3.16.26 \(\int \frac {(d+e x)^7}{(a^2+2 a b x+b^2 x^2)^3} \, dx\) [1526]

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

[Out]

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

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Rubi [A]
time = 0.16, antiderivative size = 181, 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 {21 e^5 (b d-a e)^2 \log (a+b x)}{b^8}-\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 {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^6 x (7 b d-6 a e)}{b^7}+\frac {e^7 x^2}{2 b^6} \end {gather*}

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 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)^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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(389\) vs. \(2(181)=362\).
time = 0.09, size = 389, normalized size = 2.15 \begin {gather*} \frac {459 a^7 e^7+3 a^6 b e^6 (-406 d+625 e x)+a^5 b^2 e^5 \left (959 d^2-5250 d e x+2700 e^2 x^2\right )+5 a^4 b^3 e^4 \left (-28 d^3+875 d^2 e x-1680 d e^2 x^2+260 e^3 x^3\right )-5 a^3 b^4 e^3 \left (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\right )-a^2 b^5 e^2 \left (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\right )-7 a b^6 e \left (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\right )-b^7 \left (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\right )+420 e^5 (b d-a e)^2 (a+b x)^5 \log (a+b x)}{20 b^8 (a+b x)^5} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(450\) vs. \(2(173)=346\).
time = 0.67, size = 451, normalized size = 2.49

method	result	size
default	\(-\frac {e^{6} \left (-\frac {1}{2} b e \,x^{2}+6 a e x -7 x b d \right )}{b^{7}}+\frac {35 e^{4} \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{b^{8} \left (b x +a \right )}-\frac {35 e^{3} \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{2 b^{8} \left (b x +a \right )^{2}}+\frac {21 e^{5} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \ln \left (b x +a \right )}{b^{8}}+\frac {7 e^{2} \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{b^{8} \left (b x +a \right )^{3}}-\frac {-a^{7} e^{7}+7 a^{6} b d \,e^{6}-21 a^{5} b^{2} d^{2} e^{5}+35 b^{3} d^{3} e^{4} a^{4}-35 b^{4} d^{4} e^{3} a^{3}+21 b^{5} d^{5} e^{2} a^{2}-7 b^{6} d^{6} e a +b^{7} d^{7}}{5 b^{8} \left (b x +a \right )^{5}}-\frac {7 e \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}{4 b^{8} \left (b x +a \right )^{4}}\)	\(451\)
