3.548 \(\int \frac{(d+e x)^4 (f+g x)^2}{d^2-e^2 x^2} \, dx\)

Optimal. Leaf size=141 \[ -\frac{8 d^3 (d g+e f)^2 \log (d-e x)}{e^3}-\frac{d x^2 \left (4 d^2 g^2+7 d e f g+2 e^2 f^2\right )}{e}-\frac{d^2 x \left (8 d^2 g^2+16 d e f g+7 e^2 f^2\right )}{e^2}-\frac{1}{2} e g x^4 (2 d g+e f)-\frac{1}{3} x^3 (d g+e f) (7 d g+e f)-\frac{1}{5} e^2 g^2 x^5 \]

[Out]

-((d^2*(7*e^2*f^2 + 16*d*e*f*g + 8*d^2*g^2)*x)/e^2) - (d*(2*e^2*f^2 + 7*d*e*f*g
+ 4*d^2*g^2)*x^2)/e - ((e*f + d*g)*(e*f + 7*d*g)*x^3)/3 - (e*g*(e*f + 2*d*g)*x^4
)/2 - (e^2*g^2*x^5)/5 - (8*d^3*(e*f + d*g)^2*Log[d - e*x])/e^3

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Rubi [A]  time = 0.322101, antiderivative size = 141, 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 \[ -\frac{8 d^3 (d g+e f)^2 \log (d-e x)}{e^3}-\frac{d x^2 \left (4 d^2 g^2+7 d e f g+2 e^2 f^2\right )}{e}-\frac{d^2 x \left (8 d^2 g^2+16 d e f g+7 e^2 f^2\right )}{e^2}-\frac{1}{2} e g x^4 (2 d g+e f)-\frac{1}{3} x^3 (d g+e f) (7 d g+e f)-\frac{1}{5} e^2 g^2 x^5 \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)^4*(f + g*x)^2)/(d^2 - e^2*x^2),x]

[Out]

-((d^2*(7*e^2*f^2 + 16*d*e*f*g + 8*d^2*g^2)*x)/e^2) - (d*(2*e^2*f^2 + 7*d*e*f*g
+ 4*d^2*g^2)*x^2)/e - ((e*f + d*g)*(e*f + 7*d*g)*x^3)/3 - (e*g*(e*f + 2*d*g)*x^4
)/2 - (e^2*g^2*x^5)/5 - (8*d^3*(e*f + d*g)^2*Log[d - e*x])/e^3

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{8 d^{3} \left (d g + e f\right )^{2} \log{\left (d - e x \right )}}{e^{3}} - \frac{7 d^{2} x \left (\frac{8 d g \left (d g + 2 e f\right )}{7} + e^{2} f^{2}\right )}{e^{2}} - \frac{2 d \left (4 d^{2} g^{2} + 7 d e f g + 2 e^{2} f^{2}\right ) \int x\, dx}{e} - \frac{e^{2} g^{2} x^{5}}{5} - \frac{e g x^{4} \left (2 d g + e f\right )}{2} - \frac{x^{3} \left (d g + e f\right ) \left (7 d g + e f\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**4*(g*x+f)**2/(-e**2*x**2+d**2),x)

[Out]

-8*d**3*(d*g + e*f)**2*log(d - e*x)/e**3 - 7*d**2*x*(8*d*g*(d*g + 2*e*f)/7 + e**
2*f**2)/e**2 - 2*d*(4*d**2*g**2 + 7*d*e*f*g + 2*e**2*f**2)*Integral(x, x)/e - e*
*2*g**2*x**5/5 - e*g*x**4*(2*d*g + e*f)/2 - x**3*(d*g + e*f)*(7*d*g + e*f)/3

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Mathematica [A]  time = 0.130076, size = 134, normalized size = 0.95 \[ -\frac{8 d^3 (d g+e f)^2 \log (d-e x)}{e^3}-\frac{x \left (240 d^4 g^2+120 d^3 e g (4 f+g x)+70 d^2 e^2 \left (3 f^2+3 f g x+g^2 x^2\right )+10 d e^3 x \left (6 f^2+8 f g x+3 g^2 x^2\right )+e^4 x^2 \left (10 f^2+15 f g x+6 g^2 x^2\right )\right )}{30 e^2} \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)^4*(f + g*x)^2)/(d^2 - e^2*x^2),x]

