3.1527 \(\int \frac{(d+e x)^6}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=155 \[ -\frac{15 e^4 (b d-a e)^2}{b^7 (a+b x)}-\frac{10 e^3 (b d-a e)^3}{b^7 (a+b x)^2}-\frac{5 e^2 (b d-a e)^4}{b^7 (a+b x)^3}+\frac{6 e^5 (b d-a e) \log (a+b x)}{b^7}-\frac{3 e (b d-a e)^5}{2 b^7 (a+b x)^4}-\frac{(b d-a e)^6}{5 b^7 (a+b x)^5}+\frac{e^6 x}{b^6} \]

[Out]

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

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Rubi [A]  time = 0.169131, antiderivative size = 155, 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, Rules used = {27, 43} \[ -\frac{15 e^4 (b d-a e)^2}{b^7 (a+b x)}-\frac{10 e^3 (b d-a e)^3}{b^7 (a+b x)^2}-\frac{5 e^2 (b d-a e)^4}{b^7 (a+b x)^3}+\frac{6 e^5 (b d-a e) \log (a+b x)}{b^7}-\frac{3 e (b d-a e)^5}{2 b^7 (a+b x)^4}-\frac{(b d-a e)^6}{5 b^7 (a+b x)^5}+\frac{e^6 x}{b^6} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^6/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{(d+e x)^6}{(a+b x)^6} \, dx\\ &=\int \left (\frac{e^6}{b^6}+\frac{(b d-a e)^6}{b^6 (a+b x)^6}+\frac{6 e (b d-a e)^5}{b^6 (a+b x)^5}+\frac{15 e^2 (b d-a e)^4}{b^6 (a+b x)^4}+\frac{20 e^3 (b d-a e)^3}{b^6 (a+b x)^3}+\frac{15 e^4 (b d-a e)^2}{b^6 (a+b x)^2}+\frac{6 e^5 (b d-a e)}{b^6 (a+b x)}\right ) \, dx\\ &=\frac{e^6 x}{b^6}-\frac{(b d-a e)^6}{5 b^7 (a+b x)^5}-\frac{3 e (b d-a e)^5}{2 b^7 (a+b x)^4}-\frac{5 e^2 (b d-a e)^4}{b^7 (a+b x)^3}-\frac{10 e^3 (b d-a e)^3}{b^7 (a+b x)^2}-\frac{15 e^4 (b d-a e)^2}{b^7 (a+b x)}+\frac{6 e^5 (b d-a e) \log (a+b x)}{b^7}\\ \end{align*}

Mathematica [A]  time = 0.162551, size = 300, normalized size = 1.94 \[ -\frac{5 a^2 b^4 e^2 \left (60 d^2 e^2 x^2+10 d^3 e x+d^4-180 d e^3 x^3+10 e^4 x^4\right )+10 a^3 b^3 e^3 \left (15 d^2 e x+d^3-110 d e^2 x^2+40 e^3 x^3\right )+5 a^4 b^2 e^4 \left (6 d^2-125 d e x+120 e^2 x^2\right )+a^5 b e^5 (375 e x-137 d)+87 a^6 e^6+a b^5 e \left (100 d^3 e^2 x^2+300 d^2 e^3 x^3+25 d^4 e x+3 d^5-300 d e^4 x^4-50 e^5 x^5\right )+60 e^5 (a+b x)^5 (a e-b d) \log (a+b x)+b^6 \left (50 d^4 e^2 x^2+100 d^3 e^3 x^3+150 d^2 e^4 x^4+15 d^5 e x+2 d^6-10 e^6 x^6\right )}{10 b^7 (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^6/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-(87*a^6*e^6 + a^5*b*e^5*(-137*d + 375*e*x) + 5*a^4*b^2*e^4*(6*d^2 - 125*d*e*x + 120*e^2*x^2) + 10*a^3*b^3*e^3
*(d^3 + 15*d^2*e*x - 110*d*e^2*x^2 + 40*e^3*x^3) + 5*a^2*b^4*e^2*(d^4 + 10*d^3*e*x + 60*d^2*e^2*x^2 - 180*d*e^
3*x^3 + 10*e^4*x^4) + a*b^5*e*(3*d^5 + 25*d^4*e*x + 100*d^3*e^2*x^2 + 300*d^2*e^3*x^3 - 300*d*e^4*x^4 - 50*e^5
*x^5) + b^6*(2*d^6 + 15*d^5*e*x + 50*d^4*e^2*x^2 + 100*d^3*e^3*x^3 + 150*d^2*e^4*x^4 - 10*e^6*x^6) + 60*e^5*(-
(b*d) + a*e)*(a + b*x)^5*Log[a + b*x])/(10*b^7*(a + b*x)^5)

