3.1526 \(\int \frac{(d+e x)^7}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=181 \[ \frac{e^6 x (7 b d-6 a e)}{b^7}-\frac{35 e^4 (b d-a e)^3}{b^8 (a+b x)}-\frac{35 e^3 (b d-a e)^4}{2 b^8 (a+b x)^2}-\frac{7 e^2 (b d-a e)^5}{b^8 (a+b x)^3}+\frac{21 e^5 (b d-a e)^2 \log (a+b x)}{b^8}-\frac{7 e (b d-a e)^6}{4 b^8 (a+b x)^4}-\frac{(b d-a e)^7}{5 b^8 (a+b x)^5}+\frac{e^7 x^2}{2 b^6} \]

[Out]

(e^6*(7*b*d - 6*a*e)*x)/b^7 + (e^7*x^2)/(2*b^6) - (b*d - a*e)^7/(5*b^8*(a + b*x)^5) - (7*e*(b*d - a*e)^6)/(4*b
^8*(a + b*x)^4) - (7*e^2*(b*d - a*e)^5)/(b^8*(a + b*x)^3) - (35*e^3*(b*d - a*e)^4)/(2*b^8*(a + b*x)^2) - (35*e
^4*(b*d - a*e)^3)/(b^8*(a + b*x)) + (21*e^5*(b*d - a*e)^2*Log[a + b*x])/b^8

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Rubi [A]  time = 0.234766, antiderivative size = 181, 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} \[ \frac{e^6 x (7 b d-6 a e)}{b^7}-\frac{35 e^4 (b d-a e)^3}{b^8 (a+b x)}-\frac{35 e^3 (b d-a e)^4}{2 b^8 (a+b x)^2}-\frac{7 e^2 (b d-a e)^5}{b^8 (a+b x)^3}+\frac{21 e^5 (b d-a e)^2 \log (a+b x)}{b^8}-\frac{7 e (b d-a e)^6}{4 b^8 (a+b x)^4}-\frac{(b d-a e)^7}{5 b^8 (a+b x)^5}+\frac{e^7 x^2}{2 b^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e^6*(7*b*d - 6*a*e)*x)/b^7 + (e^7*x^2)/(2*b^6) - (b*d - a*e)^7/(5*b^8*(a + b*x)^5) - (7*e*(b*d - a*e)^6)/(4*b
^8*(a + b*x)^4) - (7*e^2*(b*d - a*e)^5)/(b^8*(a + b*x)^3) - (35*e^3*(b*d - a*e)^4)/(2*b^8*(a + b*x)^2) - (35*e
^4*(b*d - a*e)^3)/(b^8*(a + b*x)) + (21*e^5*(b*d - a*e)^2*Log[a + b*x])/b^8

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)^7}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{(d+e x)^7}{(a+b x)^6} \, dx\\ &=\int \left (\frac{e^6 (7 b d-6 a e)}{b^7}+\frac{e^7 x}{b^6}+\frac{(b d-a e)^7}{b^7 (a+b x)^6}+\frac{7 e (b d-a e)^6}{b^7 (a+b x)^5}+\frac{21 e^2 (b d-a e)^5}{b^7 (a+b x)^4}+\frac{35 e^3 (b d-a e)^4}{b^7 (a+b x)^3}+\frac{35 e^4 (b d-a e)^3}{b^7 (a+b x)^2}+\frac{21 e^5 (b d-a e)^2}{b^7 (a+b x)}\right ) \, dx\\ &=\frac{e^6 (7 b d-6 a e) x}{b^7}+\frac{e^7 x^2}{2 b^6}-\frac{(b d-a e)^7}{5 b^8 (a+b x)^5}-\frac{7 e (b d-a e)^6}{4 b^8 (a+b x)^4}-\frac{7 e^2 (b d-a e)^5}{b^8 (a+b x)^3}-\frac{35 e^3 (b d-a e)^4}{2 b^8 (a+b x)^2}-\frac{35 e^4 (b d-a e)^3}{b^8 (a+b x)}+\frac{21 e^5 (b d-a e)^2 \log (a+b x)}{b^8}\\ \end{align*}

