3.1917 \(\int (a+b x) (d+e x)^5 (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=143 \[ \frac{5 e^4 (a+b x)^{12} (b d-a e)}{12 b^6}+\frac{10 e^3 (a+b x)^{11} (b d-a e)^2}{11 b^6}+\frac{e^2 (a+b x)^{10} (b d-a e)^3}{b^6}+\frac{5 e (a+b x)^9 (b d-a e)^4}{9 b^6}+\frac{(a+b x)^8 (b d-a e)^5}{8 b^6}+\frac{e^5 (a+b x)^{13}}{13 b^6} \]

[Out]

((b*d - a*e)^5*(a + b*x)^8)/(8*b^6) + (5*e*(b*d - a*e)^4*(a + b*x)^9)/(9*b^6) + (e^2*(b*d - a*e)^3*(a + b*x)^1
0)/b^6 + (10*e^3*(b*d - a*e)^2*(a + b*x)^11)/(11*b^6) + (5*e^4*(b*d - a*e)*(a + b*x)^12)/(12*b^6) + (e^5*(a +
b*x)^13)/(13*b^6)

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Rubi [A]  time = 0.35868, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 43} \[ \frac{5 e^4 (a+b x)^{12} (b d-a e)}{12 b^6}+\frac{10 e^3 (a+b x)^{11} (b d-a e)^2}{11 b^6}+\frac{e^2 (a+b x)^{10} (b d-a e)^3}{b^6}+\frac{5 e (a+b x)^9 (b d-a e)^4}{9 b^6}+\frac{(a+b x)^8 (b d-a e)^5}{8 b^6}+\frac{e^5 (a+b x)^{13}}{13 b^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

((b*d - a*e)^5*(a + b*x)^8)/(8*b^6) + (5*e*(b*d - a*e)^4*(a + b*x)^9)/(9*b^6) + (e^2*(b*d - a*e)^3*(a + b*x)^1
0)/b^6 + (10*e^3*(b*d - a*e)^2*(a + b*x)^11)/(11*b^6) + (5*e^4*(b*d - a*e)*(a + b*x)^12)/(12*b^6) + (e^5*(a +
b*x)^13)/(13*b^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 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 (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^7 (d+e x)^5 \, dx\\ &=\int \left (\frac{(b d-a e)^5 (a+b x)^7}{b^5}+\frac{5 e (b d-a e)^4 (a+b x)^8}{b^5}+\frac{10 e^2 (b d-a e)^3 (a+b x)^9}{b^5}+\frac{10 e^3 (b d-a e)^2 (a+b x)^{10}}{b^5}+\frac{5 e^4 (b d-a e) (a+b x)^{11}}{b^5}+\frac{e^5 (a+b x)^{12}}{b^5}\right ) \, dx\\ &=\frac{(b d-a e)^5 (a+b x)^8}{8 b^6}+\frac{5 e (b d-a e)^4 (a+b x)^9}{9 b^6}+\frac{e^2 (b d-a e)^3 (a+b x)^{10}}{b^6}+\frac{10 e^3 (b d-a e)^2 (a+b x)^{11}}{11 b^6}+\frac{5 e^4 (b d-a e) (a+b x)^{12}}{12 b^6}+\frac{e^5 (a+b x)^{13}}{13 b^6}\\ \end{align*}

