3.1515 \(\int \frac{(d+e x)^6}{(a^2+2 a b x+b^2 x^2)^2} \, dx\)

Optimal. Leaf size=156 \[ \frac{3 e^5 (a+b x)^2 (b d-a e)}{b^7}+\frac{15 e^4 x (b d-a e)^2}{b^6}-\frac{15 e^2 (b d-a e)^4}{b^7 (a+b x)}+\frac{20 e^3 (b d-a e)^3 \log (a+b x)}{b^7}-\frac{3 e (b d-a e)^5}{b^7 (a+b x)^2}-\frac{(b d-a e)^6}{3 b^7 (a+b x)^3}+\frac{e^6 (a+b x)^3}{3 b^7} \]

[Out]

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

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Rubi [A]  time = 0.179918, antiderivative size = 156, 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} \[ \frac{3 e^5 (a+b x)^2 (b d-a e)}{b^7}+\frac{15 e^4 x (b d-a e)^2}{b^6}-\frac{15 e^2 (b d-a e)^4}{b^7 (a+b x)}+\frac{20 e^3 (b d-a e)^3 \log (a+b x)}{b^7}-\frac{3 e (b d-a e)^5}{b^7 (a+b x)^2}-\frac{(b d-a e)^6}{3 b^7 (a+b x)^3}+\frac{e^6 (a+b x)^3}{3 b^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.11873, size = 301, normalized size = 1.93 \[ \frac{3 a^2 b^4 e^2 \left (-45 d^2 e^2 x^2+90 d^3 e x-5 d^4-63 d e^3 x^3+5 e^4 x^4\right )+a^3 b^3 e^3 \left (-405 d^2 e x+110 d^3-27 d e^2 x^2+73 e^3 x^3\right )+3 a^4 b^2 e^4 \left (-65 d^2+81 d e x+13 e^2 x^2\right )+3 a^5 b e^5 (47 d-17 e x)-37 a^6 e^6-3 a b^5 e \left (-60 d^3 e^2 x^2-45 d^2 e^3 x^3+15 d^4 e x+d^5+15 d e^4 x^4+e^5 x^5\right )-60 e^3 (a+b x)^3 (a e-b d)^3 \log (a+b x)+b^6 \left (-45 d^4 e^2 x^2+45 d^2 e^4 x^4-9 d^5 e x-d^6+9 d e^5 x^5+e^6 x^6\right )}{3 b^7 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-37*a^6*e^6 + 3*a^5*b*e^5*(47*d - 17*e*x) + 3*a^4*b^2*e^4*(-65*d^2 + 81*d*e*x + 13*e^2*x^2) + a^3*b^3*e^3*(11
0*d^3 - 405*d^2*e*x - 27*d*e^2*x^2 + 73*e^3*x^3) + 3*a^2*b^4*e^2*(-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*b^5*e*(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)
+ b^6*(-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) - 60*e^3*(-(b*d) + a*e)^3*
(a + b*x)^3*Log[a + b*x])/(3*b^7*(a + b*x)^3)

