3.16.25 \(\int \frac {(d+e x)^8}{(a^2+2 a b x+b^2 x^2)^3} \, dx\) [1525]

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

[Out]

28*e^6*(-a*e+b*d)^2*x/b^8-1/5*(-a*e+b*d)^8/b^9/(b*x+a)^5-2*e*(-a*e+b*d)^7/b^9/(b*x+a)^4-28/3*e^2*(-a*e+b*d)^6/
b^9/(b*x+a)^3-28*e^3*(-a*e+b*d)^5/b^9/(b*x+a)^2-70*e^4*(-a*e+b*d)^4/b^9/(b*x+a)+4*e^7*(-a*e+b*d)*(b*x+a)^2/b^9
+1/3*e^8*(b*x+a)^3/b^9+56*e^5*(-a*e+b*d)^3*ln(b*x+a)/b^9

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Rubi [A]
time = 0.23, 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, 45} \begin {gather*} \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} \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 45

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*}

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Mathematica [A]
time = 0.10, size = 195, normalized size = 0.94 \begin {gather*} \frac {15 b e^6 \left (28 b^2 d^2-48 a b d e+21 a^2 e^2\right ) x+15 b^2 e^7 (4 b d-3 a e) x^2+5 b^3 e^8 x^3-\frac {3 (b d-a e)^8}{(a+b x)^5}+\frac {30 e (-b d+a e)^7}{(a+b x)^4}-\frac {140 e^2 (b d-a e)^6}{(a+b x)^3}+\frac {420 e^3 (-b d+a e)^5}{(a+b x)^2}-\frac {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} \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)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(572\) vs. \(2(202)=404\).
time = 0.64, size = 573, normalized size = 2.75