Mathematica [B]  time = 0.147218, size = 493, normalized size = 3.45 \[ \frac{x \left (1287 a^5 b^2 x^2 \left (336 d^3 e^2 x^2+280 d^2 e^3 x^3+210 d^4 e x+56 d^5+120 d e^4 x^4+21 e^5 x^5\right )+715 a^4 b^3 x^3 \left (840 d^3 e^2 x^2+720 d^2 e^3 x^3+504 d^4 e x+126 d^5+315 d e^4 x^4+56 e^5 x^5\right )+286 a^3 b^4 x^4 \left (1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+1050 d^4 e x+252 d^5+700 d e^4 x^4+126 e^5 x^5\right )+78 a^2 b^5 x^5 \left (3465 d^3 e^2 x^2+3080 d^2 e^3 x^3+1980 d^4 e x+462 d^5+1386 d e^4 x^4+252 e^5 x^5\right )+1716 a^6 b x \left (105 d^3 e^2 x^2+84 d^2 e^3 x^3+70 d^4 e x+21 d^5+35 d e^4 x^4+6 e^5 x^5\right )+1716 a^7 \left (20 d^3 e^2 x^2+15 d^2 e^3 x^3+15 d^4 e x+6 d^5+6 d e^4 x^4+e^5 x^5\right )+13 a b^6 x^6 \left (6160 d^3 e^2 x^2+5544 d^2 e^3 x^3+3465 d^4 e x+792 d^5+2520 d e^4 x^4+462 e^5 x^5\right )+b^7 x^7 \left (10296 d^3 e^2 x^2+9360 d^2 e^3 x^3+5720 d^4 e x+1287 d^5+4290 d e^4 x^4+792 e^5 x^5\right )\right )}{10296} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(x*(1716*a^7*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e^4*x^4 + e^5*x^5) + 1716*a^6*b*x*(21
*d^5 + 70*d^4*e*x + 105*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5) + 1287*a^5*b^2*x^2*(56*d^5 +
210*d^4*e*x + 336*d^3*e^2*x^2 + 280*d^2*e^3*x^3 + 120*d*e^4*x^4 + 21*e^5*x^5) + 715*a^4*b^3*x^3*(126*d^5 + 504
*d^4*e*x + 840*d^3*e^2*x^2 + 720*d^2*e^3*x^3 + 315*d*e^4*x^4 + 56*e^5*x^5) + 286*a^3*b^4*x^4*(252*d^5 + 1050*d
^4*e*x + 1800*d^3*e^2*x^2 + 1575*d^2*e^3*x^3 + 700*d*e^4*x^4 + 126*e^5*x^5) + 78*a^2*b^5*x^5*(462*d^5 + 1980*d
^4*e*x + 3465*d^3*e^2*x^2 + 3080*d^2*e^3*x^3 + 1386*d*e^4*x^4 + 252*e^5*x^5) + 13*a*b^6*x^6*(792*d^5 + 3465*d^
4*e*x + 6160*d^3*e^2*x^2 + 5544*d^2*e^3*x^3 + 2520*d*e^4*x^4 + 462*e^5*x^5) + b^7*x^7*(1287*d^5 + 5720*d^4*e*x
+ 10296*d^3*e^2*x^2 + 9360*d^2*e^3*x^3 + 4290*d*e^4*x^4 + 792*e^5*x^5)))/10296

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Maple [B]  time = 0.001, size = 982, normalized size = 6.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