Mathematica [B]  time = 0.148749, size = 389, normalized size = 2.15 \[ \frac{-a^2 b^5 e^2 \left (1400 d^3 e^2 x^2-6300 d^2 e^3 x^3+175 d^4 e x+14 d^5+700 d e^4 x^4+500 e^5 x^5\right )-5 a^3 b^4 e^3 \left (-1540 d^2 e^2 x^2+140 d^3 e x+7 d^4+1120 d e^3 x^3+80 e^4 x^4\right )+5 a^4 b^3 e^4 \left (875 d^2 e x-28 d^3-1680 d e^2 x^2+260 e^3 x^3\right )+a^5 b^2 e^5 \left (959 d^2-5250 d e x+2700 e^2 x^2\right )+3 a^6 b e^6 (625 e x-406 d)+459 a^7 e^7-7 a b^6 e \left (50 d^4 e^2 x^2+200 d^3 e^3 x^3-300 d^2 e^4 x^4+10 d^5 e x+d^6-100 d e^5 x^5+10 e^6 x^6\right )+420 e^5 (a+b x)^5 (b d-a e)^2 \log (a+b x)+b^7 \left (-\left (140 d^5 e^2 x^2+350 d^4 e^3 x^3+700 d^3 e^4 x^4+35 d^6 e x+4 d^7-140 d e^6 x^6-10 e^7 x^7\right )\right )}{20 b^8 (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(459*a^7*e^7 + 3*a^6*b*e^6*(-406*d + 625*e*x) + a^5*b^2*e^5*(959*d^2 - 5250*d*e*x + 2700*e^2*x^2) + 5*a^4*b^3*
e^4*(-28*d^3 + 875*d^2*e*x - 1680*d*e^2*x^2 + 260*e^3*x^3) - 5*a^3*b^4*e^3*(7*d^4 + 140*d^3*e*x - 1540*d^2*e^2
*x^2 + 1120*d*e^3*x^3 + 80*e^4*x^4) - a^2*b^5*e^2*(14*d^5 + 175*d^4*e*x + 1400*d^3*e^2*x^2 - 6300*d^2*e^3*x^3
+ 700*d*e^4*x^4 + 500*e^5*x^5) - 7*a*b^6*e*(d^6 + 10*d^5*e*x + 50*d^4*e^2*x^2 + 200*d^3*e^3*x^3 - 300*d^2*e^4*
x^4 - 100*d*e^5*x^5 + 10*e^6*x^6) - b^7*(4*d^7 + 35*d^6*e*x + 140*d^5*e^2*x^2 + 350*d^4*e^3*x^3 + 700*d^3*e^4*
x^4 - 140*d*e^6*x^6 - 10*e^7*x^7) + 420*e^5*(b*d - a*e)^2*(a + b*x)^5*Log[a + b*x])/(20*b^8*(a + b*x)^5)

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Maple [B]  time = 0.056, size = 656, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

21/b^8*e^7*ln(b*x+a)*a^2+21/b^6*e^5*ln(b*x+a)*d^2+35/b^8*e^7/(b*x+a)*a^3-35/b^5*e^4/(b*x+a)*d^3-6*e^7/b^7*a*x+
7*e^6/b^6*x*d-35/2/b^8*e^7/(b*x+a)^2*a^4-35/2/b^4*e^3/(b*x+a)^2*d^4+1/5/b^8/(b*x+a)^5*a^7*e^7+7/b^8*e^7/(b*x+a
)^3*a^5-7/b^3*e^2/(b*x+a)^3*d^5-7/4/b^8*e^7/(b*x+a)^4*a^6-7/4/b^2*e/(b*x+a)^4*d^6-35/b^7*e^6/(b*x+a)^3*a^4*d+7
