3.13.27 \(\int \frac {(d+e x)^8}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=208 \[ \frac {4 e^7 (a+b x)^2 (b d-a e)}{b^9}+\frac {56 e^5 (b d-a e)^3 \log (a+b x)}{b^9}-\frac {70 e^4 (b d-a e)^4}{b^9 (a+b x)}-\frac {28 e^3 (b d-a e)^5}{b^9 (a+b x)^2}-\frac {28 e^2 (b d-a e)^6}{3 b^9 (a+b x)^3}-\frac {2 e (b d-a e)^7}{b^9 (a+b x)^4}-\frac {(b d-a e)^8}{5 b^9 (a+b x)^5}+\frac {e^8 (a+b x)^3}{3 b^9}+\frac {28 e^6 x (b d-a e)^2}{b^8} \]

________________________________________________________________________________________

Rubi [A]  time = 0.31, antiderivative size = 208, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {4 e^7 (a+b x)^2 (b d-a e)}{b^9}+\frac {28 e^6 x (b d-a e)^2}{b^8}-\frac {70 e^4 (b d-a e)^4}{b^9 (a+b x)}-\frac {28 e^3 (b d-a e)^5}{b^9 (a+b x)^2}-\frac {28 e^2 (b d-a e)^6}{3 b^9 (a+b x)^3}+\frac {56 e^5 (b d-a e)^3 \log (a+b x)}{b^9}-\frac {2 e (b d-a e)^7}{b^9 (a+b x)^4}-\frac {(b d-a e)^8}{5 b^9 (a+b x)^5}+\frac {e^8 (a+b x)^3}{3 b^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^8/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(28*e^6*(b*d - a*e)^2*x)/b^8 - (b*d - a*e)^8/(5*b^9*(a + b*x)^5) - (2*e*(b*d - a*e)^7)/(b^9*(a + b*x)^4) - (28
*e^2*(b*d - a*e)^6)/(3*b^9*(a + b*x)^3) - (28*e^3*(b*d - a*e)^5)/(b^9*(a + b*x)^2) - (70*e^4*(b*d - a*e)^4)/(b
^9*(a + b*x)) + (4*e^7*(b*d - a*e)*(a + b*x)^2)/b^9 + (e^8*(a + b*x)^3)/(3*b^9) + (56*e^5*(b*d - a*e)^3*Log[a
+ b*x])/b^9

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)^8}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(d+e x)^8}{(a+b x)^6} \, dx\\ &=\int \left (\frac {28 e^6 (b d-a e)^2}{b^8}+\frac {(b d-a e)^8}{b^8 (a+b x)^6}+\frac {8 e (b d-a e)^7}{b^8 (a+b x)^5}+\frac {28 e^2 (b d-a e)^6}{b^8 (a+b x)^4}+\frac {56 e^3 (b d-a e)^5}{b^8 (a+b x)^3}+\frac {70 e^4 (b d-a e)^4}{b^8 (a+b x)^2}+\frac {56 e^5 (b d-a e)^3}{b^8 (a+b x)}+\frac {8 e^7 (b d-a e) (a+b x)}{b^8}+\frac {e^8 (a+b x)^2}{b^8}\right ) \, dx\\ &=\frac {28 e^6 (b d-a e)^2 x}{b^8}-\frac {(b d-a e)^8}{5 b^9 (a+b x)^5}-\frac {2 e (b d-a e)^7}{b^9 (a+b x)^4}-\frac {28 e^2 (b d-a e)^6}{3 b^9 (a+b x)^3}-\frac {28 e^3 (b d-a e)^5}{b^9 (a+b x)^2}-\frac {70 e^4 (b d-a e)^4}{b^9 (a+b x)}+\frac {4 e^7 (b d-a e) (a+b x)^2}{b^9}+\frac {e^8 (a+b x)^3}{3 b^9}+\frac {56 e^5 (b d-a e)^3 \log (a+b x)}{b^9}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.16, size = 195, normalized size = 0.94 \begin {gather*} \frac {15 b e^6 x \left (21 a^2 e^2-48 a b d e+28 b^2 d^2\right )+15 b^2 e^7 x^2 (4 b d-3 a e)+840 e^5 (b d-a e)^3 \log (a+b x)-\frac {1050 e^4 (b d-a e)^4}{a+b x}+\frac {420 e^3 (a e-b d)^5}{(a+b x)^2}-\frac {140 e^2 (b d-a e)^6}{(a+b x)^3}+\frac {30 e (a e-b d)^7}{(a+b x)^4}-\frac {3 (b d-a e)^8}{(a+b x)^5}+5 b^3 e^8 x^3}{15 b^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^8/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(15*b*e^6*(28*b^2*d^2 - 48*a*b*d*e + 21*a^2*e^2)*x + 15*b^2*e^7*(4*b*d - 3*a*e)*x^2 + 5*b^3*e^8*x^3 - (3*(b*d
- a*e)^8)/(a + b*x)^5 + (30*e*(-(b*d) + a*e)^7)/(a + b*x)^4 - (140*e^2*(b*d - a*e)^6)/(a + b*x)^3 + (420*e^3*(
-(b*d) + a*e)^5)/(a + b*x)^2 - (1050*e^4*(b*d - a*e)^4)/(a + b*x) + 840*e^5*(b*d - a*e)^3*Log[a + b*x])/(15*b^
9)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^8}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^8/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

