3.1885 \(\int \frac{(d+e x)^{10}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx\)

Optimal. Leaf size=217 \[ \frac{e^4 x \left (10 a^2 e^4-24 a c d^2 e^2+15 c^2 d^4\right )}{c^6 d^6}-\frac{15 e^2 \left (c d^2-a e^2\right )^4}{c^7 d^7 (a e+c d x)}-\frac{3 e \left (c d^2-a e^2\right )^5}{c^7 d^7 (a e+c d x)^2}-\frac{\left (c d^2-a e^2\right )^6}{3 c^7 d^7 (a e+c d x)^3}+\frac{20 e^3 \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^7 d^7}+\frac{e^5 x^2 \left (3 c d^2-2 a e^2\right )}{c^5 d^5}+\frac{e^6 x^3}{3 c^4 d^4} \]

[Out]

(e^4*(15*c^2*d^4 - 24*a*c*d^2*e^2 + 10*a^2*e^4)*x)/(c^6*d^6) + (e^5*(3*c*d^2 - 2
*a*e^2)*x^2)/(c^5*d^5) + (e^6*x^3)/(3*c^4*d^4) - (c*d^2 - a*e^2)^6/(3*c^7*d^7*(a
*e + c*d*x)^3) - (3*e*(c*d^2 - a*e^2)^5)/(c^7*d^7*(a*e + c*d*x)^2) - (15*e^2*(c*
d^2 - a*e^2)^4)/(c^7*d^7*(a*e + c*d*x)) + (20*e^3*(c*d^2 - a*e^2)^3*Log[a*e + c*
d*x])/(c^7*d^7)

_______________________________________________________________________________________

Rubi [A]  time = 0.605854, antiderivative size = 217, 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 \[ \frac{e^4 x \left (10 a^2 e^4-24 a c d^2 e^2+15 c^2 d^4\right )}{c^6 d^6}-\frac{15 e^2 \left (c d^2-a e^2\right )^4}{c^7 d^7 (a e+c d x)}-\frac{3 e \left (c d^2-a e^2\right )^5}{c^7 d^7 (a e+c d x)^2}-\frac{\left (c d^2-a e^2\right )^6}{3 c^7 d^7 (a e+c d x)^3}+\frac{20 e^3 \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^7 d^7}+\frac{e^5 x^2 \left (3 c d^2-2 a e^2\right )}{c^5 d^5}+\frac{e^6 x^3}{3 c^4 d^4} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^10/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

(e^4*(15*c^2*d^4 - 24*a*c*d^2*e^2 + 10*a^2*e^4)*x)/(c^6*d^6) + (e^5*(3*c*d^2 - 2
*a*e^2)*x^2)/(c^5*d^5) + (e^6*x^3)/(3*c^4*d^4) - (c*d^2 - a*e^2)^6/(3*c^7*d^7*(a
*e + c*d*x)^3) - (3*e*(c*d^2 - a*e^2)^5)/(c^7*d^7*(a*e + c*d*x)^2) - (15*e^2*(c*
d^2 - a*e^2)^4)/(c^7*d^7*(a*e + c*d*x)) + (20*e^3*(c*d^2 - a*e^2)^3*Log[a*e + c*
d*x])/(c^7*d^7)

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{e^{4} \left (10 a^{2} e^{4} - 24 a c d^{2} e^{2} + 15 c^{2} d^{4}\right ) \int \frac{1}{c^{6}}\, dx}{d^{6}} + \frac{e^{6} x^{3}}{3 c^{4} d^{4}} - \frac{2 e^{5} \left (2 a e^{2} - 3 c d^{2}\right ) \int x\, dx}{c^{5} d^{5}} - \frac{20 e^{3} \left (a e^{2} - c d^{2}\right )^{3} \log{\left (a e + c d x \right )}}{c^{7} d^{7}} - \frac{15 e^{2} \left (a e^{2} - c d^{2}\right )^{4}}{c^{7} d^{7} \left (a e + c d x\right )} + \frac{3 e \left (a e^{2} - c d^{2}\right )^{5}}{c^{7} d^{7} \left (a e + c d x\right )^{2}} - \frac{\left (a e^{2} - c d^{2}\right )^{6}}{3 c^{7} d^{7} \left (a e + c d x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**10/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)