0/b^6*e^5/(b*x+a)^3*a^3*d^2+1/2*e^7*x^2/b^6-1/5/b/(b*x+a)^5*d^7-42/b^7*e^6*ln(b*x+a)*a*d+35/b^5*e^4/(b*x+a)^4*
a^3*d^3-105/4/b^4*e^3/(b*x+a)^4*a^2*d^4+21/2/b^3*e^2/(b*x+a)^4*a*d^5-7/5/b^7/(b*x+a)^5*a^6*d*e^6+21/5/b^6/(b*x
+a)^5*a^5*d^2*e^5-7/b^5/(b*x+a)^5*a^4*d^3*e^4+7/b^4/(b*x+a)^5*a^3*d^4*e^3-21/5/b^3/(b*x+a)^5*a^2*d^5*e^2+7/5/b
^2/(b*x+a)^5*a*d^6*e+70/b^7*e^6/(b*x+a)^2*a^3*d-105/b^6*e^5/(b*x+a)^2*a^2*d^2+70/b^5*e^4/(b*x+a)^2*a*d^3-105/b
^7*e^6/(b*x+a)*a^2*d+105/b^6*e^5/(b*x+a)*a*d^2-70/b^5*e^4/(b*x+a)^3*a^2*d^3+35/b^4*e^3/(b*x+a)^3*a*d^4+21/2/b^
7*e^6/(b*x+a)^4*a^5*d-105/4/b^6*e^5/(b*x+a)^4*d^2*a^4

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Maxima [B]  time = 1.29536, size = 680, normalized size = 3.76 \begin{align*} -\frac{4 \, b^{7} d^{7} + 7 \, a b^{6} d^{6} e + 14 \, a^{2} b^{5} d^{5} e^{2} + 35 \, a^{3} b^{4} d^{4} e^{3} + 140 \, a^{4} b^{3} d^{3} e^{4} - 959 \, a^{5} b^{2} d^{2} e^{5} + 1218 \, a^{6} b d e^{6} - 459 \, a^{7} e^{7} + 700 \,{\left (b^{7} d^{3} e^{4} - 3 \, a b^{6} d^{2} e^{5} + 3 \, a^{2} b^{5} d e^{6} - a^{3} b^{4} e^{7}\right )} x^{4} + 350 \,{\left (b^{7} d^{4} e^{3} + 4 \, a b^{6} d^{3} e^{4} - 18 \, a^{2} b^{5} d^{2} e^{5} + 20 \, a^{3} b^{4} d e^{6} - 7 \, a^{4} b^{3} e^{7}\right )} x^{3} + 70 \,{\left (2 \, b^{7} d^{5} e^{2} + 5 \, a b^{6} d^{4} e^{3} + 20 \, a^{2} b^{5} d^{3} e^{4} - 110 \, a^{3} b^{4} d^{2} e^{5} + 130 \, a^{4} b^{3} d e^{6} - 47 \, a^{5} b^{2} e^{7}\right )} x^{2} + 35 \,{\left (b^{7} d^{6} e + 2 \, a b^{6} d^{5} e^{2} + 5 \, a^{2} b^{5} d^{4} e^{3} + 20 \, a^{3} b^{4} d^{3} e^{4} - 125 \, a^{4} b^{3} d^{2} e^{5} + 154 \, a^{5} b^{2} d e^{6} - 57 \, a^{6} b e^{7}\right )} x}{20 \,{\left (b^{13} x^{5} + 5 \, a b^{12} x^{4} + 10 \, a^{2} b^{11} x^{3} + 10 \, a^{3} b^{10} x^{2} + 5 \, a^{4} b^{9} x + a^{5} b^{8}\right )}} + \frac{b e^{7} x^{2} + 2 \,{\left (7 \, b d e^{6} - 6 \, a e^{7}\right )} x}{2 \, b^{7}} + \frac{21 \,{\left (b^{2} d^{2} e^{5} - 2 \, a b d e^{6} + a^{2} e^{7}\right )} \log \left (b x + a\right )}{b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*(4*b^7*d^7 + 7*a*b^6*d^6*e + 14*a^2*b^5*d^5*e^2 + 35*a^3*b^4*d^4*e^3 + 140*a^4*b^3*d^3*e^4 - 959*a^5*b^2