IntegrateAlgebraic[(d + e*x)^8/(a^2 + 2*a*b*x + b^2*x^2)^3, x]

________________________________________________________________________________________

fricas [B]  time = 0.39, size = 943, normalized size = 4.53 \begin {gather*} \frac {5 \, b^{8} e^{8} x^{8} - 3 \, b^{8} d^{8} - 6 \, a b^{7} d^{7} e - 14 \, a^{2} b^{6} d^{6} e^{2} - 42 \, a^{3} b^{5} d^{5} e^{3} - 210 \, a^{4} b^{4} d^{4} e^{4} + 1918 \, a^{5} b^{3} d^{3} e^{5} - 3654 \, a^{6} b^{2} d^{2} e^{6} + 2754 \, a^{7} b d e^{7} - 743 \, a^{8} e^{8} + 20 \, {\left (3 \, b^{8} d e^{7} - a b^{7} e^{8}\right )} x^{7} + 140 \, {\left (3 \, b^{8} d^{2} e^{6} - 3 \, a b^{7} d e^{7} + a^{2} b^{6} e^{8}\right )} x^{6} + 25 \, {\left (84 \, a b^{7} d^{2} e^{6} - 120 \, a^{2} b^{6} d e^{7} + 47 \, a^{3} b^{5} e^{8}\right )} x^{5} - 25 \, {\left (42 \, b^{8} d^{4} e^{4} - 168 \, a b^{7} d^{3} e^{5} + 84 \, a^{2} b^{6} d^{2} e^{6} + 96 \, a^{3} b^{5} d e^{7} - 67 \, a^{4} b^{4} e^{8}\right )} x^{4} - 10 \, {\left (42 \, b^{8} d^{5} e^{3} + 210 \, a b^{7} d^{4} e^{4} - 1260 \, a^{2} b^{6} d^{3} e^{5} + 1680 \, a^{3} b^{5} d^{2} e^{6} - 780 \, a^{4} b^{4} d e^{7} + 85 \, a^{5} b^{3} e^{8}\right )} x^{3} - 10 \, {\left (14 \, b^{8} d^{6} e^{2} + 42 \, a b^{7} d^{5} e^{3} + 210 \, a^{2} b^{6} d^{4} e^{4} - 1540 \, a^{3} b^{5} d^{3} e^{5} + 2520 \, a^{4} b^{4} d^{2} e^{6} - 1620 \, a^{5} b^{3} d e^{7} + 365 \, a^{6} b^{2} e^{8}\right )} x^{2} - 5 \, {\left (6 \, b^{8} d^{7} e + 14 \, a b^{7} d^{6} e^{2} + 42 \, a^{2} b^{6} d^{5} e^{3} + 210 \, a^{3} b^{5} d^{4} e^{4} - 1750 \, a^{4} b^{4} d^{3} e^{5} + 3150 \, a^{5} b^{3} d^{2} e^{6} - 2250 \, a^{6} b^{2} d e^{7} + 575 \, a^{7} b e^{8}\right )} x + 840 \, {\left (a^{5} b^{3} d^{3} e^{5} - 3 \, a^{6} b^{2} d^{2} e^{6} + 3 \, a^{7} b d e^{7} - a^{8} e^{8} + {\left (b^{8} d^{3} e^{5} - 3 \, a b^{7} d^{2} e^{6} + 3 \, a^{2} b^{6} d e^{7} - a^{3} b^{5} e^{8}\right )} x^{5} + 5 \, {\left (a b^{7} d^{3} e^{5} - 3 \, a^{2} b^{6} d^{2} e^{6} + 3 \, a^{3} b^{5} d e^{7} - a^{4} b^{4} e^{8}\right )} x^{4} + 10 \, {\left (a^{2} b^{6} d^{3} e^{5} - 3 \, a^{3} b^{5} d^{2} e^{6} + 3 \, a^{4} b^{4} d e^{7} - a^{5} b^{3} e^{8}\right )} x^{3} + 10 \, {\left (a^{3} b^{5} d^{3} e^{5} - 3 \, a^{4} b^{4} d^{2} e^{6} + 3 \, a^{5} b^{3} d e^{7} - a^{6} b^{2} e^{8}\right )} x^{2} + 5 \, {\left (a^{4} b^{4} d^{3} e^{5} - 3 \, a^{5} b^{3} d^{2} e^{6} + 3 \, a^{6} b^{2} d e^{7} - a^{7} b e^{8}\right )} x\right )} \log \left (b x + a\right )}{15 \, {\left (b^{14} x^{5} + 5 \, a b^{13} x^{4} + 10 \, a^{2} b^{12} x^{3} + 10 \, a^{3} b^{11} x^{2} + 5 \, a^{4} b^{10} x + a^{5} b^{9}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^8/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

