3.570 \(\int \frac{(d+e x)^6 (f+g x)^2}{(d^2-e^2 x^2)^3} \, dx\)

Optimal. Leaf size=149 \[ -\frac{x \left (18 d^2 g^2+12 d e f g+e^2 f^2\right )}{e^2}-\frac{2 d \left (19 d^2 g^2+18 d e f g+3 e^2 f^2\right ) \log (d-e x)}{e^3}+\frac{4 d^3 (d g+e f)^2}{e^3 (d-e x)^2}-\frac{4 d^2 (d g+e f) (7 d g+3 e f)}{e^3 (d-e x)}-\frac{g x^2 (3 d g+e f)}{e}-\frac{g^2 x^3}{3} \]

[Out]

-(((e^2*f^2 + 12*d*e*f*g + 18*d^2*g^2)*x)/e^2) - (g*(e*f + 3*d*g)*x^2)/e - (g^2*x^3)/3 + (4*d^3*(e*f + d*g)^2)
/(e^3*(d - e*x)^2) - (4*d^2*(e*f + d*g)*(3*e*f + 7*d*g))/(e^3*(d - e*x)) - (2*d*(3*e^2*f^2 + 18*d*e*f*g + 19*d
^2*g^2)*Log[d - e*x])/e^3

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Rubi [A]  time = 0.197255, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {848, 88} \[ -\frac{x \left (18 d^2 g^2+12 d e f g+e^2 f^2\right )}{e^2}-\frac{2 d \left (19 d^2 g^2+18 d e f g+3 e^2 f^2\right ) \log (d-e x)}{e^3}+\frac{4 d^3 (d g+e f)^2}{e^3 (d-e x)^2}-\frac{4 d^2 (d g+e f) (7 d g+3 e f)}{e^3 (d-e x)}-\frac{g x^2 (3 d g+e f)}{e}-\frac{g^2 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^6*(f + g*x)^2)/(d^2 - e^2*x^2)^3,x]

[Out]

-(((e^2*f^2 + 12*d*e*f*g + 18*d^2*g^2)*x)/e^2) - (g*(e*f + 3*d*g)*x^2)/e - (g^2*x^3)/3 + (4*d^3*(e*f + d*g)^2)
/(e^3*(d - e*x)^2) - (4*d^2*(e*f + d*g)*(3*e*f + 7*d*g))/(e^3*(d - e*x)) - (2*d*(3*e^2*f^2 + 18*d*e*f*g + 19*d
^2*g^2)*Log[d - e*x])/e^3

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]))

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]))

Rubi steps

\begin{align*} \int \frac{(d+e x)^6 (f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx &=\int \frac{(d+e x)^3 (f+g x)^2}{(d-e x)^3} \, dx\\ &=\int \left (\frac{-e^2 f^2-12 d e f g-18 d^2 g^2}{e^2}-\frac{2 g (e f+3 d g) x}{e}-g^2 x^2+\frac{4 d^2 (-3 e f-7 d g) (e f+d g)}{e^2 (d-e x)^2}-\frac{8 d^3 (e f+d g)^2}{e^2 (-d+e x)^3}-\frac{2 d \left (3 e^2 f^2+18 d e f g+19 d^2 g^2\right )}{e^2 (-d+e x)}\right ) \, dx\\ &=-\frac{\left (e^2 f^2+12 d e f g+18 d^2 g^2\right ) x}{e^2}-\frac{g (e f+3 d g) x^2}{e}-\frac{g^2 x^3}{3}+\frac{4 d^3 (e f+d g)^2}{e^3 (d-e x)^2}-\frac{4 d^2 (e f+d g) (3 e f+7 d g)}{e^3 (d-e x)}-\frac{2 d \left (3 e^2 f^2+18 d e f g+19 d^2 g^2\right ) \log (d-e x)}{e^3}\\ \end{align*}