1/13*b^7*e^5*x^13+1/12*((a*e^5+5*b*d*e^4)*b^6+6*b^6*e^5*a)*x^12+1/11*((5*a*d*e^4+10*b*d^2*e^3)*b^6+6*(a*e^5+5*
b*d*e^4)*a*b^5+15*b^5*e^5*a^2)*x^11+1/10*((10*a*d^2*e^3+10*b*d^3*e^2)*b^6+6*(5*a*d*e^4+10*b*d^2*e^3)*a*b^5+15*
(a*e^5+5*b*d*e^4)*a^2*b^4+20*b^4*e^5*a^3)*x^10+1/9*((10*a*d^3*e^2+5*b*d^4*e)*b^6+6*(10*a*d^2*e^3+10*b*d^3*e^2)
*a*b^5+15*(5*a*d*e^4+10*b*d^2*e^3)*a^2*b^4+20*(a*e^5+5*b*d*e^4)*a^3*b^3+15*b^3*e^5*a^4)*x^9+1/8*((5*a*d^4*e+b*
d^5)*b^6+6*(10*a*d^3*e^2+5*b*d^4*e)*a*b^5+15*(10*a*d^2*e^3+10*b*d^3*e^2)*a^2*b^4+20*(5*a*d*e^4+10*b*d^2*e^3)*a
^3*b^3+15*(a*e^5+5*b*d*e^4)*a^4*b^2+6*b^2*e^5*a^5)*x^8+1/7*(a*d^5*b^6+6*(5*a*d^4*e+b*d^5)*a*b^5+15*(10*a*d^3*e
^2+5*b*d^4*e)*a^2*b^4+20*(10*a*d^2*e^3+10*b*d^3*e^2)*a^3*b^3+15*(5*a*d*e^4+10*b*d^2*e^3)*a^4*b^2+6*(a*e^5+5*b*
d*e^4)*a^5*b+b*e^5*a^6)*x^7+1/6*(6*a^2*d^5*b^5+15*(5*a*d^4*e+b*d^5)*a^2*b^4+20*(10*a*d^3*e^2+5*b*d^4*e)*a^3*b^
3+15*(10*a*d^2*e^3+10*b*d^3*e^2)*a^4*b^2+6*(5*a*d*e^4+10*b*d^2*e^3)*a^5*b+(a*e^5+5*b*d*e^4)*a^6)*x^6+1/5*(15*a
^3*d^5*b^4+20*(5*a*d^4*e+b*d^5)*a^3*b^3+15*(10*a*d^3*e^2+5*b*d^4*e)*a^4*b^2+6*(10*a*d^2*e^3+10*b*d^3*e^2)*a^5*
b+(5*a*d*e^4+10*b*d^2*e^3)*a^6)*x^5+1/4*(20*a^4*d^5*b^3+15*(5*a*d^4*e+b*d^5)*a^4*b^2+6*(10*a*d^3*e^2+5*b*d^4*e
)*a^5*b+(10*a*d^2*e^3+10*b*d^3*e^2)*a^6)*x^4+1/3*(15*a^5*d^5*b^2+6*(5*a*d^4*e+b*d^5)*a^5*b+(10*a*d^3*e^2+5*b*d
^4*e)*a^6)*x^3+1/2*(6*a^6*d^5*b+(5*a*d^4*e+b*d^5)*a^6)*x^2+a^7*d^5*x

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Maxima [B]  time = 0.994932, size = 802, normalized size = 5.61 \begin{align*} \frac{1}{13} \, b^{7} e^{5} x^{13} + a^{7} d^{5} x + \frac{1}{12} \,{\left (5 \, b^{7} d e^{4} + 7 \, a b^{6} e^{5}\right )} x^{12} + \frac{1}{11} \,{\left (10 \, b^{7} d^{2} e^{3} + 35 \, a b^{6} d e^{4} + 21 \, a^{2} b^{5} e^{5}\right )} x^{11} + \frac{1}{2} \,{\left (2 \, b^{7} d^{3} e^{2} + 14 \, a b^{6} d^{2} e^{3} + 21 \, a^{2} b^{5} d e^{4} + 7 \, a^{3} b^{4} e^{5}\right )} x^{10} + \frac{5}{9} \,{\left (b^{7} d^{4} e + 14 \, a b^{6} d^{3} e^{2} + 42 \, a^{2} b^{5} d^{2} e^{3} + 35 \, a^{3} b^{4} d e^{4} + 7 \, a^{4} b^{3} e^{5}\right )} x^{9} + \frac{1}{8} \,{\left (b^{7} d^{5} + 35 \, a b^{6} d^{4} e + 210 \, a^{2} b^{5} d^{3} e^{2} + 350 \, a^{3} b^{4} d^{2} e^{3} + 175 \, a^{4} b^{3} d e^{4} + 21 \, a^{5} b^{2} e^{5}\right )} x^{8} +{\left (a b^{6} d^{5} + 15 \, a^{2} b^{5} d^{4} e + 50 \, a^{3} b^{4} d^{3} e^{2} + 50 \, a^{4} b^{3} d^{2} e^{3} + 15 \, a^{5} b^{2} d e^{4} + a^{6} b e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (21 \, a^{2} b^{5} d^{5} + 175 \, a^{3} b^{4} d^{4} e + 350 \, a^{4} b^{3} d^{3} e^{2} + 210 \, a^{5} b^{2} d^{2} e^{3} + 35 \, a^{6} b d e^{4} + a^{7} e^{5}\right )} x^{6} +{\left (7 \, a^{3} b^{4} d^{5} + 35 \, a^{4} b^{3} d^{4} e + 42 \, a^{5} b^{2} d^{3} e^{2} + 14 \, a^{6} b d^{2} e^{3} + a^{7} d e^{4}\right )} x^{5} + \frac{5}{4} \,{\left (7 \, a^{4} b^{3} d^{5} + 21 \, a^{5} b^{2} d^{4} e + 14 \, a^{6} b d^{3} e^{2} + 2 \, a^{7} d^{2} e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (21 \, a^{5} b^{2} d^{5} + 35 \, a^{6} b d^{4} e + 10 \, a^{7} d^{3} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (7 \, a^{6} b d^{5} + 5 \, a^{7} d^{4} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

