3.1463 \(\int (d+e x)^6 (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=119 \[ -\frac {2 b^3 (d+e x)^{10} (b d-a e)}{5 e^5}+\frac {2 b^2 (d+e x)^9 (b d-a e)^2}{3 e^5}-\frac {b (d+e x)^8 (b d-a e)^3}{2 e^5}+\frac {(d+e x)^7 (b d-a e)^4}{7 e^5}+\frac {b^4 (d+e x)^{11}}{11 e^5} \]

[Out]

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

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Rubi [A]  time = 0.27, antiderivative size = 119, 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} \[ -\frac {2 b^3 (d+e x)^{10} (b d-a e)}{5 e^5}+\frac {2 b^2 (d+e x)^9 (b d-a e)^2}{3 e^5}-\frac {b (d+e x)^8 (b d-a e)^3}{2 e^5}+\frac {(d+e x)^7 (b d-a e)^4}{7 e^5}+\frac {b^4 (d+e x)^{11}}{11 e^5} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

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

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Mathematica [B]  time = 0.06, size = 398, normalized size = 3.34 \[ a^4 d^6 x+a^3 d^5 x^2 (3 a e+2 b d)+\frac {1}{3} b^2 e^4 x^9 \left (2 a^2 e^2+8 a b d e+5 b^2 d^2\right )+a^2 d^4 x^3 \left (5 a^2 e^2+8 a b d e+2 b^2 d^2\right )+\frac {1}{2} b e^3 x^8 \left (a^3 e^3+9 a^2 b d e^2+15 a b^2 d^2 e+5 b^3 d^3\right )+a d^3 x^4 \left (5 a^3 e^3+15 a^2 b d e^2+9 a b^2 d^2 e+b^3 d^3\right )+\frac {1}{7} e^2 x^7 \left (a^4 e^4+24 a^3 b d e^3+90 a^2 b^2 d^2 e^2+80 a b^3 d^3 e+15 b^4 d^4\right )+d e x^6 \left (a^4 e^4+10 a^3 b d e^3+20 a^2 b^2 d^2 e^2+10 a b^3 d^3 e+b^4 d^4\right )+\frac {1}{5} d^2 x^5 \left (15 a^4 e^4+80 a^3 b d e^3+90 a^2 b^2 d^2 e^2+24 a b^3 d^3 e+b^4 d^4\right )+\frac {1}{5} b^3 e^5 x^{10} (2 a e+3 b d)+\frac {1}{11} b^4 e^6 x^{11} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

a^4*d^6*x + a^3*d^5*(2*b*d + 3*a*e)*x^2 + a^2*d^4*(2*b^2*d^2 + 8*a*b*d*e + 5*a^2*e^2)*x^3 + a*d^3*(b^3*d^3 + 9
*a*b^2*d^2*e + 15*a^2*b*d*e^2 + 5*a^3*e^3)*x^4 + (d^2*(b^4*d^4 + 24*a*b^3*d^3*e + 90*a^2*b^2*d^2*e^2 + 80*a^3*
b*d*e^3 + 15*a^4*e^4)*x^5)/5 + d*e*(b^4*d^4 + 10*a*b^3*d^3*e + 20*a^2*b^2*d^2*e^2 + 10*a^3*b*d*e^3 + a^4*e^4)*
x^6 + (e^2*(15*b^4*d^4 + 80*a*b^3*d^3*e + 90*a^2*b^2*d^2*e^2 + 24*a^3*b*d*e^3 + a^4*e^4)*x^7)/7 + (b*e^3*(5*b^
3*d^3 + 15*a*b^2*d^2*e + 9*a^2*b*d*e^2 + a^3*e^3)*x^8)/2 + (b^2*e^4*(5*b^2*d^2 + 8*a*b*d*e + 2*a^2*e^2)*x^9)/3
+ (b^3*e^5*(3*b*d + 2*a*e)*x^10)/5 + (b^4*e^6*x^11)/11