1/15*(5*b^8*e^8*x^8 - 3*b^8*d^8 - 6*a*b^7*d^7*e - 14*a^2*b^6*d^6*e^2 - 42*a^3*b^5*d^5*e^3 - 210*a^4*b^4*d^4*e^
4 + 1918*a^5*b^3*d^3*e^5 - 3654*a^6*b^2*d^2*e^6 + 2754*a^7*b*d*e^7 - 743*a^8*e^8 + 20*(3*b^8*d*e^7 - a*b^7*e^8
)*x^7 + 140*(3*b^8*d^2*e^6 - 3*a*b^7*d*e^7 + a^2*b^6*e^8)*x^6 + 25*(84*a*b^7*d^2*e^6 - 120*a^2*b^6*d*e^7 + 47*
a^3*b^5*e^8)*x^5 - 25*(42*b^8*d^4*e^4 - 168*a*b^7*d^3*e^5 + 84*a^2*b^6*d^2*e^6 + 96*a^3*b^5*d*e^7 - 67*a^4*b^4
*e^8)*x^4 - 10*(42*b^8*d^5*e^3 + 210*a*b^7*d^4*e^4 - 1260*a^2*b^6*d^3*e^5 + 1680*a^3*b^5*d^2*e^6 - 780*a^4*b^4
*d*e^7 + 85*a^5*b^3*e^8)*x^3 - 10*(14*b^8*d^6*e^2 + 42*a*b^7*d^5*e^3 + 210*a^2*b^6*d^4*e^4 - 1540*a^3*b^5*d^3*
e^5 + 2520*a^4*b^4*d^2*e^6 - 1620*a^5*b^3*d*e^7 + 365*a^6*b^2*e^8)*x^2 - 5*(6*b^8*d^7*e + 14*a*b^7*d^6*e^2 + 4
2*a^2*b^6*d^5*e^3 + 210*a^3*b^5*d^4*e^4 - 1750*a^4*b^4*d^3*e^5 + 3150*a^5*b^3*d^2*e^6 - 2250*a^6*b^2*d*e^7 + 5
75*a^7*b*e^8)*x + 840*(a^5*b^3*d^3*e^5 - 3*a^6*b^2*d^2*e^6 + 3*a^7*b*d*e^7 - a^8*e^8 + (b^8*d^3*e^5 - 3*a*b^7*
d^2*e^6 + 3*a^2*b^6*d*e^7 - a^3*b^5*e^8)*x^5 + 5*(a*b^7*d^3*e^5 - 3*a^2*b^6*d^2*e^6 + 3*a^3*b^5*d*e^7 - a^4*b^
4*e^8)*x^4 + 10*(a^2*b^6*d^3*e^5 - 3*a^3*b^5*d^2*e^6 + 3*a^4*b^4*d*e^7 - a^5*b^3*e^8)*x^3 + 10*(a^3*b^5*d^3*e^
5 - 3*a^4*b^4*d^2*e^6 + 3*a^5*b^3*d*e^7 - a^6*b^2*e^8)*x^2 + 5*(a^4*b^4*d^3*e^5 - 3*a^5*b^3*d^2*e^6 + 3*a^6*b^
2*d*e^7 - a^7*b*e^8)*x)*log(b*x + a))/(b^14*x^5 + 5*a*b^13*x^4 + 10*a^2*b^12*x^3 + 10*a^3*b^11*x^2 + 5*a^4*b^1
0*x + a^5*b^9)