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Maple [B]  time = 0.052, size = 508, normalized size = 3.3 \begin{align*} -3\,{\frac{{d}^{2}{e}^{4}{a}^{4}}{{b}^{5} \left ( bx+a \right ) ^{5}}}+4\,{\frac{{a}^{3}{d}^{3}{e}^{3}}{{b}^{4} \left ( bx+a \right ) ^{5}}}-3\,{\frac{{a}^{2}{d}^{4}{e}^{2}}{{b}^{3} \left ( bx+a \right ) ^{5}}}+20\,{\frac{{a}^{3}{e}^{5}d}{{b}^{6} \left ( bx+a \right ) ^{3}}}+30\,{\frac{{e}^{5}ad}{{b}^{6} \left ( bx+a \right ) }}-{\frac{15\,{e}^{5}{a}^{4}d}{2\,{b}^{6} \left ( bx+a \right ) ^{4}}}+15\,{\frac{{e}^{4}{a}^{3}{d}^{2}}{{b}^{5} \left ( bx+a \right ) ^{4}}}-15\,{\frac{{a}^{2}{e}^{3}{d}^{3}}{{b}^{4} \left ( bx+a \right ) ^{4}}}+{\frac{15\,a{e}^{2}{d}^{4}}{2\,{b}^{3} \left ( bx+a \right ) ^{4}}}+{\frac{6\,a{d}^{5}e}{5\,{b}^{2} \left ( bx+a \right ) ^{5}}}-30\,{\frac{{a}^{2}{e}^{5}d}{{b}^{6} \left ( bx+a \right ) ^{2}}}+30\,{\frac{{e}^{4}a{d}^{2}}{{b}^{5} \left ( bx+a \right ) ^{2}}}+{\frac{6\,{a}^{5}d{e}^{5}}{5\,{b}^{6} \left ( bx+a \right ) ^{5}}}-6\,{\frac{{e}^{6}\ln \left ( bx+a \right ) a}{{b}^{7}}}+6\,{\frac{{e}^{5}\ln \left ( bx+a \right ) d}{{b}^{6}}}-15\,{\frac{{e}^{6}{a}^{2}}{{b}^{7} \left ( bx+a \right ) }}-15\,{\frac{{d}^{2}{e}^{4}}{{b}^{5} \left ( bx+a \right ) }}+10\,{\frac{{a}^{3}{e}^{6}}{{b}^{7} \left ( bx+a \right ) ^{2}}}-10\,{\frac{{e}^{3}{d}^{3}}{{b}^{4} \left ( bx+a \right ) ^{2}}}-{\frac{{a}^{6}{e}^{6}}{5\,{b}^{7} \left ( bx+a \right ) ^{5}}}-5\,{\frac{{e}^{6}{a}^{4}}{{b}^{7} \left ( bx+a \right ) ^{3}}}-5\,{\frac{{e}^{2}{d}^{4}}{{b}^{3} \left ( bx+a \right ) ^{3}}}+{\frac{3\,{a}^{5}{e}^{6}}{2\,{b}^{7} \left ( bx+a \right ) ^{4}}}-{\frac{3\,e{d}^{5}}{2\,{b}^{2} \left ( bx+a \right ) ^{4}}}-30\,{\frac{{a}^{2}{e}^{4}{d}^{2}}{{b}^{5} \left ( bx+a \right ) ^{3}}}+20\,{\frac{a{e}^{3}{d}^{3}}{{b}^{4} \left ( bx+a \right ) ^{3}}}-{\frac{{d}^{6}}{5\,b \left ( bx+a \right ) ^{5}}}+{\frac{{e}^{6}x}{{b}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^6/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