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Maple [B]  time = 0.051, size = 483, normalized size = 3.1 \begin{align*} -15\,{\frac{{e}^{5}{a}^{4}d}{{b}^{6} \left ( bx+a \right ) ^{2}}}+2\,{\frac{a{d}^{5}e}{{b}^{2} \left ( bx+a \right ) ^{3}}}+60\,{\frac{{e}^{5}\ln \left ( bx+a \right ){a}^{2}d}{{b}^{6}}}-60\,{\frac{{e}^{4}\ln \left ( bx+a \right ) a{d}^{2}}{{b}^{5}}}+30\,{\frac{{e}^{4}{a}^{3}{d}^{2}}{{b}^{5} \left ( bx+a \right ) ^{2}}}-30\,{\frac{{a}^{2}{e}^{3}{d}^{3}}{{b}^{4} \left ( bx+a \right ) ^{2}}}+15\,{\frac{a{e}^{2}{d}^{4}}{{b}^{3} \left ( bx+a \right ) ^{2}}}+2\,{\frac{{a}^{5}d{e}^{5}}{{b}^{6} \left ( bx+a \right ) ^{3}}}-5\,{\frac{{d}^{2}{e}^{4}{a}^{4}}{{b}^{5} \left ( bx+a \right ) ^{3}}}+{\frac{20\,{a}^{3}{d}^{3}{e}^{3}}{3\,{b}^{4} \left ( bx+a \right ) ^{3}}}-5\,{\frac{{a}^{2}{d}^{4}{e}^{2}}{{b}^{3} \left ( bx+a \right ) ^{3}}}+60\,{\frac{{a}^{3}{e}^{5}d}{{b}^{6} \left ( bx+a \right ) }}-90\,{\frac{{d}^{2}{e}^{4}{a}^{2}}{{b}^{5} \left ( bx+a \right ) }}+60\,{\frac{a{e}^{3}{d}^{3}}{{b}^{4} \left ( bx+a \right ) }}-2\,{\frac{{e}^{6}{x}^{2}a}{{b}^{5}}}+3\,{\frac{{e}^{5}{x}^{2}d}{{b}^{4}}}+10\,{\frac{{a}^{2}{e}^{6}x}{{b}^{6}}}+15\,{\frac{{d}^{2}{e}^{4}x}{{b}^{4}}}+3\,{\frac{{a}^{5}{e}^{6}}{{b}^{7} \left ( bx+a \right ) ^{2}}}-3\,{\frac{e{d}^{5}}{{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{{e}^{6}{a}^{6}}{3\,{b}^{7} \left ( bx+a \right ) ^{3}}}-20\,{\frac{{e}^{6}\ln \left ( bx+a \right ){a}^{3}}{{b}^{7}}}+20\,{\frac{{e}^{3}\ln \left ( bx+a \right ){d}^{3}}{{b}^{4}}}-15\,{\frac{{e}^{6}{a}^{4}}{{b}^{7} \left ( bx+a \right ) }}-15\,{\frac{{e}^{2}{d}^{4}}{{b}^{3} \left ( bx+a \right ) }}+{\frac{{e}^{6}{x}^{3}}{3\,{b}^{4}}}-{\frac{{d}^{6}}{3\,b \left ( bx+a \right ) ^{3}}}-24\,{\frac{ad{e}^{5}x}{{b}^{5}}} \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)^2,x)

[Out]

-15/b^6*e^5/(b*x+a)^2*a^4*d+2/b^2/(b*x+a)^3*a*d^5*e+60/b^6*e^5*ln(b*x+a)*a^2*d-60/b^5*e^4*ln(b*x+a)*a*d^2+30/b
^5*e^4/(b*x+a)^2*a^3*d^2-30/b^4*e^3/(b*x+a)^2*a^2*d^3+15/b^3*e^2/(b*x+a)^2*a*d^4+2/b^6/(b*x+a)^3*a^5*d*e^5-5/b
^5/(b*x+a)^3*d^2*e^4*a^4+20/3/b^4/(b*x+a)^3*a^3*d^3*e^3-5/b^3/(b*x+a)^3*a^2*d^4*e^2+60/b^6*e^5/(b*x+a)*a^3*d-9
0/b^5*e^4/(b*x+a)*d^2*a^2+60/b^4*e^3/(b*x+a)*a*d^3-2*e^6/b^5*x^2*a+3*e^5/b^4*x^2*d+10*e^6/b^6*a^2*x+15*e^4/b^4
*d^2*x+3/b^7*e^6/(b*x+a)^2*a^5-3/b^2*e/(b*x+a)^2*d^5-1/3/b^7/(b*x+a)^3*e^6*a^6-20/b^7*e^6*ln(b*x+a)*a^3+20/b^4
*e^3*ln(b*x+a)*d^3-15/b^7*e^6/(b*x+a)*a^4-15/b^3*e^2/(b*x+a)*d^4+1/3*e^6/b^4*x^3-1/3/b/(b*x+a)^3*d^6-24*e^5/b^
5*a*d*x

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Maxima [B]  time = 1.25855, size = 505, normalized size = 3.24 \begin{align*} -\frac{b^{6} d^{6} + 3 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 110 \, a^{3} b^{3} d^{3} e^{3} + 195 \, a^{4} b^{2} d^{2} e^{4} - 141 \, a^{5} b d e^{5} + 37 \, a^{6} e^{6} + 45 \,{\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 9 \,{\left (b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} - 30 \, a^{2} b^{4} d^{3} e^{3} + 50 \, a^{3} b^{3} d^{2} e^{4} - 35 \, a^{4} b^{2} d e^{5} + 9 \, a^{5} b e^{6}\right )} x}{3 \,{\left (b^{10} x^{3} + 3 \, a b^{9} x^{2} + 3 \, a^{2} b^{8} x + a^{3} b^{7}\right )}} + \frac{b^{2} e^{6} x^{3} + 3 \,{\left (3 \, b^{2} d e^{5} - 2 \, a b e^{6}\right )} x^{2} + 3 \,{\left (15 \, b^{2} d^{2} e^{4} - 24 \, a b d e^{5} + 10 \, a^{2} e^{6}\right )} x}{3 \, b^{6}} + \frac{20 \,{\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} 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)^2,x, algorithm="maxima")