________________________________________________________________________________________

giac [B]  time = 0.16, size = 546, normalized size = 2.62 \begin {gather*} \frac {56 \, {\left (b^{3} d^{3} e^{5} - 3 \, a b^{2} d^{2} e^{6} + 3 \, a^{2} b d e^{7} - a^{3} e^{8}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{9}} - \frac {3 \, b^{8} d^{8} + 6 \, a b^{7} d^{7} e + 14 \, a^{2} b^{6} d^{6} e^{2} + 42 \, a^{3} b^{5} d^{5} e^{3} + 210 \, a^{4} b^{4} d^{4} e^{4} - 1918 \, a^{5} b^{3} d^{3} e^{5} + 3654 \, a^{6} b^{2} d^{2} e^{6} - 2754 \, a^{7} b d e^{7} + 743 \, a^{8} e^{8} + 1050 \, {\left (b^{8} d^{4} e^{4} - 4 \, a b^{7} d^{3} e^{5} + 6 \, a^{2} b^{6} d^{2} e^{6} - 4 \, a^{3} b^{5} d e^{7} + a^{4} b^{4} e^{8}\right )} x^{4} + 420 \, {\left (b^{8} d^{5} e^{3} + 5 \, a b^{7} d^{4} e^{4} - 30 \, a^{2} b^{6} d^{3} e^{5} + 50 \, a^{3} b^{5} d^{2} e^{6} - 35 \, a^{4} b^{4} d e^{7} + 9 \, a^{5} b^{3} e^{8}\right )} x^{3} + 140 \, {\left (b^{8} d^{6} e^{2} + 3 \, a b^{7} d^{5} e^{3} + 15 \, a^{2} b^{6} d^{4} e^{4} - 110 \, a^{3} b^{5} d^{3} e^{5} + 195 \, a^{4} b^{4} d^{2} e^{6} - 141 \, a^{5} b^{3} d e^{7} + 37 \, a^{6} b^{2} e^{8}\right )} x^{2} + 10 \, {\left (3 \, b^{8} d^{7} e + 7 \, a b^{7} d^{6} e^{2} + 21 \, a^{2} b^{6} d^{5} e^{3} + 105 \, a^{3} b^{5} d^{4} e^{4} - 875 \, a^{4} b^{4} d^{3} e^{5} + 1617 \, a^{5} b^{3} d^{2} e^{6} - 1197 \, a^{6} b^{2} d e^{7} + 319 \, a^{7} b e^{8}\right )} x}{15 \, {\left (b x + a\right )}^{5} b^{9}} + \frac {b^{12} x^{3} e^{8} + 12 \, b^{12} d x^{2} e^{7} + 84 \, b^{12} d^{2} x e^{6} - 9 \, a b^{11} x^{2} e^{8} - 144 \, a b^{11} d x e^{7} + 63 \, a^{2} b^{10} x e^{8}}{3 \, b^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^8/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

56*(b^3*d^3*e^5 - 3*a*b^2*d^2*e^6 + 3*a^2*b*d*e^7 - a^3*e^8)*log(abs(b*x + a))/b^9 - 1/15*(3*b^8*d^8 + 6*a*b^7
*d^7*e + 14*a^2*b^6*d^6*e^2 + 42*a^3*b^5*d^5*e^3 + 210*a^4*b^4*d^4*e^4 - 1918*a^5*b^3*d^3*e^5 + 3654*a^6*b^2*d
^2*e^6 - 2754*a^7*b*d*e^7 + 743*a^8*e^8 + 1050*(b^8*d^4*e^4 - 4*a*b^7*d^3*e^5 + 6*a^2*b^6*d^2*e^6 - 4*a^3*b^5*
d*e^7 + a^4*b^4*e^8)*x^4 + 420*(b^8*d^5*e^3 + 5*a*b^7*d^4*e^4 - 30*a^2*b^6*d^3*e^5 + 50*a^3*b^5*d^2*e^6 - 35*a
^4*b^4*d*e^7 + 9*a^5*b^3*e^8)*x^3 + 140*(b^8*d^6*e^2 + 3*a*b^7*d^5*e^3 + 15*a^2*b^6*d^4*e^4 - 110*a^3*b^5*d^3*
e^5 + 195*a^4*b^4*d^2*e^6 - 141*a^5*b^3*d*e^7 + 37*a^6*b^2*e^8)*x^2 + 10*(3*b^8*d^7*e + 7*a*b^7*d^6*e^2 + 21*a
^2*b^6*d^5*e^3 + 105*a^3*b^5*d^4*e^4 - 875*a^4*b^4*d^3*e^5 + 1617*a^5*b^3*d^2*e^6 - 1197*a^6*b^2*d*e^7 + 319*a
^7*b*e^8)*x)/((b*x + a)^5*b^9) + 1/3*(b^12*x^3*e^8 + 12*b^12*d*x^2*e^7 + 84*b^12*d^2*x*e^6 - 9*a*b^11*x^2*e^8
- 144*a*b^11*d*x*e^7 + 63*a^2*b^10*x*e^8)/b^18