-3/b^5/(b*x+a)^5*d^2*e^4*a^4+4/b^4/(b*x+a)^5*a^3*d^3*e^3-3/b^3/(b*x+a)^5*a^2*d^4*e^2+20/b^6*e^5/(b*x+a)^3*a^3*
d+30/b^6*e^5/(b*x+a)*a*d-15/2/b^6*e^5/(b*x+a)^4*a^4*d+15/b^5*e^4/(b*x+a)^4*a^3*d^2-15/b^4*e^3/(b*x+a)^4*a^2*d^
3+15/2/b^3*e^2/(b*x+a)^4*a*d^4+6/5/b^2/(b*x+a)^5*a*d^5*e-30/b^6*e^5/(b*x+a)^2*a^2*d+30/b^5*e^4/(b*x+a)^2*a*d^2
+6/5/b^6/(b*x+a)^5*a^5*d*e^5-6/b^7*e^6*ln(b*x+a)*a+6/b^6*e^5*ln(b*x+a)*d-15/b^7*e^6/(b*x+a)*a^2-15/b^5*e^4/(b*
x+a)*d^2+10/b^7*e^6/(b*x+a)^2*a^3-10/b^4*e^3/(b*x+a)^2*d^3-1/5/b^7/(b*x+a)^5*a^6*e^6-5/b^7*e^6/(b*x+a)^3*a^4-5
/b^3*e^2/(b*x+a)^3*d^4+3/2/b^7*e^6/(b*x+a)^4*a^5-3/2/b^2*e/(b*x+a)^4*d^5-30/b^5*e^4/(b*x+a)^3*a^2*d^2+20/b^4*e
^3/(b*x+a)^3*a*d^3-1/5/b/(b*x+a)^5*d^6+e^6*x/b^6

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Maxima [B]  time = 1.25274, size = 536, normalized size = 3.46 \begin{align*} \frac{e^{6} x}{b^{6}} - \frac{2 \, b^{6} d^{6} + 3 \, a b^{5} d^{5} e + 5 \, a^{2} b^{4} d^{4} e^{2} + 10 \, a^{3} b^{3} d^{3} e^{3} + 30 \, a^{4} b^{2} d^{2} e^{4} - 137 \, a^{5} b d e^{5} + 87 \, a^{6} e^{6} + 150 \,{\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 100 \,{\left (b^{6} d^{3} e^{3} + 3 \, a b^{5} d^{2} e^{4} - 9 \, a^{2} b^{4} d e^{5} + 5 \, a^{3} b^{3} e^{6}\right )} x^{3} + 50 \,{\left (b^{6} d^{4} e^{2} + 2 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 22 \, a^{3} b^{3} d e^{5} + 13 \, a^{4} b^{2} e^{6}\right )} x^{2} + 5 \,{\left (3 \, b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} + 30 \, a^{3} b^{3} d^{2} e^{4} - 125 \, a^{4} b^{2} d e^{5} + 77 \, a^{5} b e^{6}\right )} x}{10 \,{\left (b^{12} x^{5} + 5 \, a b^{11} x^{4} + 10 \, a^{2} b^{10} x^{3} + 10 \, a^{3} b^{9} x^{2} + 5 \, a^{4} b^{8} x + a^{5} b^{7}\right )}} + \frac{6 \,{\left (b d e^{5} - a e^{6}\right )} \log \left (b x + a\right )}{b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

e^6*x/b^6 - 1/10*(2*b^6*d^6 + 3*a*b^5*d^5*e + 5*a^2*b^4*d^4*e^2 + 10*a^3*b^3*d^3*e^3 + 30*a^4*b^2*d^2*e^4 - 13
7*a^5*b*d*e^5 + 87*a^6*e^6 + 150*(b^6*d^2*e^4 - 2*a*b^5*d*e^5 + a^2*b^4*e^6)*x^4 + 100*(b^6*d^3*e^3 + 3*a*b^5*
d^2*e^4 - 9*a^2*b^4*d*e^5 + 5*a^3*b^3*e^6)*x^3 + 50*(b^6*d^4*e^2 + 2*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 - 22*a^
3*b^3*d*e^5 + 13*a^4*b^2*e^6)*x^2 + 5*(3*b^6*d^5*e + 5*a*b^5*d^4*e^2 + 10*a^2*b^4*d^3*e^3 + 30*a^3*b^3*d^2*e^4
- 125*a^4*b^2*d*e^5 + 77*a^5*b*e^6)*x)/(b^12*x^5 + 5*a*b^11*x^4 + 10*a^2*b^10*x^3 + 10*a^3*b^9*x^2 + 5*a^4*b^
8*x + a^5*b^7) + 6*(b*d*e^5 - a*e^6)*log(b*x + a)/b^7

