3.1918 \(\int (a+b x) (d+e x)^4 (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=119 \[ \frac{4 e^3 (a+b x)^{11} (b d-a e)}{11 b^5}+\frac{3 e^2 (a+b x)^{10} (b d-a e)^2}{5 b^5}+\frac{4 e (a+b x)^9 (b d-a e)^3}{9 b^5}+\frac{(a+b x)^8 (b d-a e)^4}{8 b^5}+\frac{e^4 (a+b x)^{12}}{12 b^5} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.276049, antiderivative size = 119, 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} \[ \frac{4 e^3 (a+b x)^{11} (b d-a e)}{11 b^5}+\frac{3 e^2 (a+b x)^{10} (b d-a e)^2}{5 b^5}+\frac{4 e (a+b x)^9 (b d-a e)^3}{9 b^5}+\frac{(a+b x)^8 (b d-a e)^4}{8 b^5}+\frac{e^4 (a+b x)^{12}}{12 b^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

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

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

Mathematica [B]  time = 0.130537, size = 405, normalized size = 3.4 \[ \frac{x \left (792 a^5 b^2 x^2 \left (126 d^2 e^2 x^2+105 d^3 e x+35 d^4+70 d e^3 x^3+15 e^4 x^4\right )+495 a^4 b^3 x^3 \left (280 d^2 e^2 x^2+224 d^3 e x+70 d^4+160 d e^3 x^3+35 e^4 x^4\right )+220 a^3 b^4 x^4 \left (540 d^2 e^2 x^2+420 d^3 e x+126 d^4+315 d e^3 x^3+70 e^4 x^4\right )+66 a^2 b^5 x^5 \left (945 d^2 e^2 x^2+720 d^3 e x+210 d^4+560 d e^3 x^3+126 e^4 x^4\right )+924 a^6 b x \left (45 d^2 e^2 x^2+40 d^3 e x+15 d^4+24 d e^3 x^3+5 e^4 x^4\right )+792 a^7 \left (10 d^2 e^2 x^2+10 d^3 e x+5 d^4+5 d e^3 x^3+e^4 x^4\right )+12 a b^6 x^6 \left (1540 d^2 e^2 x^2+1155 d^3 e x+330 d^4+924 d e^3 x^3+210 e^4 x^4\right )+b^7 x^7 \left (2376 d^2 e^2 x^2+1760 d^3 e x+495 d^4+1440 d e^3 x^3+330 e^4 x^4\right )\right )}{3960} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(x*(792*a^7*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4) + 924*a^6*b*x*(15*d^4 + 40*d^3*e*x +
45*d^2*e^2*x^2 + 24*d*e^3*x^3 + 5*e^4*x^4) + 792*a^5*b^2*x^2*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^2 + 70*d*e
^3*x^3 + 15*e^4*x^4) + 495*a^4*b^3*x^3*(70*d^4 + 224*d^3*e*x + 280*d^2*e^2*x^2 + 160*d*e^3*x^3 + 35*e^4*x^4) +
220*a^3*b^4*x^4*(126*d^4 + 420*d^3*e*x + 540*d^2*e^2*x^2 + 315*d*e^3*x^3 + 70*e^4*x^4) + 66*a^2*b^5*x^5*(210*
d^4 + 720*d^3*e*x + 945*d^2*e^2*x^2 + 560*d*e^3*x^3 + 126*e^4*x^4) + 12*a*b^6*x^6*(330*d^4 + 1155*d^3*e*x + 15
40*d^2*e^2*x^2 + 924*d*e^3*x^3 + 210*e^4*x^4) + b^7*x^7*(495*d^4 + 1760*d^3*e*x + 2376*d^2*e^2*x^2 + 1440*d*e^
3*x^3 + 330*e^4*x^4)))/3960