________________________________________________________________________________________

maple [B]  time = 0.06, size = 820, normalized size = 3.94 \begin {gather*} -\frac {a^{8} e^{8}}{5 \left (b x +a \right )^{5} b^{9}}+\frac {8 a^{7} d \,e^{7}}{5 \left (b x +a \right )^{5} b^{8}}-\frac {28 a^{6} d^{2} e^{6}}{5 \left (b x +a \right )^{5} b^{7}}+\frac {56 a^{5} d^{3} e^{5}}{5 \left (b x +a \right )^{5} b^{6}}-\frac {14 a^{4} d^{4} e^{4}}{\left (b x +a \right )^{5} b^{5}}+\frac {56 a^{3} d^{5} e^{3}}{5 \left (b x +a \right )^{5} b^{4}}-\frac {28 a^{2} d^{6} e^{2}}{5 \left (b x +a \right )^{5} b^{3}}+\frac {8 a \,d^{7} e}{5 \left (b x +a \right )^{5} b^{2}}-\frac {d^{8}}{5 \left (b x +a \right )^{5} b}+\frac {2 a^{7} e^{8}}{\left (b x +a \right )^{4} b^{9}}-\frac {14 a^{6} d \,e^{7}}{\left (b x +a \right )^{4} b^{8}}+\frac {42 a^{5} d^{2} e^{6}}{\left (b x +a \right )^{4} b^{7}}-\frac {70 a^{4} d^{3} e^{5}}{\left (b x +a \right )^{4} b^{6}}+\frac {70 a^{3} d^{4} e^{4}}{\left (b x +a \right )^{4} b^{5}}-\frac {42 a^{2} d^{5} e^{3}}{\left (b x +a \right )^{4} b^{4}}+\frac {14 a \,d^{6} e^{2}}{\left (b x +a \right )^{4} b^{3}}-\frac {2 d^{7} e}{\left (b x +a \right )^{4} b^{2}}-\frac {28 a^{6} e^{8}}{3 \left (b x +a \right )^{3} b^{9}}+\frac {56 a^{5} d \,e^{7}}{\left (b x +a \right )^{3} b^{8}}-\frac {140 a^{4} d^{2} e^{6}}{\left (b x +a \right )^{3} b^{7}}+\frac {560 a^{3} d^{3} e^{5}}{3 \left (b x +a \right )^{3} b^{6}}-\frac {140 a^{2} d^{4} e^{4}}{\left (b x +a \right )^{3} b^{5}}+\frac {56 a \,d^{5} e^{3}}{\left (b x +a \right )^{3} b^{4}}-\frac {28 d^{6} e^{2}}{3 \left (b x +a \right )^{3} b^{3}}+\frac {e^{8} x^{3}}{3 b^{6}}+\frac {28 a^{5} e^{8}}{\left (b x +a \right )^{2} b^{9}}-\frac {140 a^{4} d \,e^{7}}{\left (b x +a \right )^{2} b^{8}}+\frac {280 a^{3} d^{2} e^{6}}{\left (b x +a \right )^{2} b^{7}}-\frac {280 a^{2} d^{3} e^{5}}{\left (b x +a \right )^{2} b^{6}}+\frac {140 a \,d^{4} e^{4}}{\left (b x +a \right )^{2} b^{5}}-\frac {3 a \,e^{8} x^{2}}{b^{7}}-\frac {28 d^{5} e^{3}}{\left (b x +a \right )^{2} b^{4}}+\frac {4 d \,e^{7} x^{2}}{b^{6}}-\frac {70 a^{4} e^{8}}{\left (b x +a \right ) b^{9}}+\frac {280 a^{3} d \,e^{7}}{\left (b x +a \right ) b^{8}}-\frac {56 a^{3} e^{8} \ln \left (b x +a \right )}{b^{9}}-\frac {420 a^{2} d^{2} e^{6}}{\left (b x +a \right ) b^{7}}+\frac {168 a^{2} d \,e^{7} \ln \left (b x +a \right )}{b^{8}}+\frac {21 a^{2} e^{8} x}{b^{8}}+\frac {280 a \,d^{3} e^{5}}{\left (b x +a \right ) b^{6}}-\frac {168 a \,d^{2} e^{6} \ln \left (b x +a \right )}{b^{7}}-\frac {48 a d \,e^{7} x}{b^{7}}-\frac {70 d^{4} e^{4}}{\left (b x +a \right ) b^{5}}+\frac {56 d^{3} e^{5} \ln \left (b x +a \right )}{b^{6}}+\frac {28 d^{2} e^{6} x}{b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^8/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

