\(\int \frac {(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^7} \, dx\) [1496]

Optimal result

Integrand size = 26, antiderivative size = 167 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^7} \, dx=-\frac {(b d-a e)^6}{6 e^7 (d+e x)^6}+\frac {6 b (b d-a e)^5}{5 e^7 (d+e x)^5}-\frac {15 b^2 (b d-a e)^4}{4 e^7 (d+e x)^4}+\frac {20 b^3 (b d-a e)^3}{3 e^7 (d+e x)^3}-\frac {15 b^4 (b d-a e)^2}{2 e^7 (d+e x)^2}+\frac {6 b^5 (b d-a e)}{e^7 (d+e x)}+\frac {b^6 \log (d+e x)}{e^7} \]

[Out]

-1/6*(-a*e+b*d)^6/e^7/(e*x+d)^6+6/5*b*(-a*e+b*d)^5/e^7/(e*x+d)^5-15/4*b^2*(-a*e+b*d)^4/e^7/(e*x+d)^4+20/3*b^3*
(-a*e+b*d)^3/e^7/(e*x+d)^3-15/2*b^4*(-a*e+b*d)^2/e^7/(e*x+d)^2+6*b^5*(-a*e+b*d)/e^7/(e*x+d)+b^6*ln(e*x+d)/e^7

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^7} \, dx=\frac {6 b^5 (b d-a e)}{e^7 (d+e x)}-\frac {15 b^4 (b d-a e)^2}{2 e^7 (d+e x)^2}+\frac {20 b^3 (b d-a e)^3}{3 e^7 (d+e x)^3}-\frac {15 b^2 (b d-a e)^4}{4 e^7 (d+e x)^4}+\frac {6 b (b d-a e)^5}{5 e^7 (d+e x)^5}-\frac {(b d-a e)^6}{6 e^7 (d+e x)^6}+\frac {b^6 \log (d+e x)}{e^7} \]

[In]

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

[Out]

-1/6*(b*d - a*e)^6/(e^7*(d + e*x)^6) + (6*b*(b*d - a*e)^5)/(5*e^7*(d + e*x)^5) - (15*b^2*(b*d - a*e)^4)/(4*e^7
*(d + e*x)^4) + (20*b^3*(b*d - a*e)^3)/(3*e^7*(d + e*x)^3) - (15*b^4*(b*d - a*e)^2)/(2*e^7*(d + e*x)^2) + (6*b
^5*(b*d - a*e))/(e^7*(d + e*x)) + (b^6*Log[d + e*x])/e^7

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*} \text {integral}& = \int \frac {(a+b x)^6}{(d+e x)^7} \, dx \\ & = \int \left (\frac {(-b d+a e)^6}{e^6 (d+e x)^7}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^6}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^5}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^4}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^3}-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)^2}+\frac {b^6}{e^6 (d+e x)}\right ) \, dx \\ & = -\frac {(b d-a e)^6}{6 e^7 (d+e x)^6}+\frac {6 b (b d-a e)^5}{5 e^7 (d+e x)^5}-\frac {15 b^2 (b d-a e)^4}{4 e^7 (d+e x)^4}+\frac {20 b^3 (b d-a e)^3}{3 e^7 (d+e x)^3}-\frac {15 b^4 (b d-a e)^2}{2 e^7 (d+e x)^2}+\frac {6 b^5 (b d-a e)}{e^7 (d+e x)}+\frac {b^6 \log (d+e x)}{e^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^7} \, dx=\frac {\frac {(b d-a e) \left (10 a^5 e^5+2 a^4 b e^4 (11 d+36 e x)+a^3 b^2 e^3 \left (37 d^2+162 d e x+225 e^2 x^2\right )+a^2 b^3 e^2 \left (57 d^3+282 d^2 e x+525 d e^2 x^2+400 e^3 x^3\right )+a b^4 e \left (87 d^4+462 d^3 e x+975 d^2 e^2 x^2+1000 d e^3 x^3+450 e^4 x^4\right )+b^5 \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )\right )}{(d+e x)^6}+60 b^6 \log (d+e x)}{60 e^7} \]