________________________________________________________________________________________

Maple [B]  time = 0.003, size = 799, normalized size = 6.7 \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)^4*(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 0.981105, size = 660, normalized size = 5.55 \begin{align*} \frac{1}{12} \, b^{7} e^{4} x^{12} + a^{7} d^{4} x + \frac{1}{11} \,{\left (4 \, b^{7} d e^{3} + 7 \, a b^{6} e^{4}\right )} x^{11} + \frac{1}{10} \,{\left (6 \, b^{7} d^{2} e^{2} + 28 \, a b^{6} d e^{3} + 21 \, a^{2} b^{5} e^{4}\right )} x^{10} + \frac{1}{9} \,{\left (4 \, b^{7} d^{3} e + 42 \, a b^{6} d^{2} e^{2} + 84 \, a^{2} b^{5} d e^{3} + 35 \, a^{3} b^{4} e^{4}\right )} x^{9} + \frac{1}{8} \,{\left (b^{7} d^{4} + 28 \, a b^{6} d^{3} e + 126 \, a^{2} b^{5} d^{2} e^{2} + 140 \, a^{3} b^{4} d e^{3} + 35 \, a^{4} b^{3} e^{4}\right )} x^{8} +{\left (a b^{6} d^{4} + 12 \, a^{2} b^{5} d^{3} e + 30 \, a^{3} b^{4} d^{2} e^{2} + 20 \, a^{4} b^{3} d e^{3} + 3 \, a^{5} b^{2} e^{4}\right )} x^{7} + \frac{7}{6} \,{\left (3 \, a^{2} b^{5} d^{4} + 20 \, a^{3} b^{4} d^{3} e + 30 \, a^{4} b^{3} d^{2} e^{2} + 12 \, a^{5} b^{2} d e^{3} + a^{6} b e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (35 \, a^{3} b^{4} d^{4} + 140 \, a^{4} b^{3} d^{3} e + 126 \, a^{5} b^{2} d^{2} e^{2} + 28 \, a^{6} b d e^{3} + a^{7} e^{4}\right )} x^{5} + \frac{1}{4} \,{\left (35 \, a^{4} b^{3} d^{4} + 84 \, a^{5} b^{2} d^{3} e + 42 \, a^{6} b d^{2} e^{2} + 4 \, a^{7} d e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (21 \, a^{5} b^{2} d^{4} + 28 \, a^{6} b d^{3} e + 6 \, a^{7} d^{2} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (7 \, a^{6} b d^{4} + 4 \, a^{7} d^{3} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

1/12*b^7*e^4*x^12 + a^7*d^4*x + 1/11*(4*b^7*d*e^3 + 7*a*b^6*e^4)*x^11 + 1/10*(6*b^7*d^2*e^2 + 28*a*b^6*d*e^3 +
21*a^2*b^5*e^4)*x^10 + 1/9*(4*b^7*d^3*e + 42*a*b^6*d^2*e^2 + 84*a^2*b^5*d*e^3 + 35*a^3*b^4*e^4)*x^9 + 1/8*(b^
7*d^4 + 28*a*b^6*d^3*e + 126*a^2*b^5*d^2*e^2 + 140*a^3*b^4*d*e^3 + 35*a^4*b^3*e^4)*x^8 + (a*b^6*d^4 + 12*a^2*b
^5*d^3*e + 30*a^3*b^4*d^2*e^2 + 20*a^4*b^3*d*e^3 + 3*a^5*b^2*e^4)*x^7 + 7/6*(3*a^2*b^5*d^4 + 20*a^3*b^4*d^3*e
+ 30*a^4*b^3*d^2*e^2 + 12*a^5*b^2*d*e^3 + a^6*b*e^4)*x^6 + 1/5*(35*a^3*b^4*d^4 + 140*a^4*b^3*d^3*e + 126*a^5*b
^2*d^2*e^2 + 28*a^6*b*d*e^3 + a^7*e^4)*x^5 + 1/4*(35*a^4*b^3*d^4 + 84*a^5*b^2*d^3*e + 42*a^6*b*d^2*e^2 + 4*a^7
*d*e^3)*x^4 + 1/3*(21*a^5*b^2*d^4 + 28*a^6*b*d^3*e + 6*a^7*d^2*e^2)*x^3 + 1/2*(7*a^6*b*d^4 + 4*a^7*d^3*e)*x^2

