3.16.2 \(\int \frac {(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^{13}} \, dx\) [1502]

Optimal. Leaf size=173 \[ -\frac {(b d-a e)^6}{12 e^7 (d+e x)^{12}}+\frac {6 b (b d-a e)^5}{11 e^7 (d+e x)^{11}}-\frac {3 b^2 (b d-a e)^4}{2 e^7 (d+e x)^{10}}+\frac {20 b^3 (b d-a e)^3}{9 e^7 (d+e x)^9}-\frac {15 b^4 (b d-a e)^2}{8 e^7 (d+e x)^8}+\frac {6 b^5 (b d-a e)}{7 e^7 (d+e x)^7}-\frac {b^6}{6 e^7 (d+e x)^6} \]

[Out]

-1/12*(-a*e+b*d)^6/e^7/(e*x+d)^12+6/11*b*(-a*e+b*d)^5/e^7/(e*x+d)^11-3/2*b^2*(-a*e+b*d)^4/e^7/(e*x+d)^10+20/9*
b^3*(-a*e+b*d)^3/e^7/(e*x+d)^9-15/8*b^4*(-a*e+b*d)^2/e^7/(e*x+d)^8+6/7*b^5*(-a*e+b*d)/e^7/(e*x+d)^7-1/6*b^6/e^
7/(e*x+d)^6

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Rubi [A]
time = 0.08, antiderivative size = 173, 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 {6 b^5 (b d-a e)}{7 e^7 (d+e x)^7}-\frac {15 b^4 (b d-a e)^2}{8 e^7 (d+e x)^8}+\frac {20 b^3 (b d-a e)^3}{9 e^7 (d+e x)^9}-\frac {3 b^2 (b d-a e)^4}{2 e^7 (d+e x)^{10}}+\frac {6 b (b d-a e)^5}{11 e^7 (d+e x)^{11}}-\frac {(b d-a e)^6}{12 e^7 (d+e x)^{12}}-\frac {b^6}{6 e^7 (d+e x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^13,x]

[Out]

-1/12*(b*d - a*e)^6/(e^7*(d + e*x)^12) + (6*b*(b*d - a*e)^5)/(11*e^7*(d + e*x)^11) - (3*b^2*(b*d - a*e)^4)/(2*
e^7*(d + e*x)^10) + (20*b^3*(b*d - a*e)^3)/(9*e^7*(d + e*x)^9) - (15*b^4*(b*d - a*e)^2)/(8*e^7*(d + e*x)^8) +
(6*b^5*(b*d - a*e))/(7*e^7*(d + e*x)^7) - b^6/(6*e^7*(d + e*x)^6)

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 {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{13}} \, dx &=\int \frac {(a+b x)^6}{(d+e x)^{13}} \, dx\\ &=\int \left (\frac {(-b d+a e)^6}{e^6 (d+e x)^{13}}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^{12}}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^{11}}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^{10}}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^9}-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)^8}+\frac {b^6}{e^6 (d+e x)^7}\right ) \, dx\\ &=-\frac {(b d-a e)^6}{12 e^7 (d+e x)^{12}}+\frac {6 b (b d-a e)^5}{11 e^7 (d+e x)^{11}}-\frac {3 b^2 (b d-a e)^4}{2 e^7 (d+e x)^{10}}+\frac {20 b^3 (b d-a e)^3}{9 e^7 (d+e x)^9}-\frac {15 b^4 (b d-a e)^2}{8 e^7 (d+e x)^8}+\frac {6 b^5 (b d-a e)}{7 e^7 (d+e x)^7}-\frac {b^6}{6 e^7 (d+e x)^6}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 277, normalized size = 1.60 \begin {gather*} -\frac {462 a^6 e^6+252 a^5 b e^5 (d+12 e x)+126 a^4 b^2 e^4 \left (d^2+12 d e x+66 e^2 x^2\right )+56 a^3 b^3 e^3 \left (d^3+12 d^2 e x+66 d e^2 x^2+220 e^3 x^3\right )+21 a^2 b^4 e^2 \left (d^4+12 d^3 e x+66 d^2 e^2 x^2+220 d e^3 x^3+495 e^4 x^4\right )+6 a b^5 e \left (d^5+12 d^4 e x+66 d^3 e^2 x^2+220 d^2 e^3 x^3+495 d e^4 x^4+792 e^5 x^5\right )+b^6 \left (d^6+12 d^5 e x+66 d^4 e^2 x^2+220 d^3 e^3 x^3+495 d^2 e^4 x^4+792 d e^5 x^5+924 e^6 x^6\right )}{5544 e^7 (d+e x)^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^13,x]