1/13*b^7*e^5*x^13 + a^7*d^5*x + 1/12*(5*b^7*d*e^4 + 7*a*b^6*e^5)*x^12 + 1/11*(10*b^7*d^2*e^3 + 35*a*b^6*d*e^4
+ 21*a^2*b^5*e^5)*x^11 + 1/2*(2*b^7*d^3*e^2 + 14*a*b^6*d^2*e^3 + 21*a^2*b^5*d*e^4 + 7*a^3*b^4*e^5)*x^10 + 5/9*
(b^7*d^4*e + 14*a*b^6*d^3*e^2 + 42*a^2*b^5*d^2*e^3 + 35*a^3*b^4*d*e^4 + 7*a^4*b^3*e^5)*x^9 + 1/8*(b^7*d^5 + 35
*a*b^6*d^4*e + 210*a^2*b^5*d^3*e^2 + 350*a^3*b^4*d^2*e^3 + 175*a^4*b^3*d*e^4 + 21*a^5*b^2*e^5)*x^8 + (a*b^6*d^
5 + 15*a^2*b^5*d^4*e + 50*a^3*b^4*d^3*e^2 + 50*a^4*b^3*d^2*e^3 + 15*a^5*b^2*d*e^4 + a^6*b*e^5)*x^7 + 1/6*(21*a
^2*b^5*d^5 + 175*a^3*b^4*d^4*e + 350*a^4*b^3*d^3*e^2 + 210*a^5*b^2*d^2*e^3 + 35*a^6*b*d*e^4 + a^7*e^5)*x^6 + (
7*a^3*b^4*d^5 + 35*a^4*b^3*d^4*e + 42*a^5*b^2*d^3*e^2 + 14*a^6*b*d^2*e^3 + a^7*d*e^4)*x^5 + 5/4*(7*a^4*b^3*d^5
+ 21*a^5*b^2*d^4*e + 14*a^6*b*d^3*e^2 + 2*a^7*d^2*e^3)*x^4 + 1/3*(21*a^5*b^2*d^5 + 35*a^6*b*d^4*e + 10*a^7*d^
3*e^2)*x^3 + 1/2*(7*a^6*b*d^5 + 5*a^7*d^4*e)*x^2