-1/5/b/(b*x+a)^5*d^8+1/3*e^8/b^6*x^3-48*e^7/b^7*a*d*x+8/5/b^8/(b*x+a)^5*a^7*d*e^7+280/b^7*e^6/(b*x+a)^2*a^3*d^
2-280/b^6*e^5/(b*x+a)^2*a^2*d^3+140/b^5*e^4/(b*x+a)^2*a*d^4+168/b^8*e^7*ln(b*x+a)*a^2*d-168/b^7*e^6*ln(b*x+a)*
a*d^2+280/b^8*e^7/(b*x+a)*d*a^3-420/b^7*e^6/(b*x+a)*a^2*d^2+280/b^6*e^5/(b*x+a)*a*d^3-140/b^8*e^7/(b*x+a)^2*a^
4*d-28/3/b^3*e^2/(b*x+a)^3*d^6-1/5/b^9/(b*x+a)^5*a^8*e^8-3*e^8/b^7*x^2*a+4*e^7/b^6*x^2*d+21*e^8/b^8*a^2*x+28*e
^6/b^6*d^2*x-56/b^9*e^8*ln(b*x+a)*a^3+56/b^6*e^5*ln(b*x+a)*d^3-70/b^9*e^8/(b*x+a)*a^4-70/b^5*e^4/(b*x+a)*d^4+2
8/b^9*e^8/(b*x+a)^2*a^5-28/b^4*e^3/(b*x+a)^2*d^5+2/b^9*e^8/(b*x+a)^4*a^7-2/b^2*e/(b*x+a)^4*d^7-28/3/b^9*e^8/(b
*x+a)^3*a^6+14/b^3*e^2/(b*x+a)^4*a*d^6+56/b^8*e^7/(b*x+a)^3*d*a^5-140/b^7*e^6/(b*x+a)^3*d^2*a^4+560/3/b^6*e^5/
(b*x+a)^3*d^3*a^3-140/b^5*e^4/(b*x+a)^3*d^4*a^2+56/b^4*e^3/(b*x+a)^3*d^5*a-28/5/b^7/(b*x+a)^5*a^6*d^2*e^6+56/5
/b^6/(b*x+a)^5*a^5*d^3*e^5-14/b^5/(b*x+a)^5*a^4*d^4*e^4+56/5/b^4/(b*x+a)^5*a^3*d^5*e^3-28/5/b^3/(b*x+a)^5*a^2*
d^6*e^2+8/5/b^2/(b*x+a)^5*a*d^7*e-14/b^8*e^7/(b*x+a)^4*a^6*d+42/b^7*e^6/(b*x+a)^4*a^5*d^2-70/b^6*e^5/(b*x+a)^4
*a^4*d^3+70/b^5*e^4/(b*x+a)^4*a^3*d^4-42/b^4*e^3/(b*x+a)^4*a^2*d^5

________________________________________________________________________________________

maxima [B]  time = 1.97, size = 626, normalized size = 3.01 \begin {gather*} -\frac {3 \, b^{8} d^{8} + 6 \, a b^{7} d^{7} e + 14 \, a^{2} b^{6} d^{6} e^{2} + 42 \, a^{3} b^{5} d^{5} e^{3} + 210 \, a^{4} b^{4} d^{4} e^{4} - 1918 \, a^{5} b^{3} d^{3} e^{5} + 3654 \, a^{6} b^{2} d^{2} e^{6} - 2754 \, a^{7} b d e^{7} + 743 \, a^{8} e^{8} + 1050 \, {\left (b^{8} d^{4} e^{4} - 4 \, a b^{7} d^{3} e^{5} + 6 \, a^{2} b^{6} d^{2} e^{6} - 4 \, a^{3} b^{5} d e^{7} + a^{4} b^{4} e^{8}\right )} x^{4} + 420 \, {\left (b^{8} d^{5} e^{3} + 5 \, a b^{7} d^{4} e^{4} - 30 \, a^{2} b^{6} d^{3} e^{5} + 50 \, a^{3} b^{5} d^{2} e^{6} - 35 \, a^{4} b^{4} d e^{7} + 9 \, a^{5} b^{3} e^{8}\right )} x^{3} + 140 \, {\left (b^{8} d^{6} e^{2} + 3 \, a b^{7} d^{5} e^{3} + 15 \, a^{2} b^{6} d^{4} e^{4} - 110 \, a^{3} b^{5} d^{3} e^{5} + 195 \, a^{4} b^{4} d^{2} e^{6} - 141 \, a^{5} b^{3} d e^{7} + 37 \, a^{6} b^{2} e^{8}\right )} x^{2} + 10 \, {\left (3 \, b^{8} d^{7} e + 7 \, a b^{7} d^{6} e^{2} + 21 \, a^{2} b^{6} d^{5} e^{3} + 105 \, a^{3} b^{5} d^{4} e^{4} - 875 \, a^{4} b^{4} d^{3} e^{5} + 1617 \, a^{5} b^{3} d^{2} e^{6} - 1197 \, a^{6} b^{2} d e^{7} + 319 \, a^{7} b e^{8}\right )} x}{15 \, {\left (b^{14} x^{5} + 5 \, a b^{13} x^{4} + 10 \, a^{2} b^{12} x^{3} + 10 \, a^{3} b^{11} x^{2} + 5 \, a^{4} b^{10} x + a^{5} b^{9}\right )}} + \frac {b^{2} e^{8} x^{3} + 3 \, {\left (4 \, b^{2} d e^{7} - 3 \, a b e^{8}\right )} x^{2} + 3 \, {\left (28 \, b^{2} d^{2} e^{6} - 48 \, a b d e^{7} + 21 \, a^{2} e^{8}\right )} x}{3 \, b^{8}} + \frac {56 \, {\left (b^{3} d^{3} e^{5} - 3 \, a b^{2} d^{2} e^{6} + 3 \, a^{2} b d e^{7} - a^{3} e^{8}\right )} \log \left (b x + a\right )}{b^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^8/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