[Out]

-1/5544*(462*a^6*e^6 + 252*a^5*b*e^5*(d + 12*e*x) + 126*a^4*b^2*e^4*(d^2 + 12*d*e*x + 66*e^2*x^2) + 56*a^3*b^3
*e^3*(d^3 + 12*d^2*e*x + 66*d*e^2*x^2 + 220*e^3*x^3) + 21*a^2*b^4*e^2*(d^4 + 12*d^3*e*x + 66*d^2*e^2*x^2 + 220
*d*e^3*x^3 + 495*e^4*x^4) + 6*a*b^5*e*(d^5 + 12*d^4*e*x + 66*d^3*e^2*x^2 + 220*d^2*e^3*x^3 + 495*d*e^4*x^4 + 7
92*e^5*x^5) + b^6*(d^6 + 12*d^5*e*x + 66*d^4*e^2*x^2 + 220*d^3*e^3*x^3 + 495*d^2*e^4*x^4 + 792*d*e^5*x^5 + 924
*e^6*x^6))/(e^7*(d + e*x)^12)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(356\) vs. \(2(159)=318\).
time = 0.64, size = 357, normalized size = 2.06

method	result	size
risch	\(\frac {-\frac {b^{6} x^{6}}{6 e}-\frac {b^{5} \left (6 a e +b d \right ) x^{5}}{7 e^{2}}-\frac {5 b^{4} \left (21 a^{2} e^{2}+6 a b d e +b^{2} d^{2}\right ) x^{4}}{56 e^{3}}-\frac {5 b^{3} \left (56 e^{3} a^{3}+21 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{126 e^{4}}-\frac {b^{2} \left (126 e^{4} a^{4}+56 a^{3} b d \,e^{3}+21 a^{2} b^{2} d^{2} e^{2}+6 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}}{84 e^{5}}-\frac {b \left (252 a^{5} e^{5}+126 a^{4} b d \,e^{4}+56 a^{3} b^{2} d^{2} e^{3}+21 a^{2} b^{3} d^{3} e^{2}+6 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) x}{462 e^{6}}-\frac {462 a^{6} e^{6}+252 a^{5} b d \,e^{5}+126 a^{4} b^{2} d^{2} e^{4}+56 a^{3} b^{3} d^{3} e^{3}+21 a^{2} b^{4} d^{4} e^{2}+6 a \,b^{5} d^{5} e +b^{6} d^{6}}{5544 e^{7}}}{\left (e x +d \right )^{12}}\)	\(335\)
default	\(-\frac {6 b^{5} \left (a e -b d \right )}{7 e^{7} \left (e x +d \right )^{7}}-\frac {6 b \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{11 e^{7} \left (e x +d \right )^{11}}-\frac {20 b^{3} \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{9 e^{7} \left (e x +d \right )^{9}}-\frac {3 b^{2} \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{2 e^{7} \left (e x +d \right )^{10}}-\frac {a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}{12 e^{7} \left (e x +d \right )^{12}}-\frac {b^{6}}{6 e^{7} \left (e x +d \right )^{6}}-\frac {15 b^{4} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{8 e^{7} \left (e x +d \right )^{8}}\)	\(357\)