Mathematica [A]  time = 0.0875661, size = 157, normalized size = 1.05 \[ \frac{4 d^2 \left (7 d^2 g^2+10 d e f g+3 e^2 f^2\right )}{e^3 (e x-d)}-\frac{x \left (18 d^2 g^2+12 d e f g+e^2 f^2\right )}{e^2}-\frac{2 d \left (19 d^2 g^2+18 d e f g+3 e^2 f^2\right ) \log (d-e x)}{e^3}+\frac{4 d^3 (d g+e f)^2}{e^3 (d-e x)^2}-\frac{g x^2 (3 d g+e f)}{e}-\frac{g^2 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^6*(f + g*x)^2)/(d^2 - e^2*x^2)^3,x]

[Out]

-(((e^2*f^2 + 12*d*e*f*g + 18*d^2*g^2)*x)/e^2) - (g*(e*f + 3*d*g)*x^2)/e - (g^2*x^3)/3 + (4*d^3*(e*f + d*g)^2)
/(e^3*(d - e*x)^2) + (4*d^2*(3*e^2*f^2 + 10*d*e*f*g + 7*d^2*g^2))/(e^3*(-d + e*x)) - (2*d*(3*e^2*f^2 + 18*d*e*
f*g + 19*d^2*g^2)*Log[d - e*x])/e^3

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Maple [A]  time = 0.052, size = 228, normalized size = 1.5 \begin{align*} -{\frac{{g}^{2}{x}^{3}}{3}}-3\,{\frac{d{x}^{2}{g}^{2}}{e}}-{x}^{2}fg-18\,{\frac{{d}^{2}{g}^{2}x}{{e}^{2}}}-12\,{\frac{dfgx}{e}}-x{f}^{2}-38\,{\frac{{d}^{3}\ln \left ( ex-d \right ){g}^{2}}{{e}^{3}}}-36\,{\frac{{d}^{2}\ln \left ( ex-d \right ) fg}{{e}^{2}}}-6\,{\frac{d\ln \left ( ex-d \right ){f}^{2}}{e}}+4\,{\frac{{d}^{5}{g}^{2}}{{e}^{3} \left ( ex-d \right ) ^{2}}}+8\,{\frac{{d}^{4}fg}{{e}^{2} \left ( ex-d \right ) ^{2}}}+4\,{\frac{{d}^{3}{f}^{2}}{e \left ( ex-d \right ) ^{2}}}+28\,{\frac{{d}^{4}{g}^{2}}{{e}^{3} \left ( ex-d \right ) }}+40\,{\frac{{d}^{3}fg}{{e}^{2} \left ( ex-d \right ) }}+12\,{\frac{{d}^{2}{f}^{2}}{e \left ( ex-d \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^6*(g*x+f)^2/(-e^2*x^2+d^2)^3,x)

[Out]

-1/3*g^2*x^3-3/e*x^2*d*g^2-x^2*f*g-18/e^2*d^2*g^2*x-12/e*d*f*g*x-x*f^2-38*d^3/e^3*ln(e*x-d)*g^2-36*d^2/e^2*ln(
e*x-d)*f*g-6*d/e*ln(e*x-d)*f^2+4*d^5/e^3/(e*x-d)^2*g^2+8*d^4/e^2/(e*x-d)^2*f*g+4*d^3/e/(e*x-d)^2*f^2+28*d^4/e^
3/(e*x-d)*g^2+40*d^3/e^2/(e*x-d)*f*g+12*d^2/e/(e*x-d)*f^2

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Maxima [A]  time = 0.974452, size = 254, normalized size = 1.7 \begin{align*} -\frac{4 \,{\left (2 \, d^{3} e^{2} f^{2} + 8 \, d^{4} e f g + 6 \, d^{5} g^{2} -{\left (3 \, d^{2} e^{3} f^{2} + 10 \, d^{3} e^{2} f g + 7 \, d^{4} e g^{2}\right )} x\right )}}{e^{5} x^{2} - 2 \, d e^{4} x + d^{2} e^{3}} - \frac{e^{2} g^{2} x^{3} + 3 \,{\left (e^{2} f g + 3 \, d e g^{2}\right )} x^{2} + 3 \,{\left (e^{2} f^{2} + 12 \, d e f g + 18 \, d^{2} g^{2}\right )} x}{3 \, e^{2}} - \frac{2 \,{\left (3 \, d e^{2} f^{2} + 18 \, d^{2} e f g + 19 \, d^{3} g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6*(g*x+f)^2/(-e^2*x^2+d^2)^3,x, algorithm="maxima")