[Out]

-(x*(240*d^4*g^2 + 120*d^3*e*g*(4*f + g*x) + 70*d^2*e^2*(3*f^2 + 3*f*g*x + g^2*x
^2) + 10*d*e^3*x*(6*f^2 + 8*f*g*x + 3*g^2*x^2) + e^4*x^2*(10*f^2 + 15*f*g*x + 6*
g^2*x^2)))/(30*e^2) - (8*d^3*(e*f + d*g)^2*Log[d - e*x])/e^3

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Maple [A]  time = 0.007, size = 186, normalized size = 1.3 \[ -{\frac{{e}^{2}{g}^{2}{x}^{5}}{5}}-e{x}^{4}d{g}^{2}-{\frac{{e}^{2}{x}^{4}fg}{2}}-{\frac{7\,{x}^{3}{d}^{2}{g}^{2}}{3}}-{\frac{8\,e{x}^{3}dfg}{3}}-{\frac{{e}^{2}{x}^{3}{f}^{2}}{3}}-4\,{\frac{{x}^{2}{d}^{3}{g}^{2}}{e}}-7\,{x}^{2}{d}^{2}fg-2\,e{x}^{2}d{f}^{2}-8\,{\frac{{d}^{4}{g}^{2}x}{{e}^{2}}}-16\,{\frac{{d}^{3}fgx}{e}}-7\,{d}^{2}{f}^{2}x-8\,{\frac{{d}^{5}\ln \left ( ex-d \right ){g}^{2}}{{e}^{3}}}-16\,{\frac{{d}^{4}\ln \left ( ex-d \right ) fg}{{e}^{2}}}-8\,{\frac{{d}^{3}\ln \left ( ex-d \right ){f}^{2}}{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4*(g*x+f)^2/(-e^2*x^2+d^2),x)

[Out]

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

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Maxima [A]  time = 0.714868, size = 236, normalized size = 1.67 \[ -\frac{6 \, e^{4} g^{2} x^{5} + 15 \,{\left (e^{4} f g + 2 \, d e^{3} g^{2}\right )} x^{4} + 10 \,{\left (e^{4} f^{2} + 8 \, d e^{3} f g + 7 \, d^{2} e^{2} g^{2}\right )} x^{3} + 30 \,{\left (2 \, d e^{3} f^{2} + 7 \, d^{2} e^{2} f g + 4 \, d^{3} e g^{2}\right )} x^{2} + 30 \,{\left (7 \, d^{2} e^{2} f^{2} + 16 \, d^{3} e f g + 8 \, d^{4} g^{2}\right )} x}{30 \, e^{2}} - \frac{8 \,{\left (d^{3} e^{2} f^{2} + 2 \, d^{4} e f g + d^{5} g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e*x + d)^4*(g*x + f)^2/(e^2*x^2 - d^2),x, algorithm="maxima")

[Out]

-1/30*(6*e^4*g^2*x^5 + 15*(e^4*f*g + 2*d*e^3*g^2)*x^4 + 10*(e^4*f^2 + 8*d*e^3*f*
g + 7*d^2*e^2*g^2)*x^3 + 30*(2*d*e^3*f^2 + 7*d^2*e^2*f*g + 4*d^3*e*g^2)*x^2 + 30
*(7*d^2*e^2*f^2 + 16*d^3*e*f*g + 8*d^4*g^2)*x)/e^2 - 8*(d^3*e^2*f^2 + 2*d^4*e*f*
g + d^5*g^2)*log(e*x - d)/e^3

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Fricas [A]  time = 0.280468, size = 238, normalized size = 1.69 \[ -\frac{6 \, e^{5} g^{2} x^{5} + 15 \,{\left (e^{5} f g + 2 \, d e^{4} g^{2}\right )} x^{4} + 10 \,{\left (e^{5} f^{2} + 8 \, d e^{4} f g + 7 \, d^{2} e^{3} g^{2}\right )} x^{3} + 30 \,{\left (2 \, d e^{4} f^{2} + 7 \, d^{2} e^{3} f g + 4 \, d^{3} e^{2} g^{2}\right )} x^{2} + 30 \,{\left (7 \, d^{2} e^{3} f^{2} + 16 \, d^{3} e^{2} f g + 8 \, d^{4} e g^{2}\right )} x + 240 \,{\left (d^{3} e^{2} f^{2} + 2 \, d^{4} e f g + d^{5} g^{2}\right )} \log \left (e x - d\right )}{30 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e*x + d)^4*(g*x + f)^2/(e^2*x^2 - d^2),x, algorithm="fricas")