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Fricas [B]  time = 1.31593, size = 1467, normalized size = 10.26 \begin{align*} \frac{1}{13} x^{13} e^{5} b^{7} + \frac{5}{12} x^{12} e^{4} d b^{7} + \frac{7}{12} x^{12} e^{5} b^{6} a + \frac{10}{11} x^{11} e^{3} d^{2} b^{7} + \frac{35}{11} x^{11} e^{4} d b^{6} a + \frac{21}{11} x^{11} e^{5} b^{5} a^{2} + x^{10} e^{2} d^{3} b^{7} + 7 x^{10} e^{3} d^{2} b^{6} a + \frac{21}{2} x^{10} e^{4} d b^{5} a^{2} + \frac{7}{2} x^{10} e^{5} b^{4} a^{3} + \frac{5}{9} x^{9} e d^{4} b^{7} + \frac{70}{9} x^{9} e^{2} d^{3} b^{6} a + \frac{70}{3} x^{9} e^{3} d^{2} b^{5} a^{2} + \frac{175}{9} x^{9} e^{4} d b^{4} a^{3} + \frac{35}{9} x^{9} e^{5} b^{3} a^{4} + \frac{1}{8} x^{8} d^{5} b^{7} + \frac{35}{8} x^{8} e d^{4} b^{6} a + \frac{105}{4} x^{8} e^{2} d^{3} b^{5} a^{2} + \frac{175}{4} x^{8} e^{3} d^{2} b^{4} a^{3} + \frac{175}{8} x^{8} e^{4} d b^{3} a^{4} + \frac{21}{8} x^{8} e^{5} b^{2} a^{5} + x^{7} d^{5} b^{6} a + 15 x^{7} e d^{4} b^{5} a^{2} + 50 x^{7} e^{2} d^{3} b^{4} a^{3} + 50 x^{7} e^{3} d^{2} b^{3} a^{4} + 15 x^{7} e^{4} d b^{2} a^{5} + x^{7} e^{5} b a^{6} + \frac{7}{2} x^{6} d^{5} b^{5} a^{2} + \frac{175}{6} x^{6} e d^{4} b^{4} a^{3} + \frac{175}{3} x^{6} e^{2} d^{3} b^{3} a^{4} + 35 x^{6} e^{3} d^{2} b^{2} a^{5} + \frac{35}{6} x^{6} e^{4} d b a^{6} + \frac{1}{6} x^{6} e^{5} a^{7} + 7 x^{5} d^{5} b^{4} a^{3} + 35 x^{5} e d^{4} b^{3} a^{4} + 42 x^{5} e^{2} d^{3} b^{2} a^{5} + 14 x^{5} e^{3} d^{2} b a^{6} + x^{5} e^{4} d a^{7} + \frac{35}{4} x^{4} d^{5} b^{3} a^{4} + \frac{105}{4} x^{4} e d^{4} b^{2} a^{5} + \frac{35}{2} x^{4} e^{2} d^{3} b a^{6} + \frac{5}{2} x^{4} e^{3} d^{2} a^{7} + 7 x^{3} d^{5} b^{2} a^{5} + \frac{35}{3} x^{3} e d^{4} b a^{6} + \frac{10}{3} x^{3} e^{2} d^{3} a^{7} + \frac{7}{2} x^{2} d^{5} b a^{6} + \frac{5}{2} x^{2} e d^{4} a^{7} + x d^{5} a^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

1/13*x^13*e^5*b^7 + 5/12*x^12*e^4*d*b^7 + 7/12*x^12*e^5*b^6*a + 10/11*x^11*e^3*d^2*b^7 + 35/11*x^11*e^4*d*b^6*
a + 21/11*x^11*e^5*b^5*a^2 + x^10*e^2*d^3*b^7 + 7*x^10*e^3*d^2*b^6*a + 21/2*x^10*e^4*d*b^5*a^2 + 7/2*x^10*e^5*
b^4*a^3 + 5/9*x^9*e*d^4*b^7 + 70/9*x^9*e^2*d^3*b^6*a + 70/3*x^9*e^3*d^2*b^5*a^2 + 175/9*x^9*e^4*d*b^4*a^3 + 35
/9*x^9*e^5*b^3*a^4 + 1/8*x^8*d^5*b^7 + 35/8*x^8*e*d^4*b^6*a + 105/4*x^8*e^2*d^3*b^5*a^2 + 175/4*x^8*e^3*d^2*b^
4*a^3 + 175/8*x^8*e^4*d*b^3*a^4 + 21/8*x^8*e^5*b^2*a^5 + x^7*d^5*b^6*a + 15*x^7*e*d^4*b^5*a^2 + 50*x^7*e^2*d^3
*b^4*a^3 + 50*x^7*e^3*d^2*b^3*a^4 + 15*x^7*e^4*d*b^2*a^5 + x^7*e^5*b*a^6 + 7/2*x^6*d^5*b^5*a^2 + 175/6*x^6*e*d
^4*b^4*a^3 + 175/3*x^6*e^2*d^3*b^3*a^4 + 35*x^6*e^3*d^2*b^2*a^5 + 35/6*x^6*e^4*d*b*a^6 + 1/6*x^6*e^5*a^7 + 7*x
^5*d^5*b^4*a^3 + 35*x^5*e*d^4*b^3*a^4 + 42*x^5*e^2*d^3*b^2*a^5 + 14*x^5*e^3*d^2*b*a^6 + x^5*e^4*d*a^7 + 35/4*x
^4*d^5*b^3*a^4 + 105/4*x^4*e*d^4*b^2*a^5 + 35/2*x^4*e^2*d^3*b*a^6 + 5/2*x^4*e^3*d^2*a^7 + 7*x^3*d^5*b^2*a^5 +
35/3*x^3*e*d^4*b*a^6 + 10/3*x^3*e^2*d^3*a^7 + 7/2*x^2*d^5*b*a^6 + 5/2*x^2*e*d^4*a^7 + x*d^5*a^7