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Fricas [B]  time = 1.92112, size = 1095, normalized size = 7.06 \begin{align*} \frac{10 \, b^{6} e^{6} x^{6} + 50 \, a b^{5} e^{6} x^{5} - 2 \, b^{6} d^{6} - 3 \, a b^{5} d^{5} e - 5 \, a^{2} b^{4} d^{4} e^{2} - 10 \, a^{3} b^{3} d^{3} e^{3} - 30 \, a^{4} b^{2} d^{2} e^{4} + 137 \, a^{5} b d e^{5} - 87 \, a^{6} e^{6} - 50 \,{\left (3 \, b^{6} d^{2} e^{4} - 6 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} - 100 \,{\left (b^{6} d^{3} e^{3} + 3 \, a b^{5} d^{2} e^{4} - 9 \, a^{2} b^{4} d e^{5} + 4 \, a^{3} b^{3} e^{6}\right )} x^{3} - 50 \,{\left (b^{6} d^{4} e^{2} + 2 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 22 \, a^{3} b^{3} d e^{5} + 12 \, a^{4} b^{2} e^{6}\right )} x^{2} - 5 \,{\left (3 \, b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} + 30 \, a^{3} b^{3} d^{2} e^{4} - 125 \, a^{4} b^{2} d e^{5} + 75 \, a^{5} b e^{6}\right )} x + 60 \,{\left (a^{5} b d e^{5} - a^{6} e^{6} +{\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 5 \,{\left (a b^{5} d e^{5} - a^{2} b^{4} e^{6}\right )} x^{4} + 10 \,{\left (a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )} x^{3} + 10 \,{\left (a^{3} b^{3} d e^{5} - a^{4} b^{2} e^{6}\right )} x^{2} + 5 \,{\left (a^{4} b^{2} d e^{5} - a^{5} b e^{6}\right )} x\right )} \log \left (b x + a\right )}{10 \,{\left (b^{12} x^{5} + 5 \, a b^{11} x^{4} + 10 \, a^{2} b^{10} x^{3} + 10 \, a^{3} b^{9} x^{2} + 5 \, a^{4} b^{8} x + a^{5} b^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

1/10*(10*b^6*e^6*x^6 + 50*a*b^5*e^6*x^5 - 2*b^6*d^6 - 3*a*b^5*d^5*e - 5*a^2*b^4*d^4*e^2 - 10*a^3*b^3*d^3*e^3 -
30*a^4*b^2*d^2*e^4 + 137*a^5*b*d*e^5 - 87*a^6*e^6 - 50*(3*b^6*d^2*e^4 - 6*a*b^5*d*e^5 + a^2*b^4*e^6)*x^4 - 10
0*(b^6*d^3*e^3 + 3*a*b^5*d^2*e^4 - 9*a^2*b^4*d*e^5 + 4*a^3*b^3*e^6)*x^3 - 50*(b^6*d^4*e^2 + 2*a*b^5*d^3*e^3 +
6*a^2*b^4*d^2*e^4 - 22*a^3*b^3*d*e^5 + 12*a^4*b^2*e^6)*x^2 - 5*(3*b^6*d^5*e + 5*a*b^5*d^4*e^2 + 10*a^2*b^4*d^3
*e^3 + 30*a^3*b^3*d^2*e^4 - 125*a^4*b^2*d*e^5 + 75*a^5*b*e^6)*x + 60*(a^5*b*d*e^5 - a^6*e^6 + (b^6*d*e^5 - a*b
^5*e^6)*x^5 + 5*(a*b^5*d*e^5 - a^2*b^4*e^6)*x^4 + 10*(a^2*b^4*d*e^5 - a^3*b^3*e^6)*x^3 + 10*(a^3*b^3*d*e^5 - a
^4*b^2*e^6)*x^2 + 5*(a^4*b^2*d*e^5 - a^5*b*e^6)*x)*log(b*x + a))/(b^12*x^5 + 5*a*b^11*x^4 + 10*a^2*b^10*x^3 +
10*a^3*b^9*x^2 + 5*a^4*b^8*x + a^5*b^7)

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Sympy [B]  time = 37.5701, size = 420, normalized size = 2.71 \begin{align*} - \frac{87 a^{6} e^{6} - 137 a^{5} b d e^{5} + 30 a^{4} b^{2} d^{2} e^{4} + 10 a^{3} b^{3} d^{3} e^{3} + 5 a^{2} b^{4} d^{4} e^{2} + 3 a b^{5} d^{5} e + 2 b^{6} d^{6} + x^{4} \left (150 a^{2} b^{4} e^{6} - 300 a b^{5} d e^{5} + 150 b^{6} d^{2} e^{4}\right ) + x^{3} \left (500 a^{3} b^{3} e^{6} - 900 a^{2} b^{4} d e^{5} + 300 a b^{5} d^{2} e^{4} + 100 b^{6} d^{3} e^{3}\right ) + x^{2} \left (650 a^{4} b^{2} e^{6} - 1100 a^{3} b^{3} d e^{5} + 300 a^{2} b^{4} d^{2} e^{4} + 100 a b^{5} d^{3} e^{3} + 50 b^{6} d^{4} e^{2}\right ) + x \left (385 a^{5} b e^{6} - 625 a^{4} b^{2} d e^{5} + 150 a^{3} b^{3} d^{2} e^{4} + 50 a^{2} b^{4} d^{3} e^{3} + 25 a b^{5} d^{4} e^{2} + 15 b^{6} d^{5} e\right )}{10 a^{5} b^{7} + 50 a^{4} b^{8} x + 100 a^{3} b^{9} x^{2} + 100 a^{2} b^{10} x^{3} + 50 a b^{11} x^{4} + 10 b^{12} x^{5}} + \frac{e^{6} x}{b^{6}} - \frac{6 e^{5} \left (a e - b d\right ) \log{\left (a + b x \right )}}{b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**6/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