________________________________________________________________________________________

Fricas [B]  time = 1.34082, size = 1184, normalized size = 9.95 \begin{align*} \frac{1}{12} x^{12} e^{4} b^{7} + \frac{4}{11} x^{11} e^{3} d b^{7} + \frac{7}{11} x^{11} e^{4} b^{6} a + \frac{3}{5} x^{10} e^{2} d^{2} b^{7} + \frac{14}{5} x^{10} e^{3} d b^{6} a + \frac{21}{10} x^{10} e^{4} b^{5} a^{2} + \frac{4}{9} x^{9} e d^{3} b^{7} + \frac{14}{3} x^{9} e^{2} d^{2} b^{6} a + \frac{28}{3} x^{9} e^{3} d b^{5} a^{2} + \frac{35}{9} x^{9} e^{4} b^{4} a^{3} + \frac{1}{8} x^{8} d^{4} b^{7} + \frac{7}{2} x^{8} e d^{3} b^{6} a + \frac{63}{4} x^{8} e^{2} d^{2} b^{5} a^{2} + \frac{35}{2} x^{8} e^{3} d b^{4} a^{3} + \frac{35}{8} x^{8} e^{4} b^{3} a^{4} + x^{7} d^{4} b^{6} a + 12 x^{7} e d^{3} b^{5} a^{2} + 30 x^{7} e^{2} d^{2} b^{4} a^{3} + 20 x^{7} e^{3} d b^{3} a^{4} + 3 x^{7} e^{4} b^{2} a^{5} + \frac{7}{2} x^{6} d^{4} b^{5} a^{2} + \frac{70}{3} x^{6} e d^{3} b^{4} a^{3} + 35 x^{6} e^{2} d^{2} b^{3} a^{4} + 14 x^{6} e^{3} d b^{2} a^{5} + \frac{7}{6} x^{6} e^{4} b a^{6} + 7 x^{5} d^{4} b^{4} a^{3} + 28 x^{5} e d^{3} b^{3} a^{4} + \frac{126}{5} x^{5} e^{2} d^{2} b^{2} a^{5} + \frac{28}{5} x^{5} e^{3} d b a^{6} + \frac{1}{5} x^{5} e^{4} a^{7} + \frac{35}{4} x^{4} d^{4} b^{3} a^{4} + 21 x^{4} e d^{3} b^{2} a^{5} + \frac{21}{2} x^{4} e^{2} d^{2} b a^{6} + x^{4} e^{3} d a^{7} + 7 x^{3} d^{4} b^{2} a^{5} + \frac{28}{3} x^{3} e d^{3} b a^{6} + 2 x^{3} e^{2} d^{2} a^{7} + \frac{7}{2} x^{2} d^{4} b a^{6} + 2 x^{2} e d^{3} a^{7} + x d^{4} a^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

1/12*x^12*e^4*b^7 + 4/11*x^11*e^3*d*b^7 + 7/11*x^11*e^4*b^6*a + 3/5*x^10*e^2*d^2*b^7 + 14/5*x^10*e^3*d*b^6*a +
21/10*x^10*e^4*b^5*a^2 + 4/9*x^9*e*d^3*b^7 + 14/3*x^9*e^2*d^2*b^6*a + 28/3*x^9*e^3*d*b^5*a^2 + 35/9*x^9*e^4*b
^4*a^3 + 1/8*x^8*d^4*b^7 + 7/2*x^8*e*d^3*b^6*a + 63/4*x^8*e^2*d^2*b^5*a^2 + 35/2*x^8*e^3*d*b^4*a^3 + 35/8*x^8*
e^4*b^3*a^4 + x^7*d^4*b^6*a + 12*x^7*e*d^3*b^5*a^2 + 30*x^7*e^2*d^2*b^4*a^3 + 20*x^7*e^3*d*b^3*a^4 + 3*x^7*e^4
*b^2*a^5 + 7/2*x^6*d^4*b^5*a^2 + 70/3*x^6*e*d^3*b^4*a^3 + 35*x^6*e^2*d^2*b^3*a^4 + 14*x^6*e^3*d*b^2*a^5 + 7/6*
x^6*e^4*b*a^6 + 7*x^5*d^4*b^4*a^3 + 28*x^5*e*d^3*b^3*a^4 + 126/5*x^5*e^2*d^2*b^2*a^5 + 28/5*x^5*e^3*d*b*a^6 +
1/5*x^5*e^4*a^7 + 35/4*x^4*d^4*b^3*a^4 + 21*x^4*e*d^3*b^2*a^5 + 21/2*x^4*e^2*d^2*b*a^6 + x^4*e^3*d*a^7 + 7*x^3
*d^4*b^2*a^5 + 28/3*x^3*e*d^3*b*a^6 + 2*x^3*e^2*d^2*a^7 + 7/2*x^2*d^4*b*a^6 + 2*x^2*e*d^3*a^7 + x*d^4*a^7