method	result	size
norman	\(\frac {-\frac {1918 a^{8} e^{8}-5754 a^{7} b d \,e^{7}+5754 a^{6} b^{2} d^{2} e^{6}-1918 b^{3} d^{3} e^{5} a^{5}+210 b^{4} d^{4} e^{4} a^{4}+42 b^{5} d^{5} e^{3} a^{3}+14 b^{6} d^{6} e^{2} a^{2}+6 b^{7} d^{7} e a +3 b^{8} d^{8}}{15 b^{9}}+\frac {e^{8} x^{8}}{3 b}-\frac {5 \left (56 a^{4} e^{8}-168 a^{3} b d \,e^{7}+168 a^{2} b^{2} d^{2} e^{6}-56 a \,b^{3} d^{3} e^{5}+14 d^{4} e^{4} b^{4}\right ) x^{4}}{b^{5}}-\frac {2 \left (420 e^{8} a^{5}-1260 a^{4} b d \,e^{7}+1260 a^{3} b^{2} d^{2} e^{6}-420 a^{2} b^{3} d^{3} e^{5}+70 a \,b^{4} d^{4} e^{4}+14 d^{5} e^{3} b^{5}\right ) x^{3}}{b^{6}}-\frac {2 \left (1540 e^{8} a^{6}-4620 a^{5} b d \,e^{7}+4620 a^{4} b^{2} d^{2} e^{6}-1540 a^{3} b^{3} d^{3} e^{5}+210 a^{2} b^{4} d^{4} e^{4}+42 a \,b^{5} d^{5} e^{3}+14 b^{6} d^{6} e^{2}\right ) x^{2}}{3 b^{7}}-\frac {\left (1750 e^{8} a^{7}-5250 a^{6} b d \,e^{7}+5250 a^{5} b^{2} d^{2} e^{6}-1750 a^{4} b^{3} d^{3} e^{5}+210 a^{3} b^{4} d^{4} e^{4}+42 a^{2} b^{5} d^{5} e^{3}+14 a \,b^{6} d^{6} e^{2}+6 d^{7} e \,b^{7}\right ) x}{3 b^{8}}+\frac {28 e^{6} \left (a^{2} e^{2}-3 a b d e +3 b^{2} d^{2}\right ) x^{6}}{3 b^{3}}-\frac {4 e^{7} \left (a e -3 b d \right ) x^{7}}{3 b^{2}}}{\left (b x +a \right )^{5}}-\frac {56 e^{5} \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \ln \left (b x +a \right )}{b^{9}}\)	\(572\)
default	\(\frac {e^{6} \left (\frac {1}{3} b^{2} e^{2} x^{3}-3 a b \,e^{2} x^{2}+4 b^{2} d e \,x^{2}+21 a^{2} e^{2} x -48 a b d e x +28 x \,b^{2} d^{2}\right )}{b^{8}}-\frac {70 e^{4} \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{b^{9} \left (b x +a \right )}+\frac {28 e^{3} \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{b^{9} \left (b x +a \right )^{2}}-\frac {56 e^{5} \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \ln \left (b x +a \right )}{b^{9}}-\frac {28 e^{2} \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}{3 b^{9} \left (b x +a \right )^{3}}-\frac {a^{8} e^{8}-8 a^{7} b d \,e^{7}+28 a^{6} b^{2} d^{2} e^{6}-56 b^{3} d^{3} e^{5} a^{5}+70 b^{4} d^{4} e^{4} a^{4}-56 b^{5} d^{5} e^{3} a^{3}+28 b^{6} d^{6} e^{2} a^{2}-8 b^{7} d^{7} e a +b^{8} d^{8}}{5 b^{9} \left (b x +a \right )^{5}}+\frac {2 e \left (a^{7} e^{7}-7 a^{6} b d \,e^{6}+21 a^{5} b^{2} d^{2} e^{5}-35 b^{3} d^{3} e^{4} a^{4}+35 b^{4} d^{4} e^{3} a^{3}-21 b^{5} d^{5} e^{2} a^{2}+7 b^{6} d^{6} e a -b^{7} d^{7}\right )}{b^{9} \left (b x +a \right )^{4}}\)	\(573\)
risch	\(\frac {e^{8} x^{3}}{3 b^{6}}-\frac {3 e^{8} a \,x^{2}}{b^{7}}+\frac {4 e^{7} d \,x^{2}}{b^{6}}+\frac {21 e^{8} a^{2} x}{b^{8}}-\frac {48 e^{7} a d x}{b^{7}}+\frac {28 e^{6} x \,d^{2}}{b^{6}}+\frac {\left (-70 a^{4} b^{3} e^{8}+280 d \,e^{7} a^{3} b^{4}-420 d^{2} e^{6} a^{2} b^{5}+280 a \,b^{6} d^{3} e^{5}-70 d^{4} e^{4} b^{7}\right ) x^{4}-28 b^{2} e^{3} \left (9 a^{5} e^{5}-35 a^{4} b d \,e^{4}+50 a^{3} b^{2} d^{2} e^{3}-30 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) x^{3}-\frac {28 b \,e^{2} \left (37 a^{6} e^{6}-141 a^{5} b d \,e^{5}+195 a^{4} b^{2} d^{2} e^{4}-110 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}+3 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) x^{2}}{3}-\frac {2 e \left (319 a^{7} e^{7}-1197 a^{6} b d \,e^{6}+1617 a^{5} b^{2} d^{2} e^{5}-875 b^{3} d^{3} e^{4} a^{4}+105 b^{4} d^{4} e^{3} a^{3}+21 b^{5} d^{5} e^{2} a^{2}+7 b^{6} d^{6} e a +3 b^{7} d^{7}\right ) x}{3}-\frac {743 a^{8} e^{8}-2754 a^{7} b d \,e^{7}+3654 a^{6} b^{2} d^{2} e^{6}-1918 b^{3} d^{3} e^{5} a^{5}+210 b^{4} d^{4} e^{4} a^{4}+42 b^{5} d^{5} e^{3} a^{3}+14 b^{6} d^{6} e^{2} a^{2}+6 b^{7} d^{7} e a +3 b^{8} d^{8}}{15 b}}{b^{8} \left (b x +a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2}}-\frac {56 e^{8} \ln \left (b x +a \right ) a^{3}}{b^{9}}+\frac {168 e^{7} \ln \left (b x +a \right ) a^{2} d}{b^{8}}-\frac {168 e^{6} \ln \left (b x +a \right ) a \,d^{2}}{b^{7}}+\frac {56 e^{5} \ln \left (b x +a \right ) d^{3}}{b^{6}}\)	\(608\)

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,method=_RETURNVERBOSE)