*d^2*e^5 + 1218*a^6*b*d*e^6 - 459*a^7*e^7 + 700*(b^7*d^3*e^4 - 3*a*b^6*d^2*e^5 + 3*a^2*b^5*d*e^6 - a^3*b^4*e^7
)*x^4 + 350*(b^7*d^4*e^3 + 4*a*b^6*d^3*e^4 - 18*a^2*b^5*d^2*e^5 + 20*a^3*b^4*d*e^6 - 7*a^4*b^3*e^7)*x^3 + 70*(
2*b^7*d^5*e^2 + 5*a*b^6*d^4*e^3 + 20*a^2*b^5*d^3*e^4 - 110*a^3*b^4*d^2*e^5 + 130*a^4*b^3*d*e^6 - 47*a^5*b^2*e^
7)*x^2 + 35*(b^7*d^6*e + 2*a*b^6*d^5*e^2 + 5*a^2*b^5*d^4*e^3 + 20*a^3*b^4*d^3*e^4 - 125*a^4*b^3*d^2*e^5 + 154*
a^5*b^2*d*e^6 - 57*a^6*b*e^7)*x)/(b^13*x^5 + 5*a*b^12*x^4 + 10*a^2*b^11*x^3 + 10*a^3*b^10*x^2 + 5*a^4*b^9*x +
a^5*b^8) + 1/2*(b*e^7*x^2 + 2*(7*b*d*e^6 - 6*a*e^7)*x)/b^7 + 21*(b^2*d^2*e^5 - 2*a*b*d*e^6 + a^2*e^7)*log(b*x
+ a)/b^8

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Fricas [B]  time = 1.72278, size = 1517, normalized size = 8.38 \begin{align*} \frac{10 \, b^{7} e^{7} x^{7} - 4 \, b^{7} d^{7} - 7 \, a b^{6} d^{6} e - 14 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} - 140 \, a^{4} b^{3} d^{3} e^{4} + 959 \, a^{5} b^{2} d^{2} e^{5} - 1218 \, a^{6} b d e^{6} + 459 \, a^{7} e^{7} + 70 \,{\left (2 \, b^{7} d e^{6} - a b^{6} e^{7}\right )} x^{6} + 100 \,{\left (7 \, a b^{6} d e^{6} - 5 \, a^{2} b^{5} e^{7}\right )} x^{5} - 100 \,{\left (7 \, b^{7} d^{3} e^{4} - 21 \, a b^{6} d^{2} e^{5} + 7 \, a^{2} b^{5} d e^{6} + 4 \, a^{3} b^{4} e^{7}\right )} x^{4} - 50 \,{\left (7 \, b^{7} d^{4} e^{3} + 28 \, a b^{6} d^{3} e^{4} - 126 \, a^{2} b^{5} d^{2} e^{5} + 112 \, a^{3} b^{4} d e^{6} - 26 \, a^{4} b^{3} e^{7}\right )} x^{3} - 10 \,{\left (14 \, b^{7} d^{5} e^{2} + 35 \, a b^{6} d^{4} e^{3} + 140 \, a^{2} b^{5} d^{3} e^{4} - 770 \, a^{3} b^{4} d^{2} e^{5} + 840 \, a^{4} b^{3} d e^{6} - 270 \, a^{5} b^{2} e^{7}\right )} x^{2} - 5 \,{\left (7 \, b^{7} d^{6} e + 14 \, a b^{6} d^{5} e^{2} + 35 \, a^{2} b^{5} d^{4} e^{3} + 140 \, a^{3} b^{4} d^{3} e^{4} - 875 \, a^{4} b^{3} d^{2} e^{5} + 1050 \, a^{5} b^{2} d e^{6} - 375 \, a^{6} b e^{7}\right )} x + 420 \,{\left (a^{5} b^{2} d^{2} e^{5} - 2 \, a^{6} b d e^{6} + a^{7} e^{7} +{\left (b^{7} d^{2} e^{5} - 2 \, a b^{6} d e^{6} + a^{2} b^{5} e^{7}\right )} x^{5} + 5 \,{\left (a b^{6} d^{2} e^{5} - 2 \, a^{2} b^{5} d e^{6} + a^{3} b^{4} e^{7}\right )} x^{4} + 10 \,{\left (a^{2} b^{5} d^{2} e^{5} - 2 \, a^{3} b^{4} d e^{6} + a^{4} b^{3} e^{7}\right )} x^{3} + 10 \,{\left (a^{3} b^{4} d^{2} e^{5} - 2 \, a^{4} b^{3} d e^{6} + a^{5} b^{2} e^{7}\right )} x^{2} + 5 \,{\left (a^{4} b^{3} d^{2} e^{5} - 2 \, a^{5} b^{2} d e^{6} + a^{6} b e^{7}\right )} x\right )} \log \left (b x + a\right )}{20 \,{\left (b^{13} x^{5} + 5 \, a b^{12} x^{4} + 10 \, a^{2} b^{11} x^{3} + 10 \, a^{3} b^{10} x^{2} + 5 \, a^{4} b^{9} x + a^{5} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(10*b^7*e^7*x^7 - 4*b^7*d^7 - 7*a*b^6*d^6*e - 14*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^4*e^3 - 140*a^4*b^3*d^3*e
^4 + 959*a^5*b^2*d^2*e^5 - 1218*a^6*b*d*e^6 + 459*a^7*e^7 + 70*(2*b^7*d*e^6 - a*b^6*e^7)*x^6 + 100*(7*a*b^6*d*
e^6 - 5*a^2*b^5*e^7)*x^5 - 100*(7*b^7*d^3*e^4 - 21*a*b^6*d^2*e^5 + 7*a^2*b^5*d*e^6 + 4*a^3*b^4*e^7)*x^4 - 50*(
7*b^7*d^4*e^3 + 28*a*b^6*d^3*e^4 - 126*a^2*b^5*d^2*e^5 + 112*a^3*b^4*d*e^6 - 26*a^4*b^3*e^7)*x^3 - 10*(14*b^7*
d^5*e^2 + 35*a*b^6*d^4*e^3 + 140*a^2*b^5*d^3*e^4 - 770*a^3*b^4*d^2*e^5 + 840*a^4*b^3*d*e^6 - 270*a^5*b^2*e^7)*
x^2 - 5*(7*b^7*d^6*e + 14*a*b^6*d^5*e^2 + 35*a^2*b^5*d^4*e^3 + 140*a^3*b^4*d^3*e^4 - 875*a^4*b^3*d^2*e^5 + 105
0*a^5*b^2*d*e^6 - 375*a^6*b*e^7)*x + 420*(a^5*b^2*d^2*e^5 - 2*a^6*b*d*e^6 + a^7*e^7 + (b^7*d^2*e^5 - 2*a*b^6*d
*e^6 + a^2*b^5*e^7)*x^5 + 5*(a*b^6*d^2*e^5 - 2*a^2*b^5*d*e^6 + a^3*b^4*e^7)*x^4 + 10*(a^2*b^5*d^2*e^5 - 2*a^3*
b^4*d*e^6 + a^4*b^3*e^7)*x^3 + 10*(a^3*b^4*d^2*e^5 - 2*a^4*b^3*d*e^6 + a^5*b^2*e^7)*x^2 + 5*(a^4*b^3*d^2*e^5 -
2*a^5*b^2*d*e^6 + a^6*b*e^7)*x)*log(b*x + a))/(b^13*x^5 + 5*a*b^12*x^4 + 10*a^2*b^11*x^3 + 10*a^3*b^10*x^2 +
5*a^4*b^9*x + a^5*b^8)

