3.1485 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^7} \, dx\)

Optimal. Leaf size=167 \[ \frac{6 b^5 (b d-a e)}{e^7 (d+e x)}-\frac{15 b^4 (b d-a e)^2}{2 e^7 (d+e x)^2}+\frac{20 b^3 (b d-a e)^3}{3 e^7 (d+e x)^3}-\frac{15 b^2 (b d-a e)^4}{4 e^7 (d+e x)^4}+\frac{6 b (b d-a e)^5}{5 e^7 (d+e x)^5}-\frac{(b d-a e)^6}{6 e^7 (d+e x)^6}+\frac{b^6 \log (d+e x)}{e^7} \]

[Out]

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

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Rubi [A]  time = 0.378293, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{6 b^5 (b d-a e)}{e^7 (d+e x)}-\frac{15 b^4 (b d-a e)^2}{2 e^7 (d+e x)^2}+\frac{20 b^3 (b d-a e)^3}{3 e^7 (d+e x)^3}-\frac{15 b^2 (b d-a e)^4}{4 e^7 (d+e x)^4}+\frac{6 b (b d-a e)^5}{5 e^7 (d+e x)^5}-\frac{(b d-a e)^6}{6 e^7 (d+e x)^6}+\frac{b^6 \log (d+e x)}{e^7} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 90.7564, size = 153, normalized size = 0.92 \[ \frac{b^{6} \log{\left (d + e x \right )}}{e^{7}} - \frac{6 b^{5} \left (a e - b d\right )}{e^{7} \left (d + e x\right )} - \frac{15 b^{4} \left (a e - b d\right )^{2}}{2 e^{7} \left (d + e x\right )^{2}} - \frac{20 b^{3} \left (a e - b d\right )^{3}}{3 e^{7} \left (d + e x\right )^{3}} - \frac{15 b^{2} \left (a e - b d\right )^{4}}{4 e^{7} \left (d + e x\right )^{4}} - \frac{6 b \left (a e - b d\right )^{5}}{5 e^{7} \left (d + e x\right )^{5}} - \frac{\left (a e - b d\right )^{6}}{6 e^{7} \left (d + e x\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**6*log(d + e*x)/e**7 - 6*b**5*(a*e - b*d)/(e**7*(d + e*x)) - 15*b**4*(a*e - b*
d)**2/(2*e**7*(d + e*x)**2) - 20*b**3*(a*e - b*d)**3/(3*e**7*(d + e*x)**3) - 15*
b**2*(a*e - b*d)**4/(4*e**7*(d + e*x)**4) - 6*b*(a*e - b*d)**5/(5*e**7*(d + e*x)
**5) - (a*e - b*d)**6/(6*e**7*(d + e*x)**6)

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Mathematica [A]  time = 0.245043, size = 233, normalized size = 1.4 \[ \frac{\frac{(b d-a e) \left (10 a^5 e^5+2 a^4 b e^4 (11 d+36 e x)+a^3 b^2 e^3 \left (37 d^2+162 d e x+225 e^2 x^2\right )+a^2 b^3 e^2 \left (57 d^3+282 d^2 e x+525 d e^2 x^2+400 e^3 x^3\right )+a b^4 e \left (87 d^4+462 d^3 e x+975 d^2 e^2 x^2+1000 d e^3 x^3+450 e^4 x^4\right )+b^5 \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )\right )}{(d+e x)^6}+60 b^6 \log (d+e x)}{60 e^7} \]

Antiderivative was successfully verified.

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

[Out]

(((b*d - a*e)*(10*a^5*e^5 + 2*a^4*b*e^4*(11*d + 36*e*x) + a^3*b^2*e^3*(37*d^2 +
162*d*e*x + 225*e^2*x^2) + a^2*b^3*e^2*(57*d^3 + 282*d^2*e*x + 525*d*e^2*x^2 + 4
00*e^3*x^3) + a*b^4*e*(87*d^4 + 462*d^3*e*x + 975*d^2*e^2*x^2 + 1000*d*e^3*x^3 +
450*e^4*x^4) + b^5*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3
+ 1350*d*e^4*x^4 + 360*e^5*x^5)))/(d + e*x)^6 + 60*b^6*Log[d + e*x])/(60*e^7)