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fricas [B]  time = 1.19, size = 470, normalized size = 3.95 \[ \frac {1}{11} x^{11} e^{6} b^{4} + \frac {3}{5} x^{10} e^{5} d b^{4} + \frac {2}{5} x^{10} e^{6} b^{3} a + \frac {5}{3} x^{9} e^{4} d^{2} b^{4} + \frac {8}{3} x^{9} e^{5} d b^{3} a + \frac {2}{3} x^{9} e^{6} b^{2} a^{2} + \frac {5}{2} x^{8} e^{3} d^{3} b^{4} + \frac {15}{2} x^{8} e^{4} d^{2} b^{3} a + \frac {9}{2} x^{8} e^{5} d b^{2} a^{2} + \frac {1}{2} x^{8} e^{6} b a^{3} + \frac {15}{7} x^{7} e^{2} d^{4} b^{4} + \frac {80}{7} x^{7} e^{3} d^{3} b^{3} a + \frac {90}{7} x^{7} e^{4} d^{2} b^{2} a^{2} + \frac {24}{7} x^{7} e^{5} d b a^{3} + \frac {1}{7} x^{7} e^{6} a^{4} + x^{6} e d^{5} b^{4} + 10 x^{6} e^{2} d^{4} b^{3} a + 20 x^{6} e^{3} d^{3} b^{2} a^{2} + 10 x^{6} e^{4} d^{2} b a^{3} + x^{6} e^{5} d a^{4} + \frac {1}{5} x^{5} d^{6} b^{4} + \frac {24}{5} x^{5} e d^{5} b^{3} a + 18 x^{5} e^{2} d^{4} b^{2} a^{2} + 16 x^{5} e^{3} d^{3} b a^{3} + 3 x^{5} e^{4} d^{2} a^{4} + x^{4} d^{6} b^{3} a + 9 x^{4} e d^{5} b^{2} a^{2} + 15 x^{4} e^{2} d^{4} b a^{3} + 5 x^{4} e^{3} d^{3} a^{4} + 2 x^{3} d^{6} b^{2} a^{2} + 8 x^{3} e d^{5} b a^{3} + 5 x^{3} e^{2} d^{4} a^{4} + 2 x^{2} d^{6} b a^{3} + 3 x^{2} e d^{5} a^{4} + x d^{6} a^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

1/11*x^11*e^6*b^4 + 3/5*x^10*e^5*d*b^4 + 2/5*x^10*e^6*b^3*a + 5/3*x^9*e^4*d^2*b^4 + 8/3*x^9*e^5*d*b^3*a + 2/3*
x^9*e^6*b^2*a^2 + 5/2*x^8*e^3*d^3*b^4 + 15/2*x^8*e^4*d^2*b^3*a + 9/2*x^8*e^5*d*b^2*a^2 + 1/2*x^8*e^6*b*a^3 + 1
5/7*x^7*e^2*d^4*b^4 + 80/7*x^7*e^3*d^3*b^3*a + 90/7*x^7*e^4*d^2*b^2*a^2 + 24/7*x^7*e^5*d*b*a^3 + 1/7*x^7*e^6*a
^4 + x^6*e*d^5*b^4 + 10*x^6*e^2*d^4*b^3*a + 20*x^6*e^3*d^3*b^2*a^2 + 10*x^6*e^4*d^2*b*a^3 + x^6*e^5*d*a^4 + 1/
5*x^5*d^6*b^4 + 24/5*x^5*e*d^5*b^3*a + 18*x^5*e^2*d^4*b^2*a^2 + 16*x^5*e^3*d^3*b*a^3 + 3*x^5*e^4*d^2*a^4 + x^4
*d^6*b^3*a + 9*x^4*e*d^5*b^2*a^2 + 15*x^4*e^2*d^4*b*a^3 + 5*x^4*e^3*d^3*a^4 + 2*x^3*d^6*b^2*a^2 + 8*x^3*e*d^5*
b*a^3 + 5*x^3*e^2*d^4*a^4 + 2*x^2*d^6*b*a^3 + 3*x^2*e*d^5*a^4 + x*d^6*a^4