-1/15*(3*b^8*d^8 + 6*a*b^7*d^7*e + 14*a^2*b^6*d^6*e^2 + 42*a^3*b^5*d^5*e^3 + 210*a^4*b^4*d^4*e^4 - 1918*a^5*b^
3*d^3*e^5 + 3654*a^6*b^2*d^2*e^6 - 2754*a^7*b*d*e^7 + 743*a^8*e^8 + 1050*(b^8*d^4*e^4 - 4*a*b^7*d^3*e^5 + 6*a^
2*b^6*d^2*e^6 - 4*a^3*b^5*d*e^7 + a^4*b^4*e^8)*x^4 + 420*(b^8*d^5*e^3 + 5*a*b^7*d^4*e^4 - 30*a^2*b^6*d^3*e^5 +
50*a^3*b^5*d^2*e^6 - 35*a^4*b^4*d*e^7 + 9*a^5*b^3*e^8)*x^3 + 140*(b^8*d^6*e^2 + 3*a*b^7*d^5*e^3 + 15*a^2*b^6*
d^4*e^4 - 110*a^3*b^5*d^3*e^5 + 195*a^4*b^4*d^2*e^6 - 141*a^5*b^3*d*e^7 + 37*a^6*b^2*e^8)*x^2 + 10*(3*b^8*d^7*
e + 7*a*b^7*d^6*e^2 + 21*a^2*b^6*d^5*e^3 + 105*a^3*b^5*d^4*e^4 - 875*a^4*b^4*d^3*e^5 + 1617*a^5*b^3*d^2*e^6 -
1197*a^6*b^2*d*e^7 + 319*a^7*b*e^8)*x)/(b^14*x^5 + 5*a*b^13*x^4 + 10*a^2*b^12*x^3 + 10*a^3*b^11*x^2 + 5*a^4*b^
10*x + a^5*b^9) + 1/3*(b^2*e^8*x^3 + 3*(4*b^2*d*e^7 - 3*a*b*e^8)*x^2 + 3*(28*b^2*d^2*e^6 - 48*a*b*d*e^7 + 21*a
^2*e^8)*x)/b^8 + 56*(b^3*d^3*e^5 - 3*a*b^2*d^2*e^6 + 3*a^2*b*d*e^7 - a^3*e^8)*log(b*x + a)/b^9

