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

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

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Rubi [A]  time = 0.23, 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, 43} \begin {gather*} \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} \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 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*}

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Mathematica [B]  time = 0.15, size = 389, normalized size = 2.15 \begin {gather*} \frac {459 a^7 e^7+3 a^6 b e^6 (625 e x-406 d)+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 )+420 e^5 (a+b x)^5 (b d-a e)^2 \log (a+b x)-\left (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 )\right )}{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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^7}{\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)^7/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

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

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fricas [B]  time = 0.39, size = 732, normalized size = 4.04 \begin {gather*} \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 {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*(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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giac [B]  time = 0.17, 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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maple [B]  time = 0.06, size = 656, normalized size = 3.62 \begin {gather*} \frac {a^{7} e^{7}}{5 \left (b x +a \right )^{5} b^{8}}-\frac {7 a^{6} d \,e^{6}}{5 \left (b x +a \right )^{5} b^{7}}+\frac {21 a^{5} d^{2} e^{5}}{5 \left (b x +a \right )^{5} b^{6}}-\frac {7 a^{4} d^{3} e^{4}}{\left (b x +a \right )^{5} b^{5}}+\frac {7 a^{3} d^{4} e^{3}}{\left (b x +a \right )^{5} b^{4}}-\frac {21 a^{2} d^{5} e^{2}}{5 \left (b x +a \right )^{5} b^{3}}+\frac {7 a \,d^{6} e}{5 \left (b x +a \right )^{5} b^{2}}-\frac {d^{7}}{5 \left (b x +a \right )^{5} b}-\frac {7 a^{6} e^{7}}{4 \left (b x +a \right )^{4} b^{8}}+\frac {21 a^{5} d \,e^{6}}{2 \left (b x +a \right )^{4} b^{7}}-\frac {105 a^{4} d^{2} e^{5}}{4 \left (b x +a \right )^{4} b^{6}}+\frac {35 a^{3} d^{3} e^{4}}{\left (b x +a \right )^{4} b^{5}}-\frac {105 a^{2} d^{4} e^{3}}{4 \left (b x +a \right )^{4} b^{4}}+\frac {21 a \,d^{5} e^{2}}{2 \left (b x +a \right )^{4} b^{3}}-\frac {7 d^{6} e}{4 \left (b x +a \right )^{4} b^{2}}+\frac {7 a^{5} e^{7}}{\left (b x +a \right )^{3} b^{8}}-\frac {35 a^{4} d \,e^{6}}{\left (b x +a \right )^{3} b^{7}}+\frac {70 a^{3} d^{2} e^{5}}{\left (b x +a \right )^{3} b^{6}}-\frac {70 a^{2} d^{3} e^{4}}{\left (b x +a \right )^{3} b^{5}}+\frac {35 a \,d^{4} e^{3}}{\left (b x +a \right )^{3} b^{4}}-\frac {7 d^{5} e^{2}}{\left (b x +a \right )^{3} b^{3}}-\frac {35 a^{4} e^{7}}{2 \left (b x +a \right )^{2} b^{8}}+\frac {70 a^{3} d \,e^{6}}{\left (b x +a \right )^{2} b^{7}}-\frac {105 a^{2} d^{2} e^{5}}{\left (b x +a \right )^{2} b^{6}}+\frac {70 a \,d^{3} e^{4}}{\left (b x +a \right )^{2} b^{5}}-\frac {35 d^{4} e^{3}}{2 \left (b x +a \right )^{2} b^{4}}+\frac {e^{7} x^{2}}{2 b^{6}}+\frac {35 a^{3} e^{7}}{\left (b x +a \right ) b^{8}}-\frac {105 a^{2} d \,e^{6}}{\left (b x +a \right ) b^{7}}+\frac {21 a^{2} e^{7} \ln \left (b x +a \right )}{b^{8}}+\frac {105 a \,d^{2} e^{5}}{\left (b x +a \right ) b^{6}}-\frac {42 a d \,e^{6} \ln \left (b x +a \right )}{b^{7}}-\frac {6 a \,e^{7} x}{b^{7}}-\frac {35 d^{3} e^{4}}{\left (b x +a \right ) b^{5}}+\frac {21 d^{2} e^{5} \ln \left (b x +a \right )}{b^{6}}+\frac {7 d \,e^{6} x}{b^{6}} \end {gather*}

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]

-1/5/b/(b*x+a)^5*d^7-7/5/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^3*e^2/(b*x+a)^3*d^5-35/2/b
^8*e^7/(b*x+a)^2*a^4-35/2/b^4*e^3/(b*x+a)^2*d^4-7/4/b^8*e^7/(b*x+a)^4*a^6-7/4/b^2*e/(b*x+a)^4*d^6+21/b^8*e^7*l
n(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+1/5/b^8/(b*x+a)^5*a^7*e^7+7/b^8*e^7/(b*x+a)^3*a^5-105/b^7*e^6/(b*x+a)*a^2*d+105/b^6*e^5/(b*x+a)*a*d^2-42/b^7
*e^6*ln(b*x+a)*a*d-35/b^7*e^6/(b*x+a)^3*a^4*d+70/b^6*e^5/(b*x+a)^3*a^3*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*d*a^5-105/4/b^6*e^5/(b*x+a)^4*d^2*a^4+35/b^5*e^4/(b*x+a)^4*d^3*a^3
-105/4/b^4*e^3/(b*x+a)^4*d^4*a^2+21/2/b^3*e^2/(b*x+a)^4*d^5*a+1/2*e^7*x^2/b^6+70/b^7*e^6/(b*x+a)^2*d*a^3-7/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-10
5/b^6*e^5/(b*x+a)^2*a^2*d^2+70/b^5*e^4/(b*x+a)^2*a*d^3

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maxima [B]  time = 1.79, size = 504, normalized size = 2.78 \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 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 {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*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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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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sympy [B]  time = 98.97, size = 524, normalized size = 2.90 \begin {gather*} x \left (- \frac {6 a e^{7}}{b^{7}} + \frac {7 d e^{6}}{b^{6}}\right ) + \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 {21 e^{5} \left (a e - b d\right )^{2} \log {\left (a + b x \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)

[Out]

x*(-6*a*e**7/b**7 + 7*d*e**6/b**6) + (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*(329
0*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*(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))/(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) +
21*e**5*(a*e - b*d)**2*log(a + b*x)/b**8

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