3.1876 \(\int \frac{(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx\)

Optimal. Leaf size=142 \[ -\frac{4 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)}-\frac{\left (c d^2-a e^2\right )^4}{2 c^5 d^5 (a e+c d x)^2}+\frac{6 e^2 \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^5 d^5}+\frac{e^3 x \left (4 c d^2-3 a e^2\right )}{c^4 d^4}+\frac{e^4 x^2}{2 c^3 d^3} \]

[Out]

(e^3*(4*c*d^2 - 3*a*e^2)*x)/(c^4*d^4) + (e^4*x^2)/(2*c^3*d^3) - (c*d^2 - a*e^2)^
4/(2*c^5*d^5*(a*e + c*d*x)^2) - (4*e*(c*d^2 - a*e^2)^3)/(c^5*d^5*(a*e + c*d*x))
+ (6*e^2*(c*d^2 - a*e^2)^2*Log[a*e + c*d*x])/(c^5*d^5)

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Rubi [A]  time = 0.327015, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ -\frac{4 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)}-\frac{\left (c d^2-a e^2\right )^4}{2 c^5 d^5 (a e+c d x)^2}+\frac{6 e^2 \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^5 d^5}+\frac{e^3 x \left (4 c d^2-3 a e^2\right )}{c^4 d^4}+\frac{e^4 x^2}{2 c^3 d^3} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^7/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

(e^3*(4*c*d^2 - 3*a*e^2)*x)/(c^4*d^4) + (e^4*x^2)/(2*c^3*d^3) - (c*d^2 - a*e^2)^
4/(2*c^5*d^5*(a*e + c*d*x)^2) - (4*e*(c*d^2 - a*e^2)^3)/(c^5*d^5*(a*e + c*d*x))
+ (6*e^2*(c*d^2 - a*e^2)^2*Log[a*e + c*d*x])/(c^5*d^5)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{e^{3} \left (3 a e^{2} - 4 c d^{2}\right ) \int \frac{1}{c^{4}}\, dx}{d^{4}} + \frac{e^{4} \int x\, dx}{c^{3} d^{3}} + \frac{6 e^{2} \left (a e^{2} - c d^{2}\right )^{2} \log{\left (a e + c d x \right )}}{c^{5} d^{5}} + \frac{4 e \left (a e^{2} - c d^{2}\right )^{3}}{c^{5} d^{5} \left (a e + c d x\right )} - \frac{\left (a e^{2} - c d^{2}\right )^{4}}{2 c^{5} d^{5} \left (a e + c d x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**7/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

-e**3*(3*a*e**2 - 4*c*d**2)*Integral(c**(-4), x)/d**4 + e**4*Integral(x, x)/(c**
3*d**3) + 6*e**2*(a*e**2 - c*d**2)**2*log(a*e + c*d*x)/(c**5*d**5) + 4*e*(a*e**2
- c*d**2)**3/(c**5*d**5*(a*e + c*d*x)) - (a*e**2 - c*d**2)**4/(2*c**5*d**5*(a*e
+ c*d*x)**2)

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Mathematica [A]  time = 0.140985, size = 191, normalized size = 1.35 \[ \frac{7 a^4 e^8+2 a^3 c d e^6 (e x-10 d)+a^2 c^2 d^2 e^4 \left (18 d^2-16 d e x-11 e^2 x^2\right )-4 a c^3 d^3 e^2 \left (d^3-6 d^2 e x-4 d e^2 x^2+e^3 x^3\right )+12 e^2 \left (c d^2-a e^2\right )^2 (a e+c d x)^2 \log (a e+c d x)+c^4 d^4 \left (-d^4-8 d^3 e x+8 d e^3 x^3+e^4 x^4\right )}{2 c^5 d^5 (a e+c d x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^7/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

(7*a^4*e^8 + 2*a^3*c*d*e^6*(-10*d + e*x) + a^2*c^2*d^2*e^4*(18*d^2 - 16*d*e*x -
11*e^2*x^2) - 4*a*c^3*d^3*e^2*(d^3 - 6*d^2*e*x - 4*d*e^2*x^2 + e^3*x^3) + c^4*d^
4*(-d^4 - 8*d^3*e*x + 8*d*e^3*x^3 + e^4*x^4) + 12*e^2*(c*d^2 - a*e^2)^2*(a*e + c
*d*x)^2*Log[a*e + c*d*x])/(2*c^5*d^5*(a*e + c*d*x)^2)