norman	\(\frac {-\frac {b^{6} x^{6}}{6 e}-\frac {\left (6 a \,b^{5} e^{6}+b^{6} d \,e^{5}\right ) x^{5}}{7 e^{7}}-\frac {5 \left (21 a^{2} b^{4} e^{7}+6 a \,b^{5} d \,e^{6}+b^{6} d^{2} e^{5}\right ) x^{4}}{56 e^{8}}-\frac {5 \left (56 a^{3} b^{3} e^{8}+21 a^{2} b^{4} d \,e^{7}+6 a \,b^{5} e^{6} d^{2}+b^{6} e^{5} d^{3}\right ) x^{3}}{126 e^{9}}-\frac {\left (126 a^{4} b^{2} e^{9}+56 a^{3} b^{3} d \,e^{8}+21 a^{2} b^{4} d^{2} e^{7}+6 a \,b^{5} d^{3} e^{6}+b^{6} d^{4} e^{5}\right ) x^{2}}{84 e^{10}}-\frac {\left (252 a^{5} b \,e^{10}+126 a^{4} b^{2} d \,e^{9}+56 a^{3} b^{3} d^{2} e^{8}+21 a^{2} b^{4} d^{3} e^{7}+6 a \,b^{5} d^{4} e^{6}+b^{6} d^{5} e^{5}\right ) x}{462 e^{11}}-\frac {462 a^{6} e^{11}+252 a^{5} b d \,e^{10}+126 a^{4} b^{2} d^{2} e^{9}+56 a^{3} b^{3} d^{3} e^{8}+21 a^{2} b^{4} d^{4} e^{7}+6 a \,b^{5} d^{5} e^{6}+b^{6} d^{6} e^{5}}{5544 e^{12}}}{\left (e x +d \right )^{12}}\)	\(375\)
gosper	\(-\frac {924 b^{6} x^{6} e^{6}+4752 a \,b^{5} e^{6} x^{5}+792 b^{6} d \,e^{5} x^{5}+10395 a^{2} b^{4} e^{6} x^{4}+2970 a \,b^{5} d \,e^{5} x^{4}+495 b^{6} d^{2} e^{4} x^{4}+12320 a^{3} b^{3} e^{6} x^{3}+4620 a^{2} b^{4} d \,e^{5} x^{3}+1320 a \,b^{5} d^{2} e^{4} x^{3}+220 b^{6} d^{3} e^{3} x^{3}+8316 a^{4} b^{2} e^{6} x^{2}+3696 a^{3} b^{3} d \,e^{5} x^{2}+1386 a^{2} b^{4} d^{2} e^{4} x^{2}+396 a \,b^{5} d^{3} e^{3} x^{2}+66 b^{6} d^{4} e^{2} x^{2}+3024 a^{5} b \,e^{6} x +1512 a^{4} b^{2} d \,e^{5} x +672 a^{3} b^{3} d^{2} e^{4} x +252 a^{2} b^{4} d^{3} e^{3} x +72 a \,b^{5} d^{4} e^{2} x +12 b^{6} d^{5} e x +462 a^{6} e^{6}+252 a^{5} b d \,e^{5}+126 a^{4} b^{2} d^{2} e^{4}+56 a^{3} b^{3} d^{3} e^{3}+21 a^{2} b^{4} d^{4} e^{2}+6 a \,b^{5} d^{5} e +b^{6} d^{6}}{5544 e^{7} \left (e x +d \right )^{12}}\)	\(376\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^13,x,method=_RETURNVERBOSE)

[Out]

-6/7*b^5/e^7*(a*e-b*d)/(e*x+d)^7-6/11*b/e^7*(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)/(e*x+d)^11-20/9*b^3/e^7*(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)/(e*x+d)^9-3/2*b^2/e^7*
(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)/(e*x+d)^10-1/12*(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)/e^7/(e*x+d)^12-1/6*b^6/e^7/(e*x+d)^6-
15/8*b^4/e^7*(a^2*e^2-2*a*b*d*e+b^2*d^2)/(e*x+d)^8