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Sympy [B]  time = 111.638, size = 522, normalized size = 2.88 \begin{align*} \frac{459 a^{7} e^{7} - 1218 a^{6} b d e^{6} + 959 a^{5} b^{2} d^{2} e^{5} - 140 a^{4} b^{3} d^{3} e^{4} - 35 a^{3} b^{4} d^{4} e^{3} - 14 a^{2} b^{5} d^{5} e^{2} - 7 a b^{6} d^{6} e - 4 b^{7} d^{7} + x^{4} \left (700 a^{3} b^{4} e^{7} - 2100 a^{2} b^{5} d e^{6} + 2100 a b^{6} d^{2} e^{5} - 700 b^{7} d^{3} e^{4}\right ) + x^{3} \left (2450 a^{4} b^{3} e^{7} - 7000 a^{3} b^{4} d e^{6} + 6300 a^{2} b^{5} d^{2} e^{5} - 1400 a b^{6} d^{3} e^{4} - 350 b^{7} d^{4} e^{3}\right ) + x^{2} \left (3290 a^{5} b^{2} e^{7} - 9100 a^{4} b^{3} d e^{6} + 7700 a^{3} b^{4} d^{2} e^{5} - 1400 a^{2} b^{5} d^{3} e^{4} - 350 a b^{6} d^{4} e^{3} - 140 b^{7} d^{5} e^{2}\right ) + x \left (1995 a^{6} b e^{7} - 5390 a^{5} b^{2} d e^{6} + 4375 a^{4} b^{3} d^{2} e^{5} - 700 a^{3} b^{4} d^{3} e^{4} - 175 a^{2} b^{5} d^{4} e^{3} - 70 a b^{6} d^{5} e^{2} - 35 b^{7} d^{6} e\right )}{20 a^{5} b^{8} + 100 a^{4} b^{9} x + 200 a^{3} b^{10} x^{2} + 200 a^{2} b^{11} x^{3} + 100 a b^{12} x^{4} + 20 b^{13} x^{5}} + \frac{e^{7} x^{2}}{2 b^{6}} - \frac{x \left (6 a e^{7} - 7 b d e^{6}\right )}{b^{7}} + \frac{21 e^{5} \left (a e - b d\right )^{2} \log{\left (a + b x \right )}}{b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(459*a**7*e**7 - 1218*a**6*b*d*e**6 + 959*a**5*b**2*d**2*e**5 - 140*a**4*b**3*d**3*e**4 - 35*a**3*b**4*d**4*e*
*3 - 14*a**2*b**5*d**5*e**2 - 7*a*b**6*d**6*e - 4*b**7*d**7 + x**4*(700*a**3*b**4*e**7 - 2100*a**2*b**5*d*e**6
+ 2100*a*b**6*d**2*e**5 - 700*b**7*d**3*e**4) + x**3*(2450*a**4*b**3*e**7 - 7000*a**3*b**4*d*e**6 + 6300*a**2
*b**5*d**2*e**5 - 1400*a*b**6*d**3*e**4 - 350*b**7*d**4*e**3) + x**2*(3290*a**5*b**2*e**7 - 9100*a**4*b**3*d*e
**6 + 7700*a**3*b**4*d**2*e**5 - 1400*a**2*b**5*d**3*e**4 - 350*a*b**6*d**4*e**3 - 140*b**7*d**5*e**2) + x*(19
95*a**6*b*e**7 - 5390*a**5*b**2*d*e**6 + 4375*a**4*b**3*d**2*e**5 - 700*a**3*b**4*d**3*e**4 - 175*a**2*b**5*d*
*4*e**3 - 70*a*b**6*d**5*e**2 - 35*b**7*d**6*e))/(20*a**5*b**8 + 100*a**4*b**9*x + 200*a**3*b**10*x**2 + 200*a
**2*b**11*x**3 + 100*a*b**12*x**4 + 20*b**13*x**5) + e**7*x**2/(2*b**6) - x*(6*a*e**7 - 7*b*d*e**6)/b**7 + 21*
e**5*(a*e - b*d)**2*log(a + b*x)/b**8