[Out]

e^6/b^8*(1/3*b^2*e^2*x^3-3*a*b*e^2*x^2+4*b^2*d*e*x^2+21*a^2*e^2*x-48*a*b*d*e*x+28*x*b^2*d^2)-70/b^9*e^4*(a^4*e
^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)/(b*x+a)+28/b^9*e^3*(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2
*d^2*e^3-10*a^2*b^3*d^3*e^2+5*a*b^4*d^4*e-b^5*d^5)/(b*x+a)^2-56/b^9*e^5*(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b
^3*d^3)*ln(b*x+a)-28/3/b^9*e^2*(a^6*e^6-6*a^5*b*d*e^5+15*a^4*b^2*d^2*e^4-20*a^3*b^3*d^3*e^3+15*a^2*b^4*d^4*e^2
-6*a*b^5*d^5*e+b^6*d^6)/(b*x+a)^3-1/5/b^9*(a^8*e^8-8*a^7*b*d*e^7+28*a^6*b^2*d^2*e^6-56*a^5*b^3*d^3*e^5+70*a^4*
b^4*d^4*e^4-56*a^3*b^5*d^5*e^3+28*a^2*b^6*d^6*e^2-8*a*b^7*d^7*e+b^8*d^8)/(b*x+a)^5+2/b^9*e*(a^7*e^7-7*a^6*b*d*
e^6+21*a^5*b^2*d^2*e^5-35*a^4*b^3*d^3*e^4+35*a^3*b^4*d^4*e^3-21*a^2*b^5*d^5*e^2+7*a*b^6*d^6*e-b^7*d^7)/(b*x+a)
^4