[In]

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

[Out]

(((b*d - a*e)*(10*a^5*e^5 + 2*a^4*b*e^4*(11*d + 36*e*x) + a^3*b^2*e^3*(37*d^2 + 162*d*e*x + 225*e^2*x^2) + a^2
*b^3*e^2*(57*d^3 + 282*d^2*e*x + 525*d*e^2*x^2 + 400*e^3*x^3) + a*b^4*e*(87*d^4 + 462*d^3*e*x + 975*d^2*e^2*x^
2 + 1000*d*e^3*x^3 + 450*e^4*x^4) + b^5*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d*
e^4*x^4 + 360*e^5*x^5)))/(d + e*x)^6 + 60*b^6*Log[d + e*x])/(60*e^7)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(341\) vs. \(2(157)=314\).

Time = 2.39 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.05

method	result	size
risch	\(\frac {-\frac {6 b^{5} \left (a e -b d \right ) x^{5}}{e^{2}}-\frac {15 b^{4} \left (a^{2} e^{2}+2 a b d e -3 b^{2} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {10 b^{3} \left (2 a^{3} e^{3}+3 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -11 b^{3} d^{3}\right ) x^{3}}{3 e^{4}}-\frac {5 b^{2} \left (3 e^{4} a^{4}+4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}+12 a \,b^{3} d^{3} e -25 b^{4} d^{4}\right ) x^{2}}{4 e^{5}}-\frac {b \left (12 a^{5} e^{5}+15 a^{4} b d \,e^{4}+20 a^{3} b^{2} d^{2} e^{3}+30 a^{2} b^{3} d^{3} e^{2}+60 a \,b^{4} d^{4} e -137 b^{5} d^{5}\right ) x}{10 e^{6}}-\frac {10 a^{6} e^{6}+12 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}+20 a^{3} b^{3} d^{3} e^{3}+30 a^{2} b^{4} d^{4} e^{2}+60 a \,b^{5} d^{5} e -147 b^{6} d^{6}}{60 e^{7}}}{\left (e x +d \right )^{6}}+\frac {b^{6} \ln \left (e x +d \right )}{e^{7}}\)	\(342\)
norman	\(\frac {-\frac {10 a^{6} e^{6}+12 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}+20 a^{3} b^{3} d^{3} e^{3}+30 a^{2} b^{4} d^{4} e^{2}+60 a \,b^{5} d^{5} e -147 b^{6} d^{6}}{60 e^{7}}-\frac {6 \left (e a \,b^{5}-d \,b^{6}\right ) x^{5}}{e^{2}}-\frac {15 \left (e^{2} a^{2} b^{4}+2 d e a \,b^{5}-3 b^{6} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {10 \left (2 e^{3} a^{3} b^{3}+3 d \,e^{2} a^{2} b^{4}+6 d^{2} e a \,b^{5}-11 d^{3} b^{6}\right ) x^{3}}{3 e^{4}}-\frac {5 \left (3 e^{4} a^{4} b^{2}+4 d \,e^{3} a^{3} b^{3}+6 d^{2} e^{2} a^{2} b^{4}+12 d^{3} e a \,b^{5}-25 d^{4} b^{6}\right ) x^{2}}{4 e^{5}}-\frac {\left (12 a^{5} b \,e^{5}+15 d \,e^{4} a^{4} b^{2}+20 d^{2} e^{3} a^{3} b^{3}+30 d^{3} e^{2} a^{2} b^{4}+60 d^{4} e a \,b^{5}-137 d^{5} b^{6}\right ) x}{10 e^{6}}}{\left (e x +d \right )^{6}}+\frac {b^{6} \ln \left (e x +d \right )}{e^{7}}\)	\(352\)