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Maple [B]  time = 0.012, size = 302, normalized size = 2.1 \[{\frac{{e}^{4}{x}^{2}}{2\,{c}^{3}{d}^{3}}}-3\,{\frac{a{e}^{5}x}{{c}^{4}{d}^{4}}}+4\,{\frac{{e}^{3}x}{{c}^{3}{d}^{2}}}+4\,{\frac{{a}^{3}{e}^{7}}{{c}^{5}{d}^{5} \left ( cdx+ae \right ) }}-12\,{\frac{{a}^{2}{e}^{5}}{{d}^{3}{c}^{4} \left ( cdx+ae \right ) }}+12\,{\frac{a{e}^{3}}{d{c}^{3} \left ( cdx+ae \right ) }}-4\,{\frac{de}{{c}^{2} \left ( cdx+ae \right ) }}-{\frac{{a}^{4}{e}^{8}}{2\,{c}^{5}{d}^{5} \left ( cdx+ae \right ) ^{2}}}+2\,{\frac{{a}^{3}{e}^{6}}{{d}^{3}{c}^{4} \left ( cdx+ae \right ) ^{2}}}-3\,{\frac{{a}^{2}{e}^{4}}{d{c}^{3} \left ( cdx+ae \right ) ^{2}}}+2\,{\frac{ad{e}^{2}}{{c}^{2} \left ( cdx+ae \right ) ^{2}}}-{\frac{{d}^{3}}{2\,c \left ( cdx+ae \right ) ^{2}}}+6\,{\frac{{e}^{6}\ln \left ( cdx+ae \right ){a}^{2}}{{c}^{5}{d}^{5}}}-12\,{\frac{{e}^{4}\ln \left ( cdx+ae \right ) a}{{d}^{3}{c}^{4}}}+6\,{\frac{{e}^{2}\ln \left ( cdx+ae \right ) }{d{c}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^7/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)

[Out]

1/2*e^4*x^2/c^3/d^3-3*e^5/c^4/d^4*a*x+4*e^3/c^3/d^2*x+4/d^5*e^7/c^5/(c*d*x+a*e)*
a^3-12/d^3*e^5/c^4/(c*d*x+a*e)*a^2+12/d*e^3/c^3/(c*d*x+a*e)*a-4*d*e/c^2/(c*d*x+a
*e)-1/2/c^5/d^5/(c*d*x+a*e)^2*a^4*e^8+2/c^4/d^3/(c*d*x+a*e)^2*a^3*e^6-3/c^3/d/(c
*d*x+a*e)^2*a^2*e^4+2/c^2*d/(c*d*x+a*e)^2*a*e^2-1/2/c*d^3/(c*d*x+a*e)^2+6/c^5/d^
5*e^6*ln(c*d*x+a*e)*a^2-12/c^4/d^3*e^4*ln(c*d*x+a*e)*a+6/c^3/d*e^2*ln(c*d*x+a*e)

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Maxima [A]  time = 0.734897, size = 302, normalized size = 2.13 \[ -\frac{c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} - 18 \, a^{2} c^{2} d^{4} e^{4} + 20 \, a^{3} c d^{2} e^{6} - 7 \, a^{4} e^{8} + 8 \,{\left (c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x}{2 \,{\left (c^{7} d^{7} x^{2} + 2 \, a c^{6} d^{6} e x + a^{2} c^{5} d^{5} e^{2}\right )}} + \frac{c d e^{4} x^{2} + 2 \,{\left (4 \, c d^{2} e^{3} - 3 \, a e^{5}\right )} x}{2 \, c^{4} d^{4}} + \frac{6 \,{\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} \log \left (c d x + a e\right )}{c^{5} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^7/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="maxima")

[Out]

-1/2*(c^4*d^8 + 4*a*c^3*d^6*e^2 - 18*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 - 7*a^4*
e^8 + 8*(c^4*d^7*e - 3*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x)/(c^7*
d^7*x^2 + 2*a*c^6*d^6*e*x + a^2*c^5*d^5*e^2) + 1/2*(c*d*e^4*x^2 + 2*(4*c*d^2*e^3
- 3*a*e^5)*x)/(c^4*d^4) + 6*(c^2*d^4*e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*log(c*d*x +
a*e)/(c^5*d^5)