-(87*a**6*e**6 - 137*a**5*b*d*e**5 + 30*a**4*b**2*d**2*e**4 + 10*a**3*b**3*d**3*e**3 + 5*a**2*b**4*d**4*e**2 +
3*a*b**5*d**5*e + 2*b**6*d**6 + x**4*(150*a**2*b**4*e**6 - 300*a*b**5*d*e**5 + 150*b**6*d**2*e**4) + x**3*(50
0*a**3*b**3*e**6 - 900*a**2*b**4*d*e**5 + 300*a*b**5*d**2*e**4 + 100*b**6*d**3*e**3) + x**2*(650*a**4*b**2*e**
6 - 1100*a**3*b**3*d*e**5 + 300*a**2*b**4*d**2*e**4 + 100*a*b**5*d**3*e**3 + 50*b**6*d**4*e**2) + x*(385*a**5*
b*e**6 - 625*a**4*b**2*d*e**5 + 150*a**3*b**3*d**2*e**4 + 50*a**2*b**4*d**3*e**3 + 25*a*b**5*d**4*e**2 + 15*b*
*6*d**5*e))/(10*a**5*b**7 + 50*a**4*b**8*x + 100*a**3*b**9*x**2 + 100*a**2*b**10*x**3 + 50*a*b**11*x**4 + 10*b
**12*x**5) + e**6*x/b**6 - 6*e**5*(a*e - b*d)*log(a + b*x)/b**7

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Giac [B]  time = 1.15733, size = 443, normalized size = 2.86 \begin{align*} \frac{x e^{6}}{b^{6}} + \frac{6 \,{\left (b d e^{5} - a e^{6}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7}} - \frac{2 \, b^{6} d^{6} + 3 \, a b^{5} d^{5} e + 5 \, a^{2} b^{4} d^{4} e^{2} + 10 \, a^{3} b^{3} d^{3} e^{3} + 30 \, a^{4} b^{2} d^{2} e^{4} - 137 \, a^{5} b d e^{5} + 87 \, a^{6} e^{6} + 150 \,{\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 100 \,{\left (b^{6} d^{3} e^{3} + 3 \, a b^{5} d^{2} e^{4} - 9 \, a^{2} b^{4} d e^{5} + 5 \, a^{3} b^{3} e^{6}\right )} x^{3} + 50 \,{\left (b^{6} d^{4} e^{2} + 2 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 22 \, a^{3} b^{3} d e^{5} + 13 \, a^{4} b^{2} e^{6}\right )} x^{2} + 5 \,{\left (3 \, b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} + 30 \, a^{3} b^{3} d^{2} e^{4} - 125 \, a^{4} b^{2} d e^{5} + 77 \, a^{5} b e^{6}\right )} x}{10 \,{\left (b x + a\right )}^{5} b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

x*e^6/b^6 + 6*(b*d*e^5 - a*e^6)*log(abs(b*x + a))/b^7 - 1/10*(2*b^6*d^6 + 3*a*b^5*d^5*e + 5*a^2*b^4*d^4*e^2 +
10*a^3*b^3*d^3*e^3 + 30*a^4*b^2*d^2*e^4 - 137*a^5*b*d*e^5 + 87*a^6*e^6 + 150*(b^6*d^2*e^4 - 2*a*b^5*d*e^5 + a^
2*b^4*e^6)*x^4 + 100*(b^6*d^3*e^3 + 3*a*b^5*d^2*e^4 - 9*a^2*b^4*d*e^5 + 5*a^3*b^3*e^6)*x^3 + 50*(b^6*d^4*e^2 +
2*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 - 22*a^3*b^3*d*e^5 + 13*a^4*b^2*e^6)*x^2 + 5*(3*b^6*d^5*e + 5*a*b^5*d^4*e
^2 + 10*a^2*b^4*d^3*e^3 + 30*a^3*b^3*d^2*e^4 - 125*a^4*b^2*d*e^5 + 77*a^5*b*e^6)*x)/((b*x + a)^5*b^7)