default	\(-\frac {6 b \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{5 e^{7} \left (e x +d \right )^{5}}-\frac {6 b^{5} \left (a e -b d \right )}{e^{7} \left (e x +d \right )}-\frac {20 b^{3} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{3 e^{7} \left (e x +d \right )^{3}}-\frac {15 b^{2} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{4 e^{7} \left (e x +d \right )^{4}}-\frac {15 b^{4} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{2 e^{7} \left (e x +d \right )^{2}}+\frac {b^{6} \ln \left (e x +d \right )}{e^{7}}-\frac {a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}{6 e^{7} \left (e x +d \right )^{6}}\)	\(355\)
parallelrisch	\(\frac {-72 x \,a^{5} b \,e^{6}+822 x \,b^{6} d^{5} e -360 x^{5} a \,b^{5} e^{6}+360 x^{5} b^{6} d \,e^{5}-450 x^{4} a^{2} b^{4} e^{6}+1350 x^{4} b^{6} d^{2} e^{4}-400 x^{3} a^{3} b^{3} e^{6}+2200 x^{3} b^{6} d^{3} e^{3}-225 x^{2} a^{4} b^{2} e^{6}-10 a^{6} e^{6}+147 b^{6} d^{6}-60 a \,b^{5} d^{5} e -15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}-30 a^{2} b^{4} d^{4} e^{2}-12 a^{5} b d \,e^{5}+1200 \ln \left (e x +d \right ) x^{3} b^{6} d^{3} e^{3}+60 \ln \left (e x +d \right ) b^{6} d^{6}+60 \ln \left (e x +d \right ) x^{6} b^{6} e^{6}+900 \ln \left (e x +d \right ) x^{4} b^{6} d^{2} e^{4}+360 \ln \left (e x +d \right ) x^{5} b^{6} d \,e^{5}+900 \ln \left (e x +d \right ) x^{2} b^{6} d^{4} e^{2}+360 \ln \left (e x +d \right ) x \,b^{6} d^{5} e +1875 x^{2} b^{6} d^{4} e^{2}-900 x^{4} a \,b^{5} d \,e^{5}-600 x^{3} a^{2} b^{4} d \,e^{5}-1200 x^{3} a \,b^{5} d^{2} e^{4}-300 x^{2} a^{3} b^{3} d \,e^{5}-450 x^{2} a^{2} b^{4} d^{2} e^{4}-900 x^{2} a \,b^{5} d^{3} e^{3}-90 x \,a^{4} b^{2} d \,e^{5}-120 x \,a^{3} b^{3} d^{2} e^{4}-180 x \,a^{2} b^{4} d^{3} e^{3}-360 x a \,b^{5} d^{4} e^{2}}{60 e^{7} \left (e x +d \right )^{6}}\)	\(491\)