________________________________________________________________________________________

Sympy [B]  time = 0.140896, size = 549, normalized size = 4.61 \begin{align*} a^{7} d^{4} x + \frac{b^{7} e^{4} x^{12}}{12} + x^{11} \left (\frac{7 a b^{6} e^{4}}{11} + \frac{4 b^{7} d e^{3}}{11}\right ) + x^{10} \left (\frac{21 a^{2} b^{5} e^{4}}{10} + \frac{14 a b^{6} d e^{3}}{5} + \frac{3 b^{7} d^{2} e^{2}}{5}\right ) + x^{9} \left (\frac{35 a^{3} b^{4} e^{4}}{9} + \frac{28 a^{2} b^{5} d e^{3}}{3} + \frac{14 a b^{6} d^{2} e^{2}}{3} + \frac{4 b^{7} d^{3} e}{9}\right ) + x^{8} \left (\frac{35 a^{4} b^{3} e^{4}}{8} + \frac{35 a^{3} b^{4} d e^{3}}{2} + \frac{63 a^{2} b^{5} d^{2} e^{2}}{4} + \frac{7 a b^{6} d^{3} e}{2} + \frac{b^{7} d^{4}}{8}\right ) + x^{7} \left (3 a^{5} b^{2} e^{4} + 20 a^{4} b^{3} d e^{3} + 30 a^{3} b^{4} d^{2} e^{2} + 12 a^{2} b^{5} d^{3} e + a b^{6} d^{4}\right ) + x^{6} \left (\frac{7 a^{6} b e^{4}}{6} + 14 a^{5} b^{2} d e^{3} + 35 a^{4} b^{3} d^{2} e^{2} + \frac{70 a^{3} b^{4} d^{3} e}{3} + \frac{7 a^{2} b^{5} d^{4}}{2}\right ) + x^{5} \left (\frac{a^{7} e^{4}}{5} + \frac{28 a^{6} b d e^{3}}{5} + \frac{126 a^{5} b^{2} d^{2} e^{2}}{5} + 28 a^{4} b^{3} d^{3} e + 7 a^{3} b^{4} d^{4}\right ) + x^{4} \left (a^{7} d e^{3} + \frac{21 a^{6} b d^{2} e^{2}}{2} + 21 a^{5} b^{2} d^{3} e + \frac{35 a^{4} b^{3} d^{4}}{4}\right ) + x^{3} \left (2 a^{7} d^{2} e^{2} + \frac{28 a^{6} b d^{3} e}{3} + 7 a^{5} b^{2} d^{4}\right ) + x^{2} \left (2 a^{7} d^{3} e + \frac{7 a^{6} b d^{4}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**4*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

a**7*d**4*x + b**7*e**4*x**12/12 + x**11*(7*a*b**6*e**4/11 + 4*b**7*d*e**3/11) + x**10*(21*a**2*b**5*e**4/10 +
14*a*b**6*d*e**3/5 + 3*b**7*d**2*e**2/5) + x**9*(35*a**3*b**4*e**4/9 + 28*a**2*b**5*d*e**3/3 + 14*a*b**6*d**2
*e**2/3 + 4*b**7*d**3*e/9) + x**8*(35*a**4*b**3*e**4/8 + 35*a**3*b**4*d*e**3/2 + 63*a**2*b**5*d**2*e**2/4 + 7*
a*b**6*d**3*e/2 + b**7*d**4/8) + x**7*(3*a**5*b**2*e**4 + 20*a**4*b**3*d*e**3 + 30*a**3*b**4*d**2*e**2 + 12*a*
*2*b**5*d**3*e + a*b**6*d**4) + x**6*(7*a**6*b*e**4/6 + 14*a**5*b**2*d*e**3 + 35*a**4*b**3*d**2*e**2 + 70*a**3
*b**4*d**3*e/3 + 7*a**2*b**5*d**4/2) + x**5*(a**7*e**4/5 + 28*a**6*b*d*e**3/5 + 126*a**5*b**2*d**2*e**2/5 + 28
*a**4*b**3*d**3*e + 7*a**3*b**4*d**4) + x**4*(a**7*d*e**3 + 21*a**6*b*d**2*e**2/2 + 21*a**5*b**2*d**3*e + 35*a
**4*b**3*d**4/4) + x**3*(2*a**7*d**2*e**2 + 28*a**6*b*d**3*e/3 + 7*a**5*b**2*d**4) + x**2*(2*a**7*d**3*e + 7*a
**6*b*d**4/2)

