 3*(d^6*g^2*x + 2*
d^6*f*g)*e)*log(x*e - d))/(d^6*x^5*e^8 + 3*d^7*x^4*e^7 + 2*d^8*x^3*e^6 - 2*d^9*x^2*e^5 - 3*d^10*x*e^4 - d^11*e
^3)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 376 vs. \(2 (162) = 324\).
time = 0.91, size = 376, normalized size = 2.11 \begin {gather*} \frac {- 8 d^{6} g^{2} + 32 d^{4} e^{2} f^{2} + x^{4} \left (- 3 d^{2} e^{4} g^{2} - 18 d e^{5} f g - 15 e^{6} f^{2}\right ) + x^{3} \left (- 9 d^{3} e^{3} g^{2} - 54 d^{2} e^{4} f g - 45 d e^{5} f^{2}\right ) + x^{2} \left (- 7 d^{4} e^{2} g^{2} - 42 d^{3} e^{3} f g - 35 d^{2} e^{4} f^{2}\right ) + x \left (- 21 d^{5} e g^{2} + 18 d^{4} e^{2} f g + 15 d^{3} e^{3} f^{2}\right )}{- 96 d^{10} e^{3} - 288 d^{9} e^{4} x - 192 d^{8} e^{5} x^{2} + 192 d^{7} e^{6} x^{3} + 288 d^{6} e^{7} x^{4} + 96 d^{5} e^{8} x^{5}} - \frac {\left (d g + e f\right ) \left (d g + 5 e f\right ) \log {\left (- \frac {d \left (d g + e f\right ) \left (d g + 5 e f\right )}{e \left (d^{2} g^{2} + 6 d e f g + 5 e^{2} f^{2}\right )} + x \right )}}{64 d^{6} e^{3}} + \frac {\left (d g + e f\right ) \left (d g + 5 e f\right ) \log {\left (\frac {d \left (d g + e f\right ) \left (d g + 5 e f\right )}{e \left (d^{2} g^{2} + 6 d e f g + 5 e^{2} f^{2}\right )} + x \right )}}{64 d^{6} 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)**2,x)

[Out]

(-8*d**6*g**2 + 32*d**4*e**2*f**2 + x**4*(-3*d**2*e**4*g**2 - 18*d*e**5*f*g - 15*e**6*f**2) + x**3*(-9*d**3*e*
*3*g**2 - 54*d**2*e**4*f*g - 45*d*e**5*f**2) + x**2*(-7*d**4*e**2*g**2 - 42*d**3*e**3*f*g - 35*d**2*e**4*f**2)
+ x*(-21*d**5*e*g**2 + 18*d**4*e**2*f*g + 15*d**3*e**3*f**2))/(-96*d**10*e**3 - 288*d**9*e**4*x - 192*d**8*e*
*5*x**2 + 192*d**7*e**6*x**3 + 288*d**6*e**7*x**4 + 96*d**5*e**8*x**5) - (d*g + e*f)*(d*g + 5*e*f)*log(-d*(d*g
+ e*f)*(d*g + 5*e*f)/(e*(d**2*g**2 + 6*d*e*f*g + 5*e**2*f**2)) + x)/(64*d**6*e**3) + (d*g + e*f)*(d*g + 5*e*f
)*log(d*(d*g + e*f)*(d*g + 5*e*f)/(e*(d**2*g**2 + 6*d*e*f*g + 5*e**2*f**2)) + x)/(64*d**6*e**3)

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Giac [A]
time = 2.22, size = 254, normalized size = 1.43 \begin {gather*} \frac {{\left (d^{2} g^{2} + 6 \, d f g e + 5 \, f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right )}{64 \, d^{6}} - \frac {{\left (d^{2} g^{2} + 6 \, d f g e + 5 \, f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e - d \right |}\right )}{64 \, d^{6}} - \frac {{\left (8 \, d^{7} g^{2} - 32 \, d^{5} f^{2} e^{2} + 3 \, {\left (d^{3} g^{2} e^{4} + 6 \, d^{2} f g e^{5} + 5 \, d f^{2} e^{6}\right )} x^{4} + 9 \, {\left (d^{4} g^{2} e^{3} + 6 \, d^{3} f g e^{4} + 5 \, d^{2} f^{2} e^{5}\right )} x^{3} + 7 \, {\left (d^{5} g^{2} e^{2} + 6 \, d^{4} f g e^{3} + 5 \, d^{3} f^{2} e^{4}\right )} x^{2} + 3 \, {\left (7 \, d^{6} g^{2} e - 6 \, d^{5} f g e^{2} - 5 \, d^{4} f^{2} e^{3}\right )} x\right )} e^{\left (-3\right )}}{96 \, {\left (x e + d\right )}^{4} {\left (x e - d\right )} d^{6}} \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)^2,x, algorithm="giac")