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 586 vs. \(2 (204) = 408\).
time = 0.33, size = 586, normalized size = 2.82 \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} x^{3} e^{8} + 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*x^3*e^8 + 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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 870 vs. \(2 (204) = 408\).
time = 2.05, size = 870, normalized size = 4.18 \begin {gather*} -\frac {3 \, b^{8} d^{8} - {\left (5 \, b^{8} x^{8} - 20 \, a b^{7} x^{7} + 140 \, a^{2} b^{6} x^{6} + 1175 \, a^{3} b^{5} x^{5} + 1675 \, a^{4} b^{4} x^{4} - 850 \, a^{5} b^{3} x^{3} - 3650 \, a^{6} b^{2} x^{2} - 2875 \, a^{7} b x - 743 \, a^{8}\right )} e^{8} - 6 \, {\left (10 \, b^{8} d x^{7} - 70 \, a b^{7} d x^{6} - 500 \, a^{2} b^{6} d x^{5} - 400 \, a^{3} b^{5} d x^{4} + 1300 \, a^{4} b^{4} d x^{3} + 2700 \, a^{5} b^{3} d x^{2} + 1875 \, a^{6} b^{2} d x + 459 \, a^{7} b d\right )} e^{7} - 42 \, {\left (10 \, b^{8} d^{2} x^{6} + 50 \, a b^{7} d^{2} x^{5} - 50 \, a^{2} b^{6} d^{2} x^{4} - 400 \, a^{3} b^{5} d^{2} x^{3} - 600 \, a^{4} b^{4} d^{2} x^{2} - 375 \, a^{5} b^{3} d^{2} x - 87 \, a^{6} b^{2} d^{2}\right )} e^{6} - 14 \, {\left (300 \, a b^{7} d^{3} x^{4} + 900 \, a^{2} b^{6} d^{3} x^{3} + 1100 \, a^{3} b^{5} d^{3} x^{2} + 625 \, a^{4} b^{4} d^{3} x + 137 \, a^{5} b^{3} d^{3}\right )} e^{5} + 210 \, {\left (5 \, b^{8} d^{4} x^{4} + 10 \, a b^{7} d^{4} x^{3} + 10 \, a^{2} b^{6} d^{4} x^{2} + 5 \, a^{3} b^{5} d^{4} x + a^{4} b^{4} d^{4}\right )} e^{4} + 42 \, {\left (10 \, b^{8} d^{5} x^{3} + 10 \, a b^{7} d^{5} x^{2} + 5 \, a^{2} b^{6} d^{5} x + a^{3} b^{5} d^{5}\right )} e^{3} + 14 \, {\left (10 \, b^{8} d^{6} x^{2} + 5 \, a b^{7} d^{6} x + a^{2} b^{6} d^{6}\right )} e^{2} + 6 \, {\left (5 \, b^{8} d^{7} x + a b^{7} d^{7}\right )} e + 840 \, {\left ({\left (a^{3} b^{5} x^{5} + 5 \, a^{4} b^{4} x^{4} + 10 \, a^{5} b^{3} x^{3} + 10 \, a^{6} b^{2} x^{2} + 5 \, a^{7} b x + a^{8}\right )} e^{8} - 3 \, {\left (a^{2} b^{6} d x^{5} + 5 \, a^{3} b^{5} d x^{4} + 10 \, a^{4} b^{4} d x^{3} + 10 \, a^{5} b^{3} d x^{2} + 5 \, a^{6} b^{2} d x + a^{7} b d\right )} e^{7} + 3 \, {\left (a b^{7} d^{2} x^{5} + 5 \, a^{2} b^{6} d^{2} x^{4} + 10 \, a^{3} b^{5} d^{2} x^{3} + 10 \, a^{4} b^{4} d^{2} x^{2} + 5 \, a^{5} b^{3} d^{2} x + a^{6} b^{2} d^{2}\right )} e^{6} - {\left (b^{8} d^{3} x^{5} + 5 \, a b^{7} d^{3} x^{4} + 10 \, a^{2} b^{6} d^{3} x^{3} + 10 \, a^{3} b^{5} d^{3} x^{2} + 5 \, a^{4} b^{4} d^{3} x + a^{5} b^{3} d^{3}\right )} e^{5}\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*(3*b^8*d^8 - (5*b^8*x^8 - 20*a*b^7*x^7 + 140*a^2*b^6*x^6 + 1175*a^3*b^5*x^5 + 1675*a^4*b^4*x^4 - 850*a^5
*b^3*x^3 - 3650*a^6*b^2*x^2 - 2875*a^7*b*x - 743*a^8)*e^8 - 6*(10*b^8*d*x^7 - 70*a*b^7*d*x^6 - 500*a^2*b^6*d*x
^5 - 400*a^3*b^5*d*x^4 + 1300*a^4*b^4*d*x^3 + 2700*a^5*b^3*d*x^2 + 1875*a^6*b^2*d*x + 459*a^7*b*d)*e^7 - 42*(1
0*b^8*d^2*x^6 + 50*a*b^7*d^2*x^5 - 50*a^2*b^6*d^2*x^4 - 400*a^3*b^5*d^2*x^3 - 600*a^4*b^4*d^2*x^2 - 375*a^5*b^
3*d^2*x - 87*a^6*b^2*d^2)*e^6 - 14*(300*a*b^7*d^3*x^4 + 900*a^2*b^6*d^3*x^3 + 1100*a^3*b^5*d^3*x^2 + 625*a^4*b
^4*d^3*x + 137*a^5*b^3*d^3)*e^5 + 210*(5*b^8*d^4*x^4 + 10*a*b^7*d^4*x^3 + 10*a^2*b^6*d^4*x^2 + 5*a^3*b^5*d^4*x
+ a^4*b^4*d^4)*e^4 + 42*(10*b^8*d^5*x^3 + 10*a*b^7*d^5*x^2 + 5*a^2*b^6*d^5*x + a^3*b^5*d^5)*e^3 + 14*(10*b^8*
d^6*x^2 + 5*a*b^7*d^6*x + a^2*b^6*d^6)*e^2 + 6*(5*b^8*d^7*x + a*b^7*d^7)*e + 840*((a^3*b^5*x^5 + 5*a^4*b^4*x^4
+ 10*a^5*b^3*x^3 + 10*a^6*b^2*x^2 + 5*a^7*b*x + a^8)*e^8 - 3*(a^2*b^6*d*x^5 + 5*a^3*b^5*d*x^4 + 10*a^4*b^4*d*
x^3 + 10*a^5*b^3*d*x^2 + 5*a^6*b^2*d*x + a^7*b*d)*e^7 + 3*(a*b^7*d^2*x^5 + 5*a^2*b^6*d^2*x^4 + 10*a^3*b^5*d^2*
x^3 + 10*a^4*b^4*d^2*x^2 + 5*a^5*b^3*d^2*x + a^6*b^2*d^2)*e^6 - (b^8*d^3*x^5 + 5*a*b^7*d^3*x^4 + 10*a^2*b^6*d^
3*x^3 + 10*a^3*b^5*d^3*x^2 + 5*a^4*b^4*d^3*x + a^5*b^3*d^3)*e^5)*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^10*x + a^5*b^9)

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Sympy [F(-1)] Timed out
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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 546 vs. \(2 (204) = 408\).
time = 1.63, 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

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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)

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