norman	\(\frac {\frac {959 a^{7} e^{7}-1918 a^{6} b d \,e^{6}+959 a^{5} b^{2} d^{2} e^{5}-140 b^{3} d^{3} e^{4} a^{4}-35 b^{4} d^{4} e^{3} a^{3}-14 b^{5} d^{5} e^{2} a^{2}-7 b^{6} d^{6} e a -4 b^{7} d^{7}}{20 b^{8}}+\frac {e^{7} x^{7}}{2 b}+\frac {5 \left (21 e^{7} a^{3}-42 a^{2} b d \,e^{6}+21 a \,b^{2} d^{2} e^{5}-7 d^{3} e^{4} b^{3}\right ) x^{4}}{b^{4}}+\frac {5 \left (126 e^{7} a^{4}-252 a^{3} b d \,e^{6}+126 a^{2} b^{2} d^{2} e^{5}-28 a \,b^{3} d^{3} e^{4}-7 d^{4} e^{3} b^{4}\right ) x^{3}}{2 b^{5}}+\frac {\left (770 e^{7} a^{5}-1540 a^{4} b d \,e^{6}+770 a^{3} b^{2} d^{2} e^{5}-140 a^{2} b^{3} d^{3} e^{4}-35 a \,b^{4} d^{4} e^{3}-14 d^{5} e^{2} b^{5}\right ) x^{2}}{2 b^{6}}+\frac {\left (875 a^{6} e^{7}-1750 a^{5} b d \,e^{6}+875 a^{4} b^{2} d^{2} e^{5}-140 a^{3} b^{3} d^{3} e^{4}-35 a^{2} b^{4} e^{3} d^{4}-14 a \,b^{5} d^{5} e^{2}-7 b^{6} d^{6} e \right ) x}{4 b^{7}}-\frac {7 e^{6} \left (a e -2 b d \right ) x^{6}}{2 b^{2}}}{\left (b x +a \right )^{5}}+\frac {21 e^{5} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \ln \left (b x +a \right )}{b^{8}}\)	\(454\)
risch	\(\frac {e^{7} x^{2}}{2 b^{6}}-\frac {6 e^{7} a x}{b^{7}}+\frac {7 e^{6} x d}{b^{6}}+\frac {\left (35 a^{3} b^{3} e^{7}-105 a^{2} b^{4} d \,e^{6}+105 a \,b^{5} d^{2} e^{5}-35 b^{6} d^{3} e^{4}\right ) x^{4}+\frac {35 b^{2} e^{3} \left (7 e^{4} a^{4}-20 a^{3} b d \,e^{3}+18 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e -b^{4} d^{4}\right ) x^{3}}{2}+\frac {7 b \,e^{2} \left (47 a^{5} e^{5}-130 a^{4} b d \,e^{4}+110 a^{3} b^{2} d^{2} e^{3}-20 a^{2} b^{3} d^{3} e^{2}-5 a \,b^{4} d^{4} e -2 b^{5} d^{5}\right ) x^{2}}{2}+\frac {7 e \left (57 a^{6} e^{6}-154 a^{5} b d \,e^{5}+125 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}-5 a^{2} b^{4} d^{4} e^{2}-2 a \,b^{5} d^{5} e -b^{6} d^{6}\right ) x}{4}+\frac {459 a^{7} e^{7}-1218 a^{6} b d \,e^{6}+959 a^{5} b^{2} d^{2} e^{5}-140 b^{3} d^{3} e^{4} a^{4}-35 b^{4} d^{4} e^{3} a^{3}-14 b^{5} d^{5} e^{2} a^{2}-7 b^{6} d^{6} e a -4 b^{7} d^{7}}{20 b}}{b^{7} \left (b x +a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2}}+\frac {21 e^{7} \ln \left (b x +a \right ) a^{2}}{b^{8}}-\frac {42 e^{6} \ln \left (b x +a \right ) a d}{b^{7}}+\frac {21 e^{5} \ln \left (b x +a \right ) d^{2}}{b^{6}}\)	\(481\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 473 vs. \(2 (175) = 350\).
time = 0.33, size = 473, normalized size = 2.61 \begin {gather*} -\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 x^{2} e^{7} + 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 {gather*}