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giac [B]  time = 0.17, size = 450, normalized size = 3.78 \[ \frac {1}{11} \, b^{4} x^{11} e^{6} + \frac {3}{5} \, b^{4} d x^{10} e^{5} + \frac {5}{3} \, b^{4} d^{2} x^{9} e^{4} + \frac {5}{2} \, b^{4} d^{3} x^{8} e^{3} + \frac {15}{7} \, b^{4} d^{4} x^{7} e^{2} + b^{4} d^{5} x^{6} e + \frac {1}{5} \, b^{4} d^{6} x^{5} + \frac {2}{5} \, a b^{3} x^{10} e^{6} + \frac {8}{3} \, a b^{3} d x^{9} e^{5} + \frac {15}{2} \, a b^{3} d^{2} x^{8} e^{4} + \frac {80}{7} \, a b^{3} d^{3} x^{7} e^{3} + 10 \, a b^{3} d^{4} x^{6} e^{2} + \frac {24}{5} \, a b^{3} d^{5} x^{5} e + a b^{3} d^{6} x^{4} + \frac {2}{3} \, a^{2} b^{2} x^{9} e^{6} + \frac {9}{2} \, a^{2} b^{2} d x^{8} e^{5} + \frac {90}{7} \, a^{2} b^{2} d^{2} x^{7} e^{4} + 20 \, a^{2} b^{2} d^{3} x^{6} e^{3} + 18 \, a^{2} b^{2} d^{4} x^{5} e^{2} + 9 \, a^{2} b^{2} d^{5} x^{4} e + 2 \, a^{2} b^{2} d^{6} x^{3} + \frac {1}{2} \, a^{3} b x^{8} e^{6} + \frac {24}{7} \, a^{3} b d x^{7} e^{5} + 10 \, a^{3} b d^{2} x^{6} e^{4} + 16 \, a^{3} b d^{3} x^{5} e^{3} + 15 \, a^{3} b d^{4} x^{4} e^{2} + 8 \, a^{3} b d^{5} x^{3} e + 2 \, a^{3} b d^{6} x^{2} + \frac {1}{7} \, a^{4} x^{7} e^{6} + a^{4} d x^{6} e^{5} + 3 \, a^{4} d^{2} x^{5} e^{4} + 5 \, a^{4} d^{3} x^{4} e^{3} + 5 \, a^{4} d^{4} x^{3} e^{2} + 3 \, a^{4} d^{5} x^{2} e + a^{4} d^{6} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

1/11*b^4*x^11*e^6 + 3/5*b^4*d*x^10*e^5 + 5/3*b^4*d^2*x^9*e^4 + 5/2*b^4*d^3*x^8*e^3 + 15/7*b^4*d^4*x^7*e^2 + b^
4*d^5*x^6*e + 1/5*b^4*d^6*x^5 + 2/5*a*b^3*x^10*e^6 + 8/3*a*b^3*d*x^9*e^5 + 15/2*a*b^3*d^2*x^8*e^4 + 80/7*a*b^3
*d^3*x^7*e^3 + 10*a*b^3*d^4*x^6*e^2 + 24/5*a*b^3*d^5*x^5*e + a*b^3*d^6*x^4 + 2/3*a^2*b^2*x^9*e^6 + 9/2*a^2*b^2
*d*x^8*e^5 + 90/7*a^2*b^2*d^2*x^7*e^4 + 20*a^2*b^2*d^3*x^6*e^3 + 18*a^2*b^2*d^4*x^5*e^2 + 9*a^2*b^2*d^5*x^4*e
+ 2*a^2*b^2*d^6*x^3 + 1/2*a^3*b*x^8*e^6 + 24/7*a^3*b*d*x^7*e^5 + 10*a^3*b*d^2*x^6*e^4 + 16*a^3*b*d^3*x^5*e^3 +
15*a^3*b*d^4*x^4*e^2 + 8*a^3*b*d^5*x^3*e + 2*a^3*b*d^6*x^2 + 1/7*a^4*x^7*e^6 + a^4*d*x^6*e^5 + 3*a^4*d^2*x^5*
e^4 + 5*a^4*d^3*x^4*e^3 + 5*a^4*d^4*x^3*e^2 + 3*a^4*d^5*x^2*e + a^4*d^6*x

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maple [B]  time = 0.04, size = 427, normalized size = 3.59 \[ \frac {b^{4} e^{6} x^{11}}{11}+a^{4} d^{6} x +\frac {\left (4 e^{6} a \,b^{3}+6 d \,e^{5} b^{4}\right ) x^{10}}{10}+\frac {\left (6 e^{6} b^{2} a^{2}+24 d \,e^{5} a \,b^{3}+15 d^{2} e^{4} b^{4}\right ) x^{9}}{9}+\frac {\left (4 e^{6} a^{3} b +36 d \,e^{5} b^{2} a^{2}+60 d^{2} e^{4} a \,b^{3}+20 d^{3} e^{3} b^{4}\right ) x^{8}}{8}+\frac {\left (e^{6} a^{4}+24 d \,e^{5} a^{3} b +90 d^{2} e^{4} b^{2} a^{2}+80 d^{3} e^{3} a \,b^{3}+15 d^{4} e^{2} b^{4}\right ) x^{7}}{7}+\frac {\left (6 d \,e^{5} a^{4}+60 d^{2} e^{4} a^{3} b +120 d^{3} e^{3} b^{2} a^{2}+60 d^{4} e^{2} a \,b^{3}+6 d^{5} e \,b^{4}\right ) x^{6}}{6}+\frac {\left (15 d^{2} e^{4} a^{4}+80 d^{3} e^{3} a^{3} b +90 d^{4} e^{2} b^{2} a^{2}+24 d^{5} e a \,b^{3}+d^{6} b^{4}\right ) x^{5}}{5}+\frac {\left (20 d^{3} e^{3} a^{4}+60 d^{4} e^{2} a^{3} b +36 d^{5} e \,b^{2} a^{2}+4 d^{6} a \,b^{3}\right ) x^{4}}{4}+\frac {\left (15 d^{4} e^{2} a^{4}+24 d^{5} e \,a^{3} b +6 d^{6} b^{2} a^{2}\right ) x^{3}}{3}+\frac {\left (6 d^{5} e \,a^{4}+4 d^{6} a^{3} b \right ) x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