[Out]

e**4*(10*a**2*e**4 - 24*a*c*d**2*e**2 + 15*c**2*d**4)*Integral(c**(-6), x)/d**6
+ e**6*x**3/(3*c**4*d**4) - 2*e**5*(2*a*e**2 - 3*c*d**2)*Integral(x, x)/(c**5*d*
*5) - 20*e**3*(a*e**2 - c*d**2)**3*log(a*e + c*d*x)/(c**7*d**7) - 15*e**2*(a*e**
2 - c*d**2)**4/(c**7*d**7*(a*e + c*d*x)) + 3*e*(a*e**2 - c*d**2)**5/(c**7*d**7*(
a*e + c*d*x)**2) - (a*e**2 - c*d**2)**6/(3*c**7*d**7*(a*e + c*d*x)**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.261504, size = 335, normalized size = 1.54 \[ \frac{-37 a^6 e^{12}+3 a^5 c d e^{10} (47 d-17 e x)+3 a^4 c^2 d^2 e^8 \left (-65 d^2+81 d e x+13 e^2 x^2\right )+a^3 c^3 d^3 e^6 \left (110 d^3-405 d^2 e x-27 d e^2 x^2+73 e^3 x^3\right )-3 a^2 c^4 d^4 e^4 \left (5 d^4-90 d^3 e x+45 d^2 e^2 x^2+63 d e^3 x^3-5 e^4 x^4\right )-3 a c^5 d^5 e^2 \left (d^5+15 d^4 e x-60 d^3 e^2 x^2-45 d^2 e^3 x^3+15 d e^4 x^4+e^5 x^5\right )-60 e^3 \left (a e^2-c d^2\right )^3 (a e+c d x)^3 \log (a e+c d x)+c^6 d^6 \left (-d^6-9 d^5 e x-45 d^4 e^2 x^2+45 d^2 e^4 x^4+9 d e^5 x^5+e^6 x^6\right )}{3 c^7 d^7 (a e+c d x)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^10/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

(-37*a^6*e^12 + 3*a^5*c*d*e^10*(47*d - 17*e*x) + 3*a^4*c^2*d^2*e^8*(-65*d^2 + 81
*d*e*x + 13*e^2*x^2) + a^3*c^3*d^3*e^6*(110*d^3 - 405*d^2*e*x - 27*d*e^2*x^2 + 7
3*e^3*x^3) - 3*a^2*c^4*d^4*e^4*(5*d^4 - 90*d^3*e*x + 45*d^2*e^2*x^2 + 63*d*e^3*x
^3 - 5*e^4*x^4) - 3*a*c^5*d^5*e^2*(d^5 + 15*d^4*e*x - 60*d^3*e^2*x^2 - 45*d^2*e^
3*x^3 + 15*d*e^4*x^4 + e^5*x^5) + c^6*d^6*(-d^6 - 9*d^5*e*x - 45*d^4*e^2*x^2 + 4
5*d^2*e^4*x^4 + 9*d*e^5*x^5 + e^6*x^6) - 60*e^3*(-(c*d^2) + a*e^2)^3*(a*e + c*d*
x)^3*Log[a*e + c*d*x])/(3*c^7*d^7*(a*e + c*d*x)^3)