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Sympy [B]  time = 0.157246, size = 673, normalized size = 4.71 \begin{align*} a^{7} d^{5} x + \frac{b^{7} e^{5} x^{13}}{13} + x^{12} \left (\frac{7 a b^{6} e^{5}}{12} + \frac{5 b^{7} d e^{4}}{12}\right ) + x^{11} \left (\frac{21 a^{2} b^{5} e^{5}}{11} + \frac{35 a b^{6} d e^{4}}{11} + \frac{10 b^{7} d^{2} e^{3}}{11}\right ) + x^{10} \left (\frac{7 a^{3} b^{4} e^{5}}{2} + \frac{21 a^{2} b^{5} d e^{4}}{2} + 7 a b^{6} d^{2} e^{3} + b^{7} d^{3} e^{2}\right ) + x^{9} \left (\frac{35 a^{4} b^{3} e^{5}}{9} + \frac{175 a^{3} b^{4} d e^{4}}{9} + \frac{70 a^{2} b^{5} d^{2} e^{3}}{3} + \frac{70 a b^{6} d^{3} e^{2}}{9} + \frac{5 b^{7} d^{4} e}{9}\right ) + x^{8} \left (\frac{21 a^{5} b^{2} e^{5}}{8} + \frac{175 a^{4} b^{3} d e^{4}}{8} + \frac{175 a^{3} b^{4} d^{2} e^{3}}{4} + \frac{105 a^{2} b^{5} d^{3} e^{2}}{4} + \frac{35 a b^{6} d^{4} e}{8} + \frac{b^{7} d^{5}}{8}\right ) + x^{7} \left (a^{6} b e^{5} + 15 a^{5} b^{2} d e^{4} + 50 a^{4} b^{3} d^{2} e^{3} + 50 a^{3} b^{4} d^{3} e^{2} + 15 a^{2} b^{5} d^{4} e + a b^{6} d^{5}\right ) + x^{6} \left (\frac{a^{7} e^{5}}{6} + \frac{35 a^{6} b d e^{4}}{6} + 35 a^{5} b^{2} d^{2} e^{3} + \frac{175 a^{4} b^{3} d^{3} e^{2}}{3} + \frac{175 a^{3} b^{4} d^{4} e}{6} + \frac{7 a^{2} b^{5} d^{5}}{2}\right ) + x^{5} \left (a^{7} d e^{4} + 14 a^{6} b d^{2} e^{3} + 42 a^{5} b^{2} d^{3} e^{2} + 35 a^{4} b^{3} d^{4} e + 7 a^{3} b^{4} d^{5}\right ) + x^{4} \left (\frac{5 a^{7} d^{2} e^{3}}{2} + \frac{35 a^{6} b d^{3} e^{2}}{2} + \frac{105 a^{5} b^{2} d^{4} e}{4} + \frac{35 a^{4} b^{3} d^{5}}{4}\right ) + x^{3} \left (\frac{10 a^{7} d^{3} e^{2}}{3} + \frac{35 a^{6} b d^{4} e}{3} + 7 a^{5} b^{2} d^{5}\right ) + x^{2} \left (\frac{5 a^{7} d^{4} e}{2} + \frac{7 a^{6} b d^{5}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**5*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