1/11*e^6*b^4*x^11+1/10*(4*a*b^3*e^6+6*b^4*d*e^5)*x^10+1/9*(6*a^2*b^2*e^6+24*a*b^3*d*e^5+15*b^4*d^2*e^4)*x^9+1/
8*(4*a^3*b*e^6+36*a^2*b^2*d*e^5+60*a*b^3*d^2*e^4+20*b^4*d^3*e^3)*x^8+1/7*(a^4*e^6+24*a^3*b*d*e^5+90*a^2*b^2*d^
2*e^4+80*a*b^3*d^3*e^3+15*b^4*d^4*e^2)*x^7+1/6*(6*a^4*d*e^5+60*a^3*b*d^2*e^4+120*a^2*b^2*d^3*e^3+60*a*b^3*d^4*
e^2+6*b^4*d^5*e)*x^6+1/5*(15*a^4*d^2*e^4+80*a^3*b*d^3*e^3+90*a^2*b^2*d^4*e^2+24*a*b^3*d^5*e+b^4*d^6)*x^5+1/4*(
20*a^4*d^3*e^3+60*a^3*b*d^4*e^2+36*a^2*b^2*d^5*e+4*a*b^3*d^6)*x^4+1/3*(15*a^4*d^4*e^2+24*a^3*b*d^5*e+6*a^2*b^2
*d^6)*x^3+1/2*(6*a^4*d^5*e+4*a^3*b*d^6)*x^2+d^6*a^4*x

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maxima [B]  time = 1.36, size = 418, normalized size = 3.51 \[ \frac {1}{11} \, b^{4} e^{6} x^{11} + a^{4} d^{6} x + \frac {1}{5} \, {\left (3 \, b^{4} d e^{5} + 2 \, a b^{3} e^{6}\right )} x^{10} + \frac {1}{3} \, {\left (5 \, b^{4} d^{2} e^{4} + 8 \, a b^{3} d e^{5} + 2 \, a^{2} b^{2} e^{6}\right )} x^{9} + \frac {1}{2} \, {\left (5 \, b^{4} d^{3} e^{3} + 15 \, a b^{3} d^{2} e^{4} + 9 \, a^{2} b^{2} d e^{5} + a^{3} b e^{6}\right )} x^{8} + \frac {1}{7} \, {\left (15 \, b^{4} d^{4} e^{2} + 80 \, a b^{3} d^{3} e^{3} + 90 \, a^{2} b^{2} d^{2} e^{4} + 24 \, a^{3} b d e^{5} + a^{4} e^{6}\right )} x^{7} + {\left (b^{4} d^{5} e + 10 \, a b^{3} d^{4} e^{2} + 20 \, a^{2} b^{2} d^{3} e^{3} + 10 \, a^{3} b d^{2} e^{4} + a^{4} d e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{6} + 24 \, a b^{3} d^{5} e + 90 \, a^{2} b^{2} d^{4} e^{2} + 80 \, a^{3} b d^{3} e^{3} + 15 \, a^{4} d^{2} e^{4}\right )} x^{5} + {\left (a b^{3} d^{6} + 9 \, a^{2} b^{2} d^{5} e + 15 \, a^{3} b d^{4} e^{2} + 5 \, a^{4} d^{3} e^{3}\right )} x^{4} + {\left (2 \, a^{2} b^{2} d^{6} + 8 \, a^{3} b d^{5} e + 5 \, a^{4} d^{4} e^{2}\right )} x^{3} + {\left (2 \, a^{3} b d^{6} + 3 \, a^{4} d^{5} e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