_______________________________________________________________________________________

Maple [B]  time = 0.017, size = 578, normalized size = 2.7 \[ -{\frac{{d}^{5}}{3\,c \left ( cdx+ae \right ) ^{3}}}+{\frac{{e}^{6}{x}^{3}}{3\,{c}^{4}{d}^{4}}}-60\,{\frac{{e}^{5}\ln \left ( cdx+ae \right ) a}{{c}^{5}{d}^{3}}}-{\frac{{a}^{6}{e}^{12}}{3\,{c}^{7}{d}^{7} \left ( cdx+ae \right ) ^{3}}}+2\,{\frac{{a}^{5}{e}^{10}}{{c}^{6}{d}^{5} \left ( cdx+ae \right ) ^{3}}}-5\,{\frac{{a}^{4}{e}^{8}}{{c}^{5}{d}^{3} \left ( cdx+ae \right ) ^{3}}}+{\frac{20\,{a}^{3}{e}^{6}}{3\,{c}^{4}d \left ( cdx+ae \right ) ^{3}}}-5\,{\frac{{a}^{2}d{e}^{4}}{{c}^{3} \left ( cdx+ae \right ) ^{3}}}-24\,{\frac{a{e}^{6}x}{{c}^{5}{d}^{4}}}+2\,{\frac{{d}^{3}{e}^{2}a}{{c}^{2} \left ( cdx+ae \right ) ^{3}}}-2\,{\frac{{e}^{7}{x}^{2}a}{{c}^{5}{d}^{5}}}+10\,{\frac{{a}^{2}{e}^{8}x}{{c}^{6}{d}^{6}}}-15\,{\frac{{e}^{10}{a}^{4}}{{c}^{7}{d}^{7} \left ( cdx+ae \right ) }}+60\,{\frac{{e}^{8}{a}^{3}}{{c}^{6}{d}^{5} \left ( cdx+ae \right ) }}-20\,{\frac{{e}^{9}\ln \left ( cdx+ae \right ){a}^{3}}{{c}^{7}{d}^{7}}}+3\,{\frac{{a}^{5}{e}^{11}}{{c}^{7}{d}^{7} \left ( cdx+ae \right ) ^{2}}}-15\,{\frac{{e}^{9}{a}^{4}}{{c}^{6}{d}^{5} \left ( cdx+ae \right ) ^{2}}}+30\,{\frac{{e}^{7}{a}^{3}}{{c}^{5}{d}^{3} \left ( cdx+ae \right ) ^{2}}}-30\,{\frac{{a}^{2}{e}^{5}}{{c}^{4}d \left ( cdx+ae \right ) ^{2}}}+15\,{\frac{ad{e}^{3}}{{c}^{3} \left ( cdx+ae \right ) ^{2}}}-15\,{\frac{d{e}^{2}}{{c}^{3} \left ( cdx+ae \right ) }}-3\,{\frac{{d}^{3}e}{{c}^{2} \left ( cdx+ae \right ) ^{2}}}+20\,{\frac{{e}^{3}\ln \left ( cdx+ae \right ) }{{c}^{4}d}}+3\,{\frac{{e}^{5}{x}^{2}}{{c}^{4}{d}^{3}}}+15\,{\frac{{e}^{4}x}{{c}^{4}{d}^{2}}}+60\,{\frac{{e}^{7}\ln \left ( cdx+ae \right ){a}^{2}}{{c}^{6}{d}^{5}}}-90\,{\frac{{e}^{6}{a}^{2}}{{c}^{5}{d}^{3} \left ( cdx+ae \right ) }}+60\,{\frac{{e}^{4}a}{{c}^{4}d \left ( cdx+ae \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^10/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x)

[Out]

-1/3/c*d^5/(c*d*x+a*e)^3+1/3*e^6*x^3/c^4/d^4-60/c^5/d^3*e^5*ln(c*d*x+a*e)*a-1/3/
c^7/d^7/(c*d*x+a*e)^3*a^6*e^12+2/c^6/d^5/(c*d*x+a*e)^3*a^5*e^10-5/c^5/d^3/(c*d*x
+a*e)^3*a^4*e^8+20/3/c^4/d/(c*d*x+a*e)^3*a^3*e^6-5/c^3*d/(c*d*x+a*e)^3*a^2*e^4-2
4*e^6/c^5/d^4*a*x+2/c^2*d^3/(c*d*x+a*e)^3*a*e^2-2*e^7/c^5/d^5*x^2*a+10*e^8/c^6/d
^6*a^2*x-15/c^7/d^7*e^10/(c*d*x+a*e)*a^4+60/c^6/d^5*e^8/(c*d*x+a*e)*a^3-20/c^7/d
^7*e^9*ln(c*d*x+a*e)*a^3+3/d^7*e^11/c^7/(c*d*x+a*e)^2*a^5-15/d^5*e^9/c^6/(c*d*x+
a*e)^2*a^4+30/d^3*e^7/c^5/(c*d*x+a*e)^2*a^3-30/d*e^5/c^4/(c*d*x+a*e)^2*a^2+15*d*
e^3/c^3/(c*d*x+a*e)^2*a-15/c^3*d*e^2/(c*d*x+a*e)-3*d^3*e/c^2/(c*d*x+a*e)^2+20/c^
4/d*e^3*ln(c*d*x+a*e)+3*e^5/c^4/d^3*x^2+15*e^4/c^4/d^2*x+60/c^6/d^5*e^7*ln(c*d*x
+a*e)*a^2-90/c^5/d^3*e^6/(c*d*x+a*e)*a^2+60/c^4/d*e^4/(c*d*x+a*e)*a