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 438 vs. \(2 (165) = 330\).
time = 0.31, size = 438, normalized size = 2.53 \begin {gather*} -\frac {924 \, b^{6} x^{6} e^{6} + b^{6} d^{6} + 6 \, a b^{5} d^{5} e + 21 \, a^{2} b^{4} d^{4} e^{2} + 56 \, a^{3} b^{3} d^{3} e^{3} + 126 \, a^{4} b^{2} d^{2} e^{4} + 252 \, a^{5} b d e^{5} + 462 \, a^{6} e^{6} + 792 \, {\left (b^{6} d e^{5} + 6 \, a b^{5} e^{6}\right )} x^{5} + 495 \, {\left (b^{6} d^{2} e^{4} + 6 \, a b^{5} d e^{5} + 21 \, a^{2} b^{4} e^{6}\right )} x^{4} + 220 \, {\left (b^{6} d^{3} e^{3} + 6 \, a b^{5} d^{2} e^{4} + 21 \, a^{2} b^{4} d e^{5} + 56 \, a^{3} b^{3} e^{6}\right )} x^{3} + 66 \, {\left (b^{6} d^{4} e^{2} + 6 \, a b^{5} d^{3} e^{3} + 21 \, a^{2} b^{4} d^{2} e^{4} + 56 \, a^{3} b^{3} d e^{5} + 126 \, a^{4} b^{2} e^{6}\right )} x^{2} + 12 \, {\left (b^{6} d^{5} e + 6 \, a b^{5} d^{4} e^{2} + 21 \, a^{2} b^{4} d^{3} e^{3} + 56 \, a^{3} b^{3} d^{2} e^{4} + 126 \, a^{4} b^{2} d e^{5} + 252 \, a^{5} b e^{6}\right )} x}{5544 \, {\left (x^{12} e^{19} + 12 \, d x^{11} e^{18} + 66 \, d^{2} x^{10} e^{17} + 220 \, d^{3} x^{9} e^{16} + 495 \, d^{4} x^{8} e^{15} + 792 \, d^{5} x^{7} e^{14} + 924 \, d^{6} x^{6} e^{13} + 792 \, d^{7} x^{5} e^{12} + 495 \, d^{8} x^{4} e^{11} + 220 \, d^{9} x^{3} e^{10} + 66 \, d^{10} x^{2} e^{9} + 12 \, d^{11} x e^{8} + d^{12} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^13,x, algorithm="maxima")

[Out]