1/11*b^4*e^6*x^11 + a^4*d^6*x + 1/5*(3*b^4*d*e^5 + 2*a*b^3*e^6)*x^10 + 1/3*(5*b^4*d^2*e^4 + 8*a*b^3*d*e^5 + 2*
a^2*b^2*e^6)*x^9 + 1/2*(5*b^4*d^3*e^3 + 15*a*b^3*d^2*e^4 + 9*a^2*b^2*d*e^5 + a^3*b*e^6)*x^8 + 1/7*(15*b^4*d^4*
e^2 + 80*a*b^3*d^3*e^3 + 90*a^2*b^2*d^2*e^4 + 24*a^3*b*d*e^5 + a^4*e^6)*x^7 + (b^4*d^5*e + 10*a*b^3*d^4*e^2 +
20*a^2*b^2*d^3*e^3 + 10*a^3*b*d^2*e^4 + a^4*d*e^5)*x^6 + 1/5*(b^4*d^6 + 24*a*b^3*d^5*e + 90*a^2*b^2*d^4*e^2 +
80*a^3*b*d^3*e^3 + 15*a^4*d^2*e^4)*x^5 + (a*b^3*d^6 + 9*a^2*b^2*d^5*e + 15*a^3*b*d^4*e^2 + 5*a^4*d^3*e^3)*x^4
+ (2*a^2*b^2*d^6 + 8*a^3*b*d^5*e + 5*a^4*d^4*e^2)*x^3 + (2*a^3*b*d^6 + 3*a^4*d^5*e)*x^2

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mupad [B]  time = 0.62, size = 402, normalized size = 3.38 \[ x^5\,\left (3\,a^4\,d^2\,e^4+16\,a^3\,b\,d^3\,e^3+18\,a^2\,b^2\,d^4\,e^2+\frac {24\,a\,b^3\,d^5\,e}{5}+\frac {b^4\,d^6}{5}\right )+x^7\,\left (\frac {a^4\,e^6}{7}+\frac {24\,a^3\,b\,d\,e^5}{7}+\frac {90\,a^2\,b^2\,d^2\,e^4}{7}+\frac {80\,a\,b^3\,d^3\,e^3}{7}+\frac {15\,b^4\,d^4\,e^2}{7}\right )+x^4\,\left (5\,a^4\,d^3\,e^3+15\,a^3\,b\,d^4\,e^2+9\,a^2\,b^2\,d^5\,e+a\,b^3\,d^6\right )+x^8\,\left (\frac {a^3\,b\,e^6}{2}+\frac {9\,a^2\,b^2\,d\,e^5}{2}+\frac {15\,a\,b^3\,d^2\,e^4}{2}+\frac {5\,b^4\,d^3\,e^3}{2}\right )+x^6\,\left (a^4\,d\,e^5+10\,a^3\,b\,d^2\,e^4+20\,a^2\,b^2\,d^3\,e^3+10\,a\,b^3\,d^4\,e^2+b^4\,d^5\,e\right )+a^4\,d^6\,x+\frac {b^4\,e^6\,x^{11}}{11}+a^3\,d^5\,x^2\,\left (3\,a\,e+2\,b\,d\right )+\frac {b^3\,e^5\,x^{10}\,\left (2\,a\,e+3\,b\,d\right )}{5}+a^2\,d^4\,x^3\,\left (5\,a^2\,e^2+8\,a\,b\,d\,e+2\,b^2\,d^2\right )+\frac {b^2\,e^4\,x^9\,\left (2\,a^2\,e^2+8\,a\,b\,d\,e+5\,b^2\,d^2\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^6*(a^2 + b^2*x^2 + 2*a*b*x)^2,x)

[Out]

x^5*((b^4*d^6)/5 + 3*a^4*d^2*e^4 + 16*a^3*b*d^3*e^3 + 18*a^2*b^2*d^4*e^2 + (24*a*b^3*d^5*e)/5) + x^7*((a^4*e^6
)/7 + (15*b^4*d^4*e^2)/7 + (80*a*b^3*d^3*e^3)/7 + (90*a^2*b^2*d^2*e^4)/7 + (24*a^3*b*d*e^5)/7) + x^4*(a*b^3*d^
6 + 5*a^4*d^3*e^3 + 9*a^2*b^2*d^5*e + 15*a^3*b*d^4*e^2) + x^8*((a^3*b*e^6)/2 + (5*b^4*d^3*e^3)/2 + (15*a*b^3*d
^2*e^4)/2 + (9*a^2*b^2*d*e^5)/2) + x^6*(a^4*d*e^5 + b^4*d^5*e + 10*a*b^3*d^4*e^2 + 10*a^3*b*d^2*e^4 + 20*a^2*b
^2*d^3*e^3) + a^4*d^6*x + (b^4*e^6*x^11)/11 + a^3*d^5*x^2*(3*a*e + 2*b*d) + (b^3*e^5*x^10*(2*a*e + 3*b*d))/5 +
a^2*d^4*x^3*(5*a^2*e^2 + 2*b^2*d^2 + 8*a*b*d*e) + (b^2*e^4*x^9*(2*a^2*e^2 + 5*b^2*d^2 + 8*a*b*d*e))/3