[Out]

-1/30*(6*e^5*g^2*x^5 + 15*(e^5*f*g + 2*d*e^4*g^2)*x^4 + 10*(e^5*f^2 + 8*d*e^4*f*
g + 7*d^2*e^3*g^2)*x^3 + 30*(2*d*e^4*f^2 + 7*d^2*e^3*f*g + 4*d^3*e^2*g^2)*x^2 +
30*(7*d^2*e^3*f^2 + 16*d^3*e^2*f*g + 8*d^4*e*g^2)*x + 240*(d^3*e^2*f^2 + 2*d^4*e
*f*g + d^5*g^2)*log(e*x - d))/e^3

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Sympy [A]  time = 2.60131, size = 156, normalized size = 1.11 \[ - \frac{8 d^{3} \left (d g + e f\right )^{2} \log{\left (- d + e x \right )}}{e^{3}} - \frac{e^{2} g^{2} x^{5}}{5} - x^{4} \left (d e g^{2} + \frac{e^{2} f g}{2}\right ) - x^{3} \left (\frac{7 d^{2} g^{2}}{3} + \frac{8 d e f g}{3} + \frac{e^{2} f^{2}}{3}\right ) - \frac{x^{2} \left (4 d^{3} g^{2} + 7 d^{2} e f g + 2 d e^{2} f^{2}\right )}{e} - \frac{x \left (8 d^{4} g^{2} + 16 d^{3} e f g + 7 d^{2} e^{2} f^{2}\right )}{e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**4*(g*x+f)**2/(-e**2*x**2+d**2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.275882, size = 336, normalized size = 2.38 \[ -4 \,{\left (d^{5} g^{2} e^{3} + 2 \, d^{4} f g e^{4} + d^{3} f^{2} e^{5}\right )} e^{\left (-6\right )}{\rm ln}\left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) - \frac{1}{30} \,{\left (6 \, g^{2} x^{5} e^{12} + 30 \, d g^{2} x^{4} e^{11} + 70 \, d^{2} g^{2} x^{3} e^{10} + 120 \, d^{3} g^{2} x^{2} e^{9} + 240 \, d^{4} g^{2} x e^{8} + 15 \, f g x^{4} e^{12} + 80 \, d f g x^{3} e^{11} + 210 \, d^{2} f g x^{2} e^{10} + 480 \, d^{3} f g x e^{9} + 10 \, f^{2} x^{3} e^{12} + 60 \, d f^{2} x^{2} e^{11} + 210 \, d^{2} f^{2} x e^{10}\right )} e^{\left (-10\right )} - \frac{4 \,{\left (d^{6} g^{2} e^{4} + 2 \, d^{5} f g e^{5} + d^{4} f^{2} e^{6}\right )} e^{\left (-7\right )}{\rm ln}\left (\frac{{\left | 2 \, x e^{2} - 2 \,{\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \,{\left | d \right |} e \right |}}\right )}{{\left | d \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e*x + d)^4*(g*x + f)^2/(e^2*x^2 - d^2),x, algorithm="giac")

[Out]

-4*(d^5*g^2*e^3 + 2*d^4*f*g*e^4 + d^3*f^2*e^5)*e^(-6)*ln(abs(x^2*e^2 - d^2)) - 1
/30*(6*g^2*x^5*e^12 + 30*d*g^2*x^4*e^11 + 70*d^2*g^2*x^3*e^10 + 120*d^3*g^2*x^2*
e^9 + 240*d^4*g^2*x*e^8 + 15*f*g*x^4*e^12 + 80*d*f*g*x^3*e^11 + 210*d^2*f*g*x^2*
e^10 + 480*d^3*f*g*x*e^9 + 10*f^2*x^3*e^12 + 60*d*f^2*x^2*e^11 + 210*d^2*f^2*x*e
^10)*e^(-10) - 4*(d^6*g^2*e^4 + 2*d^5*f*g*e^5 + d^4*f^2*e^6)*e^(-7)*ln(abs(2*x*e
^2 - 2*abs(d)*e)/abs(2*x*e^2 + 2*abs(d)*e))/abs(d)