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Giac [B]  time = 1.1476, size = 583, normalized size = 3.22 \begin{align*} \frac{21 \,{\left (b^{2} d^{2} e^{5} - 2 \, a b d e^{6} + a^{2} e^{7}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{8}} + \frac{b^{6} x^{2} e^{7} + 14 \, b^{6} d x e^{6} - 12 \, a b^{5} x e^{7}}{2 \, b^{12}} - \frac{4 \, b^{7} d^{7} + 7 \, a b^{6} d^{6} e + 14 \, a^{2} b^{5} d^{5} e^{2} + 35 \, a^{3} b^{4} d^{4} e^{3} + 140 \, a^{4} b^{3} d^{3} e^{4} - 959 \, a^{5} b^{2} d^{2} e^{5} + 1218 \, a^{6} b d e^{6} - 459 \, a^{7} e^{7} + 700 \,{\left (b^{7} d^{3} e^{4} - 3 \, a b^{6} d^{2} e^{5} + 3 \, a^{2} b^{5} d e^{6} - a^{3} b^{4} e^{7}\right )} x^{4} + 350 \,{\left (b^{7} d^{4} e^{3} + 4 \, a b^{6} d^{3} e^{4} - 18 \, a^{2} b^{5} d^{2} e^{5} + 20 \, a^{3} b^{4} d e^{6} - 7 \, a^{4} b^{3} e^{7}\right )} x^{3} + 70 \,{\left (2 \, b^{7} d^{5} e^{2} + 5 \, a b^{6} d^{4} e^{3} + 20 \, a^{2} b^{5} d^{3} e^{4} - 110 \, a^{3} b^{4} d^{2} e^{5} + 130 \, a^{4} b^{3} d e^{6} - 47 \, a^{5} b^{2} e^{7}\right )} x^{2} + 35 \,{\left (b^{7} d^{6} e + 2 \, a b^{6} d^{5} e^{2} + 5 \, a^{2} b^{5} d^{4} e^{3} + 20 \, a^{3} b^{4} d^{3} e^{4} - 125 \, a^{4} b^{3} d^{2} e^{5} + 154 \, a^{5} b^{2} d e^{6} - 57 \, a^{6} b e^{7}\right )} x}{20 \,{\left (b x + a\right )}^{5} b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

21*(b^2*d^2*e^5 - 2*a*b*d*e^6 + a^2*e^7)*log(abs(b*x + a))/b^8 + 1/2*(b^6*x^2*e^7 + 14*b^6*d*x*e^6 - 12*a*b^5*
x*e^7)/b^12 - 1/20*(4*b^7*d^7 + 7*a*b^6*d^6*e + 14*a^2*b^5*d^5*e^2 + 35*a^3*b^4*d^4*e^3 + 140*a^4*b^3*d^3*e^4
- 959*a^5*b^2*d^2*e^5 + 1218*a^6*b*d*e^6 - 459*a^7*e^7 + 700*(b^7*d^3*e^4 - 3*a*b^6*d^2*e^5 + 3*a^2*b^5*d*e^6
- a^3*b^4*e^7)*x^4 + 350*(b^7*d^4*e^3 + 4*a*b^6*d^3*e^4 - 18*a^2*b^5*d^2*e^5 + 20*a^3*b^4*d*e^6 - 7*a^4*b^3*e^
7)*x^3 + 70*(2*b^7*d^5*e^2 + 5*a*b^6*d^4*e^3 + 20*a^2*b^5*d^3*e^4 - 110*a^3*b^4*d^2*e^5 + 130*a^4*b^3*d*e^6 -
47*a^5*b^2*e^7)*x^2 + 35*(b^7*d^6*e + 2*a*b^6*d^5*e^2 + 5*a^2*b^5*d^4*e^3 + 20*a^3*b^4*d^3*e^4 - 125*a^4*b^3*d
^2*e^5 + 154*a^5*b^2*d*e^6 - 57*a^6*b*e^7)*x)/((b*x + a)^5*b^8)