[In]

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

[Out]

(-6*b^5*(a*e-b*d)/e^2*x^5-15/2*b^4*(a^2*e^2+2*a*b*d*e-3*b^2*d^2)/e^3*x^4-10/3*b^3*(2*a^3*e^3+3*a^2*b*d*e^2+6*a
*b^2*d^2*e-11*b^3*d^3)/e^4*x^3-5/4*b^2*(3*a^4*e^4+4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2+12*a*b^3*d^3*e-25*b^4*d^4)/e
^5*x^2-1/10*b*(12*a^5*e^5+15*a^4*b*d*e^4+20*a^3*b^2*d^2*e^3+30*a^2*b^3*d^3*e^2+60*a*b^4*d^4*e-137*b^5*d^5)/e^6
*x-1/60*(10*a^6*e^6+12*a^5*b*d*e^5+15*a^4*b^2*d^2*e^4+20*a^3*b^3*d^3*e^3+30*a^2*b^4*d^4*e^2+60*a*b^5*d^5*e-147
*b^6*d^6)/e^7)/(e*x+d)^6+b^6*ln(e*x+d)/e^7

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 492 vs. \(2 (157) = 314\).

Time = 0.30 (sec) , antiderivative size = 492, normalized size of antiderivative = 2.95 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^7} \, dx=\frac {147 \, b^{6} d^{6} - 60 \, a b^{5} d^{5} e - 30 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6} + 360 \, {\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 450 \, {\left (3 \, b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} - a^{2} b^{4} e^{6}\right )} x^{4} + 200 \, {\left (11 \, b^{6} d^{3} e^{3} - 6 \, a b^{5} d^{2} e^{4} - 3 \, a^{2} b^{4} d e^{5} - 2 \, a^{3} b^{3} e^{6}\right )} x^{3} + 75 \, {\left (25 \, b^{6} d^{4} e^{2} - 12 \, a b^{5} d^{3} e^{3} - 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} - 3 \, a^{4} b^{2} e^{6}\right )} x^{2} + 6 \, {\left (137 \, b^{6} d^{5} e - 60 \, a b^{5} d^{4} e^{2} - 30 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 15 \, a^{4} b^{2} d e^{5} - 12 \, a^{5} b e^{6}\right )} x + 60 \, {\left (b^{6} e^{6} x^{6} + 6 \, b^{6} d e^{5} x^{5} + 15 \, b^{6} d^{2} e^{4} x^{4} + 20 \, b^{6} d^{3} e^{3} x^{3} + 15 \, b^{6} d^{4} e^{2} x^{2} + 6 \, b^{6} d^{5} e x + b^{6} d^{6}\right )} \log \left (e x + d\right )}{60 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} \]

[In]

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

[Out]

1/60*(147*b^6*d^6 - 60*a*b^5*d^5*e - 30*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 - 15*a^4*b^2*d^2*e^4 - 12*a^5*b*d
*e^5 - 10*a^6*e^6 + 360*(b^6*d*e^5 - a*b^5*e^6)*x^5 + 450*(3*b^6*d^2*e^4 - 2*a*b^5*d*e^5 - a^2*b^4*e^6)*x^4 +
200*(11*b^6*d^3*e^3 - 6*a*b^5*d^2*e^4 - 3*a^2*b^4*d*e^5 - 2*a^3*b^3*e^6)*x^3 + 75*(25*b^6*d^4*e^2 - 12*a*b^5*d
^3*e^3 - 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 - 3*a^4*b^2*e^6)*x^2 + 6*(137*b^6*d^5*e - 60*a*b^5*d^4*e^2 - 30*a
^2*b^4*d^3*e^3 - 20*a^3*b^3*d^2*e^4 - 15*a^4*b^2*d*e^5 - 12*a^5*b*e^6)*x + 60*(b^6*e^6*x^6 + 6*b^6*d*e^5*x^5 +
15*b^6*d^2*e^4*x^4 + 20*b^6*d^3*e^3*x^3 + 15*b^6*d^4*e^2*x^2 + 6*b^6*d^5*e*x + b^6*d^6)*log(e*x + d))/(e^13*x
^6 + 6*d*e^12*x^5 + 15*d^2*e^11*x^4 + 20*d^3*e^10*x^3 + 15*d^4*e^9*x^2 + 6*d^5*e^8*x + d^6*e^7)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^7} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (157) = 314\).

Time = 0.24 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.49 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^7} \, dx=\frac {147 \, b^{6} d^{6} - 60 \, a b^{5} d^{5} e - 30 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6} + 360 \, {\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 450 \, {\left (3 \, b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} - a^{2} b^{4} e^{6}\right )} x^{4} + 200 \, {\left (11 \, b^{6} d^{3} e^{3} - 6 \, a b^{5} d^{2} e^{4} - 3 \, a^{2} b^{4} d e^{5} - 2 \, a^{3} b^{3} e^{6}\right )} x^{3} + 75 \, {\left (25 \, b^{6} d^{4} e^{2} - 12 \, a b^{5} d^{3} e^{3} - 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} - 3 \, a^{4} b^{2} e^{6}\right )} x^{2} + 6 \, {\left (137 \, b^{6} d^{5} e - 60 \, a b^{5} d^{4} e^{2} - 30 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 15 \, a^{4} b^{2} d e^{5} - 12 \, a^{5} b e^{6}\right )} x}{60 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} + \frac {b^{6} \log \left (e x + d\right )}{e^{7}} \]