________________________________________________________________________________________

Giac [B]  time = 1.12921, size = 716, normalized size = 6.02 \begin{align*} \frac{1}{12} \, b^{7} x^{12} e^{4} + \frac{4}{11} \, b^{7} d x^{11} e^{3} + \frac{3}{5} \, b^{7} d^{2} x^{10} e^{2} + \frac{4}{9} \, b^{7} d^{3} x^{9} e + \frac{1}{8} \, b^{7} d^{4} x^{8} + \frac{7}{11} \, a b^{6} x^{11} e^{4} + \frac{14}{5} \, a b^{6} d x^{10} e^{3} + \frac{14}{3} \, a b^{6} d^{2} x^{9} e^{2} + \frac{7}{2} \, a b^{6} d^{3} x^{8} e + a b^{6} d^{4} x^{7} + \frac{21}{10} \, a^{2} b^{5} x^{10} e^{4} + \frac{28}{3} \, a^{2} b^{5} d x^{9} e^{3} + \frac{63}{4} \, a^{2} b^{5} d^{2} x^{8} e^{2} + 12 \, a^{2} b^{5} d^{3} x^{7} e + \frac{7}{2} \, a^{2} b^{5} d^{4} x^{6} + \frac{35}{9} \, a^{3} b^{4} x^{9} e^{4} + \frac{35}{2} \, a^{3} b^{4} d x^{8} e^{3} + 30 \, a^{3} b^{4} d^{2} x^{7} e^{2} + \frac{70}{3} \, a^{3} b^{4} d^{3} x^{6} e + 7 \, a^{3} b^{4} d^{4} x^{5} + \frac{35}{8} \, a^{4} b^{3} x^{8} e^{4} + 20 \, a^{4} b^{3} d x^{7} e^{3} + 35 \, a^{4} b^{3} d^{2} x^{6} e^{2} + 28 \, a^{4} b^{3} d^{3} x^{5} e + \frac{35}{4} \, a^{4} b^{3} d^{4} x^{4} + 3 \, a^{5} b^{2} x^{7} e^{4} + 14 \, a^{5} b^{2} d x^{6} e^{3} + \frac{126}{5} \, a^{5} b^{2} d^{2} x^{5} e^{2} + 21 \, a^{5} b^{2} d^{3} x^{4} e + 7 \, a^{5} b^{2} d^{4} x^{3} + \frac{7}{6} \, a^{6} b x^{6} e^{4} + \frac{28}{5} \, a^{6} b d x^{5} e^{3} + \frac{21}{2} \, a^{6} b d^{2} x^{4} e^{2} + \frac{28}{3} \, a^{6} b d^{3} x^{3} e + \frac{7}{2} \, a^{6} b d^{4} x^{2} + \frac{1}{5} \, a^{7} x^{5} e^{4} + a^{7} d x^{4} e^{3} + 2 \, a^{7} d^{2} x^{3} e^{2} + 2 \, a^{7} d^{3} x^{2} e + a^{7} d^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

1/12*b^7*x^12*e^4 + 4/11*b^7*d*x^11*e^3 + 3/5*b^7*d^2*x^10*e^2 + 4/9*b^7*d^3*x^9*e + 1/8*b^7*d^4*x^8 + 7/11*a*
b^6*x^11*e^4 + 14/5*a*b^6*d*x^10*e^3 + 14/3*a*b^6*d^2*x^9*e^2 + 7/2*a*b^6*d^3*x^8*e + a*b^6*d^4*x^7 + 21/10*a^
2*b^5*x^10*e^4 + 28/3*a^2*b^5*d*x^9*e^3 + 63/4*a^2*b^5*d^2*x^8*e^2 + 12*a^2*b^5*d^3*x^7*e + 7/2*a^2*b^5*d^4*x^
6 + 35/9*a^3*b^4*x^9*e^4 + 35/2*a^3*b^4*d*x^8*e^3 + 30*a^3*b^4*d^2*x^7*e^2 + 70/3*a^3*b^4*d^3*x^6*e + 7*a^3*b^
4*d^4*x^5 + 35/8*a^4*b^3*x^8*e^4 + 20*a^4*b^3*d*x^7*e^3 + 35*a^4*b^3*d^2*x^6*e^2 + 28*a^4*b^3*d^3*x^5*e + 35/4
*a^4*b^3*d^4*x^4 + 3*a^5*b^2*x^7*e^4 + 14*a^5*b^2*d*x^6*e^3 + 126/5*a^5*b^2*d^2*x^5*e^2 + 21*a^5*b^2*d^3*x^4*e
+ 7*a^5*b^2*d^4*x^3 + 7/6*a^6*b*x^6*e^4 + 28/5*a^6*b*d*x^5*e^3 + 21/2*a^6*b*d^2*x^4*e^2 + 28/3*a^6*b*d^3*x^3*
e + 7/2*a^6*b*d^4*x^2 + 1/5*a^7*x^5*e^4 + a^7*d*x^4*e^3 + 2*a^7*d^2*x^3*e^2 + 2*a^7*d^3*x^2*e + a^7*d^4*x