________________________________________________________________________________________

mupad [B]  time = 0.20, size = 644, normalized size = 3.10 \begin {gather*} x\,\left (\frac {6\,a\,\left (\frac {6\,a\,e^8}{b^7}-\frac {8\,d\,e^7}{b^6}\right )}{b}-\frac {15\,a^2\,e^8}{b^8}+\frac {28\,d^2\,e^6}{b^6}\right )-\frac {x^4\,\left (70\,a^4\,b^3\,e^8-280\,a^3\,b^4\,d\,e^7+420\,a^2\,b^5\,d^2\,e^6-280\,a\,b^6\,d^3\,e^5+70\,b^7\,d^4\,e^4\right )+\frac {743\,a^8\,e^8-2754\,a^7\,b\,d\,e^7+3654\,a^6\,b^2\,d^2\,e^6-1918\,a^5\,b^3\,d^3\,e^5+210\,a^4\,b^4\,d^4\,e^4+42\,a^3\,b^5\,d^5\,e^3+14\,a^2\,b^6\,d^6\,e^2+6\,a\,b^7\,d^7\,e+3\,b^8\,d^8}{15\,b}+x\,\left (\frac {638\,a^7\,e^8}{3}-798\,a^6\,b\,d\,e^7+1078\,a^5\,b^2\,d^2\,e^6-\frac {1750\,a^4\,b^3\,d^3\,e^5}{3}+70\,a^3\,b^4\,d^4\,e^4+14\,a^2\,b^5\,d^5\,e^3+\frac {14\,a\,b^6\,d^6\,e^2}{3}+2\,b^7\,d^7\,e\right )+x^3\,\left (252\,a^5\,b^2\,e^8-980\,a^4\,b^3\,d\,e^7+1400\,a^3\,b^4\,d^2\,e^6-840\,a^2\,b^5\,d^3\,e^5+140\,a\,b^6\,d^4\,e^4+28\,b^7\,d^5\,e^3\right )+x^2\,\left (\frac {1036\,a^6\,b\,e^8}{3}-1316\,a^5\,b^2\,d\,e^7+1820\,a^4\,b^3\,d^2\,e^6-\frac {3080\,a^3\,b^4\,d^3\,e^5}{3}+140\,a^2\,b^5\,d^4\,e^4+28\,a\,b^6\,d^5\,e^3+\frac {28\,b^7\,d^6\,e^2}{3}\right )}{a^5\,b^8+5\,a^4\,b^9\,x+10\,a^3\,b^{10}\,x^2+10\,a^2\,b^{11}\,x^3+5\,a\,b^{12}\,x^4+b^{13}\,x^5}-x^2\,\left (\frac {3\,a\,e^8}{b^7}-\frac {4\,d\,e^7}{b^6}\right )-\frac {\ln \left (a+b\,x\right )\,\left (56\,a^3\,e^8-168\,a^2\,b\,d\,e^7+168\,a\,b^2\,d^2\,e^6-56\,b^3\,d^3\,e^5\right )}{b^9}+\frac {e^8\,x^3}{3\,b^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^8/(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

x*((6*a*((6*a*e^8)/b^7 - (8*d*e^7)/b^6))/b - (15*a^2*e^8)/b^8 + (28*d^2*e^6)/b^6) - (x^4*(70*a^4*b^3*e^8 + 70*
b^7*d^4*e^4 - 280*a*b^6*d^3*e^5 - 280*a^3*b^4*d*e^7 + 420*a^2*b^5*d^2*e^6) + (743*a^8*e^8 + 3*b^8*d^8 + 14*a^2
*b^6*d^6*e^2 + 42*a^3*b^5*d^5*e^3 + 210*a^4*b^4*d^4*e^4 - 1918*a^5*b^3*d^3*e^5 + 3654*a^6*b^2*d^2*e^6 + 6*a*b^
7*d^7*e - 2754*a^7*b*d*e^7)/(15*b) + x*((638*a^7*e^8)/3 + 2*b^7*d^7*e + (14*a*b^6*d^6*e^2)/3 + 14*a^2*b^5*d^5*
e^3 + 70*a^3*b^4*d^4*e^4 - (1750*a^4*b^3*d^3*e^5)/3 + 1078*a^5*b^2*d^2*e^6 - 798*a^6*b*d*e^7) + x^3*(252*a^5*b
^2*e^8 + 28*b^7*d^5*e^3 + 140*a*b^6*d^4*e^4 - 980*a^4*b^3*d*e^7 - 840*a^2*b^5*d^3*e^5 + 1400*a^3*b^4*d^2*e^6)
+ x^2*((1036*a^6*b*e^8)/3 + (28*b^7*d^6*e^2)/3 + 28*a*b^6*d^5*e^3 - 1316*a^5*b^2*d*e^7 + 140*a^2*b^5*d^4*e^4 -
(3080*a^3*b^4*d^3*e^5)/3 + 1820*a^4*b^3*d^2*e^6))/(a^5*b^8 + b^13*x^5 + 5*a^4*b^9*x + 5*a*b^12*x^4 + 10*a^3*b
^10*x^2 + 10*a^2*b^11*x^3) - x^2*((3*a*e^8)/b^7 - (4*d*e^7)/b^6) - (log(a + b*x)*(56*a^3*e^8 - 56*b^3*d^3*e^5
+ 168*a*b^2*d^2*e^6 - 168*a^2*b*d*e^7))/b^9 + (e^8*x^3)/(3*b^6)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**8/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

________________________________________________________________________________________