[Out]

-4*(2*d^3*e^2*f^2 + 8*d^4*e*f*g + 6*d^5*g^2 - (3*d^2*e^3*f^2 + 10*d^3*e^2*f*g + 7*d^4*e*g^2)*x)/(e^5*x^2 - 2*d
*e^4*x + d^2*e^3) - 1/3*(e^2*g^2*x^3 + 3*(e^2*f*g + 3*d*e*g^2)*x^2 + 3*(e^2*f^2 + 12*d*e*f*g + 18*d^2*g^2)*x)/
e^2 - 2*(3*d*e^2*f^2 + 18*d^2*e*f*g + 19*d^3*g^2)*log(e*x - d)/e^3

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Fricas [A]  time = 1.73908, size = 620, normalized size = 4.16 \begin{align*} -\frac{e^{5} g^{2} x^{5} + 24 \, d^{3} e^{2} f^{2} + 96 \, d^{4} e f g + 72 \, d^{5} g^{2} +{\left (3 \, e^{5} f g + 7 \, d e^{4} g^{2}\right )} x^{4} +{\left (3 \, e^{5} f^{2} + 30 \, d e^{4} f g + 37 \, d^{2} e^{3} g^{2}\right )} x^{3} - 3 \,{\left (2 \, d e^{4} f^{2} + 23 \, d^{2} e^{3} f g + 33 \, d^{3} e^{2} g^{2}\right )} x^{2} - 3 \,{\left (11 \, d^{2} e^{3} f^{2} + 28 \, d^{3} e^{2} f g + 10 \, d^{4} e g^{2}\right )} x + 6 \,{\left (3 \, d^{3} e^{2} f^{2} + 18 \, d^{4} e f g + 19 \, d^{5} g^{2} +{\left (3 \, d e^{4} f^{2} + 18 \, d^{2} e^{3} f g + 19 \, d^{3} e^{2} g^{2}\right )} x^{2} - 2 \,{\left (3 \, d^{2} e^{3} f^{2} + 18 \, d^{3} e^{2} f g + 19 \, d^{4} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{3 \,{\left (e^{5} x^{2} - 2 \, d e^{4} x + d^{2} e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6*(g*x+f)^2/(-e^2*x^2+d^2)^3,x, algorithm="fricas")

[Out]

-1/3*(e^5*g^2*x^5 + 24*d^3*e^2*f^2 + 96*d^4*e*f*g + 72*d^5*g^2 + (3*e^5*f*g + 7*d*e^4*g^2)*x^4 + (3*e^5*f^2 +
30*d*e^4*f*g + 37*d^2*e^3*g^2)*x^3 - 3*(2*d*e^4*f^2 + 23*d^2*e^3*f*g + 33*d^3*e^2*g^2)*x^2 - 3*(11*d^2*e^3*f^2
+ 28*d^3*e^2*f*g + 10*d^4*e*g^2)*x + 6*(3*d^3*e^2*f^2 + 18*d^4*e*f*g + 19*d^5*g^2 + (3*d*e^4*f^2 + 18*d^2*e^3
*f*g + 19*d^3*e^2*g^2)*x^2 - 2*(3*d^2*e^3*f^2 + 18*d^3*e^2*f*g + 19*d^4*e*g^2)*x)*log(e*x - d))/(e^5*x^2 - 2*d
*e^4*x + d^2*e^3)

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Sympy [A]  time = 1.50012, size = 180, normalized size = 1.21 \begin{align*} - \frac{2 d \left (19 d^{2} g^{2} + 18 d e f g + 3 e^{2} f^{2}\right ) \log{\left (- d + e x \right )}}{e^{3}} - \frac{g^{2} x^{3}}{3} + \frac{- 24 d^{5} g^{2} - 32 d^{4} e f g - 8 d^{3} e^{2} f^{2} + x \left (28 d^{4} e g^{2} + 40 d^{3} e^{2} f g + 12 d^{2} e^{3} f^{2}\right )}{d^{2} e^{3} - 2 d e^{4} x + e^{5} x^{2}} - \frac{x^{2} \left (3 d g^{2} + e f g\right )}{e} - \frac{x \left (18 d^{2} g^{2} + 12 d e f g + e^{2} f^{2}\right )}{e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**6*(g*x+f)**2/(-e**2*x**2+d**2)**3,x)