[Out]

-1/3*(b^6*d^6 + 3*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 110*a^3*b^3*d^3*e^3 + 195*a^4*b^2*d^2*e^4 - 141*a^5*b*d*e
^5 + 37*a^6*e^6 + 45*(b^6*d^4*e^2 - 4*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 +
9*(b^6*d^5*e + 5*a*b^5*d^4*e^2 - 30*a^2*b^4*d^3*e^3 + 50*a^3*b^3*d^2*e^4 - 35*a^4*b^2*d*e^5 + 9*a^5*b*e^6)*x)
/(b^10*x^3 + 3*a*b^9*x^2 + 3*a^2*b^8*x + a^3*b^7) + 1/3*(b^2*e^6*x^3 + 3*(3*b^2*d*e^5 - 2*a*b*e^6)*x^2 + 3*(15
*b^2*d^2*e^4 - 24*a*b*d*e^5 + 10*a^2*e^6)*x)/b^6 + 20*(b^3*d^3*e^3 - 3*a*b^2*d^2*e^4 + 3*a^2*b*d*e^5 - a^3*e^6
)*log(b*x + a)/b^7

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Fricas [B]  time = 1.79495, size = 1161, normalized size = 7.44 \begin{align*} \frac{b^{6} e^{6} x^{6} - b^{6} d^{6} - 3 \, a b^{5} d^{5} e - 15 \, a^{2} b^{4} d^{4} e^{2} + 110 \, a^{3} b^{3} d^{3} e^{3} - 195 \, a^{4} b^{2} d^{2} e^{4} + 141 \, a^{5} b d e^{5} - 37 \, a^{6} e^{6} + 3 \,{\left (3 \, b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 15 \,{\left (3 \, b^{6} d^{2} e^{4} - 3 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} +{\left (135 \, a b^{5} d^{2} e^{4} - 189 \, a^{2} b^{4} d e^{5} + 73 \, a^{3} b^{3} e^{6}\right )} x^{3} - 3 \,{\left (15 \, b^{6} d^{4} e^{2} - 60 \, a b^{5} d^{3} e^{3} + 45 \, a^{2} b^{4} d^{2} e^{4} + 9 \, a^{3} b^{3} d e^{5} - 13 \, a^{4} b^{2} e^{6}\right )} x^{2} - 3 \,{\left (3 \, b^{6} d^{5} e + 15 \, a b^{5} d^{4} e^{2} - 90 \, a^{2} b^{4} d^{3} e^{3} + 135 \, a^{3} b^{3} d^{2} e^{4} - 81 \, a^{4} b^{2} d e^{5} + 17 \, a^{5} b e^{6}\right )} x + 60 \,{\left (a^{3} b^{3} d^{3} e^{3} - 3 \, a^{4} b^{2} d^{2} e^{4} + 3 \, a^{5} b d e^{5} - a^{6} e^{6} +{\left (b^{6} d^{3} e^{3} - 3 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )} x^{3} + 3 \,{\left (a b^{5} d^{3} e^{3} - 3 \, a^{2} b^{4} d^{2} e^{4} + 3 \, a^{3} b^{3} d e^{5} - a^{4} b^{2} e^{6}\right )} x^{2} + 3 \,{\left (a^{2} b^{4} d^{3} e^{3} - 3 \, a^{3} b^{3} d^{2} e^{4} + 3 \, a^{4} b^{2} d e^{5} - a^{5} b e^{6}\right )} x\right )} \log \left (b x + a\right )}{3 \,{\left (b^{10} x^{3} + 3 \, a b^{9} x^{2} + 3 \, a^{2} b^{8} x + a^{3} 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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 5.61823, size = 364, normalized size = 2.33 \begin{align*} - \frac{37 a^{6} e^{6} - 141 a^{5} b d e^{5} + 195 a^{4} b^{2} d^{2} e^{4} - 110 a^{3} b^{3} d^{3} e^{3} + 15 a^{2} b^{4} d^{4} e^{2} + 3 a b^{5} d^{5} e + b^{6} d^{6} + x^{2} \left (45 a^{4} b^{2} e^{6} - 180 a^{3} b^{3} d e^{5} + 270 a^{2} b^{4} d^{2} e^{4} - 180 a b^{5} d^{3} e^{3} + 45 b^{6} d^{4} e^{2}\right ) + x \left (81 a^{5} b e^{6} - 315 a^{4} b^{2} d e^{5} + 450 a^{3} b^{3} d^{2} e^{4} - 270 a^{2} b^{4} d^{3} e^{3} + 45 a b^{5} d^{4} e^{2} + 9 b^{6} d^{5} e\right )}{3 a^{3} b^{7} + 9 a^{2} b^{8} x + 9 a b^{9} x^{2} + 3 b^{10} x^{3}} + \frac{e^{6} x^{3}}{3 b^{4}} - \frac{x^{2} \left (2 a e^{6} - 3 b d e^{5}\right )}{b^{5}} + \frac{x \left (10 a^{2} e^{6} - 24 a b d e^{5} + 15 b^{2} d^{2} e^{4}\right )}{b^{6}} - \frac{20 e^{3} \left (a e - b d\right )^{3} \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)**2,x)