_______________________________________________________________________________________

Maxima [A]  time = 0.748391, size = 572, normalized size = 2.64 \[ -\frac{c^{6} d^{12} + 3 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 110 \, a^{3} c^{3} d^{6} e^{6} + 195 \, a^{4} c^{2} d^{4} e^{8} - 141 \, a^{5} c d^{2} e^{10} + 37 \, a^{6} e^{12} + 45 \,{\left (c^{6} d^{10} e^{2} - 4 \, a c^{5} d^{8} e^{4} + 6 \, a^{2} c^{4} d^{6} e^{6} - 4 \, a^{3} c^{3} d^{4} e^{8} + a^{4} c^{2} d^{2} e^{10}\right )} x^{2} + 9 \,{\left (c^{6} d^{11} e + 5 \, a c^{5} d^{9} e^{3} - 30 \, a^{2} c^{4} d^{7} e^{5} + 50 \, a^{3} c^{3} d^{5} e^{7} - 35 \, a^{4} c^{2} d^{3} e^{9} + 9 \, a^{5} c d e^{11}\right )} x}{3 \,{\left (c^{10} d^{10} x^{3} + 3 \, a c^{9} d^{9} e x^{2} + 3 \, a^{2} c^{8} d^{8} e^{2} x + a^{3} c^{7} d^{7} e^{3}\right )}} + \frac{c^{2} d^{2} e^{6} x^{3} + 3 \,{\left (3 \, c^{2} d^{3} e^{5} - 2 \, a c d e^{7}\right )} x^{2} + 3 \,{\left (15 \, c^{2} d^{4} e^{4} - 24 \, a c d^{2} e^{6} + 10 \, a^{2} e^{8}\right )} x}{3 \, c^{6} d^{6}} + \frac{20 \,{\left (c^{3} d^{6} e^{3} - 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} - a^{3} e^{9}\right )} \log \left (c d x + a e\right )}{c^{7} d^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^10/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^4,x, algorithm="maxima")

[Out]

-1/3*(c^6*d^12 + 3*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 110*a^3*c^3*d^6*e^6 + 1
95*a^4*c^2*d^4*e^8 - 141*a^5*c*d^2*e^10 + 37*a^6*e^12 + 45*(c^6*d^10*e^2 - 4*a*c
^5*d^8*e^4 + 6*a^2*c^4*d^6*e^6 - 4*a^3*c^3*d^4*e^8 + a^4*c^2*d^2*e^10)*x^2 + 9*(
c^6*d^11*e + 5*a*c^5*d^9*e^3 - 30*a^2*c^4*d^7*e^5 + 50*a^3*c^3*d^5*e^7 - 35*a^4*
c^2*d^3*e^9 + 9*a^5*c*d*e^11)*x)/(c^10*d^10*x^3 + 3*a*c^9*d^9*e*x^2 + 3*a^2*c^8*
d^8*e^2*x + a^3*c^7*d^7*e^3) + 1/3*(c^2*d^2*e^6*x^3 + 3*(3*c^2*d^3*e^5 - 2*a*c*d
*e^7)*x^2 + 3*(15*c^2*d^4*e^4 - 24*a*c*d^2*e^6 + 10*a^2*e^8)*x)/(c^6*d^6) + 20*(
c^3*d^6*e^3 - 3*a*c^2*d^4*e^5 + 3*a^2*c*d^2*e^7 - a^3*e^9)*log(c*d*x + a*e)/(c^7
*d^7)