-1/5544*(924*b^6*x^6*e^6 + b^6*d^6 + 6*a*b^5*d^5*e + 21*a^2*b^4*d^4*e^2 + 56*a^3*b^3*d^3*e^3 + 126*a^4*b^2*d^2
*e^4 + 252*a^5*b*d*e^5 + 462*a^6*e^6 + 792*(b^6*d*e^5 + 6*a*b^5*e^6)*x^5 + 495*(b^6*d^2*e^4 + 6*a*b^5*d*e^5 +
21*a^2*b^4*e^6)*x^4 + 220*(b^6*d^3*e^3 + 6*a*b^5*d^2*e^4 + 21*a^2*b^4*d*e^5 + 56*a^3*b^3*e^6)*x^3 + 66*(b^6*d^
4*e^2 + 6*a*b^5*d^3*e^3 + 21*a^2*b^4*d^2*e^4 + 56*a^3*b^3*d*e^5 + 126*a^4*b^2*e^6)*x^2 + 12*(b^6*d^5*e + 6*a*b
^5*d^4*e^2 + 21*a^2*b^4*d^3*e^3 + 56*a^3*b^3*d^2*e^4 + 126*a^4*b^2*d*e^5 + 252*a^5*b*e^6)*x)/(x^12*e^19 + 12*d
*x^11*e^18 + 66*d^2*x^10*e^17 + 220*d^3*x^9*e^16 + 495*d^4*x^8*e^15 + 792*d^5*x^7*e^14 + 924*d^6*x^6*e^13 + 79
2*d^7*x^5*e^12 + 495*d^8*x^4*e^11 + 220*d^9*x^3*e^10 + 66*d^10*x^2*e^9 + 12*d^11*x*e^8 + d^12*e^7)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 437 vs. \(2 (165) = 330\).
time = 3.30, size = 437, normalized size = 2.53 \begin {gather*} -\frac {b^{6} d^{6} + {\left (924 \, b^{6} x^{6} + 4752 \, a b^{5} x^{5} + 10395 \, a^{2} b^{4} x^{4} + 12320 \, a^{3} b^{3} x^{3} + 8316 \, a^{4} b^{2} x^{2} + 3024 \, a^{5} b x + 462 \, a^{6}\right )} e^{6} + 6 \, {\left (132 \, b^{6} d x^{5} + 495 \, a b^{5} d x^{4} + 770 \, a^{2} b^{4} d x^{3} + 616 \, a^{3} b^{3} d x^{2} + 252 \, a^{4} b^{2} d x + 42 \, a^{5} b d\right )} e^{5} + 3 \, {\left (165 \, b^{6} d^{2} x^{4} + 440 \, a b^{5} d^{2} x^{3} + 462 \, a^{2} b^{4} d^{2} x^{2} + 224 \, a^{3} b^{3} d^{2} x + 42 \, a^{4} b^{2} d^{2}\right )} e^{4} + 4 \, {\left (55 \, b^{6} d^{3} x^{3} + 99 \, a b^{5} d^{3} x^{2} + 63 \, a^{2} b^{4} d^{3} x + 14 \, a^{3} b^{3} d^{3}\right )} e^{3} + 3 \, {\left (22 \, b^{6} d^{4} x^{2} + 24 \, a b^{5} d^{4} x + 7 \, a^{2} b^{4} d^{4}\right )} e^{2} + 6 \, {\left (2 \, b^{6} d^{5} x + a b^{5} d^{5}\right )} e}{5544 \, {\left (x^{12} e^{19} + 12 \, d x^{11} e^{18} + 66 \, d^{2} x^{10} e^{17} + 220 \, d^{3} x^{9} e^{16} + 495 \, d^{4} x^{8} e^{15} + 792 \, d^{5} x^{7} e^{14} + 924 \, d^{6} x^{6} e^{13} + 792 \, d^{7} x^{5} e^{12} + 495 \, d^{8} x^{4} e^{11} + 220 \, d^{9} x^{3} e^{10} + 66 \, d^{10} x^{2} e^{9} + 12 \, d^{11} x e^{8} + d^{12} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^13,x, algorithm="fricas")

[Out]

-1/5544*(b^6*d^6 + (924*b^6*x^6 + 4752*a*b^5*x^5 + 10395*a^2*b^4*x^4 + 12320*a^3*b^3*x^3 + 8316*a^4*b^2*x^2 +
3024*a^5*b*x + 462*a^6)*e^6 + 6*(132*b^6*d*x^5 + 495*a*b^5*d*x^4 + 770*a^2*b^4*d*x^3 + 616*a^3*b^3*d*x^2 + 252
*a^4*b^2*d*x + 42*a^5*b*d)*e^5 + 3*(165*b^6*d^2*x^4 + 440*a*b^5*d^2*x^3 + 462*a^2*b^4*d^2*x^2 + 224*a^3*b^3*d^
2*x + 42*a^4*b^2*d^2)*e^4 + 4*(55*b^6*d^3*x^3 + 99*a*b^5*d^3*x^2 + 63*a^2*b^4*d^3*x + 14*a^3*b^3*d^3)*e^3 + 3*
(22*b^6*d^4*x^2 + 24*a*b^5*d^4*x + 7*a^2*b^4*d^4)*e^2 + 6*(2*b^6*d^5*x + a*b^5*d^5)*e)/(x^12*e^19 + 12*d*x^11*
e^18 + 66*d^2*x^10*e^17 + 220*d^3*x^9*e^16 + 495*d^4*x^8*e^15 + 792*d^5*x^7*e^14 + 924*d^6*x^6*e^13 + 792*d^7*
x^5*e^12 + 495*d^8*x^4*e^11 + 220*d^9*x^3*e^10 + 66*d^10*x^2*e^9 + 12*d^11*x*e^8 + d^12*e^7)