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Fricas [A]  time = 0.205857, size = 456, normalized size = 3.21 \[ \frac{c^{4} d^{4} e^{4} x^{4} - c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 18 \, a^{2} c^{2} d^{4} e^{4} - 20 \, a^{3} c d^{2} e^{6} + 7 \, a^{4} e^{8} + 4 \,{\left (2 \, c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{3} +{\left (16 \, a c^{3} d^{4} e^{4} - 11 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{2} - 2 \,{\left (4 \, c^{4} d^{7} e - 12 \, a c^{3} d^{5} e^{3} + 8 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x + 12 \,{\left (a^{2} c^{2} d^{4} e^{4} - 2 \, a^{3} c d^{2} e^{6} + a^{4} e^{8} +{\left (c^{4} d^{6} e^{2} - 2 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \,{\left (a c^{3} d^{5} e^{3} - 2 \, a^{2} c^{2} d^{3} e^{5} + a^{3} c d e^{7}\right )} x\right )} \log \left (c d x + a e\right )}{2 \,{\left (c^{7} d^{7} x^{2} + 2 \, a c^{6} d^{6} e x + a^{2} c^{5} d^{5} e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^7/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="fricas")

[Out]

1/2*(c^4*d^4*e^4*x^4 - c^4*d^8 - 4*a*c^3*d^6*e^2 + 18*a^2*c^2*d^4*e^4 - 20*a^3*c
*d^2*e^6 + 7*a^4*e^8 + 4*(2*c^4*d^5*e^3 - a*c^3*d^3*e^5)*x^3 + (16*a*c^3*d^4*e^4
- 11*a^2*c^2*d^2*e^6)*x^2 - 2*(4*c^4*d^7*e - 12*a*c^3*d^5*e^3 + 8*a^2*c^2*d^3*e
^5 - a^3*c*d*e^7)*x + 12*(a^2*c^2*d^4*e^4 - 2*a^3*c*d^2*e^6 + a^4*e^8 + (c^4*d^6
*e^2 - 2*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 + 2*(a*c^3*d^5*e^3 - 2*a^2*c^2*d^3
*e^5 + a^3*c*d*e^7)*x)*log(c*d*x + a*e))/(c^7*d^7*x^2 + 2*a*c^6*d^6*e*x + a^2*c^
5*d^5*e^2)

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Sympy [A]  time = 8.37958, size = 224, normalized size = 1.58 \[ \frac{7 a^{4} e^{8} - 20 a^{3} c d^{2} e^{6} + 18 a^{2} c^{2} d^{4} e^{4} - 4 a c^{3} d^{6} e^{2} - c^{4} d^{8} + x \left (8 a^{3} c d e^{7} - 24 a^{2} c^{2} d^{3} e^{5} + 24 a c^{3} d^{5} e^{3} - 8 c^{4} d^{7} e\right )}{2 a^{2} c^{5} d^{5} e^{2} + 4 a c^{6} d^{6} e x + 2 c^{7} d^{7} x^{2}} + \frac{e^{4} x^{2}}{2 c^{3} d^{3}} - \frac{x \left (3 a e^{5} - 4 c d^{2} e^{3}\right )}{c^{4} d^{4}} + \frac{6 e^{2} \left (a e^{2} - c d^{2}\right )^{2} \log{\left (a e + c d x \right )}}{c^{5} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**7/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

(7*a**4*e**8 - 20*a**3*c*d**2*e**6 + 18*a**2*c**2*d**4*e**4 - 4*a*c**3*d**6*e**2
- c**4*d**8 + x*(8*a**3*c*d*e**7 - 24*a**2*c**2*d**3*e**5 + 24*a*c**3*d**5*e**3
- 8*c**4*d**7*e))/(2*a**2*c**5*d**5*e**2 + 4*a*c**6*d**6*e*x + 2*c**7*d**7*x**2
) + e**4*x**2/(2*c**3*d**3) - x*(3*a*e**5 - 4*c*d**2*e**3)/(c**4*d**4) + 6*e**2*
(a*e**2 - c*d**2)**2*log(a*e + c*d*x)/(c**5*d**5)

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GIAC/XCAS [A]  time = 8.95607, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^7/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="giac")

[Out]

Done