[In]

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

[Out]

1/60*(147*b^6*d^6 - 60*a*b^5*d^5*e - 30*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 - 15*a^4*b^2*d^2*e^4 - 12*a^5*b*d
*e^5 - 10*a^6*e^6 + 360*(b^6*d*e^5 - a*b^5*e^6)*x^5 + 450*(3*b^6*d^2*e^4 - 2*a*b^5*d*e^5 - a^2*b^4*e^6)*x^4 +
200*(11*b^6*d^3*e^3 - 6*a*b^5*d^2*e^4 - 3*a^2*b^4*d*e^5 - 2*a^3*b^3*e^6)*x^3 + 75*(25*b^6*d^4*e^2 - 12*a*b^5*d
^3*e^3 - 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 - 3*a^4*b^2*e^6)*x^2 + 6*(137*b^6*d^5*e - 60*a*b^5*d^4*e^2 - 30*a
^2*b^4*d^3*e^3 - 20*a^3*b^3*d^2*e^4 - 15*a^4*b^2*d*e^5 - 12*a^5*b*e^6)*x)/(e^13*x^6 + 6*d*e^12*x^5 + 15*d^2*e^
11*x^4 + 20*d^3*e^10*x^3 + 15*d^4*e^9*x^2 + 6*d^5*e^8*x + d^6*e^7) + b^6*log(e*x + d)/e^7

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (157) = 314\).

Time = 0.26 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.15 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^7} \, dx=\frac {b^{6} \log \left ({\left | e x + d \right |}\right )}{e^{7}} + \frac {360 \, {\left (b^{6} d e^{4} - a b^{5} e^{5}\right )} x^{5} + 450 \, {\left (3 \, b^{6} d^{2} e^{3} - 2 \, a b^{5} d e^{4} - a^{2} b^{4} e^{5}\right )} x^{4} + 200 \, {\left (11 \, b^{6} d^{3} e^{2} - 6 \, a b^{5} d^{2} e^{3} - 3 \, a^{2} b^{4} d e^{4} - 2 \, a^{3} b^{3} e^{5}\right )} x^{3} + 75 \, {\left (25 \, b^{6} d^{4} e - 12 \, a b^{5} d^{3} e^{2} - 6 \, a^{2} b^{4} d^{2} e^{3} - 4 \, a^{3} b^{3} d e^{4} - 3 \, a^{4} b^{2} e^{5}\right )} x^{2} + 6 \, {\left (137 \, b^{6} d^{5} - 60 \, a b^{5} d^{4} e - 30 \, a^{2} b^{4} d^{3} e^{2} - 20 \, a^{3} b^{3} d^{2} e^{3} - 15 \, a^{4} b^{2} d e^{4} - 12 \, a^{5} b e^{5}\right )} x + \frac {147 \, b^{6} d^{6} - 60 \, a b^{5} d^{5} e - 30 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6}}{e}}{60 \, {\left (e x + d\right )}^{6} e^{6}} \]

[In]

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

[Out]

b^6*log(abs(e*x + d))/e^7 + 1/60*(360*(b^6*d*e^4 - a*b^5*e^5)*x^5 + 450*(3*b^6*d^2*e^3 - 2*a*b^5*d*e^4 - a^2*b
^4*e^5)*x^4 + 200*(11*b^6*d^3*e^2 - 6*a*b^5*d^2*e^3 - 3*a^2*b^4*d*e^4 - 2*a^3*b^3*e^5)*x^3 + 75*(25*b^6*d^4*e
- 12*a*b^5*d^3*e^2 - 6*a^2*b^4*d^2*e^3 - 4*a^3*b^3*d*e^4 - 3*a^4*b^2*e^5)*x^2 + 6*(137*b^6*d^5 - 60*a*b^5*d^4*
e - 30*a^2*b^4*d^3*e^2 - 20*a^3*b^3*d^2*e^3 - 15*a^4*b^2*d*e^4 - 12*a^5*b*e^5)*x + (147*b^6*d^6 - 60*a*b^5*d^5
*e - 30*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 - 15*a^4*b^2*d^2*e^4 - 12*a^5*b*d*e^5 - 10*a^6*e^6)/e)/((e*x + d)
^6*e^6)