[Out]

-2*d*(19*d**2*g**2 + 18*d*e*f*g + 3*e**2*f**2)*log(-d + e*x)/e**3 - g**2*x**3/3 + (-24*d**5*g**2 - 32*d**4*e*f
*g - 8*d**3*e**2*f**2 + x*(28*d**4*e*g**2 + 40*d**3*e**2*f*g + 12*d**2*e**3*f**2))/(d**2*e**3 - 2*d*e**4*x + e
**5*x**2) - x**2*(3*d*g**2 + e*f*g)/e - x*(18*d**2*g**2 + 12*d*e*f*g + e**2*f**2)/e**2

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Giac [B]  time = 1.16219, size = 437, normalized size = 2.93 \begin{align*} -{\left (19 \, d^{3} g^{2} e^{5} + 18 \, d^{2} f g e^{6} + 3 \, d f^{2} e^{7}\right )} e^{\left (-8\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) - \frac{1}{3} \,{\left (g^{2} x^{3} e^{18} + 9 \, d g^{2} x^{2} e^{17} + 54 \, d^{2} g^{2} x e^{16} + 3 \, f g x^{2} e^{18} + 36 \, d f g x e^{17} + 3 \, f^{2} x e^{18}\right )} e^{\left (-18\right )} - \frac{{\left (19 \, d^{4} g^{2} e^{6} + 18 \, d^{3} f g e^{7} + 3 \, d^{2} f^{2} e^{8}\right )} e^{\left (-9\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 )}{{\left | d \right |}} - \frac{4 \,{\left (6 \, d^{7} g^{2} e^{5} + 8 \, d^{6} f g e^{6} + 2 \, d^{5} f^{2} e^{7} -{\left (7 \, d^{4} g^{2} e^{8} + 10 \, d^{3} f g e^{9} + 3 \, d^{2} f^{2} e^{10}\right )} x^{3} - 4 \,{\left (2 \, d^{5} g^{2} e^{7} + 3 \, d^{4} f g e^{8} + d^{3} f^{2} e^{9}\right )} x^{2} +{\left (5 \, d^{6} g^{2} e^{6} + 6 \, d^{5} f g e^{7} + d^{4} f^{2} e^{8}\right )} x\right )} e^{\left (-8\right )}}{{\left (x^{2} e^{2} - d^{2}\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6*(g*x+f)^2/(-e^2*x^2+d^2)^3,x, algorithm="giac")

[Out]

-(19*d^3*g^2*e^5 + 18*d^2*f*g*e^6 + 3*d*f^2*e^7)*e^(-8)*log(abs(x^2*e^2 - d^2)) - 1/3*(g^2*x^3*e^18 + 9*d*g^2*
x^2*e^17 + 54*d^2*g^2*x*e^16 + 3*f*g*x^2*e^18 + 36*d*f*g*x*e^17 + 3*f^2*x*e^18)*e^(-18) - (19*d^4*g^2*e^6 + 18
*d^3*f*g*e^7 + 3*d^2*f^2*e^8)*e^(-9)*log(abs(2*x*e^2 - 2*abs(d)*e)/abs(2*x*e^2 + 2*abs(d)*e))/abs(d) - 4*(6*d^
7*g^2*e^5 + 8*d^6*f*g*e^6 + 2*d^5*f^2*e^7 - (7*d^4*g^2*e^8 + 10*d^3*f*g*e^9 + 3*d^2*f^2*e^10)*x^3 - 4*(2*d^5*g
^2*e^7 + 3*d^4*f*g*e^8 + d^3*f^2*e^9)*x^2 + (5*d^6*g^2*e^6 + 6*d^5*f*g*e^7 + d^4*f^2*e^8)*x)*e^(-8)/(x^2*e^2 -
d^2)^2