[Out]

1/64*(d^2*g^2 + 6*d*f*g*e + 5*f^2*e^2)*e^(-3)*log(abs(x*e + d))/d^6 - 1/64*(d^2*g^2 + 6*d*f*g*e + 5*f^2*e^2)*e
^(-3)*log(abs(x*e - d))/d^6 - 1/96*(8*d^7*g^2 - 32*d^5*f^2*e^2 + 3*(d^3*g^2*e^4 + 6*d^2*f*g*e^5 + 5*d*f^2*e^6)
*x^4 + 9*(d^4*g^2*e^3 + 6*d^3*f*g*e^4 + 5*d^2*f^2*e^5)*x^3 + 7*(d^5*g^2*e^2 + 6*d^4*f*g*e^3 + 5*d^3*f^2*e^4)*x
^2 + 3*(7*d^6*g^2*e - 6*d^5*f*g*e^2 - 5*d^4*f^2*e^3)*x)*e^(-3)/((x*e + d)^4*(x*e - d)*d^6)

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Mupad [B]
time = 2.70, size = 274, normalized size = 1.54 \begin {gather*} \frac {\frac {d^2\,g^2-4\,e^2\,f^2}{12\,d\,e^3}+\frac {3\,x^3\,\left (d^2\,g^2+6\,d\,e\,f\,g+5\,e^2\,f^2\right )}{32\,d^4}+\frac {e\,x^4\,\left (d^2\,g^2+6\,d\,e\,f\,g+5\,e^2\,f^2\right )}{32\,d^5}-\frac {x\,\left (-7\,d^2\,g^2+6\,d\,e\,f\,g+5\,e^2\,f^2\right )}{32\,d^2\,e^2}+\frac {7\,x^2\,\left (d^2\,g^2+6\,d\,e\,f\,g+5\,e^2\,f^2\right )}{96\,d^3\,e}}{d^5+3\,d^4\,e\,x+2\,d^3\,e^2\,x^2-2\,d^2\,e^3\,x^3-3\,d\,e^4\,x^4-e^5\,x^5}+\frac {\mathrm {atanh}\left (\frac {e\,x\,\left (d\,g+e\,f\right )\,\left (d\,g+5\,e\,f\right )}{d\,\left (d^2\,g^2+6\,d\,e\,f\,g+5\,e^2\,f^2\right )}\right )\,\left (d\,g+e\,f\right )\,\left (d\,g+5\,e\,f\right )}{32\,d^6\,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)^3),x)

[Out]

((d^2*g^2 - 4*e^2*f^2)/(12*d*e^3) + (3*x^3*(d^2*g^2 + 5*e^2*f^2 + 6*d*e*f*g))/(32*d^4) + (e*x^4*(d^2*g^2 + 5*e
^2*f^2 + 6*d*e*f*g))/(32*d^5) - (x*(5*e^2*f^2 - 7*d^2*g^2 + 6*d*e*f*g))/(32*d^2*e^2) + (7*x^2*(d^2*g^2 + 5*e^2
*f^2 + 6*d*e*f*g))/(96*d^3*e))/(d^5 - e^5*x^5 - 3*d*e^4*x^4 + 2*d^3*e^2*x^2 - 2*d^2*e^3*x^3 + 3*d^4*e*x) + (at
anh((e*x*(d*g + e*f)*(d*g + 5*e*f))/(d*(d^2*g^2 + 5*e^2*f^2 + 6*d*e*f*g)))*(d*g + e*f)*(d*g + 5*e*f))/(32*d^6*
e^3)

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