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Maple [B]  time = 0.015, size = 513, normalized size = 3.1 \[{\frac{{b}^{6}\ln \left ( ex+d \right ) }{{e}^{7}}}-{\frac{{a}^{6}}{6\,e \left ( ex+d \right ) ^{6}}}-{\frac{15\,{b}^{2}{a}^{4}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{15\,{d}^{4}{b}^{6}}{4\,{e}^{7} \left ( ex+d \right ) ^{4}}}-{\frac{15\,{a}^{2}{b}^{4}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{15\,{b}^{6}{d}^{2}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}-6\,{\frac{a{b}^{5}}{{e}^{6} \left ( ex+d \right ) }}+6\,{\frac{d{b}^{6}}{{e}^{7} \left ( ex+d \right ) }}-{\frac{{d}^{6}{b}^{6}}{6\,{e}^{7} \left ( ex+d \right ) ^{6}}}-{\frac{20\,{a}^{3}{b}^{3}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}+{\frac{20\,{d}^{3}{b}^{6}}{3\,{e}^{7} \left ( ex+d \right ) ^{3}}}-12\,{\frac{{d}^{2}{a}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{5}}}+12\,{\frac{{d}^{3}{a}^{2}{b}^{4}}{{e}^{5} \left ( ex+d \right ) ^{5}}}-6\,{\frac{{d}^{4}a{b}^{5}}{{e}^{6} \left ( ex+d \right ) ^{5}}}+6\,{\frac{d{b}^{2}{a}^{4}}{{e}^{3} \left ( ex+d \right ) ^{5}}}-20\,{\frac{{d}^{2}a{b}^{5}}{{e}^{6} \left ( ex+d \right ) ^{3}}}+15\,{\frac{da{b}^{5}}{{e}^{6} \left ( ex+d \right ) ^{2}}}+{\frac{d{a}^{5}b}{{e}^{2} \left ( ex+d \right ) ^{6}}}-{\frac{5\,{d}^{2}{b}^{2}{a}^{4}}{2\,{e}^{3} \left ( ex+d \right ) ^{6}}}-{\frac{6\,{a}^{5}b}{5\,{e}^{2} \left ( ex+d \right ) ^{5}}}+{\frac{6\,{d}^{5}{b}^{6}}{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}+{\frac{10\,{d}^{3}{a}^{3}{b}^{3}}{3\,{e}^{4} \left ( ex+d \right ) ^{6}}}-{\frac{5\,{d}^{4}{a}^{2}{b}^{4}}{2\,{e}^{5} \left ( ex+d \right ) ^{6}}}+{\frac{{d}^{5}a{b}^{5}}{{e}^{6} \left ( ex+d \right ) ^{6}}}+15\,{\frac{d{a}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{45\,{d}^{2}{a}^{2}{b}^{4}}{2\,{e}^{5} \left ( ex+d \right ) ^{4}}}+15\,{\frac{{d}^{3}a{b}^{5}}{{e}^{6} \left ( ex+d \right ) ^{4}}}+20\,{\frac{d{a}^{2}{b}^{4}}{{e}^{5} \left ( ex+d \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^6*ln(e*x+d)/e^7-1/6/e/(e*x+d)^6*a^6-15/4*b^2/e^3/(e*x+d)^4*a^4-15/4*b^6/e^7/(e
*x+d)^4*d^4-15/2*b^4/e^5/(e*x+d)^2*a^2-15/2*b^6/e^7/(e*x+d)^2*d^2-6*b^5/e^6/(e*x
+d)*a+6*b^6/e^7/(e*x+d)*d-1/6/e^7/(e*x+d)^6*d^6*b^6-20/3*b^3/e^4/(e*x+d)^3*a^3+2
0/3*b^6/e^7/(e*x+d)^3*d^3-12*b^3/e^4/(e*x+d)^5*a^3*d^2+12*b^4/e^5/(e*x+d)^5*a^2*
d^3-6*b^5/e^6/(e*x+d)^5*a*d^4+6*b^2/e^3/(e*x+d)^5*a^4*d-20*b^5/e^6/(e*x+d)^3*a*d
^2+15*b^5/e^6/(e*x+d)^2*d*a+1/e^2/(e*x+d)^6*d*a^5*b-5/2/e^3/(e*x+d)^6*d^2*b^2*a^
4-6/5*b/e^2/(e*x+d)^5*a^5+6/5*b^6/e^7/(e*x+d)^5*d^5+10/3/e^4/(e*x+d)^6*d^3*a^3*b
^3-5/2/e^5/(e*x+d)^6*d^4*a^2*b^4+1/e^6/(e*x+d)^6*d^5*a*b^5+15*b^3/e^4/(e*x+d)^4*
a^3*d-45/2*b^4/e^5/(e*x+d)^4*d^2*a^2+15*b^5/e^6/(e*x+d)^4*a*d^3+20*b^4/e^5/(e*x+
d)^3*a^2*d