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*x^2*e^7 + 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] Leaf count of result is larger than twice the leaf count of optimal. 671 vs. \(2 (175) = 350\).
time = 3.40, size = 671, normalized size = 3.71 \begin {gather*} -\frac {4 \, b^{7} d^{7} - {\left (10 \, b^{7} x^{7} - 70 \, a b^{6} x^{6} - 500 \, a^{2} b^{5} x^{5} - 400 \, a^{3} b^{4} x^{4} + 1300 \, a^{4} b^{3} x^{3} + 2700 \, a^{5} b^{2} x^{2} + 1875 \, a^{6} b x + 459 \, a^{7}\right )} e^{7} - 14 \, {\left (10 \, b^{7} d x^{6} + 50 \, a b^{6} d x^{5} - 50 \, a^{2} b^{5} d x^{4} - 400 \, a^{3} b^{4} d x^{3} - 600 \, a^{4} b^{3} d x^{2} - 375 \, a^{5} b^{2} d x - 87 \, a^{6} b d\right )} e^{6} - 7 \, {\left (300 \, a b^{6} d^{2} x^{4} + 900 \, a^{2} b^{5} d^{2} x^{3} + 1100 \, a^{3} b^{4} d^{2} x^{2} + 625 \, a^{4} b^{3} d^{2} x + 137 \, a^{5} b^{2} d^{2}\right )} e^{5} + 140 \, {\left (5 \, b^{7} d^{3} x^{4} + 10 \, a b^{6} d^{3} x^{3} + 10 \, a^{2} b^{5} d^{3} x^{2} + 5 \, a^{3} b^{4} d^{3} x + a^{4} b^{3} d^{3}\right )} e^{4} + 35 \, {\left (10 \, b^{7} d^{4} x^{3} + 10 \, a b^{6} d^{4} x^{2} + 5 \, a^{2} b^{5} d^{4} x + a^{3} b^{4} d^{4}\right )} e^{3} + 14 \, {\left (10 \, b^{7} d^{5} x^{2} + 5 \, a b^{6} d^{5} x + a^{2} b^{5} d^{5}\right )} e^{2} + 7 \, {\left (5 \, b^{7} d^{6} x + a b^{6} d^{6}\right )} e - 420 \, {\left ({\left (a^{2} b^{5} x^{5} + 5 \, a^{3} b^{4} x^{4} + 10 \, a^{4} b^{3} x^{3} + 10 \, a^{5} b^{2} x^{2} + 5 \, a^{6} b x + a^{7}\right )} e^{7} - 2 \, {\left (a b^{6} d x^{5} + 5 \, a^{2} b^{5} d x^{4} + 10 \, a^{3} b^{4} d x^{3} + 10 \, a^{4} b^{3} d x^{2} + 5 \, a^{5} b^{2} d x + a^{6} b d\right )} e^{6} + {\left (b^{7} d^{2} x^{5} + 5 \, a b^{6} d^{2} x^{4} + 10 \, a^{2} b^{5} d^{2} x^{3} + 10 \, a^{3} b^{4} d^{2} x^{2} + 5 \, a^{4} b^{3} d^{2} x + a^{5} b^{2} d^{2}\right )} e^{5}\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 {gather*}