_______________________________________________________________________________________

Fricas [A]  time = 0.215411, size = 869, normalized size = 4. \[ \frac{c^{6} d^{6} e^{6} x^{6} - c^{6} d^{12} - 3 \, a c^{5} d^{10} e^{2} - 15 \, a^{2} c^{4} d^{8} e^{4} + 110 \, a^{3} c^{3} d^{6} e^{6} - 195 \, a^{4} c^{2} d^{4} e^{8} + 141 \, a^{5} c d^{2} e^{10} - 37 \, a^{6} e^{12} + 3 \,{\left (3 \, c^{6} d^{7} e^{5} - a c^{5} d^{5} e^{7}\right )} x^{5} + 15 \,{\left (3 \, c^{6} d^{8} e^{4} - 3 \, a c^{5} d^{6} e^{6} + a^{2} c^{4} d^{4} e^{8}\right )} x^{4} +{\left (135 \, a c^{5} d^{7} e^{5} - 189 \, a^{2} c^{4} d^{5} e^{7} + 73 \, a^{3} c^{3} d^{3} e^{9}\right )} x^{3} - 3 \,{\left (15 \, c^{6} d^{10} e^{2} - 60 \, a c^{5} d^{8} e^{4} + 45 \, a^{2} c^{4} d^{6} e^{6} + 9 \, a^{3} c^{3} d^{4} e^{8} - 13 \, a^{4} c^{2} d^{2} e^{10}\right )} x^{2} - 3 \,{\left (3 \, c^{6} d^{11} e + 15 \, a c^{5} d^{9} e^{3} - 90 \, a^{2} c^{4} d^{7} e^{5} + 135 \, a^{3} c^{3} d^{5} e^{7} - 81 \, a^{4} c^{2} d^{3} e^{9} + 17 \, a^{5} c d e^{11}\right )} x + 60 \,{\left (a^{3} c^{3} d^{6} e^{6} - 3 \, a^{4} c^{2} d^{4} e^{8} + 3 \, a^{5} c d^{2} e^{10} - a^{6} e^{12} +{\left (c^{6} d^{9} e^{3} - 3 \, a c^{5} d^{7} e^{5} + 3 \, a^{2} c^{4} d^{5} e^{7} - a^{3} c^{3} d^{3} e^{9}\right )} x^{3} + 3 \,{\left (a c^{5} d^{8} e^{4} - 3 \, a^{2} c^{4} d^{6} e^{6} + 3 \, a^{3} c^{3} d^{4} e^{8} - a^{4} c^{2} d^{2} e^{10}\right )} x^{2} + 3 \,{\left (a^{2} c^{4} d^{7} e^{5} - 3 \, a^{3} c^{3} d^{5} e^{7} + 3 \, a^{4} c^{2} d^{3} e^{9} - a^{5} c d e^{11}\right )} x\right )} \log \left (c d x + a e\right )}{3 \,{\left (c^{10} d^{10} x^{3} + 3 \, a c^{9} d^{9} e x^{2} + 3 \, a^{2} c^{8} d^{8} e^{2} x + a^{3} c^{7} d^{7} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^10/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^4,x, algorithm="fricas")

[Out]

1/3*(c^6*d^6*e^6*x^6 - c^6*d^12 - 3*a*c^5*d^10*e^2 - 15*a^2*c^4*d^8*e^4 + 110*a^
3*c^3*d^6*e^6 - 195*a^4*c^2*d^4*e^8 + 141*a^5*c*d^2*e^10 - 37*a^6*e^12 + 3*(3*c^
6*d^7*e^5 - a*c^5*d^5*e^7)*x^5 + 15*(3*c^6*d^8*e^4 - 3*a*c^5*d^6*e^6 + a^2*c^4*d
^4*e^8)*x^4 + (135*a*c^5*d^7*e^5 - 189*a^2*c^4*d^5*e^7 + 73*a^3*c^3*d^3*e^9)*x^3
- 3*(15*c^6*d^10*e^2 - 60*a*c^5*d^8*e^4 + 45*a^2*c^4*d^6*e^6 + 9*a^3*c^3*d^4*e^
8 - 13*a^4*c^2*d^2*e^10)*x^2 - 3*(3*c^6*d^11*e + 15*a*c^5*d^9*e^3 - 90*a^2*c^4*d
^7*e^5 + 135*a^3*c^3*d^5*e^7 - 81*a^4*c^2*d^3*e^9 + 17*a^5*c*d*e^11)*x + 60*(a^3
*c^3*d^6*e^6 - 3*a^4*c^2*d^4*e^8 + 3*a^5*c*d^2*e^10 - a^6*e^12 + (c^6*d^9*e^3 -
3*a*c^5*d^7*e^5 + 3*a^2*c^4*d^5*e^7 - a^3*c^3*d^3*e^9)*x^3 + 3*(a*c^5*d^8*e^4 -
3*a^2*c^4*d^6*e^6 + 3*a^3*c^3*d^4*e^8 - a^4*c^2*d^2*e^10)*x^2 + 3*(a^2*c^4*d^7*e
^5 - 3*a^3*c^3*d^5*e^7 + 3*a^4*c^2*d^3*e^9 - a^5*c*d*e^11)*x)*log(c*d*x + a*e))/
(c^10*d^10*x^3 + 3*a*c^9*d^9*e*x^2 + 3*a^2*c^8*d^8*e^2*x + a^3*c^7*d^7*e^3)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**10/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^10/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^4,x, algorithm="giac")

[Out]

Timed out