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Maxima [A]  time = 0.706621, size = 562, normalized size = 3.37 \[ \frac{147 \, b^{6} d^{6} - 60 \, a b^{5} d^{5} e - 30 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6} + 360 \,{\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 450 \,{\left (3 \, b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} - a^{2} b^{4} e^{6}\right )} x^{4} + 200 \,{\left (11 \, b^{6} d^{3} e^{3} - 6 \, a b^{5} d^{2} e^{4} - 3 \, a^{2} b^{4} d e^{5} - 2 \, a^{3} b^{3} e^{6}\right )} x^{3} + 75 \,{\left (25 \, b^{6} d^{4} e^{2} - 12 \, a b^{5} d^{3} e^{3} - 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} - 3 \, a^{4} b^{2} e^{6}\right )} x^{2} + 6 \,{\left (137 \, b^{6} d^{5} e - 60 \, a b^{5} d^{4} e^{2} - 30 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 15 \, a^{4} b^{2} d e^{5} - 12 \, a^{5} b e^{6}\right )} x}{60 \,{\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} + \frac{b^{6} \log \left (e x + d\right )}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*(147*b^6*d^6 - 60*a*b^5*d^5*e - 30*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 - 1
5*a^4*b^2*d^2*e^4 - 12*a^5*b*d*e^5 - 10*a^6*e^6 + 360*(b^6*d*e^5 - a*b^5*e^6)*x^
5 + 450*(3*b^6*d^2*e^4 - 2*a*b^5*d*e^5 - a^2*b^4*e^6)*x^4 + 200*(11*b^6*d^3*e^3
- 6*a*b^5*d^2*e^4 - 3*a^2*b^4*d*e^5 - 2*a^3*b^3*e^6)*x^3 + 75*(25*b^6*d^4*e^2 -
12*a*b^5*d^3*e^3 - 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 - 3*a^4*b^2*e^6)*x^2 + 6*
(137*b^6*d^5*e - 60*a*b^5*d^4*e^2 - 30*a^2*b^4*d^3*e^3 - 20*a^3*b^3*d^2*e^4 - 15
*a^4*b^2*d*e^5 - 12*a^5*b*e^6)*x)/(e^13*x^6 + 6*d*e^12*x^5 + 15*d^2*e^11*x^4 + 2
0*d^3*e^10*x^3 + 15*d^4*e^9*x^2 + 6*d^5*e^8*x + d^6*e^7) + b^6*log(e*x + d)/e^7

