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

Optimal. Leaf size=122 \[ \frac {(d g+3 e f) (e f-d g) \tanh ^{-1}\left (\frac {e x}{d}\right )}{8 d^4 e^3}-\frac {(e f-d g)^2}{8 d^3 e^3 (d+e x)}+\frac {(d g+e f)^2}{8 d^2 e^3 (d-e x)^2}+\frac {e^2 f^2-d^2 g^2}{4 d^3 e^3 (d-e x)} \]

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {799, 88, 208} \begin {gather*} \frac {e^2 f^2-d^2 g^2}{4 d^3 e^3 (d-e x)}-\frac {(e f-d g)^2}{8 d^3 e^3 (d+e x)}+\frac {(d g+e f)^2}{8 d^2 e^3 (d-e x)^2}+\frac {(d g+3 e f) (e f-d g) \tanh ^{-1}\left (\frac {e x}{d}\right )}{8 d^4 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(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)) - (e*f - d*g)^2/(8*d^3*e^3*(
d + e*x)) + ((e*f - d*g)*(3*e*f + d*g)*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 799

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^m*(
f + g*x)^(p + 1)*(a/f + (c*x)/g)^p, x] /; FreeQ[{a, c, d, e, f, g, m}, x] && EqQ[c*f^2 + a*g^2, 0] && (Integer
Q[p] || (GtQ[a, 0] && GtQ[f, 0] && EqQ[p, -1]))

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.10, size = 140, normalized size = 1.15 \begin {gather*} \frac {\frac {4 d e^2 f^2-4 d^3 g^2}{d-e x}+\left (d^2 g^2+2 d e f g-3 e^2 f^2\right ) \log (d-e x)+\left (-d^2 g^2-2 d e f g+3 e^2 f^2\right ) \log (d+e x)+\frac {2 d^2 (d g+e f)^2}{(d-e x)^2}-\frac {2 d (e f-d g)^2}{d+e x}}{16 d^4 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.40, size = 417, normalized size = 3.42 \begin {gather*} \frac {4 \, d^{3} e^{2} f^{2} + 8 \, d^{4} e f g - 4 \, d^{5} g^{2} - 2 \, {\left (3 \, d e^{4} f^{2} - 2 \, d^{2} e^{3} f g - d^{3} e^{2} g^{2}\right )} x^{2} + 2 \, {\left (3 \, d^{2} e^{3} f^{2} - 2 \, d^{3} e^{2} f g + 3 \, d^{4} e g^{2}\right )} x + {\left (3 \, d^{3} e^{2} f^{2} - 2 \, d^{4} e f g - d^{5} g^{2} + {\left (3 \, e^{5} f^{2} - 2 \, d e^{4} f g - d^{2} e^{3} g^{2}\right )} x^{3} - {\left (3 \, d e^{4} f^{2} - 2 \, d^{2} e^{3} f g - d^{3} e^{2} g^{2}\right )} x^{2} - {\left (3 \, 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 ) - {\left (3 \, d^{3} e^{2} f^{2} - 2 \, d^{4} e f g - d^{5} g^{2} + {\left (3 \, e^{5} f^{2} - 2 \, d e^{4} f g - d^{2} e^{3} g^{2}\right )} x^{3} - {\left (3 \, d e^{4} f^{2} - 2 \, d^{2} e^{3} f g - d^{3} e^{2} g^{2}\right )} x^{2} - {\left (3 \, 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 )}{16 \, {\left (d^{4} e^{6} x^{3} - d^{5} e^{5} x^{2} - d^{6} e^{4} x + d^{7} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.17, size = 191, normalized size = 1.57 \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 (\frac {{\left | 2 \, x e^{2} - 2 \, {\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \, {\left | d \right |} e \right |}}\right )}{16 \, d^{3} {\left | d \right |}} + \frac {{\left (d^{2} g^{2} x^{3} e^{4} + 4 \, d^{3} g^{2} x^{2} e^{3} + d^{4} g^{2} x e^{2} - 2 \, d^{5} g^{2} e + 2 \, d f g x^{3} e^{5} + 2 \, d^{3} f g x e^{3} + 4 \, d^{4} f g e^{2} - 3 \, f^{2} x^{3} e^{6} + 5 \, d^{2} f^{2} x e^{4} + 2 \, d^{3} f^{2} e^{3}\right )} e^{\left (-4\right )}}{8 \, {\left (x^{2} e^{2} - d^{2}\right )}^{2} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.03, size = 257, normalized size = 2.11 \begin {gather*} \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 {g^{2}}{8 \left (e x -d \right )^{2} e^{3}}+\frac {g^{2}}{4 \left (e x -d \right ) d \,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}}{4 \left (e x -d \right ) d^{3} e}-\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 {3 f^{2} \ln \left (e x -d \right )}{16 d^{4} e}+\frac {3 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((e*x+d)*(g*x+f)^2/(-e^2*x^2+d^2)^3,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.47, size = 211, normalized size = 1.73 \begin {gather*} \frac {2 \, d^{2} e^{2} f^{2} + 4 \, d^{3} e f g - 2 \, d^{4} g^{2} - {\left (3 \, e^{4} f^{2} - 2 \, d e^{3} f g - d^{2} e^{2} g^{2}\right )} x^{2} + {\left (3 \, d e^{3} f^{2} - 2 \, d^{2} e^{2} f g + 3 \, d^{3} e g^{2}\right )} x}{8 \, {\left (d^{3} e^{6} x^{3} - d^{4} e^{5} x^{2} - d^{5} e^{4} x + d^{6} e^{3}\right )}} + \frac {{\left (3 \, 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 (3 \, 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((e*x+d)*(g*x+f)^2/(-e^2*x^2+d^2)^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 2.64, size = 198, normalized size = 1.62 \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 + e*x))/(d^2 - e^2*x^2)^3,x)

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 1.32, size = 277, normalized size = 2.27 \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 - e f\right ) \left (d g + 3 e f\right ) \log {\left (- \frac {d \left (d g - e f\right ) \left (d g + 3 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 - e f\right ) \left (d g + 3 e f\right ) \log {\left (\frac {d \left (d g - e f\right ) \left (d g + 3 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((e*x+d)*(g*x+f)**2/(-e**2*x**2+d**2)**3,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 - e*f)*(d*g + 3*e*f)*log(-d*(d*g - e*f)*(d*g + 3*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 - e*f)*(d*g + 3*e*f)*log(d*(d*g - e*f)*(d*g + 3*e*f)/(e*(d**2*g**2 + 2*d*e*f*g - 3*e
**2*f**2)) + x)/(16*d**4*e**3)

________________________________________________________________________________________