a**7*d**5*x + b**7*e**5*x**13/13 + x**12*(7*a*b**6*e**5/12 + 5*b**7*d*e**4/12) + x**11*(21*a**2*b**5*e**5/11 +
35*a*b**6*d*e**4/11 + 10*b**7*d**2*e**3/11) + x**10*(7*a**3*b**4*e**5/2 + 21*a**2*b**5*d*e**4/2 + 7*a*b**6*d*
*2*e**3 + b**7*d**3*e**2) + x**9*(35*a**4*b**3*e**5/9 + 175*a**3*b**4*d*e**4/9 + 70*a**2*b**5*d**2*e**3/3 + 70
*a*b**6*d**3*e**2/9 + 5*b**7*d**4*e/9) + x**8*(21*a**5*b**2*e**5/8 + 175*a**4*b**3*d*e**4/8 + 175*a**3*b**4*d*
*2*e**3/4 + 105*a**2*b**5*d**3*e**2/4 + 35*a*b**6*d**4*e/8 + b**7*d**5/8) + x**7*(a**6*b*e**5 + 15*a**5*b**2*d
*e**4 + 50*a**4*b**3*d**2*e**3 + 50*a**3*b**4*d**3*e**2 + 15*a**2*b**5*d**4*e + a*b**6*d**5) + x**6*(a**7*e**5
/6 + 35*a**6*b*d*e**4/6 + 35*a**5*b**2*d**2*e**3 + 175*a**4*b**3*d**3*e**2/3 + 175*a**3*b**4*d**4*e/6 + 7*a**2
*b**5*d**5/2) + x**5*(a**7*d*e**4 + 14*a**6*b*d**2*e**3 + 42*a**5*b**2*d**3*e**2 + 35*a**4*b**3*d**4*e + 7*a**
3*b**4*d**5) + x**4*(5*a**7*d**2*e**3/2 + 35*a**6*b*d**3*e**2/2 + 105*a**5*b**2*d**4*e/4 + 35*a**4*b**3*d**5/4
) + x**3*(10*a**7*d**3*e**2/3 + 35*a**6*b*d**4*e/3 + 7*a**5*b**2*d**5) + x**2*(5*a**7*d**4*e/2 + 7*a**6*b*d**5
/2)

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Giac [B]  time = 1.10771, size = 872, normalized size = 6.1 \begin{align*} \frac{1}{13} \, b^{7} x^{13} e^{5} + \frac{5}{12} \, b^{7} d x^{12} e^{4} + \frac{10}{11} \, b^{7} d^{2} x^{11} e^{3} + b^{7} d^{3} x^{10} e^{2} + \frac{5}{9} \, b^{7} d^{4} x^{9} e + \frac{1}{8} \, b^{7} d^{5} x^{8} + \frac{7}{12} \, a b^{6} x^{12} e^{5} + \frac{35}{11} \, a b^{6} d x^{11} e^{4} + 7 \, a b^{6} d^{2} x^{10} e^{3} + \frac{70}{9} \, a b^{6} d^{3} x^{9} e^{2} + \frac{35}{8} \, a b^{6} d^{4} x^{8} e + a b^{6} d^{5} x^{7} + \frac{21}{11} \, a^{2} b^{5} x^{11} e^{5} + \frac{21}{2} \, a^{2} b^{5} d x^{10} e^{4} + \frac{70}{3} \, a^{2} b^{5} d^{2} x^{9} e^{3} + \frac{105}{4} \, a^{2} b^{5} d^{3} x^{8} e^{2} + 15 \, a^{2} b^{5} d^{4} x^{7} e + \frac{7}{2} \, a^{2} b^{5} d^{5} x^{6} + \frac{7}{2} \, a^{3} b^{4} x^{10} e^{5} + \frac{175}{9} \, a^{3} b^{4} d x^{9} e^{4} + \frac{175}{4} \, a^{3} b^{4} d^{2} x^{8} e^{3} + 50 \, a^{3} b^{4} d^{3} x^{7} e^{2} + \frac{175}{6} \, a^{3} b^{4} d^{4} x^{6} e + 7 \, a^{3} b^{4} d^{5} x^{5} + \frac{35}{9} \, a^{4} b^{3} x^{9} e^{5} + \frac{175}{8} \, a^{4} b^{3} d x^{8} e^{4} + 50 \, a^{4} b^{3} d^{2} x^{7} e^{3} + \frac{175}{3} \, a^{4} b^{3} d^{3} x^{6} e^{2} + 35 \, a^{4} b^{3} d^{4} x^{5} e + \frac{35}{4} \, a^{4} b^{3} d^{5} x^{4} + \frac{21}{8} \, a^{5} b^{2} x^{8} e^{5} + 15 \, a^{5} b^{2} d x^{7} e^{4} + 35 \, a^{5} b^{2} d^{2} x^{6} e^{3} + 42 \, a^{5} b^{2} d^{3} x^{5} e^{2} + \frac{105}{4} \, a^{5} b^{2} d^{4} x^{4} e + 7 \, a^{5} b^{2} d^{5} x^{3} + a^{6} b x^{7} e^{5} + \frac{35}{6} \, a^{6} b d x^{6} e^{4} + 14 \, a^{6} b d^{2} x^{5} e^{3} + \frac{35}{2} \, a^{6} b d^{3} x^{4} e^{2} + \frac{35}{3} \, a^{6} b d^{4} x^{3} e + \frac{7}{2} \, a^{6} b d^{5} x^{2} + \frac{1}{6} \, a^{7} x^{6} e^{5} + a^{7} d x^{5} e^{4} + \frac{5}{2} \, a^{7} d^{2} x^{4} e^{3} + \frac{10}{3} \, a^{7} d^{3} x^{3} e^{2} + \frac{5}{2} \, a^{7} d^{4} x^{2} e + a^{7} d^{5} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