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Fricas [A]  time = 0.205577, size = 664, normalized size = 3.98 \[ \frac{147 \, b^{6} d^{6} - 60 \, a b^{5} d^{5} e - 30 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6} + 360 \,{\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 450 \,{\left (3 \, b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} - a^{2} b^{4} e^{6}\right )} x^{4} + 200 \,{\left (11 \, b^{6} d^{3} e^{3} - 6 \, a b^{5} d^{2} e^{4} - 3 \, a^{2} b^{4} d e^{5} - 2 \, a^{3} b^{3} e^{6}\right )} x^{3} + 75 \,{\left (25 \, b^{6} d^{4} e^{2} - 12 \, a b^{5} d^{3} e^{3} - 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} - 3 \, a^{4} b^{2} e^{6}\right )} x^{2} + 6 \,{\left (137 \, b^{6} d^{5} e - 60 \, a b^{5} d^{4} e^{2} - 30 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 15 \, a^{4} b^{2} d e^{5} - 12 \, a^{5} b e^{6}\right )} x + 60 \,{\left (b^{6} e^{6} x^{6} + 6 \, b^{6} d e^{5} x^{5} + 15 \, b^{6} d^{2} e^{4} x^{4} + 20 \, b^{6} d^{3} e^{3} x^{3} + 15 \, b^{6} d^{4} e^{2} x^{2} + 6 \, b^{6} d^{5} e x + b^{6} d^{6}\right )} \log \left (e x + d\right )}{60 \,{\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*(147*b^6*d^6 - 60*a*b^5*d^5*e - 30*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 - 1
5*a^4*b^2*d^2*e^4 - 12*a^5*b*d*e^5 - 10*a^6*e^6 + 360*(b^6*d*e^5 - a*b^5*e^6)*x^
5 + 450*(3*b^6*d^2*e^4 - 2*a*b^5*d*e^5 - a^2*b^4*e^6)*x^4 + 200*(11*b^6*d^3*e^3
- 6*a*b^5*d^2*e^4 - 3*a^2*b^4*d*e^5 - 2*a^3*b^3*e^6)*x^3 + 75*(25*b^6*d^4*e^2 -
12*a*b^5*d^3*e^3 - 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 - 3*a^4*b^2*e^6)*x^2 + 6*
(137*b^6*d^5*e - 60*a*b^5*d^4*e^2 - 30*a^2*b^4*d^3*e^3 - 20*a^3*b^3*d^2*e^4 - 15
*a^4*b^2*d*e^5 - 12*a^5*b*e^6)*x + 60*(b^6*e^6*x^6 + 6*b^6*d*e^5*x^5 + 15*b^6*d^
2*e^4*x^4 + 20*b^6*d^3*e^3*x^3 + 15*b^6*d^4*e^2*x^2 + 6*b^6*d^5*e*x + b^6*d^6)*l
og(e*x + d))/(e^13*x^6 + 6*d*e^12*x^5 + 15*d^2*e^11*x^4 + 20*d^3*e^10*x^3 + 15*d
^4*e^9*x^2 + 6*d^5*e^8*x + d^6*e^7)

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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((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.212664, size = 458, normalized size = 2.74 \[ b^{6} e^{\left (-7\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{{\left (360 \,{\left (b^{6} d e^{4} - a b^{5} e^{5}\right )} x^{5} + 450 \,{\left (3 \, b^{6} d^{2} e^{3} - 2 \, a b^{5} d e^{4} - a^{2} b^{4} e^{5}\right )} x^{4} + 200 \,{\left (11 \, b^{6} d^{3} e^{2} - 6 \, a b^{5} d^{2} e^{3} - 3 \, a^{2} b^{4} d e^{4} - 2 \, a^{3} b^{3} e^{5}\right )} x^{3} + 75 \,{\left (25 \, b^{6} d^{4} e - 12 \, a b^{5} d^{3} e^{2} - 6 \, a^{2} b^{4} d^{2} e^{3} - 4 \, a^{3} b^{3} d e^{4} - 3 \, a^{4} b^{2} e^{5}\right )} x^{2} + 6 \,{\left (137 \, b^{6} d^{5} - 60 \, a b^{5} d^{4} e - 30 \, a^{2} b^{4} d^{3} e^{2} - 20 \, a^{3} b^{3} d^{2} e^{3} - 15 \, a^{4} b^{2} d e^{4} - 12 \, a^{5} b e^{5}\right )} x +{\left (147 \, b^{6} d^{6} - 60 \, a b^{5} d^{5} e - 30 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6}\right )} e^{\left (-1\right )}\right )} e^{\left (-6\right )}}{60 \,{\left (x e + d\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^6*e^(-7)*ln(abs(x*e + d)) + 1/60*(360*(b^6*d*e^4 - a*b^5*e^5)*x^5 + 450*(3*b^6
*d^2*e^3 - 2*a*b^5*d*e^4 - a^2*b^4*e^5)*x^4 + 200*(11*b^6*d^3*e^2 - 6*a*b^5*d^2*
e^3 - 3*a^2*b^4*d*e^4 - 2*a^3*b^3*e^5)*x^3 + 75*(25*b^6*d^4*e - 12*a*b^5*d^3*e^2
- 6*a^2*b^4*d^2*e^3 - 4*a^3*b^3*d*e^4 - 3*a^4*b^2*e^5)*x^2 + 6*(137*b^6*d^5 - 6
0*a*b^5*d^4*e - 30*a^2*b^4*d^3*e^2 - 20*a^3*b^3*d^2*e^3 - 15*a^4*b^2*d*e^4 - 12*
a^5*b*e^5)*x + (147*b^6*d^6 - 60*a*b^5*d^5*e - 30*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d
^3*e^3 - 15*a^4*b^2*d^2*e^4 - 12*a^5*b*d*e^5 - 10*a^6*e^6)*e^(-1))*e^(-6)/(x*e +
d)^6