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*(4*b^7*d^7 - (10*b^7*x^7 - 70*a*b^6*x^6 - 500*a^2*b^5*x^5 - 400*a^3*b^4*x^4 + 1300*a^4*b^3*x^3 + 2700*a^
5*b^2*x^2 + 1875*a^6*b*x + 459*a^7)*e^7 - 14*(10*b^7*d*x^6 + 50*a*b^6*d*x^5 - 50*a^2*b^5*d*x^4 - 400*a^3*b^4*d
*x^3 - 600*a^4*b^3*d*x^2 - 375*a^5*b^2*d*x - 87*a^6*b*d)*e^6 - 7*(300*a*b^6*d^2*x^4 + 900*a^2*b^5*d^2*x^3 + 11
00*a^3*b^4*d^2*x^2 + 625*a^4*b^3*d^2*x + 137*a^5*b^2*d^2)*e^5 + 140*(5*b^7*d^3*x^4 + 10*a*b^6*d^3*x^3 + 10*a^2
*b^5*d^3*x^2 + 5*a^3*b^4*d^3*x + a^4*b^3*d^3)*e^4 + 35*(10*b^7*d^4*x^3 + 10*a*b^6*d^4*x^2 + 5*a^2*b^5*d^4*x +
a^3*b^4*d^4)*e^3 + 14*(10*b^7*d^5*x^2 + 5*a*b^6*d^5*x + a^2*b^5*d^5)*e^2 + 7*(5*b^7*d^6*x + a*b^6*d^6)*e - 420
*((a^2*b^5*x^5 + 5*a^3*b^4*x^4 + 10*a^4*b^3*x^3 + 10*a^5*b^2*x^2 + 5*a^6*b*x + a^7)*e^7 - 2*(a*b^6*d*x^5 + 5*a
^2*b^5*d*x^4 + 10*a^3*b^4*d*x^3 + 10*a^4*b^3*d*x^2 + 5*a^5*b^2*d*x + a^6*b*d)*e^6 + (b^7*d^2*x^5 + 5*a*b^6*d^2
*x^4 + 10*a^2*b^5*d^2*x^3 + 10*a^3*b^4*d^2*x^2 + 5*a^4*b^3*d^2*x + a^5*b^2*d^2)*e^5)*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 [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)**7/(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. 432 vs. \(2 (175) = 350\).
time = 1.85, size = 432, normalized size = 2.39 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 0.64, size = 508, normalized size = 2.81 \begin {gather*} \frac {e^7\,x^2}{2\,b^6}-\frac {\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}{20\,b}+x\,\left (-\frac {399\,a^6\,e^7}{4}+\frac {539\,a^5\,b\,d\,e^6}{2}-\frac {875\,a^4\,b^2\,d^2\,e^5}{4}+35\,a^3\,b^3\,d^3\,e^4+\frac {35\,a^2\,b^4\,d^4\,e^3}{4}+\frac {7\,a\,b^5\,d^5\,e^2}{2}+\frac {7\,b^6\,d^6\,e}{4}\right )+x^3\,\left (-\frac {245\,a^4\,b^2\,e^7}{2}+350\,a^3\,b^3\,d\,e^6-315\,a^2\,b^4\,d^2\,e^5+70\,a\,b^5\,d^3\,e^4+\frac {35\,b^6\,d^4\,e^3}{2}\right )+x^2\,\left (-\frac {329\,a^5\,b\,e^7}{2}+455\,a^4\,b^2\,d\,e^6-385\,a^3\,b^3\,d^2\,e^5+70\,a^2\,b^4\,d^3\,e^4+\frac {35\,a\,b^5\,d^4\,e^3}{2}+7\,b^6\,d^5\,e^2\right )-x^4\,\left (35\,a^3\,b^3\,e^7-105\,a^2\,b^4\,d\,e^6+105\,a\,b^5\,d^2\,e^5-35\,b^6\,d^3\,e^4\right )}{a^5\,b^7+5\,a^4\,b^8\,x+10\,a^3\,b^9\,x^2+10\,a^2\,b^{10}\,x^3+5\,a\,b^{11}\,x^4+b^{12}\,x^5}-x\,\left (\frac {6\,a\,e^7}{b^7}-\frac {7\,d\,e^6}{b^6}\right )+\frac {\ln \left (a+b\,x\right )\,\left (21\,a^2\,e^7-42\,a\,b\,d\,e^6+21\,b^2\,d^2\,e^5\right )}{b^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e^7*x^2)/(2*b^6) - ((4*b^7*d^7 - 459*a^7*e^7 + 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 + 7*a*b^6*d^6*e + 1218*a^6*b*d*e^6)/(20*b) + x*((7*b^6*d^6*e)/4 - (399*a^6*e^7)/4 + (7*a
*b^5*d^5*e^2)/2 + (35*a^2*b^4*d^4*e^3)/4 + 35*a^3*b^3*d^3*e^4 - (875*a^4*b^2*d^2*e^5)/4 + (539*a^5*b*d*e^6)/2)
+ x^3*((35*b^6*d^4*e^3)/2 - (245*a^4*b^2*e^7)/2 + 70*a*b^5*d^3*e^4 + 350*a^3*b^3*d*e^6 - 315*a^2*b^4*d^2*e^5)
+ x^2*(7*b^6*d^5*e^2 - (329*a^5*b*e^7)/2 + (35*a*b^5*d^4*e^3)/2 + 455*a^4*b^2*d*e^6 + 70*a^2*b^4*d^3*e^4 - 38
5*a^3*b^3*d^2*e^5) - x^4*(35*a^3*b^3*e^7 - 35*b^6*d^3*e^4 + 105*a*b^5*d^2*e^5 - 105*a^2*b^4*d*e^6))/(a^5*b^7 +
b^12*x^5 + 5*a^4*b^8*x + 5*a*b^11*x^4 + 10*a^3*b^9*x^2 + 10*a^2*b^10*x^3) - x*((6*a*e^7)/b^7 - (7*d*e^6)/b^6)
+ (log(a + b*x)*(21*a^2*e^7 + 21*b^2*d^2*e^5 - 42*a*b*d*e^6))/b^8

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