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sympy [B]  time = 0.14, size = 462, normalized size = 3.88 \[ a^{4} d^{6} x + \frac {b^{4} e^{6} x^{11}}{11} + x^{10} \left (\frac {2 a b^{3} e^{6}}{5} + \frac {3 b^{4} d e^{5}}{5}\right ) + x^{9} \left (\frac {2 a^{2} b^{2} e^{6}}{3} + \frac {8 a b^{3} d e^{5}}{3} + \frac {5 b^{4} d^{2} e^{4}}{3}\right ) + x^{8} \left (\frac {a^{3} b e^{6}}{2} + \frac {9 a^{2} b^{2} d e^{5}}{2} + \frac {15 a b^{3} d^{2} e^{4}}{2} + \frac {5 b^{4} d^{3} e^{3}}{2}\right ) + x^{7} \left (\frac {a^{4} e^{6}}{7} + \frac {24 a^{3} b d e^{5}}{7} + \frac {90 a^{2} b^{2} d^{2} e^{4}}{7} + \frac {80 a b^{3} d^{3} e^{3}}{7} + \frac {15 b^{4} d^{4} e^{2}}{7}\right ) + x^{6} \left (a^{4} d e^{5} + 10 a^{3} b d^{2} e^{4} + 20 a^{2} b^{2} d^{3} e^{3} + 10 a b^{3} d^{4} e^{2} + b^{4} d^{5} e\right ) + x^{5} \left (3 a^{4} d^{2} e^{4} + 16 a^{3} b d^{3} e^{3} + 18 a^{2} b^{2} d^{4} e^{2} + \frac {24 a b^{3} d^{5} e}{5} + \frac {b^{4} d^{6}}{5}\right ) + x^{4} \left (5 a^{4} d^{3} e^{3} + 15 a^{3} b d^{4} e^{2} + 9 a^{2} b^{2} d^{5} e + a b^{3} d^{6}\right ) + x^{3} \left (5 a^{4} d^{4} e^{2} + 8 a^{3} b d^{5} e + 2 a^{2} b^{2} d^{6}\right ) + x^{2} \left (3 a^{4} d^{5} e + 2 a^{3} b d^{6}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**6*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

a**4*d**6*x + b**4*e**6*x**11/11 + x**10*(2*a*b**3*e**6/5 + 3*b**4*d*e**5/5) + x**9*(2*a**2*b**2*e**6/3 + 8*a*
b**3*d*e**5/3 + 5*b**4*d**2*e**4/3) + x**8*(a**3*b*e**6/2 + 9*a**2*b**2*d*e**5/2 + 15*a*b**3*d**2*e**4/2 + 5*b
**4*d**3*e**3/2) + x**7*(a**4*e**6/7 + 24*a**3*b*d*e**5/7 + 90*a**2*b**2*d**2*e**4/7 + 80*a*b**3*d**3*e**3/7 +
15*b**4*d**4*e**2/7) + x**6*(a**4*d*e**5 + 10*a**3*b*d**2*e**4 + 20*a**2*b**2*d**3*e**3 + 10*a*b**3*d**4*e**2
+ b**4*d**5*e) + x**5*(3*a**4*d**2*e**4 + 16*a**3*b*d**3*e**3 + 18*a**2*b**2*d**4*e**2 + 24*a*b**3*d**5*e/5 +
b**4*d**6/5) + x**4*(5*a**4*d**3*e**3 + 15*a**3*b*d**4*e**2 + 9*a**2*b**2*d**5*e + a*b**3*d**6) + x**3*(5*a**
4*d**4*e**2 + 8*a**3*b*d**5*e + 2*a**2*b**2*d**6) + x**2*(3*a**4*d**5*e + 2*a**3*b*d**6)

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