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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((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**13,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (165) = 330\).
time = 2.03, size = 352, normalized size = 2.03 \begin {gather*} -\frac {{\left (924 \, b^{6} x^{6} e^{6} + 792 \, b^{6} d x^{5} e^{5} + 495 \, b^{6} d^{2} x^{4} e^{4} + 220 \, b^{6} d^{3} x^{3} e^{3} + 66 \, b^{6} d^{4} x^{2} e^{2} + 12 \, b^{6} d^{5} x e + b^{6} d^{6} + 4752 \, a b^{5} x^{5} e^{6} + 2970 \, a b^{5} d x^{4} e^{5} + 1320 \, a b^{5} d^{2} x^{3} e^{4} + 396 \, a b^{5} d^{3} x^{2} e^{3} + 72 \, a b^{5} d^{4} x e^{2} + 6 \, a b^{5} d^{5} e + 10395 \, a^{2} b^{4} x^{4} e^{6} + 4620 \, a^{2} b^{4} d x^{3} e^{5} + 1386 \, a^{2} b^{4} d^{2} x^{2} e^{4} + 252 \, a^{2} b^{4} d^{3} x e^{3} + 21 \, a^{2} b^{4} d^{4} e^{2} + 12320 \, a^{3} b^{3} x^{3} e^{6} + 3696 \, a^{3} b^{3} d x^{2} e^{5} + 672 \, a^{3} b^{3} d^{2} x e^{4} + 56 \, a^{3} b^{3} d^{3} e^{3} + 8316 \, a^{4} b^{2} x^{2} e^{6} + 1512 \, a^{4} b^{2} d x e^{5} + 126 \, a^{4} b^{2} d^{2} e^{4} + 3024 \, a^{5} b x e^{6} + 252 \, a^{5} b d e^{5} + 462 \, a^{6} e^{6}\right )} e^{\left (-7\right )}}{5544 \, {\left (x e + d\right )}^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^13,x, algorithm="giac")

[Out]

-1/5544*(924*b^6*x^6*e^6 + 792*b^6*d*x^5*e^5 + 495*b^6*d^2*x^4*e^4 + 220*b^6*d^3*x^3*e^3 + 66*b^6*d^4*x^2*e^2
+ 12*b^6*d^5*x*e + b^6*d^6 + 4752*a*b^5*x^5*e^6 + 2970*a*b^5*d*x^4*e^5 + 1320*a*b^5*d^2*x^3*e^4 + 396*a*b^5*d^
3*x^2*e^3 + 72*a*b^5*d^4*x*e^2 + 6*a*b^5*d^5*e + 10395*a^2*b^4*x^4*e^6 + 4620*a^2*b^4*d*x^3*e^5 + 1386*a^2*b^4
*d^2*x^2*e^4 + 252*a^2*b^4*d^3*x*e^3 + 21*a^2*b^4*d^4*e^2 + 12320*a^3*b^3*x^3*e^6 + 3696*a^3*b^3*d*x^2*e^5 + 6
72*a^3*b^3*d^2*x*e^4 + 56*a^3*b^3*d^3*e^3 + 8316*a^4*b^2*x^2*e^6 + 1512*a^4*b^2*d*x*e^5 + 126*a^4*b^2*d^2*e^4
+ 3024*a^5*b*x*e^6 + 252*a^5*b*d*e^5 + 462*a^6*e^6)*e^(-7)/(x*e + d)^12