Mupad [B] (verification not implemented)

Time = 9.81 (sec) , antiderivative size = 353, normalized size of antiderivative = 2.11 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^7} \, dx=\frac {b^6\,\ln \left (d+e\,x\right )}{e^7}-\frac {x^5\,\left (6\,a\,b^5\,e^6-6\,b^6\,d\,e^5\right )+x^2\,\left (\frac {15\,a^4\,b^2\,e^6}{4}+5\,a^3\,b^3\,d\,e^5+\frac {15\,a^2\,b^4\,d^2\,e^4}{2}+15\,a\,b^5\,d^3\,e^3-\frac {125\,b^6\,d^4\,e^2}{4}\right )+x^4\,\left (\frac {15\,a^2\,b^4\,e^6}{2}+15\,a\,b^5\,d\,e^5-\frac {45\,b^6\,d^2\,e^4}{2}\right )+x\,\left (\frac {6\,a^5\,b\,e^6}{5}+\frac {3\,a^4\,b^2\,d\,e^5}{2}+2\,a^3\,b^3\,d^2\,e^4+3\,a^2\,b^4\,d^3\,e^3+6\,a\,b^5\,d^4\,e^2-\frac {137\,b^6\,d^5\,e}{10}\right )+\frac {a^6\,e^6}{6}-\frac {49\,b^6\,d^6}{20}+x^3\,\left (\frac {20\,a^3\,b^3\,e^6}{3}+10\,a^2\,b^4\,d\,e^5+20\,a\,b^5\,d^2\,e^4-\frac {110\,b^6\,d^3\,e^3}{3}\right )+\frac {a^2\,b^4\,d^4\,e^2}{2}+\frac {a^3\,b^3\,d^3\,e^3}{3}+\frac {a^4\,b^2\,d^2\,e^4}{4}+a\,b^5\,d^5\,e+\frac {a^5\,b\,d\,e^5}{5}}{e^7\,{\left (d+e\,x\right )}^6} \]

[In]

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

[Out]

(b^6*log(d + e*x))/e^7 - (x^5*(6*a*b^5*e^6 - 6*b^6*d*e^5) + x^2*((15*a^4*b^2*e^6)/4 - (125*b^6*d^4*e^2)/4 + 15
*a*b^5*d^3*e^3 + 5*a^3*b^3*d*e^5 + (15*a^2*b^4*d^2*e^4)/2) + x^4*((15*a^2*b^4*e^6)/2 - (45*b^6*d^2*e^4)/2 + 15
*a*b^5*d*e^5) + x*((6*a^5*b*e^6)/5 - (137*b^6*d^5*e)/10 + 6*a*b^5*d^4*e^2 + (3*a^4*b^2*d*e^5)/2 + 3*a^2*b^4*d^
3*e^3 + 2*a^3*b^3*d^2*e^4) + (a^6*e^6)/6 - (49*b^6*d^6)/20 + x^3*((20*a^3*b^3*e^6)/3 - (110*b^6*d^3*e^3)/3 + 2
0*a*b^5*d^2*e^4 + 10*a^2*b^4*d*e^5) + (a^2*b^4*d^4*e^2)/2 + (a^3*b^3*d^3*e^3)/3 + (a^4*b^2*d^2*e^4)/4 + a*b^5*
d^5*e + (a^5*b*d*e^5)/5)/(e^7*(d + e*x)^6)