[Out]

-(37*a**6*e**6 - 141*a**5*b*d*e**5 + 195*a**4*b**2*d**2*e**4 - 110*a**3*b**3*d**3*e**3 + 15*a**2*b**4*d**4*e**
2 + 3*a*b**5*d**5*e + b**6*d**6 + x**2*(45*a**4*b**2*e**6 - 180*a**3*b**3*d*e**5 + 270*a**2*b**4*d**2*e**4 - 1
80*a*b**5*d**3*e**3 + 45*b**6*d**4*e**2) + x*(81*a**5*b*e**6 - 315*a**4*b**2*d*e**5 + 450*a**3*b**3*d**2*e**4
- 270*a**2*b**4*d**3*e**3 + 45*a*b**5*d**4*e**2 + 9*b**6*d**5*e))/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2
+ 3*b**10*x**3) + e**6*x**3/(3*b**4) - x**2*(2*a*e**6 - 3*b*d*e**5)/b**5 + x*(10*a**2*e**6 - 24*a*b*d*e**5 +
15*b**2*d**2*e**4)/b**6 - 20*e**3*(a*e - b*d)**3*log(a + b*x)/b**7

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Giac [B]  time = 1.15243, size = 450, normalized size = 2.88 \begin{align*} \frac{20 \,{\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7}} - \frac{b^{6} d^{6} + 3 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 110 \, a^{3} b^{3} d^{3} e^{3} + 195 \, a^{4} b^{2} d^{2} e^{4} - 141 \, a^{5} b d e^{5} + 37 \, a^{6} e^{6} + 45 \,{\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 9 \,{\left (b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} - 30 \, a^{2} b^{4} d^{3} e^{3} + 50 \, a^{3} b^{3} d^{2} e^{4} - 35 \, a^{4} b^{2} d e^{5} + 9 \, a^{5} b e^{6}\right )} x}{3 \,{\left (b x + a\right )}^{3} b^{7}} + \frac{b^{8} x^{3} e^{6} + 9 \, b^{8} d x^{2} e^{5} + 45 \, b^{8} d^{2} x e^{4} - 6 \, a b^{7} x^{2} e^{6} - 72 \, a b^{7} d x e^{5} + 30 \, a^{2} b^{6} x e^{6}}{3 \, b^{12}} \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)^2,x, algorithm="giac")

[Out]

20*(b^3*d^3*e^3 - 3*a*b^2*d^2*e^4 + 3*a^2*b*d*e^5 - a^3*e^6)*log(abs(b*x + a))/b^7 - 1/3*(b^6*d^6 + 3*a*b^5*d^
5*e + 15*a^2*b^4*d^4*e^2 - 110*a^3*b^3*d^3*e^3 + 195*a^4*b^2*d^2*e^4 - 141*a^5*b*d*e^5 + 37*a^6*e^6 + 45*(b^6*
d^4*e^2 - 4*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 9*(b^6*d^5*e + 5*a*b^5*d^
4*e^2 - 30*a^2*b^4*d^3*e^3 + 50*a^3*b^3*d^2*e^4 - 35*a^4*b^2*d*e^5 + 9*a^5*b*e^6)*x)/((b*x + a)^3*b^7) + 1/3*(
b^8*x^3*e^6 + 9*b^8*d*x^2*e^5 + 45*b^8*d^2*x*e^4 - 6*a*b^7*x^2*e^6 - 72*a*b^7*d*x*e^5 + 30*a^2*b^6*x*e^6)/b^12