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Mupad [B]
time = 0.96, size = 456, normalized size = 2.64 \begin {gather*} -\frac {\frac {462\,a^6\,e^6+252\,a^5\,b\,d\,e^5+126\,a^4\,b^2\,d^2\,e^4+56\,a^3\,b^3\,d^3\,e^3+21\,a^2\,b^4\,d^4\,e^2+6\,a\,b^5\,d^5\,e+b^6\,d^6}{5544\,e^7}+\frac {b^6\,x^6}{6\,e}+\frac {5\,b^3\,x^3\,\left (56\,a^3\,e^3+21\,a^2\,b\,d\,e^2+6\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{126\,e^4}+\frac {b\,x\,\left (252\,a^5\,e^5+126\,a^4\,b\,d\,e^4+56\,a^3\,b^2\,d^2\,e^3+21\,a^2\,b^3\,d^3\,e^2+6\,a\,b^4\,d^4\,e+b^5\,d^5\right )}{462\,e^6}+\frac {b^5\,x^5\,\left (6\,a\,e+b\,d\right )}{7\,e^2}+\frac {b^2\,x^2\,\left (126\,a^4\,e^4+56\,a^3\,b\,d\,e^3+21\,a^2\,b^2\,d^2\,e^2+6\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{84\,e^5}+\frac {5\,b^4\,x^4\,\left (21\,a^2\,e^2+6\,a\,b\,d\,e+b^2\,d^2\right )}{56\,e^3}}{d^{12}+12\,d^{11}\,e\,x+66\,d^{10}\,e^2\,x^2+220\,d^9\,e^3\,x^3+495\,d^8\,e^4\,x^4+792\,d^7\,e^5\,x^5+924\,d^6\,e^6\,x^6+792\,d^5\,e^7\,x^7+495\,d^4\,e^8\,x^8+220\,d^3\,e^9\,x^9+66\,d^2\,e^{10}\,x^{10}+12\,d\,e^{11}\,x^{11}+e^{12}\,x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^2 + 2*a*b*x)^3/(d + e*x)^13,x)

[Out]

-((462*a^6*e^6 + b^6*d^6 + 21*a^2*b^4*d^4*e^2 + 56*a^3*b^3*d^3*e^3 + 126*a^4*b^2*d^2*e^4 + 6*a*b^5*d^5*e + 252
*a^5*b*d*e^5)/(5544*e^7) + (b^6*x^6)/(6*e) + (5*b^3*x^3*(56*a^3*e^3 + b^3*d^3 + 6*a*b^2*d^2*e + 21*a^2*b*d*e^2
))/(126*e^4) + (b*x*(252*a^5*e^5 + b^5*d^5 + 21*a^2*b^3*d^3*e^2 + 56*a^3*b^2*d^2*e^3 + 6*a*b^4*d^4*e + 126*a^4
*b*d*e^4))/(462*e^6) + (b^5*x^5*(6*a*e + b*d))/(7*e^2) + (b^2*x^2*(126*a^4*e^4 + b^4*d^4 + 21*a^2*b^2*d^2*e^2
+ 6*a*b^3*d^3*e + 56*a^3*b*d*e^3))/(84*e^5) + (5*b^4*x^4*(21*a^2*e^2 + b^2*d^2 + 6*a*b*d*e))/(56*e^3))/(d^12 +
e^12*x^12 + 12*d*e^11*x^11 + 66*d^10*e^2*x^2 + 220*d^9*e^3*x^3 + 495*d^8*e^4*x^4 + 792*d^7*e^5*x^5 + 924*d^6*
e^6*x^6 + 792*d^5*e^7*x^7 + 495*d^4*e^8*x^8 + 220*d^3*e^9*x^9 + 66*d^2*e^10*x^10 + 12*d^11*e*x)

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