1/13*b^7*x^13*e^5 + 5/12*b^7*d*x^12*e^4 + 10/11*b^7*d^2*x^11*e^3 + b^7*d^3*x^10*e^2 + 5/9*b^7*d^4*x^9*e + 1/8*
b^7*d^5*x^8 + 7/12*a*b^6*x^12*e^5 + 35/11*a*b^6*d*x^11*e^4 + 7*a*b^6*d^2*x^10*e^3 + 70/9*a*b^6*d^3*x^9*e^2 + 3
5/8*a*b^6*d^4*x^8*e + a*b^6*d^5*x^7 + 21/11*a^2*b^5*x^11*e^5 + 21/2*a^2*b^5*d*x^10*e^4 + 70/3*a^2*b^5*d^2*x^9*
e^3 + 105/4*a^2*b^5*d^3*x^8*e^2 + 15*a^2*b^5*d^4*x^7*e + 7/2*a^2*b^5*d^5*x^6 + 7/2*a^3*b^4*x^10*e^5 + 175/9*a^
3*b^4*d*x^9*e^4 + 175/4*a^3*b^4*d^2*x^8*e^3 + 50*a^3*b^4*d^3*x^7*e^2 + 175/6*a^3*b^4*d^4*x^6*e + 7*a^3*b^4*d^5
*x^5 + 35/9*a^4*b^3*x^9*e^5 + 175/8*a^4*b^3*d*x^8*e^4 + 50*a^4*b^3*d^2*x^7*e^3 + 175/3*a^4*b^3*d^3*x^6*e^2 + 3
5*a^4*b^3*d^4*x^5*e + 35/4*a^4*b^3*d^5*x^4 + 21/8*a^5*b^2*x^8*e^5 + 15*a^5*b^2*d*x^7*e^4 + 35*a^5*b^2*d^2*x^6*
e^3 + 42*a^5*b^2*d^3*x^5*e^2 + 105/4*a^5*b^2*d^4*x^4*e + 7*a^5*b^2*d^5*x^3 + a^6*b*x^7*e^5 + 35/6*a^6*b*d*x^6*
e^4 + 14*a^6*b*d^2*x^5*e^3 + 35/2*a^6*b*d^3*x^4*e^2 + 35/3*a^6*b*d^4*x^3*e + 7/2*a^6*b*d^5*x^2 + 1/6*a^7*x^6*e
^5 + a^7*d*x^5*e^4 + 5/2*a^7*d^2*x^4*e^3 + 10/3*a^7*d^3*x^3*e^2 + 5/2*a^7*d^4*x^2*e + a^7*d^5*x